(Blog&~Blog)

Beall Against Pinocchio

2011-08-31T20:25:00.000-07:00

It's a fun paradox, and Peter Eldridge-Smith argues (convincingly, by my lights) that it creates problems for the claim that 'semantic' contradictions can be true, but not 'metaphysically' substantive ones. JC Beall's half of the exchange is available for free here.

(Eldridge Smith's half is available for free too, but if you're reading this on a computer at an institution with an online subscription to Analysis.)

One problem I have with Beall's response is that it's far from clear what sort of "impossibility" he has the resources to assign to base-language contradictions. It's one thing to say that the actual world lacks them--it certainly seems to!--but Beall, of course, can hardly claim that non-trivial worlds containing base-language contradictions are *logically* impossible. As he himself convincingly argues in Spandrels of Truth, dialetheists can hardly go around claiming that some falsehoods are 'more false' than others, such that contradictions involving them really would be explosive, since one could always construct a paradoxical 'spandrel' which attributed precisely this sort of extra-special-super-falseness to itself. If the claim is that they're metaphysically-but-not-logically impossible, I think that requires considerable fleshing out. *Why* would they be metaphysically impossible?

Someone with orthodox views would say that they're metaphysically impossible *because* they're logically impossible. Once we've blocked off that route by accepting (even "purely semantic") true contradictions, an alternative explanation is required.

Check It Out!

2011-08-29T11:44:00.000-07:00

My Analysis paper is online.

Analysis

2011-05-11T00:01:00.000-07:00

I just had a paper, entitled "Paracompleteness and Revenge," accepted for publication at Analysis.

It's about revenge problems for "paracomplete" solutions to the Liar Paradox, a topic which of course I've discussed in this space before.

Singapore

2011-04-27T00:01:00.000-07:00

I just gave a talk at the National University of Singapore, entitled "Liar Paradox II: Revenge of the Liar Paradox." Singapore, by the way, is lovely, and I'm going to be sorry to leave on Thursday. On the surface it feels enough like Miami to make me feel nostalgic--humid, windy, full of palm trees and outdoor bars and a mishmash of different languages and cultures--while having a lot of appealing un-Miami-ish traits, like being chock-full of restaurants serving delicious Indian food. (The strangest thing I've seen here to date has been Haw Par Villa, a "moral instruction" theme park based on Confucianism and Chinese mythology put up by some early-twentieth-century Chinese millionaires who'd made a killing in the tiger balm trade. The main attraction is the Ten Courts of Hell, where lurid statues and signs depict the punishments sinners are sentenced to by the Emperors of Hell. For example, "cheating on examinations" gets you your intestines ripped out by demons.) While I've been in Singapore, I've been staying with NUS prof Neil Sinhababu, of Possible Girls fame, who will henceforth always have a special place in my heart for saying, when I came in on Saturday night, "I made sure to save some Laphroaig for your visit."

As far as the talk itself, here's the abstract:

Dialetheists like Graham Priest and JC Beall conclude from the Liar Paradox that sentences like “This sentence is not true” are fact both true and untrue, and that we must therefore revise our logic to accommodate the existence of true contradictions. Similarly, “paracomplete” theorists like Hartry Field avoid the contradiction posed by the Liar Paradox by rejecting one of the central elements of classical logic, the Law of the Excluded Middle. A more conservative solution starts from the claim that sentences that attempt to attribute truth or untruth to themselves are meaningless, and therefore simply not the kinds of things we can logically symbolize or apply truth talk to without committing a nonsensical category mistake. The most common objections to this move are (1) that the “meaninglessness solution” is refuted by the existence of “revenge paradoxes” like the one revolving around the sentence “This sentence is either false or meaningless”, and that (2) the sentences involved are so obviously meaningful that it’s just not possible to take seriously the claim that they’re literally meaningless in any ordinary sense, like “Blorks geblork” or “Colorless green ideas sleep furiously,” whereas the dialetheist and paracomplete approaches have the advantages that they (1*) make room for the perfectly obvious fact that, in any language with normal expressive resources, we can construct perfectly meaningful sentences that attribute untruth to themselves, and (2*) are immune to refutation by means of “revenge paradoxes.” I will argue that (1), (2), (1*) and (2*) are all completely wrong.

In terms of the talk itself, I'm never 100% sure what to think about the ethics of blogging in-person discussions, given that I'm sure I wouldn't want to be represented by someone else's half-clear recollection of what I said on the spur of the moment, so I'll pretty well stick to representing what I said myself, with one exception (one hopes, a benign one): In the talk, I spent a few minutes hammering the standard Priest/Beall/Field sort line on Curry's Paradox. In the Q&A, NUS prof Ben Blumson took issue with some of that, and later in the day I ended up spending a couple of hours in his office genially arguing about Curry and related issues, and if I'm still not utterly convinced, I will definitely say that he did a better job of presenting a fairly plausible defense of the approach to Curry I was criticizing than any other defense of that approach I've seen or read before, and in future I will be scaling back at least some of my initial objections in light of some of the points he made.*

In any case, Curry aside, a lot of the ground I covered should be familiar to regular readers here. While I briefly presented my disquotationalist argument for the claim that ungrounded truth talk is literally meaningless, including the Greenness Paradox as a way of defusing the worry that, since we are able to reason about the Liar, we know what follows from it, thus what it means and thus that it means something, my main focus was on revenge paradoxes. (A paper I'm working on making some of these points is tentatively entitled 'Who Among Us Is Safest From The Liar's Revenge?') Conventional wisdom says that the dialetheist and paracomplete approaches to the Liar, given their willingness to engage in radical surgery to our basic logical notions, gain immunity from the revenge paradoxes that typically plague classical solutions to the paradoxes, of which a particularly clear case is supposed to be the problem posed by (1) for those of us who take these sentences to be meaningless:

(1) Sentence (1) is either false or meaningless.

Since paracompletists don't assert anything about the semantic status of such sentences, but rather reject the relevant instances of Excluded Middle, reject the negation of those instances and so on 'all the way down the line,' they seem to be immune from danger from sentences that attribute to themselves the status paracompletists attribute to such sentences. Even more so, dialetheists seem to be immune from any revenge problems, because any 'revenge' liar would at worst just generate yet another true contradiction, and true contradictions don't generate triviality, given the dialetheist's claim that Disjunctive Syllogism isn't universally truth-preserving.

My claim is that (1) is not a problem for the meaninglessness solution at all. A meaningless sentence does not become meaningful once we attach the word "or" to it and paste (what would otherwise be) a meaningful sentence to its tail. Just because a meaningless sentence has the syntactic form of a disjunction and a true second disjunct does not mean that it's meaningful, much less true.

On the other hand, I argue that the paracompletist has a real problem about sentence (2) and that the dialetheist has a real problem about sentence (3).

(2) An ideally rational being who did not lack any relevant information would not accept sentence (2).

(3) It is not the case that sentence (3) is related to truth.

Regular readers will recognize that (2) is the latest form of a revenge paradox for paracompletism I've been tinkering with for some time. The problem, as I see it, is that, if (2) is true, we have the starkly counter-intuitive result that an ideally rational being would not accept a sentence it knew to be true, if (2) is false, we have the equally counter-intuitive result that an ideally rational being would accept a sentence it knew to be false, and if (2) is one of the sentences about which the most rational option is to 'go paracomplete' and reject both the sentence and its negation, then its a sentence that any ideally rational being would not accept (it would reject the sentence instead of accepting it!) and the sentence is true, and, once again, we have the conclusion that an ideally rational being would fail to accept a sentence it knew to be true.

(3) is a familiar problem, but as I argued here, the exact nature of the biggest problem it poses doesn't seem to be widely realized. If being-related-to-truth and not-being-related-to-truth overlap, just as being-related-to-truth and being-related-to-falsehood overlap, then when the dialetheist shows that (given the assumption of dialetheism) there are cases in which all the premises of Disjunctive Syllogism are related to truth and the conclusion is not, they have no more shown that DS is not universally truth-preserving than they would if they'd 'just' showed that all the premises of DS were true and the conclusion was false. No one thinks that "all the premises of argument A are true and the conclusion is false" is a dialetheistically-acceptable way of establishing a failure of truth-preservation. Why should "all the premises of A are related to truth and the conclusion is not" be even a little bit different? Without a better answer to that question, the dialetheist claim that Liars can be both true and false without triviality following simply doesn't hold up.

*On a similar note, I should include a quick shout-out to Brandon Watson for giving me a hard time recently about my error theory about the mistakenness of ordinary competent speakers who take Liars to be meaningful. My preferred way of putting the point now, which I used in the talk, goes about like this: Self-referential sentences are often meaningful--e.g. 'This sentence has seven words in it'--and sentences with precisely the same wording as the Liar sentence are meaningful in other context--'This sentence is false' said while pointing at a sentence about some substantive subject written on a chalkboard. To realize that it's meaningless when the intended reference of the 'this' is that very sentence is a conclusion that takes careful philosophical argumentation. Given those two facts, its quite natural that most people don't realize that it's true. By analogy, if an object is far away and looks a certain (misleading) way from a great distance, and out of a whole crowd of people watching the object, only Bob has a telescope, it's utterly unsurprising that most competent-users-of-functioning-human-eyes end up having a mistaken impression about the object. All Bob needs to do by way of explanation of the disconnect is to say, "yeah, you don't have a telescope. But, hey, look through it yourselves and you'll see where you went wrong here."

Some Objections to the Meaninglessness Solution to the Liar Paradox, Part IV of IV

2011-04-24T18:21:00.000-07:00

(Is it Wednesday already? Oh well, better late than never....)

I've argued that, if (as I think) the truth predicate/operator is just a device used to assert things (just as the falsehood predicate/operator is just a device used to assert their negations), it can pretty clearly only be meaningfully applied when there is something there to be asserted--thus, it can only be meaningfully applied to claims about something other than truth. Thus, for example, in a Yablo-like series of sentences where each sentence ascribes truth to the next sentence in the series,

T1: T2 is true.
T2: T3 is true.
T3: T4 is true.

....and so on forever, all the sentences in the infinite series are literally as devoid of meaning as strings of nonsense syllables, or 'Colorless green ideas sleep furiously.' If, on the other hand, sentence T1000000 is "Snow is white," the rest of the sentences inherit their meanings (and, thus, truth-values) from that.

A semantic property pretty clearly *unlike* truth in this respect is meaningfulness itself. If the meaningfulness predicate only applied to meaningful sentences, it wouldn't fulfill its sole communicative function of separating out the meaningful sentences from the meaningless ones. This is important when we consider (18), which, by analogy to the Liar, we can call The Babbler:

(18) Sentence (18) is meaningless.

If (18) is true, it's both true and meaningless, therefore both meaningful and meaningless, and, of course, if it's meaningless, it's both true and meaningless, therefore both meaningful and meaningless. As such, on pain of contradiction, (18) had better just be false.

Fortunately, in light of the above, we have a good principled reason to think that this is indeed the case. If the function of the meaningfulness predicate is to separate out the meaningful from the meaningless sentences, it has to apply to all sentences. Therefore, it's meaningful to say of any sentence that it's meaningful or meaningless, regardless of the nature of the sentence we're talking about. As such, if all a sentence does is assert a view about the meaningfulness of some sentence, even itself, there's no reason for it not to be meaningful. Thus, (18) is false and (19):

(19) Sentence (19) is meaningful.

...is true.

One important principle, underlying the whole business of revenge-paradoxology, is worth calling attention to here, since I've been implicitly using it a lot. Given these sorts of examples, or, better yet, cased like (20) and (21):

(20) This sentence has seven words in it.
(21) This sentence has twenty words in it.

....where it would be clearly absurd to assert about sentence (20), for example, that is seven words long, without granting that sentence (21) is true, we have what we can call the Meaningfulness of Self-Reference Principle: "If Sentence X has property Y, and Sentence X *states* that Sentence X has property Y, then Sentence X is true (and thus, of course, meaningful)."

With all that in mind, and the demonstrations in Parts II and III that it clearly is possible to engage in apparent reasoning about even the most clearly meaningless sentences--meaning that it's not a problem for meaninglessness solutions to the Liar that it's "clearly possible to reason about it, and we all know what does and doesn't follow from it"--let's turn to the apparently troubling revenge paradox for my view that I ended with last time:

(17) Sentence (17) is one that one would have to ultimately label as "false" if one treated it as being meaningful and went through the motions of "reasoning" about it without making the sort of mistake we would regard in normal contexts as a mistake in reasoning.

So, playing along with the game of treating it as meaningful for a moment, an obvious first question is this:

Does (17) take a stand on the question of its own meaningfulness? In other words, does it (a) say of itself that it's meaningful, (b) say of itself that it's meaningless, or (c) remain neutral on that topic?

The wording strongly suggests that (a) would be the wrong gloss--talk of treating it 'as meaningful' and 'going through the motions' strongly suggests that the point is to, at the very least, keep open the possibility that it's meaningless, if not to actively assert it. That said, if (a) is right--it's taking a stand on its own meaningfulness in the directing of asserting it--then to say that, if one went through the motions of reasoning about it, one would make something we would regard in other contexts--i.e. really reasoning about meaningful things--as a mistake, is to say that, if one reasoned about it and failed to come to the conclusion that it was false, one would be making a real, full-fledged mistake in reasoning--a factual mistake, landing us with the wrong answer. In other words, given (a), (17) is just a normal if un-usually phrased Liar sentence, the normal meaninglesness solution applies to it, and the Principle of the Meaningfulness of Self-Reference is not violated if we simultanneously say of it that, although meaningless, going through the motions seems to get us the result that it's false (and true), given that what it's saying is that this isn't a matter of going through the motions in an empty context, because it really is false. (It can be neutral about its own falsehood, given that it asserts its own meaningfulness and thus converts the neutral-sounding language about apparent mistakes into, in effect, the positive claim that one would be making a substantive mistake and getting the wrong result.)

If (b) is the case, then we have a disguised conjunction of two claims: (i) a claim about its alleged meaninglessness, and (ii) a claim about whether any possible analysis of it that (1) took it as meaningful and (2) failed to include any mistakes unrelated to the meaningfulness question would therefore (3) diagnose (17) as false. There's a lot to untangle here, but suffice to say that if it is meaningless, then the true 'first conjunct' asserting as much doesn't make the whole thing meaningful, for reasons examined when we looked at (2), above, and if it's not meaningless, the falsity of the first conjunct guarantees the falsity of the whole thing without fear of contradiction. Really, though, I think the most natural reading is (c), and that's where the real problem seems to be.

If (c) is, then, the case, as should be clear by now, (17) really amounts to a disguised disjunction between the claim that (i*) reasoning about (17) and failing to come to the conclusion that it's false would be a *factual* mistake, and (ii*) that 'reasoning' about (17) leads us to the apparent conclusion that it is false, but only because we're indulging a nonsensical category mistake. In other words, given (c), what we end up with is a disguised version of sentence (2), above:

(2) The sentence marked (2) is either false or meaningless.

...which we already dealt with in Part I. Since I enjoy the circularity of ending by directing back to the first post in the series, I think I'll just leave off there and throw open the floor to questions, comments and devastating objections.

Fatalism, Part I of II (Diet Soap Nit-Pickery)

2011-04-13T23:59:00.000-07:00

A little while ago, I mentioned that I was interviewed for a second appearance on philosophy-oriented podcast Diet Soap. (First one is here.) Anyway, I think the episode with the second appearance on it is going to be coming out within the next couple weeks.

As I emphasized then, I'm happy to be on there. Diet Soap is a podcast I listen to regularly--in fact, lately, it's been the one I've listened to the most regularly that isn't about hockey--and I almost always find it interesting, entertaining and thought-provoking. You should listen.

Anyway, despite having been a normal weekly listener for most of the last bunch of weeks, I somehow missed Episode 91, which was (among other things) about Deluze. Since the most recent episode was a different perspective on Deluze, and the show notes referred to the previous one, I thought I might as well go back and listen to the first Deluze discussion.

...most of which was interesting enough, but towards the end, the host and his guest touched on fatalism and free will in a way that made me want to rip my iPod out of my ears and throw it against the nearest wall in frustration. (I didn't. It's an expensive iPod--one of those tiny little "nano touch" thingies--and what with me being in Korea and all, it'd be even more expensive to replace. Plus, of course, despite my frustration on this particular point, I was still interested to hear the rest of what they had to say.) I've touched on this complaint before, after host Doug Lain brought it up in a previous episode, but I want to take another, more careful crack at it here. Here's what I said about it before:

(2) In the discussions about free will and fatalism, there's a lot of running together of two quite distinct claims:

(i) That there are facts about what will happen in the future, such that some statements about the future are true and some are false, and
(ii) That some being knows which statements about the future are true and which ones are false.

Clearly (at least given the orthodox assumption that truth is a necessary condition for knowledge), (ii) entails (i), but (i) can absolutely and obviously be true without (ii) being true. By analogy, consider Claim C (about the past, rather than the future):

C: "Alexander the great's maternal grandmother's paternal grandmother accidentally cut her toe on a rock when she was six years old."

C is pretty clearly either true or false. Whatever one thinks about reference failures and all of that (i.e. whether a statement like "the present King of France is bald" is true, false or neither, given that there is no present King of France), Alexander the Great clearly had a maternal grandmother, and she clearly had a paternal grandmother, and at one point she was six years old. During that year, that lady either did accidentally cut her toe on a rock--in which case C is true--or she didn't (in which case the negation of C is true), and none of this is remotely philosophically controversial. Given atheism (and the absence of time machines) no one is in any position to have epistemic access to the fact of the matter here, but no one thinks that there isn't a fact of the matter about this issue. Why on earth should it be any different, re: future facts and the absence of any being with epistemic access to those facts?

(In the comments on that post, a friend of mine who's working on free will for his doctoral dissertation put the point rather more vehemently than I would.)

Expanding a bit now:

Lain expresses the point in terms of "truth claims", which I find slightly confusing, just because it's not terminology that I'm used to, but I think it's fairly clear in context that it just means "claims that are true." (For the sake of simplicitly, let's make that "true statements.") He's responding to Taylor's classic argument for fatalism (the idea that the future is "fixed" in some way that gets in the way of some important intuitive idea of free will). That argument is spelled out formally by Taylor, and Lain is looking for a way out. So far, so good.

On the philosophical substance here, my own very strong view is that (a) there are lots of ways out, even if Lain's favored one is as problematic as I think it is, but that (b) no matter what your favorite conception is of free will, "fatalism" shouldn't be a problem. We can come back to that in a bit. Meanwhile....

Lain's move is, basically, to leverage atheism against fatalism. If Taylor's picture has it that there is an infinite set of every true statement about the future--and therefore that, for every prediction you could make about the future, either its in that set or its not, but one way or the other, there's a fixed fact of the matter about whether the prediction's going to come true--Lain wants to dispute the claim that there is such a set of true statements. After all, in the absence of an omniscient God, no one is in a position to claim all the infinitely many true things about the future.

In his most recent statement of all this, in Episode 91, Lain made a special point of saying that something can only be a statement if someone has said it, written it down or thought it. I think this might have been a way of side-stepping the way I'd previously expressed my objection, quoted above--in terms of "some statements being true and others being false" vs. "some entity being in a position to know which statements are true and which are false"--and I guess it does, but in a way that I think misses the point.

Think about the past. That's definitely "fixed" and at this point unchangeable in just the sense that anyone worried about fatalism is worried that the future is "fixed", right? Well, even if the past is finite (different physical cosmologies have importantly different results on that point), in a universe without any God-like entities, surely no one is in any position to know, or state, every true statement about the past, right? That is, however, just obviously utterly irrelevant to the pasts' "fixed"-ness.

The reason its irrelevant is that the issue isn't so much about statements as it is about facts. Even if I hadn't come up with the particular example I used in the comments--C: "Alexander the great's maternal grandmother's paternal grandmother accidentally cut her toe on a rock when she was six years old."--and indeed if no one had ever said or thought of that statement (as is extremely likely that no one would have) the lady in question, and all events in her life, would still exist, and either include or fail to include the described incident. Even if one thinks that sentences per se rather, than say, propositions, are the only things that can be "true" or "false", and even if a sentence describing the incident doesn't exist, either the incident occurred or it didn't.

Now, one could make a really radical move here and just deny the existence of un-described facts--if no one has ever commented on or thought about the number of empty bottles on the floor of the basement of the frat house, then there isn't a certain, definite, objective number of bottles there!--and that would sort-of-help here, but, in the end, it wouldn't help much. Not only would this move distance you so much from any remotely recognizable sense of the meaning of the word "truth" as it's used pre-philosophically in ordinary everyday conversations that it's no longer clear to me what we're talking about when we talk about whether some statement is "true" or "false", but even if we make this move, it won't get us off the fatalist hook.

For one thing, we can always reconstruct the fatalistic stuff in terms of hypothetical statements--e.g. "for any possible future event, if one were to make a prediction about it, that prediction would either be a true prediction or it wouldn't be"--and for another, even if we couldn't (and, again, we pretty clearly can), that wouldn't matter very much.

Here's why:

Forget "the future" as a vast (possibly infinitely extended) category, and re-ask yourself why you're concerned with fatalism in the first place. Presumably, it's because we want to think we have the power to change things with our idividual or collective choices, or at the very least that (even if we don't think it's a matter of choice) certain future possibilities we care about are still "open."

The problem is that, for any given future possibility we care about the openness of, we can just construct a sentence about it. For example:

"Doug Lain's great-grandchildren will live under precisely the sort of anarchist-socialist utopia he advocates."

or

"An anarchist-socialist utopia will never come about."

or

"Doug Lain will murder someone on July 15th, 2058, and be executed for that crime."

or

"Doug Lain will never kill anyone."

....or whatever. Whether or not there's an infinite set of true statements for any of these these statements to be part of (if any of them are true) or to be fail to be part of, we hardly need an infinite, omniscient mind for these particular statements to exist. (Check them out! I just wrote them up!) Given that they exist, they're either true or they're not, which is presumably just as much (or, of course, just as little) of a problem for the "openness" or undecidness or whatever of these future possibilities as them being or not being part of some infinite set of true statements would be.

Now, like I said earlier, I tend to think that both (a) if one thinks that its important to avoid this fatalistic result, there are plenty of moves you could make that do so, even if the move under consideration has prospects as dim as I think they are, and that (b) it's actually not important to avoid it (the future is every bit as much "up to us" and to our free choices with or without the kind of 'openness' anti-fatalists tend to be concerned with). That is, though, as they say, a-whole-nother discussion, and one probably best reserved for a blog post of its own, this one being as long as it is already.

So let's do that in a couple weeks. Meanwhile, stay tuned for the long-delayed Part IV of the Liar Pardox posts next Wednesday!

Switching Up The Schedule

2011-04-11T23:18:00.001-07:00

Scaling back to updating every Wednesday for a while.

So it goes.

Some Objections to the Meaninglessness Solution to the Liar Paradox, Part III of IV

2011-04-06T00:01:00.000-07:00

My view is that, as my boy Willard Van put it, truth is disquotation. When I prefix the words 'it's true that' to a quoted sentence, the effect of what I've done is to remove the quotation marks. Moreover, I see no principled distinction between this way of ascribing truth to a sentence (quoting it within the larger sentence in which one applies the truth operator to it), or other standard ways of ascribing truth, like saying "that's true" in response to someone else's statement, or the more formal device of writing down a pair of numbered sentences like (14) and (15):

(14) Snow is white.
(15) Sentence (14) is true.

In all of these, the function of the truth predicate/operator is exactly the same.

A good analogy in contemporary informal English is "What he said."

Imagine the following, fairly mundane interaction:

An evolutionary biologist, Jane, is drinking at a bar with her boyfriend John (a humanities major with a shaky but more or less accurate grasp on her field) and her loveable-but-frustrating cousin Jack, a fundamentalist Christian (who's just ordering coca cola and bar nachos while his heathen cousin and the man she's living in sin booze up). At some point, Jack brings up evolution and runs through some creationist talking point about missing links in the fossil record or some such. Jane sighs, orders another drink and carefully runs through the scientific explanation of what Jack's talking about. In the end, both cousins stop talking and turn to John, who just tilts his head towards his girlfriend and says "What she said."

This is a normal and immediately familiar usage--note, BTW, that John's sentence isn't syntactically "well-formed", but it's meaningful all the same, "well-formed"-ness in natural language contexts not being necessary for, sufficient for, or even especially relevant t meaningfulness--and we all get what's going on here. "What she said" is a linguistic device John is using as a shorthand method of asserting exactly what Jane just asserted. He could be using it for a variety of reasons--most obviously, using this handy abbreviation is far easier than repeating the entire explanation Jane just gave, but it could also (in this case quite plausibly) be that he doesn't remember every detail of what he means to be asserting. Certainly, there's very little temptation here to think that John means anything above and beyond, or different from, what Jane said. The point of the phrasing is, in fact, to draw attention to the fact that he means to say exactly what, well, "she said."

The word 'true' has various advantages over its functionally kindred linguistic device 'what she said'--for one thing, one can use it on written sentences, and more tellingly, on sentences whose source is unclear--but the point, I think, is the same. Instances of "what she said" are presumably meaningful if the sentence it's applied to are meaningful and meaningless otherwise (John: "Colorless green ideas sleep furiously", Jane: "What he said"), and the same goes, I would argue for truth. The general principle is that truth talk is only *parasitically* meaningful. Prefixing "it's true that" to a *quoted* sentence has the effect of removing the quotation marks--i.e. "it's true that 'snow is white'" means precisely the same thing as "snow is white"--but putting "it's true that" at the beginning of a sentence already outside of quotation marks has no semantic impact at all. (It might add emphasis, but it doesn't change the meaning.) "(16) is true" has no independent meaning not supplied by whatever sentence (16) means. An obvious consequence of this view is that if "(16) is true" inherits no meaning from sentence (16), then it means nothing at all. Hence, if sentence (16) is "snow is white", "(16) is true" does nothing but attribute whiteness to snow, and if (16) is (16) is true, it means nothing at all. Putting a "not" in the mix is never, of course, enough to change a meaningless jumble of words into a meaningful one.

At the end of Part I, I considered one of the most worrying objections to this position. Isn't it manifestly possible to 'reason' about these sentences? Doesn't someone like me, who takes Liar sentences to be meaningless, come to this position on the basis of careful consideration of various other approaches to the paradox? Isn't part of the process of arguing for this solution going to be a matter of arguing against competing solutions, and won't that, in turn, be to a considerable extent a matter of arguing about "what follows" from various Liar sentences, in combination with added premises taken from a proposed solution? (For example: "You say that standard Liar sentences are meaningful but that they do not express propositions. What, then, about a sentence that says of itself that it does not express a true proposition? Surely, if it doesn't express a proposition at all, it doesn't express a true one, right?" or "I don't see how a gap theorist can get around the revenge paradox about a sentence that says of itself that it's either false or gappy" or "How does the dialetheist deal with a sentence that says of itself that it is just false and not true?") If one plays this game as well as anyone, doesn't that show that, like everyone else one grasps the meaning of the sentences in question? After all, isn't this game of generating unappetizing inferences from alternate solutions a matter of drawing out the entailments of the content of these sentences?

In Part II, I responded to this objection by pointing out that the same problem could arise for sentences that everyone takes to be meaningless, drawing out a scenario in which many people might infer contradictions from the sentence "Colorless green ideas sleep furiously" and ways that the hypothetical philosophical debate about this "Greenness Paradox" could closely mimic the actual philosophical debate about the Liar Paradox.

One possible problem here might have to do with distinctions among meaningless sentences. In the comments on Part I, ParisW suggested that there might be quantitative degrees of, or qualitatively different types of meaningfulness, and that different meaningless sentences might interact with logic in different ways. Now, I don't words in his mouth, and the line of thought would have to be developed a bit first anyway, but the general idea could be that it's not legitimate to pick out a meaningless sentence, show how it interacts with logic, and make sweeping generalizations about how Liars interact with logic or fail to do so if they are meaningless.

Now, personally I have trouble seeing how this could get off the ground--"means something"/"means nothing" look pretty clearly binary to me--but maybe the possibility of the "Greenness Paradox" is a consequence of "Colorless green ideas sleep furiously", despite its distinguished history as a stock example of a meaningless sentence, just isn't meaningless enough. How about "blorks geblork"? We could have the following conversation:

Person 1: "Blorks geblork!"
Person 2: "That's false."
Person 1: "So you think blorks don't geblork?"
Person 3: "That doesn't follow from what 2's point that it's not the case that blorks geblork. It could be that there are no blorks."

Clearly, there's something absurd about these three people going through the motions of reasoning about a combination of nonsense syllables. Equally, clearly, Person 1's gloss on Person 2's statement commits some sort of further mistake, correctly highlighted by Person 3's comment, and this further mistake is something that goes above beyond the basic category mistake of applying truth talk to a meaningless string of nonsense syllables and going on to "reason" about them.

Again, one might think this has something to do with degrees or kinds of meaninglessness. The string of nonsense syllables in question, after all, still has an apparent subject-predicate form, which is the entry point for all this.

Take a different example. Let's say I just sneezed. I've made a sound that doesn't sound at all subject-predicate-ish, and that no one who heard it and understood what it was would mistake for a claim about anything. Now, if some strange person did say that my sneeze was "true", a tempting way to correct them would be to say, "no no no, it wasn't true or false. Sneezes are just the wrong kind of thing to be able to count as either."

Once we combine a natural way of symbolizing the first sentence of my "correction" with two banally orthodox assumptions about truth and logic, we have all the ingredients of what we can call the Sneeze Paradox:

Premise 1: For every P, T(P) iff P..
Premise 2: For every P, T(P) or F(P).
Premise 3: Ben's sneeze (S) is neither true nor false. [~T(S) & ~F(S)]

By a series of relative simple steps I'll leave as an exercise for the reader, we get to:

Conclusion: S & ~S

One could imagine an (unlikely) scenario where no one ever figured out what was wrong here, and dialetheists used the Sneeze Paradox to argue for true contradictions, gap theorists used it to argue against Bivalence, sophisticated logicians with mostly orthodox premises found all kinds of ingenious ways of twiddling with or conceptually re-thinking the rational role of the logical architecture to avoid the conclusion and so on. Inevitably, various participants in this debate would make various reasoning mistakes.

Of course, we know that the at-bottom mistake underlying the whole debate is a nonsensical category mistake, not a factual mistake of any kind. If my statement "my sneeze wasn't true or false" is true, it's because I don't mean to literally assert the negation of the disjunction of the claim that it is true and the claim that it is false. If I'm talking sense in any sense, it's because what I really mean by the sloppy shorthand "it's not true or false" is "it's not the kind of thing to which 'true' and 'false' can be meaninfully applied, which is a different kettle of fish entirely. Still, on he way to realizing this, we'd doubtless want to nit-pick the arguments of the normal participants in the debate, catch them out on 'errors.'

So, how can we possibly conceptualize these 'errors'? And won't any analysis we give of the nature of these apparent "errors in reasoning" inevitably spawn new revenge problems for the meaningless solution, along the lines of (17)?

(17) Sentence (17) is one that one would have to ultimately label as "false" if one treated it as being meaningful and went through the motions of "reasoning" about it without making the sort of mistake we would regard in normal contexts as a mistake in reasoning.

To which all I can say is, stay tuned for the exciting conclusion of our quadrilogy to find out!

Singapore Talk & Revenge Problems for Dialetheism

2011-04-04T06:22:00.001-07:00

So I just bought a roundtrip ticket to Singapore.

Three weeks from tomorrow, I'll be delivering a talk at the Philosophy Department at the National University of Singapore (NUS) entitled "Liar Paradox II: Revenge of the Liar Paradox." In lieu of Part III of the ongoing series on the same subject, here's the abstract I sent them for the talk (and then a quick explanation of where I'm going with it):

Dialetheists like Graham Priest and JC Beall conclude from the Liar Paradox that sentences like “This sentence is not true” are fact both true and untrue, and that we must therefore revise our logic to accommodate the existence of true contradictions. Similarly, “paracomplete” theorists like Hartry Field avoid the contradiction posed by the Liar Paradox by rejecting one of the central elements of classical logic, the Law of the Excluded Middle. A more conservative solution starts from the claim that sentences that attempt to attribute truth or untruth to themselves are meaningless, and therefore simply not the kinds of things we can logically symbolize or apply truth talk to without committing a nonsensical category mistake. The most common objections to this move are (1) that the “meaninglessness solution” is refuted by the existence of “revenge paradoxes” like the one revolving around the sentence “This sentence is either false or meaningless”, and that (2) the sentences involved are so obviously meaningful that it’s just not possible to take seriously the claim that they’re literally meaningless in any ordinary sense, like “Blorks geblork” or “Colorless green ideas sleep furiously,” whereas the dialetheist and paracomplete approaches have the advantages that they (1*) make room for the perfectly obvious fact that, in any language with normal expressive resources, we can construct perfectly meaningful sentences that attribute untruth to themselves, and (2*) are immune to refutation by means of “revenge paradoxes.” I will argue that (1), (2), (1*) and (2*) are all completely wrong.

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On (2)/(1), of course, see Part I and Part II of the series of posts I've been doing on that here. (And, of course, stay tuned for Part III on Wednesday!) To get a sense of what I'm talking about on (1)/(2*), see (on the paracomplete side) here. On the dialetheist side, the problem, as I see it, is this*:

The dialetheist wants to argue that Disjunctive Syllogism fails to be universally truth-preserving (given true contradictions), and so it cannot be used to infer triviality after the dialetheist has embraced the contradictions entailed by the various paradoxes.

What does it mean to fail to be universally truth-preserving?

Given logical orthodoxy, the obvious answer is that an inference fails to be universally truth preserving iff there are possibilities on which the premises are all true and the conclusion is false. The dialetheist, obviously, can't conceive of it that way. Given the possibility of true contradictions, we can have possibilities where the premises of an argument are all true and the conclusion is false....but also true. On dialetheist assumptions, the mere fact that we've inferred a false conclusion from true premises is insufficient to establish that the the inference form fails to be universally truth-preserving.

Graham Priest's solution is to conceive of failures-of-truth-preservation as cases where the premises all 'relate to truth' and the conclusion fails to do so. (For technical reasons of his own, he prefers to think of truth as a relation rather than a function--so sentences 'relate' to truth or 'relate' to falsehood or both--but that's not really relevant right now. Talk of sentences relating to truth can translate to talk of them being true with no loss of nuance relevant to our discussion here.) The problem is that, obviously, it's always possible to come up with a Liar that says of itself that it fails to relate to truth. If normal Liars establish the possibility of sentences being simultaneously true and false, these anti-dialetheist revenge Liars should equally well establish the possibility of sentences simultaneously relating to truth and failing to relate to truth. Given this, establishing the possibility of all of the premises of an instance of Disjunctive Syllogism relate to truth while the conclusion fails to relate to truth should be not a single bit more relevant to showing that it fails to be universally truth-preserving than establishing the possibility of all the premises of an instance of Disjunctive Syllogism being true while the conclusion is false.

More generally, it looks like for any Status S such that the dialetheist could try to turn to in order to say "an argument fails to be universally truth preserving iff the premises are all true and the conclusion has Status S", we can always construct a revenge Liar of the form "This sentence has Status S" that will, on dialetheist assumptions, establish that true sentences can also have Status S. Once we've established this, it's not clear why Status S is a better candidate for a definition of failures-of-truth-preservation than mere falsity.

Many people have noticed the possibility of such sentences, shredding up distinctions dialetheists seem to find important--e.g. between sentences that are false-and-also-true and those that are "just" false ("This sentence is just false and not true")--and felt that in some way this was a problem for the dialetheist position on the Liar. Graham Priest, at least, has a standard response, found in multiple books and papers. He'll analyze the Liar sentence in question, show how the contradiction is derived from it, show how triviality fails to be entailed by the contradiction in his favored system of paraconsistent logic, and make some variation of a quip about how his point was never to avoid contradictions, but merely to contain them.

The problem, if I'm right about this, is that dialetheism's revenge problems don't just deliver more contradictions, they tear down the bars dialetheists want to use to contain them.

Some Objections to the Meaninglessness Solution to the Liar Paradox, Part II of IV

2011-03-30T00:12:00.000-07:00

Imagine a world where the predicates 'is green' and 'is colored' were considered much more philosophically interesting than it is in the actual world, interesting enough that philosophers and logicians worried about what formal rules related these predicates. One fairly crushingly obvious rule about them would what we can call the G-out rule, allowing us to infer 'X is Colored' from any instance of 'X is Green.'

Now, imagine that there was one other big difference between that world and this world. In our world, the classical example of a syntactically correct but clearly meaningless sentence is C:

C: "Colorless green ideas sleep furiously."

In the imaginary world, this sentence is treated rather different. The humans in this world have made contact with unfathomably intelligent alien entity, capable of speaking English (perhaps with the aid of a universal translator). Every time the entity has been asked a question, and it has deigned to answer, its answer has been proven correct. Sometimes it has taken humans many years, and full-fledged scientific revolutions, to understand *how* what the entity said could have been true, but in the end, there's never been any room for serious doubt. The entity has never once been shown to have (or even been widely suspsect to have) misunderstood a question. At some point, for some strange reason, someone asks the entity about C and it points to the paper where the questioner has written down C and says "this is true."

As often happens when the alien entity says something interesting, ripples of immediate change go through entire fields of study. A few philosophers think that in this case the entity got confused and make a strange sort of category mistake--after all, as in our world, any position, no matter how odd, always has a few philosophical backers--but there's a wide consensus now that C must be true (and therefore meaningful) after all. Almost immediately, some clever theorists notice that this revelation has created a new problem, which they call the "Greenness Paradox." Pretty soon, the dialetheists in this world seize on the Greenness Paradox as an argument for the existence of true contradictions. Here's how it goes:

Start with the formalization of C, given classical logic and orthodox assumptions about how to read the existential quantifier:

1: "There exists an X such that X is green and it is not the case that X is colored and X is an idea and X sleeps furiously."

It clearly follows from 1 that:

2: "There exists an X such that X is green and it is not the case that X is colored."

Apply existential instantiation to 2 to get:

3: "P is green and it is not the case that P is colored."

Apply conjunction elimination to 3 to get:

4: "P is green."

Apply our G-out rule to 4 to get:

5: "P is colored."

Apply conjunction elimination to 3 once again to get:

6: "It is not the case that P is colored."

Apply conjunction addition to 5 & 6 and we get:

7: "P is colored and it is not the case that P is colored."

.....which, the dialetheists of this world argue, is a true contradiction! Viola.

Of course, the dialetheist take on the Greenness Paradox isn't the only game in town. For example, one would imagine that a more conservative solution to the Greenness Paradox would be to deny "the naive theory of greenness" and to restrict G-out in some way. An obvious non-classical but non-dialetheist solution would be to deny that the existential quantifier is ontologically loaded after all. Proponents of this Meinongian solution to the Greenness Paradox would argue that some things can be true of colorless green things ideas without there being colorless green ideas. The Hofweber of this world will argue that, while the existential quantifier is ontologically loaded, and classical logic and the naive theory of greenness are true, and we shouldn't be so arrogant as to reject the superior wisdom of the alien entity by denying C, truth preservation should be understood in a generic rather than universal way. Just as "bears are dangerous" can be true without every bear being dangerous, "valid logical inferences are truth preserving" can be true even if not every valid inference from a true premise preserves truth. The Greenness Paradox, Bizarro-Hofweber would argue, shows us that the universal reading of the notion of truth preservation represents an airy "ideal of validity" that has an obvious appeal, but that the paradox falsifies the ideal.

At this point, it should be pretty easy to come up with a variety of other such philosophically sophisticated solutions to the paradox and to have a pretty good idea of how the argument between proponents of various competing solutions would proceed. Inevitably, some solutions would seem to work better than others, to contain hidden inconsistencies, and so on, and everyone, including the few extreme skeptics who didn't think the unfathomably intelligent alien entity at the source of all this was on the level when it uttered the words "this is true" about C, would be able to do so perfectly easily. "You try to solve it by saying that the colorless, furiously sleeping ideas are red rather than green, but red things are just as colored as green ideas, so you haven't gotten around the original problem." "You forgot a negation sign in Step 5. Once you add it in, you can see that a contradiction is entailed later, when you say...." Etc., etc., etc.

Now, imagine that they were right, and that the entity actually had the same take on C as Chomsky and most of the rest of us citizens of the actual world. He was simply messing with the puny humans out of boredom by pointing to a meaningless sentence and saying the words "this is true." He'd never done this before--he'd always given good and helpful responses to the rest of their inane little queries--but there's a first time for everything. Certainly, from the perspective of the humans, it's understandable that they would never catch on. Having been shown so many amazing things by the entity--remember, scientific revolutions are sparked off by it's statements on a regular basis--it seems utterly plausible to them that a sentence they thought was definitely meaningless actually has a meaning that their puny monkey minds cannot fully grasp. From there, given the function of phrases like 'is green' and 'is colored' in meaningful sentences (G-out is clearly a good rule), the equivalence of 'colorless' with the negation of 'colored', and the ways that we translate into logical lingo sentences of the form "X-ish things do Y", the apparent possibility of reasoning from a contradiction to Y.

Now, assuming that we and the imaginary aliens are right about C, we now have a problem. It is, in fact, the same problem we ended Part I with. We know that nothing "follows" from C. It's meaningless, not the kind of thing we can logically symbolize or apply truth-talk to without committing a nonsensical category mistake. The idea that anything really "follows" from C is deeply confused, like saying that something 'follows' from a string of nonsense syllables, or a bit of burning candle wax. Somehow, though, we seem to be perfectly capable of 'reasoning' about it, as we've been doing for the last few paragraphs.

In the beginning of Part I, I argued that the diquotationalist "nothing above and beyond" principle about truth--"to say that 'P' is true is to say nothing above and beyond P", or to put it differently, "to prefix a quoted sentence with the words 'it is true that' has the semantic effect of simply removing the quotation marks" (the claim, remember, from which the word "disquotationalism" is derived)-is best explained by a general view that the truth predicate/operator is only parasitically meaningful. Of course, the original claim is about sentences that ascribe truth to sentences quoted within them and my claim broadens this to all ascriptions of truth, but I would argue that the former claim, in the absence of the latter, has some awkward consequences. For example, consider the following three sentences*:

(11) "It's true that 'snow is white.'"
(12) Sentence (13) is true.
(13) Snow is white.

I'd submit that there's something a bit strange about arguing that (11) and (12) have distinct meanings. If one asserts meaning-parasiticalness for sentences that ascribe truth to an internally quoted sentence and rejects meaning-parasiticalness for sentences that ascribe truth to other sentences in other ways, one has to explain what substantive difference the *method* of applying the predicate/operator to the claim to which truth is being ascribed makes. Moreover, the obvious explanations of *why* the "nothing above and beyond" principle would be true--most obviously, general philosophical stories like "the word 'true' doesn't pick out some substantive feature of the world, but rather functions as a time-saving way of saying other things, especially useful for cases where we aren't entirely sure *what* we're saying (i.e. blind endorsements)"--would seem to apply equally well, to sentences like (11), sentences like (12), to sentences like "everything John just said is true", to the one-word exclamation "true!" uttered in response to something one's friend has just said, and so on. The syntactic form the truth-ascription takes seems to make no difference. All sentences that do nothing but ascribe truth to a sentence inherit their meanings from the meanings of the sentences to which they ascribe truth. If a sentence S1 tries to ascribe truth to another sentence S2 that has no meaning, S1 will have no meaning either. It has nowhere to get it.

A happy consequence of this view is that, given some other plausible assumptions (e.g. that adding the word "not" to a meaningless sentence does not convert it into a meaningful one), it entails that sentences like "this sentence is not true" are meaningless. This lets us solve the Liar Paradox without having to give up on "the naive theory of truth"--a unitary truth predicate obeying all the standard rules about truth, etc.--or the unrestricted power of classical logic, or much of anything else except many people's initial intuition that the sentences involved are meaningless. At the end of Part I, though, we confronted what sounds like a serious problem:

Someone like me, who says that Liars are meaningless, has presumably been convinced of it by prolonged reflection on the paradox. In the course of this, they've sifted through various possible diagnoses of the sentences in question, thinking about consequences of various approaches, objections to failed solutions and so on. Right? Well, then, wait a damn second. Doesn't all of this involve reasoning about what does and doesn't follow from these supposedly meaningless sentences, in conjunction with various other claims. For example, to embrace the meaninglessness analysis is to reject the analysis that says that Liar sentences are meaningful but that they don't express propositions. Presumably, in explaining why the meaninglessness analysis is superior, its partisans want to bring up "revenge paradoxes" like (8). (At any rate, I certainly want to bring it up!)

(8) The sentence marked as (8) does not express a true proposition.

If (8) doesn't express a proposition, it doesn't express a true one, just as if a cat isn't a dog, it isn't a black dog. And anyone who endorsed the meaningful-but-not-expressing-a-proposition analysis presumably doesn't think a sentence can be true without expressing a true proposition--after all, if truth can exist without propositions, why clutter one's ontology with them? Thus, the solution under consideration collapses into contradiction.

Now, while I tend to lean skeptical on the subject, I'm officially agnostic about the existence of propositions. I take its neutrality on this topic to be a big selling point of my preferred approach. (For the sake of simplicity, I usually talk about "sentences", but wherever I talk about "sentences" being true or false, an enthusiast for propositions can always mentally subsitute some phrase about the propositions expressed by those sentences being true or false...and, of course, presumably, if propositions exist at all, only meaningful sentences can express them, so if I'm right that Liars are meaningless, it follows that they don't express propositions any more than bits of burning candlewax express propositions.) If, however, I abandoned my agnosticism in favor of a full-throated embrace of propositions, I'd presumably be forced to classify (8) as meaningless as well. (If I abandoned it in the opposite direction, matters would be quite different. After all, if there are no such things as propositions, it's true of every sentence that it doesn't express one!) Certainly, I view more common revenge paradoxes, like (9):

(9) The sentence marked as (9) has some status other than 'true.'

....or the familiar anti-dialetheist revenge paradox (10):

(10) This sentence is just false, rather than being both true and false.

.....as being meaningless, and still deploy them against the approaches to the paradoxes that I reject, using standard Liar reasoning, like everyone else does. Doesn't the fact that I'm able to play this game as well as anyone else, that we all understand and can use the rules against each other, proof that the sentences are meaningful, that, after all, we all understand what they mean?

Now, there's a lot more to be said about all this--particularly about the thorny question of what sort of mistake someone can be accused of when they 'reason' about something meaningless in a 'bad' way and 'contradict' themselves about it, above and beyond the original sin of treating the something in question as if it were meaningful--but I take the example at the beginning of this post to pretty definitively answer the question I ended the last post with in the negative. Someone who (as we would all agree here in the actual world, correctly) characterized C as meaningless would be faced with precisely the same problem that a pardadox-solver who takes the Liar to be meaningless is faced with in our world. Although it's still somewhat unclear *why* the objection doesn't work in either case--we'll say more about that--it's failure in the closely parallel imaginary Greenness Paradox case would seem to show that it fails when it comes to the actual Liar Paradox as well.

Diet Soap Interview & Apologies

2011-03-28T02:10:00.000-07:00

I'm once again putting off Part II. I've written about half of it, but (i) I have tests to write and classes to prep, and (ii) I just finished writing what, on copying and pasting all of my comments into a Word file and running a word count, turned out to be a bit over 2500 words in response to Colin, Brandon and ParisW's thoughts and objections to Part I. If you're desperate for more material on Liars and meaninglessness, I'll direct you to that discussion-in-progress. Meanwhile, I'll mention that this last weekend I was interviewed for a second appearance on the philosophy-themed Diet Soap podcast. Sounds like I might Episode 100. We didn't really skip to this script (plenty of questions not on the list, not all of the list questions asked), and in any case the interview lasted long enough that only a fraction of it should survive the cutting process and make it into the podcast, but to give at least an approximate flavor of the interview, here are the questions that host Doug Lain sent me in advance...

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You have a doctorate in philosophy and you specialize in philosophical systems of logic. As an American philosopher and a logician it strikes me that you'd fall in with Analytic philosophers. Is this correct?

How do you consider the division or distinction between continental and analytic philosophy?

How important is Frege, Russell and Wittgenstein to you and your philosophical work.

Do you hold to a deflationary account of truth claims?

It seems to me that Analytic philosophy might share something in common with instrumental reason. That is, that the deflationary accounts of truth claims have the impact of limiting our ability to challenge the logic of our historical moment or culture, whereas Continental philosophers like Hegel and Neitzche were primarily interested in thinking about how philosophy was tied to culture and history.

A More Succinct Proof

2011-03-23T23:58:00.000-07:00

No time for a second post in the series on the meaninglessness solution to the Liar Paradox just now--my apologies, dear reader, but you'll have to wait until Monday for that--but, to reward you for checking back for one, here's a link to a comic that the sort of people who read this blog might enjoy.

(And, on the other end of P.F. Snow's 'Two Cultures', this one's good too.)

Some Objections to the Meaninglessness Solution to the Liar Paradox, Part I of IV

2011-03-21T23:59:00.000-07:00

Elsewhere (and in my dissertation), I've argued at length that "Liar sentences", like:

(1) The sentence marked (1) is not true.

or

(2) The sentence marked (2) is either false or meaningless.

...and, for precisely, the same reason, "Truth-Teller" sentences, like:

(3) The sentences marked (3) is true.

....and, of course, conditionalized truth-tellers (better known as "Curry sentences"), like

(4) If this sentence is true, the author of the blog post it appears in is a dialetheist.

.....are quite literally meaningless. "Wait," I can hear you asking, "doesn't that make (2) true?" I've written extensively about that question in the past, but the short answer is "no." A sentence with the grammatical form of a disjunction and a "second disjunct" that, if the same words in the same order were split off into a sentence of their own, would constitute a meaningful-and-true sentence, does not thereby become a meaningful sentence, much less a true one. For example, take (5), adapted from the classical example of a meaningless-but-"well-formed" sentence:

(5) Either colorless green ideas sleep furiously or snow is white.

There is, clearly, no contradiction in asserting both (6):

(6) (5) is meaningless.

....and (7):

(7) Snow is white.

....at the same time. Now, this is a very unpopular solution to the paradoxes--which is part of what makes it interesting enough to spend years developing arguments for!--but one which there are few extensive arguments against. Many theorists interested in the paradoxes--especially those interested in non-classical approaches--just brush it off out of hand as not worth taking seriously. Graham Priest derisively refers to it in In Contradiction as "the heroic solution." Hartry Field says in the introduction to Saving Truth From Paradox that people who endorse meaninglessness solutions must mean the term "meaningless" in "some special technical way", so that what they're saying must amount to a strangely-expressed version of his own paracomplete solution.

(I've always tried to be clear that I mean the word "meaningless" is precisely the ordinary mundane sense. As a result of my version of extreme deflationism about truth, I take the sentences that JC Beall calls "TTruth-inelimable" to be literally meaningless in precisely the same sense as a string of nonsense syllables, or "Colorless green ideas sleep furiously." Click on the link above for a less abbreviated explanation, but, basically, I agree with and take literally Quine's claim that sentences that ascribe truth to other sentences mean nothing above and beyond what the original sentences mean--that's the original metaphor behind the term "disquotationalism," that the upshot of prefixing a quoted sentence with the words 'it is true that' is to "remove the quotation marks"--and I generalize this to the claim that all truth-ascribing sentences necessarily inherit their meaning from the sentences to which they ascribe it. Thus, for example, "'colorless green ideas sleep furiously' is true" ends up being meaningless, because it inherits no meaning from the sentence to which it tries to ascribe truth. For precisely the same reason, "this sentence is true" is meaningless. And, of course, as Carnap was fond of pointing out, the negation of nonsense is nonsense.)

In the same spirit as Field's disguised-paracompleteness objection, when I met a regular reader of this blog, at the Eastern APA before last, we chatted about the Liar Paradox and he said he'd have to wait to "see the technical details" before he knew if it would "work."

I have, of course, a philosophical argument for the claim, and a lot of responses to various actual and potential objections, by the very nature of the solution, there aren't and can't be any "technical details." (There's plenty of nit-picky precision work--particularly when it comes to formulating and responding to "revenge paradoxes"--but that's not what most Liar specialist mean when they talk about "technical details.") The necessary absence of technical details strike right at the heart of the difference between the meaninglessness solution and more standard ones--that nothing technical needs to be revised in any way, shape or form on account of the semantic pardoxes is one of the chief selling points of the solution! We get to keep "the naive theory of truth" rather than any of the elaborate 'technical' theories that have proliferated in the post-Tarski/post-Kripke era. We get to keep classical logic, classical T-in and T-out rules, and, in short, we get to keep everything except for the intuition that many professional philosophers report having about the semantic status of the sentences in question.

So no, no "technical details" of the kind fashionable in theories of the Liar. There are not and could not be special rules (whether thought of as logicially revisionary or placed 'on top of' the logical edifice regulating particular predicates or operators related to truth or meaninglessness) about, say, the precise behavior of M(P) and ~M(P), because, if a sentence is meaningless, to symbolize it with a letter and trying to perform logical operations on it is to commit the same nonsensical category mistake which would be committed if some very confused logician tried to do the same to a cough or a string of nonsense syllables or a bit of burning candle wax.

The most common argument against the meaninglessness sentence is a simple foot-stamping appeal to intuition. Sadly, X-phi has not yet provided us with any empirical evidence about how widely shared the intuitions in question are, so it's hard to know whether those who take it as obvious that such sentences are meaningful are right when they assert that it's generally obvious to everyone pre-philosophically, but whether they're right or wrong, it's clearly possible for competent speakers of a natural language to be mistaken about questions of meaningfulness. For example, the philosophers of the Vienna Circle were competent speakers of German, but they mistakenly took many perfectly meaningful German sentences about metaphysical subjects to be meaningless. In fact, even if we *wanted* to be semantic Cartesians, holding idealized views about the privileged access of competent speakers to the status of sentences as meaningful or meaningless, we couldn't, because there are disputes in which, whoever is right, someone is a competent speaker making this mistake. For example, Graham Priest and I are both competent speakers of English, and we disagree about the meaningfulness of Liar sentences. Whichever one of us is right, the other one is a competent speaker of a natural language who has made a mistake about meaningfulness.

Of course, there's nothing wrong with appeals to intuition--we can hardly do without them entirely--but, given a good argument and a good error theory, initial intuitive assessments are often shown to be false. Arrogantly enough, of course, I take myself to have both.

What about, however, the following more sophisticated variant on this sort of objection? (It was presented to me by a junior faculty member at the University of Miami a year or so ago, and I don't think I took it seriously enough at the time.) Someone like me, who says that Liars are meaningless, has presumably been convinced of it by prolonged reflection on the paradox. In the course of this, they've sifted through various possible diagnoses of the sentences in question, thinking about consequences of various approaches, objections to failed solutions and so on. Right? Well, then, wait a damn second. Doesn't all of this involve reasoning about what does and doesn't follow from these supposedly meaningless sentences, in conjunction with various other claims. For example, to embrace the meaninglessness analysis is to reject the analysis that says that Liar sentences are meaningful but that they don't express propositions. Presumably, in explaining why the meaninglessness analysis is superior, its partisans want to bring up "revenge paradoxes" like (8). (At any rate, I certainly want to bring it up!)

(8) The sentence marked as (8) does not express a true proposition.

If (8) doesn't express a proposition, it doesn't express a true one, just as if a cat isn't a dog, it isn't a black dog. And anyone who endorsed the meaningful-but-not-expressing-a-proposition analysis presumably doesn't think a sentence can be true without expressing a true proposition--after all, if truth can exist without propositions, why clutter one's ontology with them? Thus, the solution under consideration collapses into contradiction.

Now, while I tend to lean skeptical on the subject, I'm officially agnostic about the existence of propositions. I take its neutrality on this topic to be a big selling point of my preferred approach. (For the sake of simplicity, I usually talk about "sentences", but wherever I talk about "sentences" being true or false, an enthusiast for propositions can always mentally subsitute some phrase about the propositions expressed by those sentences being true or false...and, of course, presumably, if propositions exist at all, only meaningful sentences can express them, so if I'm right that Liars are meaningless, it follows that they don't express propositions any more than bits of burning candlewax express propositions.) If, however, I abandoned my agnosticism in favor of a full-throated embrace of propositions, I'd presumably be forced to classify (8) as meaningless as well. (If I abandoned it in the opposite direction, matters would be quite different. After all, if there are no such things as propositions, it's true of every sentence that it doesn't express one!) Certainly, I view more common revenge paradoxes, like (9):

(9) The sentence marked as (9) has some status other than 'true.'

....or the familiar anti-dialetheist revenge paradox (10):

(10) This sentence is just false, rather than being both true and false.

.....as being meaningless, and still deploy them against the approaches to the paradoxes that I reject, using standard Liar reasoning, like everyone else does. Doesn't the fact that I'm able to play this game as well as anyone else, that we all understand and can use the rules against each other, proof that the sentences are meaningful, that, after all, we all understand what they mean?

To which I say.......

Good question. Tune in on Wednesday!

Michael Sandel

2011-03-16T00:01:00.000-07:00

One of the classes I'm teaching this year is, basically, a political philosophy class for Sociology majors. Following the Sociology Department's recommendation, I'm assigning Michael Sandel's book "Justice: What's The Right Thing To Do?"

It has some important secondary advantages--e.g. it's available in Korean translation--and, to be fair, it's reasonably well-written. Sandel uses lots of nice, vivid historical examples. But in some ways.....Jesus.

To steal a line from Jay Rosenberg, Sandel's critique of utilitarianism commits genocide against an entire race of straw men.

What's Wrong With Orange County

2011-03-07T03:59:00.000-08:00

Here.

Graham Priest Interview, Part II

2011-03-02T01:07:00.000-08:00

I talked about Part I on Monday. Part II just went up. My questions were the last four included. I also contributed the clarification to the next-to-last question, obviously.

There's a lot of interesting stuff here, a good bit of which I haven't really had a chance to digest yet. One thing, however, does jump out at me immediately as a problem:

His answer to my clarification on the next-to-last question would seem to fly in the face of any intuitive understanding of the notion of 'truth-preservation.'

For background, click through to the interview. The question provides a lot of detailed background on this. "ArT" means "A relates to Truth", which is a fancy way of saying "A is True." The idea here, as Priest has explained in other contexts, such as his article What is so bad about contradictions?, is that truth is conceived, not as a function, as classical logicians understand it, but as a relation, such that a proposition can be related to truth, to falsity or to both. In What is so bad about contradictions?, he includes a fourth option--A is related to neither truth nor falsity--but that option would seem to made superfluous by his arguments against the possibility of truth-value gaps in In Contradiction, and in any case the existence or non-existence of the fourth option isn't relevant to this discussion. DS is, of course, Disjunctive Syllogism, the classical inference from ~p and (pvq) to q. Since DS, plus the dialetheist's claim that p and ~p can sometimes both be true, quickly generates triviality, Priest and other dialetheists reject it. Priest's argument is, basically, that it isn't universally truth-preserving (and hence, isn't valid) because, given the assumption that some (but not all) contradictions are true, there can be cases in which ~p is true and (pvq) is true but in which q is not.

As he says in his response to me, "The DS can be show to be invalid is the semantics of LP as follows. (The semantics has many presentations. Let us use the version in which evaluations are relations, R, between formulas and the values t and f.

"Consider the inference ~p, pvq / q. Take an interpretation where pRt, pRf, qRf, and it is not the case that qRt. By the truth and falsity conditions for negation and disjunction, (~p)Rt and (pvq)Rt. Hence there is an evaluation where the premises of the inference relate to t and the conclusion does not. Hence the inference is invalid."

"Note that this argument...[is not] undercut if it turns out that there are formulas, A, such that ARt and it is not the case that ARt - even if you could show by some argument (goodness knows what), that this held when A is the p in question. Deductive reasoning is, after all, monotonic. (Valid arguments are never made invalid by the addition of extra premises.)"

So, why do I think all of this flies in the face of any intuitive notion of 'truth-preservation'?

Well, first of all, it seems to me that Professor Priest is being a bit coy when he speaks neutrally about the possibility of "ArT" and "it is not the case that ArT" being shown to be compatible--"if it turns out that there are formulas, A, such that ARt and it is not the case that ARt..." Given Priest's assumptions, *of course* there are such formulas! After all, we can always construct a sentence A such that A="It is not the case that ArT."

Secondly--and to the point--given that such formulas would seem to have to exist on Priest's account, when he's told us that there's an interpretation on which (~p)rT and (pvq)rT but it's not the case that qrT, he hasn't precluded the possibility that qrT--in other words, he hasn't precluded the possibility that, in this case, as in all other cases, true premises, fed into DS, generate a true conclusion!

Think of it this way--Priest would not claim that the mere existence of a case in which (~p)rT and (pvq)rT but in which qrF constituted a counter-example to DS, right? Given that it's as easy to generate a formula that both does and does not relate to truth as it is to generate one that relates to both truth and falsehood, why should truth-preservation be any more violated by the existence of a case in which q doesn't relate to truth than by a case in which it does relate to falsehood?

Graham Priest Interview, Part I

2011-02-28T00:06:00.000-08:00

A few weeks back, Edgar Aroutiounian told me on Facebook that he was planning to interview Graham Priest for the Florida Student Philosophy Blog, and he asked me if I had any questions I'd like asked. I gave him some, then blogged my questions here, since I figured they were detailed enough to double as a pretty decent (if incomplete) snapshot of a lot of my objections to Priest's version of the dialetheist project. Anyway, the interview's been split into two parts, and my questions are all in the second part, which hasn't been posted yet, but Part I is available here.

Most of the questions in Part I are relatively light and biographical in nature (nothing wrong with that--some of his answers are quite interesting), but the most philosophically interesting question was the last one, a somewhat confusingly-worded question about "consistent physicalism." Priests answer included the following passage:

"Functionalism, and materialist views of the mind in general, of course have problems. The most obvious is what to say about 'raw feels' (though the problem of intentionality is also a hard one). There are different possibilities about what to say about this. I guess that most of them are consistent, but how adequate they are is much debated. (I’ve never heard anyone suggest that dialetheism might help with the matter.)"

........which, of course, amused me because I do know someone who has publicly suggested just that!

More seriously, though, Ryan's comic raises a good point:

Why on earth hasn't Priest or anyone else floated a dialetheic theory of mind and the (ir)reducibility of raw feels to functional states? The arguments for both halves of the relevant contradiction are independently extremely powerful and compelling (and often felt to be that way even by philosophers who unambiguously put themselves in one or the other camp), the problem has been with us for a long time, and it seems to exhibit much the same sort of intuitive intractability as Priest's favorite paradoxes.

Happy Valentine's Day!

2011-02-14T17:33:00.000-08:00

Cheerful V-Day information, via Brian Leiter:

"[A]ccording to a new report approximately 60 percent of all flowers sold in the United States come from Colombia. A third of Ecuador’s yearly production is exported to the U.S. for Valentine’s Day. Flower workers in these countries earn poverty-level wages, work long hours, and suffer significant health problems due to pesticides. The report also finds that over half of women workers in the flower industry in Colombia and Ecuador have been subjected to sexual harassment...."

Now that you're in a nice romantic mood from reading about that, enjoy the rest of your night!

Getting Back On Track/Questions for Graham Priest

2011-02-09T19:59:00.001-08:00

Well, I won't bore anyone with excuses for the long unplanned blogging hiatus. First year as a full-time prof, adjusting to life in the far East, yadayada. You know the drill. Anyway, I'm going to try like hell to get back to a regular Monday/Wednesday schedule here.

Anyway, last week, a grad student I'm friends with on Facebook told me he was going to be interviewing Graham Priest soon, and asked if I had any questions to suggest. It occurred to me that the questions I came up with pretty much double as explanations of a lot of my main objections to Priest's version of the dialetheist project, so I could do worse than just re-post them here by way of new content. Here goes, copied and pasted from my Facebook message:

#

I'm not sure what kind of interview we're talking about here, but if it's OK that they be a bit long-winded (I'm trying to be very careful about spelling out the ...assumptions to maximize the chances of getting philosophically interesting answers), here are my top 4 questions:

#

Dr. Priest,

(1) When it comes to giving similar paradoxes "uniform solution," you've endorsed five different claims that seem to be in tension with each other:

(a) The Principle Of Uniform solution dictates that all paradoxes of the same "type" be solved in a uniform fashion, &

(b) That the Inclosure Schema delineates a "type," and indeed

(c) That, if someone were to embrace one of the standard consistent solutions to the Liar Paradox but get around Russell's Paradox by an appeal to mathematical nominalism, then the POUS would be violated. Moreover, you've granted that:

(d) The Barber Paradox can be seen to fall under the Inclosure Schema. (It would be surprising if this were not so, given that it was invented to illustrate the structure of Russell's Paradox, which is in turn one of your favorite IS paradoxes!) Despite this, you've argued that:

(e) The POUS does not dictate that we solve Barber in the same way as we solve the main IS paradoxes.

You have justified (e) by saying that it is not enough that a proposed paradox structurally conform to the IS, but also that we have good reason to think that all of its premises are true. (You very reasonably deny that we have any good reason to believe in the existence of a barber who succeeds in shaving everyone in the town in which he lives who does not shave himself.) Why, however, couldn't the mathematical nominalist say precisely the same thing about the Russell Set (since the nominalist denies the existence of sets in general!), use the various standard arguments for nominalism--Benacerraf, etc.--to deny the Existence component of Russell's Paradox in a non-question-begging matter, and thus be perfectly entitled by your own standards to solve the Liar Paradox in a different way, without thus violating the POUS?

(2) On the same subject--Let's assume that the IS does delineate which paradoxes are "of a type" and thus must be given uniform solution. You've argued (quite plausibly) that "evading the Schema" isn't sufficiently fine-grained to satisfy the requirement of uniform solution, while your own dialetheist solution does. On the other hand, on the level of abstraction at which the Schema operates, wouldn't someone who denied the Existence component of Russell's Paradox for nominalist reasons, the Existence component of the Liar Paradox on the basis of considerations derived from their favored views about the philosophy of language and so on be just as "unified" as the dialetheist, who, after wading through various arguments about the particulars of each case, embraced all three Schema components (Existence, Closure and Transcendence) in every case?

(3) You have argued in various places that Disjunctive Syllogism is not universally truth-preserving, because it has counter-examples--cases where P is both true and false, making (P v Q) and ~P true, but in which Q fails to be true. Given the importance of rejecting rules like Disjunctive Syllogism to your overall case for dialetheism (after all, a dialetheist who thought Disjunctive Syllogism *was* universally truth-preserving would be a trivialist!), it might seem to be a a problem for your view that (a) the argument just sketched out relies on a distinction between false claims that are also true and false claims that are just false, but (b) as you are, of course, aware, many critics have pointed out that any phrase that one devises to express this distinction can be recycled in fresh paradoxes (e.g. "this sentence is just false and fails to be true", etc.) Some dialetheists, like JC Beall, lean heavily on the vocabulary of acceptance and rejection to get around these sorts of problems. (For example, in "Spandrels of Truth," he constantly uses the language of rejection to distinguish dialetheias from ordinary falsehoods.) This move is, however, not available to you, given your argument in "Doubt Truth To Be A Liar" that dialetheists should accept that the grounds for rational rejection and rational acceptance might sometimes overlap. One might think this concession deprives you of your last available tool for expressing the distinction needed for your argument against the validity of Disjunctive Syllogism. Do you see this as a problem?

(4) Your argument for the "classical re-capture" in "In Contradiction" relies on the notion that the statistical frequency of true contradictions is very low, and in particular that few statements that arise in ordinary contexts can reasonably be thought to be dialetheias. Elsewhere in the same book, however, you argue for a paraconsistent theory of change, whereby (a) as in standard tense logic, statements truth-values change over time, and more radically that (b) at any point where the subject of a statement is changing from being the way the statement asserts that it is to not being that way or vice versa, the statement is both true and false. (You formally express (b) as Zeno's Principle.) Given that theory of change, and the fact that, as Heraclitus and Engels are quick to remind us, change is a constant, pervasive feature of practically all discernible reality, doesn't it suddenly seem quite plausible that ordinary statements are dialetheic, not just in slightly contrived cases like contingent Liars or Kriple's Nixon case, but in a wide variety of contexts? If I say "the cat is on the mat" while the cat is on the mat, won't that statement be both true and false at the inevitable moment when the cat is in the process of departing from the mat? Won't, indeed, a large, stastically significant number of ordinary statements be both true and false at any given time? (One might think that, given all this, the one domain of reliably contradiction-free statements would be the domain of statements about changeless things. Historically, perhaps, the most popular candidate for changeless truths would be the mathematical one, but of course, you postulate all sorts of contradictions there as well!) In light of all this, how can we be confident that the frequency of true contradictions is very low?

Graham Priest in Yesterday's New York Times

2010-11-29T00:01:00.000-08:00

So I was delighted to see my dissertation topic in yesterday's New York Times, in the form of Graham Priest's article Paradoxical Truth.

Further comments to follow.

Political Chaospet Comic

2010-11-17T00:01:00.000-08:00

Ryan's write-up of our post-election Facebook discussion is awesome, and I say that even though I was pretty clearly on the losing end of the argument.

Definite Descriptions and Problems about the Philosophy of Time

2010-11-15T00:01:00.000-08:00

The relational theory of time says that a moment is just a bundle of co-occuring events. 4:20 PM on April 20th of the year 420 B.C.E. was nothing above and beyond the events that occurred then. The substantival theory says that times are something like independently existing abstract entities that happen to contain various events but could have contained entirely different events, or perhaps even no events at all.

(The suggestion that there could be "empty times," where nothing changed and nothing happened is incoherent on the relational view.)

The standard Benaceraff-style objections to abstract objects would all seem to apply to substantival moments. (If they all disappeared tomorrow, how would we ever know the difference? Given that a human being having an intuition is surely ultimately a certain sort of neurological event, explainable in terms of some mixture of biological hard-wiring, socialization, and so on, and that an abstract object could play no role in the causal chains leading up to such events, why on earth should we think that our intuitions track its true properties?, and so on.) Given that, what possible argument could there be for the substantival view?

Quentin Smith has argued in many places for the substantival view on the grounds that some counterfactual statements about time seem to be true. If I arrived on time to the meeting at 4:20 PM, it still seems to be true that, for example, if I had gotten pulled over by a traffic cop on my way to the meeting, I would have been late. Put differently, if I had been pulled over by a traffic cop on my way to the meeting, the moment 4:20 PM would not have included the event of my showing up at the meeting. It would have, indeed, included entirely different events, like people asking each other why I wasn't there. Intuitively, all of this would have happened at 4:20. But wait! If 4:20 just *is* a set of co-occurring events, then it wouldn't have existed in the possible world where I didn't show up on time to the meeting. For some such counterfactuals to be true, it seems, moments need to have a separate existence above and beyond the things that happened at them.

However unattractive the idea of substantival moments might be to me, this has always seemed to me like a pretty tricky problem for the relational theory. Now I'm not so sure.

Notice, though, that if one sees time-terms ("4:20," "yesterday afternoon" and so on) as shorthand definite descriptions, like "that bundle of events which includes Event E" or even "that bundle of events that's related in such-and-such order to the bundle of events that includes Event E," the problem goes away. Smith, of course, who's written extensively on phil of language, realizes that this is a way out. I don't have the link handy--his website seems to be down at the moment--but in a short article on this a while back, he presents the objection as a dilemma between rejecting the relational theory of time and rejecting the "New Theory of Reference" (i.e. the Kripkean* theory of proper names as rigid designators). To become a full-fledged argument for a substantival view, one needs compelling arguments for the Kripkean view and against the old Russellian descriptivist view. Smith, of course, like a great many people, takes it that we have such comeplling arguments.

Fair enough, but, the more I think about this, the less convinced I am that the dilemma really exists. There's only a dilemma if one thinks that *all* proper names and proper-name-like terms rigidly designate, and one can certainly be a Kripkean about reference without making that sweeping universal claim. For example, Kripke himself never claimed it and has always explicitly disowned it, being careful to emphasize all along that he was making a general claim about most normal uses of most normal proper names and nothing more.

Of course, Kripke could be mistaken in holding back from the universal claim, but there are good reasons, quite independent of the Phil of Time issues under discussion, to think that he isn't. For one thing, taking names that don't pick out any existing object--e.g. "Santa Clause"--as rigid designators (of what?) raises all kinds of sticky problems. Of course, that itself raises all kinds of deeply controversial philosophical issues, but more banal and uncontroversial examples are plentiful. To pick an obvious one (not original to me), "Jack the Ripper" is a lot more like an archetypal ordinary proper name than "4:20 PM" is, but "Jack the Ripper" pretty clearly really is a shorthand definite description, along the lines of "whoever murdered and mutilated the bodies of those prostitutes in London." Given (a) the incompatibility of the package of the relational view and the assumption that time words rigidly designate on the one hand and the apparent truth of some temporal counterfactuals on the other hand, and (b) the obvious epistemic and ontological parsimony-based objects to the substantival view, it seems to me that (c) even if one adopts a rigid designator view of most ordinary proper names--"Quentin Smith", "Saul Kripke", etc.--it's a mistake to extend that analysis to time terms.

*Smith himself probably wouldn't call it that. He has heterodox views on history of the philosophy of language that would lead him to think of it as the Marcusian theory--he and Scott Soames have killed a lot of trees with their essays arguing back and forth on this issue--but we can put that historical dispute to one side for the moment.

The content of the body of this post is true.

2010-11-10T23:45:00.000-08:00

The content of the title of this post is false.

XKCD

2010-11-08T22:22:00.000-08:00

Funny

....particularly the roll-over.

Reading the Principia Mathematica

2010-11-03T09:20:00.001-07:00

....I'm remembering that, every time I go a while without reading much Russell, I forget just how smart Russell was. (The classic example is that most people don't notice that Russell anticipated and responded to the Gettier problem as far back as Problems of Philosophy.) In any case, predictably, given my interests, the most interesting part of the early chapters, for me, is his take on the Liar. People often tend to discuss his (and Whitehead's) type theory only as a solution to the set-theoretic paradoxes, but even if one rejects it for those purposes, their type-theoretic solution to the semantic paradoxes remains independently interesting.

I think this solution is actually fairly interesting. I don't think he's right about the roots of the problem, but the solution has a lot going for it. (In some way, in fact, I find it a whole lot more attractive and plausible than some still-popular approaches to the Liar, like Kripke's.) It's extremely "unified," solving a lot of other problems at the same time, it doesn't require the rejection of any intuitively compelling logical principles, or any instances of the T-Schema (or the Capture and Release rules), it doesn't require any weird principles about certain sentences not being "contructable," it fits everything together in a tight, coherent framework, and it has interesting consequences for such apparently distant subjects as epistemic skepticism.

Again, all that said, I don't entirely buy it.

....but do expect a series of posts on Russell's Take On The Liar Paradox in the next week or two.

How JC Beall's Version of Dialetheism Tries (And Fails) To Escape Triviality (And How the Account Could Be Fixed)

2010-11-01T00:01:00.000-07:00

I've had this written up for a different purpose for a while now, so I thought I might as well stick it up here.

A couple of quick points before getting into the meat of this:

Some posts here (e.g. the anti-theist polemics, or last week's post about Zizek) are intended to be accessible to readers with a fairly minimal philosophical background. Others (e.g. my post a couple weeks ago about revenge paradoxes for paracomplete solutions to the Liar Paradox), engaging narrower issues, tend to assume intimate knowledge of the debates I'm most interested in (e.g. dialetheism, paracompleteness, logical pluralism, and so on)--in the case of those sorts of posts, I'm generally trying to make a quick point about something fairly specific, and starting with a general overview of the relevant debates each time would quickly get repetitive, and would probably be annoying for those readers most likely to be interested in the quick technical point. Sometimes I might not always clear enough about flagging which are which.

In any case, because of the nature of the specific objection to Beall I'm making here, this post is somewhere in between these two categories, but it's definitely closer to the latter than it is to the former. If that's not your cup of tea, be forewarned.

A more philosophically substantive note:

Also because of the specific nature of the criticism being lobbed here, I'm willing to just follow Beall in a lot of his philosophical machinery for these purposes. Some of that machinery (a generally disquotationalist attitude toward truth complete with unrestricted Capture and Release, a commitment to the revisability of logic, etc.) is stuff I agree with him on in any case. On the other hand, there are some important points of disagreement you could miss from all of this. One is that, at least in Spandrels of Truth, he seems to take the laws of logic as something like deep rules of language--his precise conception is often a bit unclear--while I take a more traditional approach, seeing them as laws of universal truth-preservation, meaning that one's take on which laws of logic are valid ultimately amounts to something like an overall theory of reality in general (at the level of generality and abstraction at which logical laws operate). Another important difference I have with him is that I'm deeply skeptical that one of the key distinctions that Beall makes a big fuss about when it comes to truth--"natural" properties vs. "constructed" properties--tracks any sort of real distinction whatsoever, or illuminates much of anything.

In any case, putting those larger issues aside, let's get to the part about Beall's trivialism problem:

I. The Shape of the Problem

In Spandrels of Truth, JC Beall endorses a dialetheist solution to the semantic paradoxes on the basis of a “transparent disquotationalist” account of truth. According to him, truth always plays Capture and Release. (Capture is the inference from P to Tr(P) and Release is the inference from Tr(P) to P. Given that it satisfies these intersubstitutivity rules, the truth predicate is “transparent,” letting us see through to the claims to which we are attributing truth. He signals this by continually referring to truth as "ttruth" for "transparent truth.") On the basis of familiar reasoning, this leads him to contradictions in the face of sentences like S1.

S1: This sentence is not true.

Beall is willing to pay the price of accepting true contradictions in exchange for unrestricted Capture and Release. Indeed, he regards this price as unavoidable, given his account of truth.

"Spandrels of x are inevitable, and frequently unintended, by-products of introducing x into some environment… Language has its own spandrels. This is particularly the case when a given bit of the language is introduced for a particular role, much like ttruth. The guiding metaphor, as above, has us introducing ‘ttrue’ not to name some property in the world but, rather, to enable generalizations about the world and its features….But ‘ttrue’ is a predicate, and introducing it into the grammatical environment of English yields spandrels, unintended byproducts of the device." (p. 5)

One might wonder, however, whether this result actually compromises the ‘transparency’ of his truth predicate.

"Given that the device [i.e. the truth predicate] is constructed to be entirely transparent, one expects a sort of supervenience to hold. In particular, one expects ttruth to supervene on the base-language facts—the base-language ttruths.

"The expectation of such supervenience, I think, is natural. If one were to insist on such supervenience across the board, then one would need to reject that some of the given spandrels are gluts, since such sentences are ttrue without their ttruth ‘depending’ on base-language ttruths. But…insistence seems to me to be misplaced, at least given the going conception of truth. Why insist as much when the constructed device might yield spandrels that buck supervenience? That the base language serves as a supervenient base for ttruth makes sense, I admit, for those sentences in which ttruth is eliminable via the fundamental intersubstitutivity rules—for the ‘normal’ sentences. What, though, of the inevitable spandrels? …As far as I can see, there is no reason to think as much. If anything is to determine the ttruth or tfalsity of such sentences, it’s at most the overall logic (or rules) of the language. Extending the otherwise sensible supervenience constraint to such sentences seems, as I said, to be simply misplaced." (pp. 15-16)

It is important to note that Beall’s careful formulation that the truth-values of ttruth-ineliminable sentences are determined “at most by the overall logic” (p. 4) [emphasis added] papers over a sticky problem for his account. What could possibly determine the truth-value of a non-paradoxical spandrel like S2?

S2: This sentence is true.

Certainly, the truth-value of S2 can’t supervene on any base-language fact. Moreover, unlike S1, whose gluttiness is delivered by standard Liar reasoning, Beall’s overall logic doesn’t seem to deliver the result that S2 is true, or the result that it is false, much less the result that it is both true and false. One might think that, since S2’s truth-value is not determined by either of those sources, it simply lacks a truth-value. (Indeed, an earlier version of Beall’s version of dialetheism—e.g. in his 2005 paper "Transparent Disquotationalism" (in the collection Deflationism and Paradox—he did leave room for truth-value gaps.) Unfortunately, elsewhere in Spandrels of Truth, he rules out the possibility of gaps, declaring that “negation is exhaustive.” (p. 5)

What, then, can be done about cases like S2? Initially, Beall endorses a completely unified approach to all ttruth-ineliminable sentences.

"For present purposes, I shall follow the simplest approach: I treat all such sentences as gluts." (pp. 14-15)

This approach, however, would lead to triviality if it were applied to ‘Curry sentences’ like S3, i.e. conditionals whose antecedents say of the whole sentence that it is true.

S3: If this sentence is true, JC Beall is fourteen feet tall.

Beall solves Curry in the usual dialetheist way, with conditionals too weak to enable Contraction (the inference from "if A, then if A, then B" to "if A, then B.") His criteria for “suitable conditionals” are that they obey Modus Ponens, that they honor Identity (the universal truth of "if A, then A"), and that they do not deliver triviality. However, for obvious reasons, another component of his view is that such sentences are (just) false. “On my account, Curry sentences are false; I reject that they’re ttrue.” (p. 33) In light of this position, Beall qualifies his earlier simple approach.

"In Chapter 1, I noted my openness to an asymmetric treatment of such sentences (e.g., treating some ttruth-ineliminable sentences as gluts, some classically), but officially embraced the simple approach according to which all such sentences are gluts—transparently true with transparently true negations. This position remains, but only for the conditional-free fragment." (p. 34)

This is Beall’s final, considered position in Spandrels of Truth. Unfortunately, while it may take care of Curry, it can be shown that this position still generates triviality. Consider S4.

S4: This sentence is true, and the moon is made of green cheese.

S4 is ttruth-ineliminable. The truth predicate can’t be removed from it through application of the intersubstitutivity rules, since the first conjunct refers to the whole conjunction, not just the (ttruth-eliminable) second conjunct. As such, it follows from Beall’s policy of treating all ttruth-ineliminable sentences in the conditional-free fragment of his language as gluts that S5 is both true and false, which means that it is true, which means that the second conjunct is true and that the moon is made of green cheese.

II. The Persistence of the Problem

Having shown that Beall’s account (as stated) leads to triviality, one might wonder if this problem can be solved through some extension of the move Beall employs to get around Curry. Perhaps he could just say that all ttruth-ineliminable sentences in the conditional-free and conjunction-free fragment of his language are gluts, and that, like Curry sentences, ttruth-ineliminable conjunctions are (just) false.

Unfortunately, given S5, this won’t work.

S5: This sentence is false, and it is not true.

S5 fails to be conjunction-free, but Beall’s overall logic will still get us the result that it is a glut by means of familiar Liar reasoning. Fishing around for a principled, unified policy that could help Beall out of these difficulties, one might postulate that all ttruth-ineliminable complex statements with ttruth-eliminable atomic statements within them are (just) false, whereas all sentences that are ttruth-ineliminable “straight through” are gluts. One advantage of this approach would be that it would simultaneously save him from S4-generated trivialization and from Curry-generated trivialization, without having to make his current unqualified (and thus untenable) claim that Curry sentences are (just) false. To see why this claim is untenable, consider S6.

S6: If this sentence is true, then it is true.

Remember that one of Beall’s criteria for “suitable conditionals” is that they validate every instance of Identity. S6 is an instance of Identity. Thus, there is at least one Curry sentence whose truth-value is determined by Beall’s overall logic to be true. On the proposal we are considering for patching up Beall’s views, however, S6 is not a problem. If the only spandrels that get classified as glutty are those that are ttruth-ineliminable “straight through,” S6 is true, S5 is glutty, and a simple, unified policy prevents us from claiming that any triviality-generating conditionals or conjunctions are true.

Unfortunately, even this version of Beall’s account doesn’t work. After all, consider cases like S7 and S8.

S7: Either this sentence is false or it is meaningless.

S8: Either this sentence is false or Hitler won World War II.

Given the historical facts, and Beall’s assumption that the spandrels are meaningful, his overall logic will deliver the result that both S7 and S8 are gluts. Moreover, it would be strange if it failed to deliver that result, given the prominence of examples like S7 and S8 in dialetheist arguments against various consistent solutions to the semantic paradoxes.

So, to sum up, (a) Beall’s official position generates triviality, and (b) the most obvious ways of getting around this problem (i.e. the ones closest to Beall’s own approach to getting around Curry) fail to respect Beall’s foundational position that, when the overall logic delivers gluts, we shouldn’t quarrel with the result.

III. A Better Solution

Fortunately, there is a fairly simple, principled way to patch up Beall’s account. He could simply say that the mechanisms he lists for determining the truth-values of sentences in the passage quoted above—(i) supervenience on base-language facts, and (ii) the unaided results of the overall logic—are jointly exhaustive of the possible ways in which sentences can become true. Given this move, and Beall’s principle that “negation is exhaustive,” it follows that all sentences that don’t become true in one of these two ways are (just) false. Thus, S6 is made true by (ii), S1, S5, S7 and S8 are all made true (and false) by (ii), S2, S3 and S4 remain (just) false, and triviality is avoided.

Of course, even on this patched-up version of Beall’s account, the only reason the “overall logic” doesn’t generate triviality as a result of sentences like S3 (despite their falsity) is that traditional (contracting) conditionals have been rejected in favor of “suitable conditionals.” One might wonder why, if such otherwise unmotivated tinkering with the formal rules is an acceptable way of preserving non-triviality in the face of Curry, similar tinkering wouldn’t be an acceptable way of preserving consistency in the face of the Liar. (With even weaker conditionals, we could, for example, continue to accept that (1) is either true or false, continue to accept that if it’s true, it’s false, and continue to accept that if it’s false, it’s true, but reject the conclusion that it’s both true and false.) If this would be an acceptable solution to the Liar, we have no need to postulate true contradictions. If it wouldn’t be an acceptable solution to the Liar, one might wonder why it counts as an acceptable solution to Curry.

If one wondered about these things, I would wonder with them. In terms of Beall’s more immediate problem, however, assigning the truth-values in the way I suggest should plug the hole and save his account from triviality.

Diet Soap on Zizek's Ontology

2010-10-27T23:55:00.000-07:00

Two of the last three episodes of the Diet Soap podcast are about Slavo Zizek.

It's interesting stuff, worth listening to. (Part I is here and Part II is here. Alternately, of course, you can just download them for free from iTunes--just look for the "Diet Soap" podcast, and get episodes 79 and 80.) Actually, despite the title, the discussion barely touches on Zizek's views on ontological issues. The focus is, rather, on a broad-ranging discussion of the history of western philosophy from Descartes to Hegel, with some stuff interspersed about Zizek's views about all of that, some commentary on Zizek's style and output, and some clips of the man himself. (There's also host Doug Lain's thoughts about "The Blue Beam Conspiracy." People who are easily irritated by conspiracy theories shouldn't be too quick to stop listening when that part comes around. Doug's thoughts about it aren't going where they initially seem to be.) It's all good stuff, I found some of the explanations of Hegel refreshingly clear, and it's always interesting to hear from people that far outside what we sometimes call "the analytic tradition."

Two major caveats:

(1) Zizek doesn't seem to be aware of the existence of applied ethics, much less aware that its a large and thriving part of contemporary philosophy. Zizek talks in a clip about how philosophy per se isn't going to have anything directly to say about environmental problems, abortion, gay marriage, etc., and no one on the podcast corrects this extremely strange statement. Moreover, when Doug Lain and Adrian Johnson (the author of the book "Zizek's Ontology," and the subject of Lain's interviews) start talking about philosophy's relevance or irrelevance to everyday life, and whether the conclusions of philosophical arguments should ever cause anyone to move away from a conventional "bourgeois lifestyle," it would have been really nice if Professor Johnson had mentioned the existence of Peter Singer, who is, after all (a) one of the very most prominent anglophone philosophers in the world today, and (b) a long-time advocate of making dramatic lifestyle adjustments for philosophical reasons.

(2) In the discussions about free will and fatalism, there's a lot of running together of two quite distinct claims:

(i) That there are facts about what will happen in the future, such that some statements about the future are true and some are false, and
(ii) That some being knows which statements about the future are true and which ones are false.

Clearly (at least given the orthodox assumption that truth is a necessary condition for knowledge), (ii) entails (i), but (i) can absolutely and obviously be true without (ii) being true. By analogy, consider Claim C (about the past, rather than the future):

C: "Alexander the great's maternal grandmother's paternal grandmother accidentally cut her toe on a rock when she was six years old."

C is pretty clearly either true or false. Whatever one thinks about reference failures and all of that (i.e. whether a statement like "the present King of France is bald" is true, false or neither, given that there is no present King of France), Alexander the Great clearly had a maternal grandmother, and she clearly had a paternal grandmother, and at one point she was six years old. During that year, that lady either did accidentally cut her toe on a rock--in which case C is true--or she didn't (in which case the negation of C is true), and none of this is remotely philosophically controversial. Given atheism (and the absence of time machines) no one is in any position to have epistemic access to the fact of the matter here, but no one thinks that there isn't a fact of the matter about this issue. Why on earth should it be any different, re: future facts and the absence of any being with epistemic access to those facts?

Fun With Midterms

2010-10-25T23:58:00.000-07:00

This is a question on a midterm I just gave.

#

2. When Simon Frith says that "Value arguments....aren't simply rituals of 'I like/you like'", what does he mean? What else does he think they are?

a. He thinks, in such discussions, we all make claims about what's good and bad is if this were some objective quality that's there in the music (or whatever) that's being discussed.

b. He doesn't mean anything. The whole passage was a long typo that resulted from his cat walking over his computer keyboard. It's amazing that the random keys pressed down by the cat's paws happened to result in a complete, grammatical sentence.

#

Two students got it wrong.

Now, granted, that means that 22 of them got it right, but still.....

Really?

A Quick Question For People Who Might Know More Than Me About Russell, Whitehead, or the History of Geometry

2010-10-20T20:11:00.001-07:00

I just got to this paragraph in the first chapter of The Principia Mathematica:

"Definition and real variables. When the defiens contains one or more real variables, the definiendum must also contain them. For in this case we have a function of the real variables, and the definiendum must have the same meaning as the defiens for all values of these variables, which requires that the symbol which is the defiendum should contain the letters representing the real variables. This rule is not always observed by mathematicians, and its infringement has sometimes caused important confusions of thought, notably in geometry and the philosophy of space."

Anyone have any idea about what Russell and Whitehead might be talking about in that last sentence?

Fun With Kant

2010-10-18T20:19:00.000-07:00

Here.

(I'm not necessarily endorsing any of Schwitzgebel's conclusions about withholding charitable assumptions from our interpretation of Kant's more philosophically significant work or any of that, but the information itself is interesting. And, y'know, horrifying.)

A Simplified Version of My Revenge Paradox for Paracomplete Solutions to the Liar Paradox

2010-10-13T00:01:00.000-07:00

I've played with versions of this sort of thing here a couple of times before, but right now this new version of the sentence seems best to me (in terms of denying the paracomplete theorist escape routes that may have been afforded by earlier, sloppier versions):

Sentence S: An omniscient and ideally rational being would not accept this sentence.

#

If you take S to be true, you're committed to the uncomfortable claim that an ideally rational being would reject a sentence that they knew to be true. If you take S to be false, then you're faced with an unattractive choice between being committed to saying that (a) it's also true, or that (b) sometimes an ideally rational being would accept a sentence they knew to be false. If you take S to be one of those sentences about which a good paracompletist recommends rejecting the relevant instance of the Law of the Excluded Middle, then you are committed to saying that some such sentences are also true, which would seem to defeat the whole point of the paracomplete manuever of switching from making claims about truth-values to making claims about acceptance and rejection.

Moreover, S also seems to pose a big problem for dialetheists who lean on acceptance/rejection talk to get around their difficulties with the notion of "just false." (The problem, of course, being that for whatever formulation you use to differentiate false statements that are also true from other false statements, you can always construct a sentence that says of itself that it is in the latter category--e.g. "this sentence is just false and not true.") At one time, Graham Priest was in this category--in the first edition of In Contradiction, he claims that dialetheists, despite their rejection of Disjunctive Syllogism, are still entitled to a Disjunctive Syllogism-like rule of reasoning expressed in acceptance-and-rejection talk rather than negation-talk. In the second edition, he renounces this claim, and in Doubt Truth To Be A Liar he has an extended discussion where he says that good dialetheists should accept that the ground for rational acceptance might sometimes overlap with the ground for rational rejection. JC Beall, on the other hand, still seems to think that he can avoid this conclusion. In Spandrels of Truth, he uses acceptance and rejection talk almost as much as Field does in Saving Truth From Paradox and for many of the same sorts of purposes, for example saying that he thinks that Curry sentences should all be rejected. (Basically, it looks to me paracompletists want to use "accept", "reject" and "neither accept nor reject" pretty much exactly the way truth-value-gap theorists use "true," false," and "neither true nor false," but without running into the revenge paradoxes, and that dialetheists like Beall or the previous version of Priest want to use "accept" and "reject" as stand-ins for the classical behavior of "true" and "false." In either case, the psychological concepts are substituted for the semantic ones, so that one can have one's cake and eat it too in terms of one's preferred non-classical solution to the paradoxes.) These purposes, it seems to me, are pretty well foiled if it turns out that (given his committment to thinking that paradoxical sentences are meaningful, have their apparent truth conditions and so on) he has no choice but to admit that either (a) some sentences both should and shouldn't be accepted, or (b) some true sentences shouldn't be accepted, or (c) some sentences should be accepted despite failing to be true.

A Test Question I Just Wrote

2010-10-11T00:01:00.000-07:00

5. When one is thinking about the more basic form of the Liar Paradox, called the Simple Liar—“This sentence is false”—one might think that the contradiction can be avoided by saying that the sentence is “neither true nor false.” The problem is that, when one tries to apply this solution to the version of the Liar Paradox called the Strengthened Liar—“This sentence is not true”—it generates a contradiction. How?

a. It just does.

b. If the sentence “this sentence is not true” is neither true nor false, then it’s not true, which is what it says of itself, and if what it says of itself is right, it’s true. Thus, if it’s neither true nor false, it’s both true and not true.

c. If you say that it is neither true nor false, the corner of the page on which it is written begins to smolder and burn, and deep, ominous laughter can be heard in the background.

Once More On The Stone Paradox (This Time With Symbols)

2010-10-06T02:25:00.000-07:00

Last Wednesday, I blogged about the Stone Paradox.

A quick summary of that post:

I think the reductio argument against theism goes through, that the standard response (watering down the definition omnipotence) is ad hoc and unconvincing, and that the initially-more-promising-sounding response one ocassionally hears (that God could create such a stone, and if He did so, He wouldn't be omnipotent any more, but so long as he happens to contingently continue to choose not to do so, He's still omnipotent) is completely hopeless on closer examination.

In any case, for the ease of quick reference, we can refer to the former move as the Standard Defense (SD) and the latter as the Mere Possibility Defense (MPD).

In the comments, Emil pointed me towards this formulation of the Mere Possibility Defense, from Norman Swartz. At first, I was a bit annoyed at reading it, since he simply asserts that the MPD works, without bothering to argue for it.

On second thought, I actually find Swartz's formulation useful, since his symbolization of versions of the argument helps clarify nicely where the philosophical fault lines are. Here's his symbolization (where "God is omnipotent" and "M" = "God makes an immovable stone"), re-formatted slightly because of the limits of easy symolization on the Korean computer I'm typing this on at the moment:

1: G → ◊M
2: ◊M → ~G
-----------------------
3 (from 1 & 2): G → ~G
4 (from 3): ~G

Swartz admits, of course, that this is a valid argument, but he thinks that it's unsound, because he denies Premise 2. I think both premises are true, so I take the argument to be sound.

This is a nice, useful symbolic formulation, though, for several reasons. It certainly captures the way I think of the anti-theistic argument from the Stone Paradox, and also nicely demonstrates why the Frankfurt-type response ("God could create such a stone, and He could lift it! After all, if His omnipotence lets him do one impossible thing, why not another?") that seems to strike so many people as being so clever is actually quite silly and irrelevant. The whole point is that the notion of an omnipotent being existing is inconsistent. Frankfurt's recommendation that the theist joyfully embrace the inconsistency simply underlines the point. One can do that--just as one can respond to Russell's Pardox by continuing to embrace naive set theory but paraconsistentizing one's logic--but what of it? As an argument (assuming the principle of non-contradiction) against theism, it goes through.

Formulated in these terms, the Standard Defense revolves around denying the first premise. This requires adding an inconsistency-avoidance epicycle to the definition of omnipotence. In last week's post, I argued that there's no particular reason why such a move should be more plausible here than in naive set theory, or any other instance of a general principle producing contradictions. (Indeed, I argued that it would have been far less ad hoc if naive set theorists had done this than it is for theists do so.) Of course, as I said then, if a theist has sufficiently compelling external reasons to think that God exists, this move might still result in the overall best explanation of the data, despite the ad-hocness. That's fine. But the argument does give us a reason (albeit a defeasible one) to reject theism.

The Mere Possibility Defense revolves around rejecting Step 2.

(It looks like, if one is both a theist and a classical logician, and thus unwilling to follow Frankfurt's advice and simply accept the contradictions, one had better make at least one of these two moves.)

In last week's post, I argued that, if one thinks that God contingently hasn't happened to create an unliftable stone, His inability to lift it is still a limit on his powers. After all, "powers" are always and everywhere counterfactual. I gave two analogies, one quite long and developed, but the point was this:

If Object X contingently doesn't happen to exist, that's quite irrelevant to whether Agent Y could perform Action Z to Object X. If Object X exists, that's epistemically relevant--we get to test whether Agent Y has the ability to perform Action Z--but the non-existence of the test doesn't normally entail anything one way or the other about the power.

For anyone who doesn't feel like clicking through to last week's post , here's the relevant passage:

*******************

Take an obese chain-smoking alcoholic named John. Despite his many health problems, he has the ability to climb a few flights of stairs without having a heart attack and dying. It seems fair, though, to say that John's stair-climbing abilities are limited. He could not, for example, climb a hundred flights of stairs without having a heart attack. Whether or not either of these situations will ever actually come up--e.g. whether John ever climbs stairs or he exclusively frequents buildings with elevators and escalators, whether John lives close enough to a city with a hundred-story building in it that he could attempt this feat if he were unwise enough to try it, etc.--seems quite irrelevant to our talk about John's powers. If John is a North Korean whose government will never allow him to travel to a place with hundred-story buildings, that doesn't seem to impact the truth of our statement about the limitations on his stair-climbing abilities. Nor would it, indeed, matter if, as a matter of contingent fact, the tallest building in the world happened to be ninety-eight floors tall.

To brings things closer to the God case, imagine a possible world where John--still an obese chain-smoking alcoholic--is the undisputed absolute ruler of the planet. Nothing can get built without his say-so, and he refuses to allow any building on earth to be constructed higher than four stories.

One day, two of his subjects--Jim and Jerry--are having a quiet conversation, perhaps in a quiet stairwell in one of the many four-story buildings where, as far as they know, John's secret police hasn't bothered to install any CCTV cameras or listening devices. They like to go there sometimes to hold the kind of private conversations that Winston Smith and Julia enjoyed in the early parts of Nineteen Eighty-Four.

At one point, Jim boldly speculates that, based on how pudgy and red-faced and out-of-breath Emperor John looks in the newsreels, the reason why he never allows buildings to be built over four feet high is that he doesn't have the ability to climb more flights of stairs than that and he wants to avoid the embarrassment. Jerry responds that, well, he could imagine John climbing as many as five or six flights of stairs without having heart attack, but there's no way he's healthy enough to climb, say, a hundred flights of stairs without collapsing.

At this point, of course, just like the capture scene in Nineteen Eighty-Four, Jim and Jerry find out that the secret police was listening all along, and both are tortured with rats in Room 101 until they admit that two and three make six if the Party says they do, and that Emperor John has the power to climb thousands of flights of stairs without physical setback.

Now, how would we evaluate Jerry's original claim about the limit's on John's stair-climbing abilities?

Given the innately counterfactual nature of all ability/power/powerfulness talk, the fact that John hasn't happened to create any such stairs, and has thus deprived himself of the opportunity to expose this particular limitation on his stair-climbing powers, seems quite irrelevant to the truth of Jerry's claim. Just so for God and unliftable stones.

************

So....for anyone who wants to argue that the argument symbolized above valid but unsound, here's the challenge:

Either (a) explain the relevant disanalogy between the move made by theists who employ the SD and put a logical-consistency epicycle in their new, watered-down definition of omnipotence, and other cases where a general principle generates contradictions, and we all think that the rational response is to reject the general principle rather than stick in a consistency epicycle, (b) provide an argument for theism so devestatingly convincing that it justifies the SD as the overall best explanation even in the face of the ad hocness, (c) explain the relevant disanalogy between the contingent absence of hundred-floor buildings in the John-ruled world (which seems irrelevant to the limits on John's stair-climbing powers) and the contingent absence of an unliftable stone in the God-ruled universe (which, according to partisans of the MPD, is relevant to the limits of God's powers), or (d) provide an alternative way out, thus showing that (a)-(c) aren't jointly exhaustive of the options for defenders of the doctrine of divine omnipotence who want their beliefs to be closed under some sort of (non-paraconsistent) logical consequence relation.

Kant?

2010-10-04T18:16:00.000-07:00

Next week, I'm assigning an extract from Kant's "Critique of Judgment" to my Philosophy of Art class. Anyone out there have any suggestions for good secondary stuff to go with it, or to help explicate it to the (predominantly non-native-English-speaking) students?

Some Further Points About The Stone Paradox

2010-09-29T00:01:00.000-07:00

In the "Analytic Philosophy" class I'm teaching in Korea this semester, this week we're covering the two chapters of Bertrand Russell's book My Philosophical Development concerning his and Whitehead's work writing Principia Mathematica. I always find the following fact fairly striking:

Following the discovery of Russell's Paradox, Russell got around it by means of his somewhat awkward and complicated theory of types. ZFC and other orthodox set theories got around it by the "hierarchical conception of sets" and similar means. In more recent decades, a few logicians on the radical fringe have argued for rehabilitating naive set theory at the expense of the Law of Non-Contradiction. Practically no one, though, seems to have thought of responding to the paradox by simply amending Frege's Basic Law V to something like Basic Law 5.1:

"Wherever it is logically possible for there to be a set of all and only the objects matching some description, there actually is such a set."

On a slightly different but closely related note, here's a true story about a friend of mine, D.:

One day in Hebrew school when D. was 12 or 13, the Rabbi was talking about how God can do anything. D. responded by pointing to the door of the classroom, which was always sticking and never quite closed. D. (who grew up to be a lawyer, and fondly refers to this as his "first cross-examination") pointed at the door and asked the Rabbi, "could God make that door closed all the way?" The Rabbi said "of course." Then D. asked, "could God make that door closed so that no one would open it?" Finally, D. asked, "could God make that door closed so that even God couldn't open it?" The Rabbi hemmed and hawed and never gave D. a good answer, and D. promptly gave up on belief in God.*

If, however, D. had gone into a Philosophy PhD program instead of law school, he might have discovered, through the writings of folks like Richard Swinburne, the standard theistic response to his point from Hebrew School, and that--again, the standard theistic response, indeed, almost the only response one ever encounters from philosophically trained theists to the worry--is, transparently, just a theological version of Basic Law 5.1 above. The previously mentioned Professor Swinburne tries to argue, in The Coherence of Theism, that God can perform any action, but that things like "create a stone an omnipotent being could not cause to rise" don't count as "actions." Other rather more intellectually honest theists just clarify that by "all-powerful" they mean that God can do anything that's logically possible given His omnipotence, not that God can do anything at all.

So, what, on earth, is the difference between the two cases?

Well, in terms of concern about ad-hocness, Basic Law 5.1 actually fares rather better than its theological counterpart, as it would represent a reversion to Cantor's original definition of "set," at the very dawn of naive set theory**, whereas the level of intellectual sophistication necessary for subtle caveats about "anything that's logically possible" comes rather late in the overall history of the Judeo-Christian religious tradition.

Other than that, I really have no idea. The uncharitable suggestion that rather forces itself on one after thinking about it for too long is that, by and large, set theorists are simply a bit more scrupulous than theistic philosophers when it comes to paying attention to epistemic standards like "general principles that need to have 'except when this produces a contradiction' caveats at the end of them are a lot less likely to be true than those that organically fail to produce contradictions." If one wanted to follow this thought to its natural conclusion, the sociological and psychological factors involved in filling out this story (even when it comes to very bright and otherwise epistemically scrupulous theists) aren't terribly hard to come up with.....bluntly, no one is, as a small child, indoctrinated by all the adult authority figures in their life to believe in naive set theory, so that, as an adult, they have an enormous antecedent emotional need to preserve their belief in Basic Law V.

Alternately, here's a (much more charitable) explanation:

The fact that the move from Basic Law V to Basic Law 5.1 is a bit ad hoc, that general principles are more plausible when they lack such epicycles, gives us some reason to think that naive set theory so amended gets things wrong. Similarly for the move from "naive omnipotence" to "God can do anything as long as no contradiction would follow from an omnipotent being performing the action." Still, the fact that the need for the epicycle at the end of that last sentence gives us some reason to believe that the theory in question is false isn't decisive if we have some tremendous independent reason to believe that the theory is true. Put crudely, you can get a point deducted for ad-hocness but still get more points overall than the alternative views.

If a theist had this view--they believed they had some excellent argument for the existence of God in their back pocket*** and that this justified belief in omnipotence-with-an-epicycle--that would be fair enough. Of course, that would involve acknowledging that the Stone Paradox gives us some reason to disbelieve in the existence of God, even if not a decisive one, and, at least anecdotally, that kind of attitude seems surprisingly rare.

In any case, though, it might be objected (indeed, commenters on previous posts where I've brought this sort of thing up have objected) that this whole discussion only applies to theists who take God to be omnipotent in all possible worlds, whereas a viable alternate version (we can call it Omnipotence 3.0) holds that God is omnipotent-without-epicycles in the actual case, and that He could indeed create an unliftable stone. If he chose to do so, He would, at that point, no longer be omnipotent--having just limited his power by creating a stone he couldn't lift--but that this counterfactual point doesn't bear on His actual omnipotence.

For a while, I thought this was a pretty good defense--certainly a lot more promising than the near-universal Standard Theistic Response--but, on second thought, I don't actually find it very plausible.

Here's why:

Talk of "abilities" or "powers" is counterfactual through and through.

Even if a man is part of a distant rainforest tribe that has never heard of baseballs, it can still be true of them that they have a powerful enough throwing arm to toss a baseball across a large field. This is an *actual* attribute of his, despite the fact that it doesn't come up in the circumstances of their life. If the tribesman in question was flown to America, shown a baseball, and he did indeed toss is across a large field, this would decisively confirm that he had the ability, but his having the ability is unaffected by the absence of the test.

Another example:

Take an obese chain-smoking alcoholic named John. Despite his many health problems, he has the ability to climb a few flights of stairs without having a heart attack and dying. It seems fair, though, to say that John's stair-climbing abilities are limited. He could not, for example, climb a hundred flights of stairs without having a heart attack. Whether or not either of these situations will ever actually come up--e.g. whether John ever climbs stairs or he exclusively frequents buildings with elevators and escalators, whether John lives close enough to a city with a hundred-story building in it that he could attempt this feat if he were unwise enough to try it, etc.--seems quite irrelevant to our talk about John's powers. If John is a North Korean whose government will never allow him to travel to a place with hundred-story buildings, that doesn't seem to impact the truth of our statement about the limitations on his stair-climbing abilities. Nor would it, indeed, matter if, as a matter of contingent fact, the tallest building in the world happened to be ninety-eight floors tall.

To brings things closer to the God case, imagine a possible world where John--still an obese chain-smoking alcoholic--is the undisputed absolute ruler of the planet. Nothing can get built without his say-so, and he refuses to allow any building on earth to be constructed higher than four stories.

One day, two of his subjects--Jim and Jerry--are having a quiet conversation, perhaps in a quiet stairwell in one of the many four-story buildings where, as far as they know, John's secret police hasn't bothered to install any CCTV cameras or listening devices. They like to go there sometimes to hold the kind of private conversations that Winston Smith and Julia enjoyed in the early parts of Nineteen Eighty-Four.

At one point, Jim boldly speculates that, based on how pudgy and red-faced and out-of-breath Emperor John looks in the newsreels, the reason why he never allows buildings to be built over four feet high is that he doesn't have the ability to climb more flights of stairs than that and he wants to avoid the embarrassment. Jerry responds that, well, he could imagine John climbing as many as five or six flights of stairs without having heart attack, but there's no way he's healthy enough to climb, say, a hundred flights of stairs without collapsing.

At this point, of course, just like the capture scene in Nineteen Eighty-Four, Jim and Jerry find out that the secret police was listening all along, and both are tortured with rats in Room 101 until they admit that two and three make six if the Party says they do, and that Emperor John has the power to climb thousands of flights of stairs without physical setback.

Now, how would we evaluate Jerry's original claim about the limit's on John's stair-climbing abilities?

Given the innately counterfactual nature of all ability/power/powerfulness talk, the fact that John hasn't happened to create any such stairs, and has thus deprived himself of the opportunity to expose this particular limitation on his stair-climbing powers, seems quite irrelevant to the truth of Jerry's claim. Just so for God and unliftable stones.

*With D.'s permission, I used this incident in my short story Dark Coffee, Bright Light and the Paradoxes of Omnipotence, which is going to be reprinted this winter in the Prime Books' anthology People of the Book. (I'll admit to being pretty excited about that, since I get to share a Table of Contents with the likes of Neil Gaiman and Michael Chabon.) At the time, I asked him, "is it OK with you if I steal some of your life story, and portray it as part of the life story of a character who's (a) gay and (b) a terrorist?" His response was, "geez, Ben, how do you know I'm not either of those things? Also, yeah, sure."

**Cantor distinguished between "sets" whose members could be consistently jointly thought of as one thing, and "inconsistent multiplicities" that could not. I suspect that part of the reason that people didn't respond to Russell's Paradox with, "oh well, I guess Frege's wrong, but Cantor's right," is that, while he was certainly a brilliant mathematician who contributed many still-interesting proofs, Cantor's foundational ideas about the nature of set theory were never clearly and systematically laid out the axiomatic way that Frege's were, and a lot of his extant writings that touch on it (essays, letters to Dadekind, etc.) are full of unclear assumptions, weird religious baggage, quasi-mystical beliefs about "true infinity" and so on. For anyone interested in plunging into those waters, though, a good place to start is Michael Hallett's book Cantorian set theory and limitation of size.

***Given the large flaws in the standard contenders (cosmological, teleological and so on), I'm extremely skeptical that such an argument exists, but that, of course, is quite outside the subject matter of this post. At any rate, my (largely hypothetical) problem-acknowledging theist merely has to sincerely take themselves to have such a good argument, not to actually have one.

Quantum Logic, Part IV of IV

2010-09-27T05:19:00.000-07:00

In Miami last year, I sat in on a few weeks of a seminar on philosophical issues about quantum mechanics, before I got too swamped with last-minute dissertation edits and whatnot to make the time. (Later, the professor did join me and some friends for an evening of drinking single malt and watching and making fun of "What The Bleep Do We Know?") On the first day, the professor went through a long and funny list of nonsense topics that might come to mind when one hears the phrase "philosophical issues about quantum mechanics" and announced that we wouldn't be talking about any of those. He then proceeded to list off a few "actually serious topics" we also wouldn't be going over, purely because of time constraints. One of the topics he listed off there was that of whether quantum results create a problem for classical logic.

A few minutes later, he moved on to describe the two path experiment we'd read about in a slightly whimsical way in the first chapter of David Albert's excellent book Quantum Mechanics and Experience. (The assignment was sent around by e-mail before the first day of class.) The punch line of the experiment, which establishes "superposition" (a term that often feels, at least from my non-physicist's perspective, more like a label slapped on the weirdness than anything particularly illuminating), goes like this:

Given the experimental evidence, it seems like we can absolutely rule out the possibility that the electron is passing through path A. Or path B.

Or neither.

Or both.

Which, um....

....would seem to me to kind of create a problem for classical logic.

#

That's such a good note to end on that I was tempted to it there, but I should probably say some more.

As I've indicated before, the Hartry-Field-style "paracomplete" approach to the Liar Paradox--whereby we set up an elaborate formal apparatus and use it to reject that Liar sentences are true, reject that they're false, reject that they're neither, and so on--isn't particularly attractive to me, both because I think that better options are on the table and because of the difficulties its proponents face in saying anything particularly intuitively plausible about sentences like this one:

"This sentence would not be accepted by a being who accepted all true sentences."

...and, of course, my standard complaint about non-classical solutions to the Liar, which is that the Liar and Curry are obviously instances of the same phenomenon, and a solution to the Liar that doesn't apply to Curry is no better than a solution to the Simple Liar that doesn't apply to the Strengthened Liar.

All of that said, quantum superposition state weirdness does strike me as a much more promising application of parcompleteness. I think that Priest & Routley have an old paper suggesting a dialetheic approach to quantum mechanics, but that seems to get the intuitive situation exactly wrong. It's not that all the possibilities can be jointly ruled *in*, it's that they can all be jointly ruled *out.*

Moreover, the quantum analogy to my view of the Liar Paradox would fall completely flat. Claiming that any of the statements involved commit category mistakes is (a) incompatible with the claim that they can be empirically ruled out, and (b) hard to square with the way we talk about electrons that *aren't* in superposition states. It's meaningless to say that some ideas are yellow, or to deny it, since color talk just doesn't apply to ideas, but position talk clearly *does* apply to particles.

Now, I'm certainly not endorsing a paracomplete approach to quantum weirdness--I'm not ready to give up on classical logic just yet, and the empirical and conceptual issues involved in arguing about this one way or the other get pretty murky pretty fast--but I do think there's an obvious prima facie case for some such logical revision. (Even if I hold out hope for its defeat.) Notice, though, that, as I've been arguing in the last few posts, none of this remotely threatens Distribution.

Quantum Logic, Part III of IV

2010-09-22T00:01:00.000-07:00

Quantum logicians argue that, when it comes to statements about the behavior of subatomic particles, Distribution--the inference from [P & (Q v R)] to [(P & Q) v (P & R)]--fails.

In Part II, I suggested that it's very hard to make sense of this claim if you take validity to be a matter of universal truth-preservation. If one extends the classical truth tables for conjunction and disjunction with a third truth value, it's hard to see which one would or could "get the job done." I quickly surveyed various obvious candidates for a third truth-value--gaps, gluts, undecidedness, on-the-border-between-true-and-falsiness and so on. In none of these cases does it seem plausible that any of these additions to the classical truth tables would make Distribution invalid.

Of course, the quantum logician could revise the classical truth tables instead of extending them--they could, for example, argue that it's sometimes possible for (P v Q) to be true even if neither P nor Q is true--but this makes them vulnerable to "change of meaning" charges in a way that, for example, as we saw in Part II, even as extreme a heterodox logician as the dialetheist is not vulnerable.

(In his argument against Putnam's early quantum logic stuff, Dummett claims that the whole notion of truth tables and truth-functionality subtly relies on Distribution and that, as such, the quantum logician is not entitled to it. This charge seems to me to rely on an illegitimate leap from the notion that Distribution fails in some contexts to the notion that it fails in all contexts. It is, in other words, like anti-dialetheist arguments that rely on equating dialetheism with trivialism.)

Now, it's certainly possible to concoct truth tables that fit the bill--retaining the classical lines while invalidating Distribution and keeping most of the rest of the classical laws we care about. I wrote a paper last year suggesting one scheme for that--treated as a formal exercise, because I've never been convinced by the quantum case against Distribution--and in the comments on Part II Brandon suggests another, while making clear that it's just an example, and raising the concern about his example that the "true if..." truth-values might be a bit too crudely rigged to get the desired result but speculating that there were probably more sophisticated ways to go about it. (In so far as I might have come off too strongly last time, sounding like there was literally no way to set up truth tables to get the desired results, Brandon's point was well-taken.) For the sake of elegance at least, my favorite way of invalidatng Distribution like this:

Assume there are three truth-values, 1, 0 and .5. For disjunctions to have value 1, the value of their conjuncts must add up to at least 1, for them to have value 0, the sum up of the conjuncts must be 0, and otherwise, they have value .5. Conjunctions have value 1 if the sum of their conjuncts is 2 and 0 otherwise. The value of the negation of any claim is the absolute difference between 1 and the value of the claim. Validity is preservation of value 1.

In terms of the most important rules about Disjunction and Conjunction, Disjunction-Addition and Conjunction-Elimination obviously both fall neatly out of that scheme. Disjunctive Syllogism is also safe. Here's the truth table for that:

( P v Q ) / ~ P // Q
1 1 1 0 1 1
1 1 0 0 1 0
1 1 .5 0 1 .5
.5 1 1 .5 .5 1
.5 1 .5 .5 .5 .5
.5 0 0 .5 .5 0
0 1 1 1 0 1
0 .5 .5 1 0 .5
0 0 0 1 0 0

Distribution, though, is invalid, the relevant line of the truth table being:

P & ( Q v R) // ( P & Q) v ( P & R )
1 1 .5 1 .5 1 0 .5 0 1 0 .5

A slightly less elegant way to do it is just to call the third value "O," not say much about it's relationship to T and F, and just choose a bunch of classical claims about truth-functionality while ignoring others--e.g. it will still be true that "conjunctions are only T if their conjuncts are both T", but it won't still be true that "conjunctions are only F if their conjuncts are both F," it'll still be true that "disjunctions are only F if their disjuncts are both F", but it won't still be true that "disjunctions are only T if their disjuncts are both T," etc.

So it's certainly possible to put together such truth tables, invalidating Distribution while retaining some plausible-sounding principles linking them to bivalent truth tables. The difficulty in each case is about putting conceptual meat on the bones, philosophically justifying the strange behavior of the new truth-value.

Intuitively, if .5 is read as "sort-of-true" or "half-true" or something, then, if P is completely true and Q is "sort-of-true" or "half-true" or something, then the joint statement of both should be "sort-of-true" or "half-true" or something as well. That won't do, though, in terms of the mathematicized truth tables given above, because if, when P is 1, Q is .5 and R is .5., (P & Q) and (P & R) both get value .5, their disjunction gets value 1 and Distribution goes through.

In general, however one sets things up, the sticky question is this:

What could we possibly say about the third truth-value that justifies its strange behavior here? How can we make sense of saying that the value is close enough to truth that two disjuncts that each have it add up to a disjunction which is fully true, but far enough from truth that one conjunct with the value, in combination with one fully true conjunct, add up to a conjunction that's not only untrue but fully false? Without this awkward combination, after all, Distribution goes through.

Again--if anyone has anything to offer in the comments by way of a plausible story about the nature of the proposed value, I'm all ears. I certainly don't take the limits of my creativity here to be the last word, and one of the joys of philosophy blogging is getting to brainstorm with people about this stuff, so seriously, if you've got something, I'd love to hear it. From my perspective, though, I don't see how it could work.

Of course, it could be that the crucial mistake happened at the beginning, when we assumed that validity was about truth-preservation, and so, for Distribution to be invalid, we had to find an instance of it where a true premise delivered a non-true conclusion. As widespread as this view of validity is--it certainly the official dogma of introductory logic textbooks, and it's a subject that Graham Priest and I agree about, which should indicate a certain kind of consensus--it's certainly not the only view out there. There is, for example, the truth-preservation-plus view often assumed by relevance logicians, according to which truth-preserving inferences can still fail to be valid if the premises and conclusion aren't sufficiently connected given some sort of relevance constraint (e.g. they don't share any non-logical terms in common). That clearly won't do, here, though, since instances of Distribution concerned with the behavior of quantum particles will pass those sorts of relevance tests with flying colors.

What, though, if we switched over to an important secondary tradition, the inferentialist one according to which certain inference rules are basic and analytically constitutive of the meaning of the relevant logical connectives?

One problem here will be a general one afflicting all change-of-logic proposals made by inferentialists--"given that you think that the inference rules are constitutive of the meaning of the connectives, and that you're changing your mind about the inference rules, aren't you just talking about new connectives, not making heterodox claims about the behavior of the old connectives?"

Now, I think this is a serious problem, and I don't want to dismiss it too quickly, but there is a way around it that's at least plausible enough to tentatively assume for the purposes of this discussion (as I assume for the sake of argument that--contrary to my views on the matter--inferentialism is correct):

We can bite half the bullet, in a way that makes room for arguing about logic, but saying that, for example, quantum-logical disjunction and classical disjunction really are two different connectives with two different meanings, but both represent analyses of our ordinary intuitive notion of (inclusive) disjunction. In general, we all start out with intuitive proto-versions of all the logical concepts, and the meaning-constituting inference rules of various formal systems all represent something like conceptual analyses of those proto-concepts.

(That was a bit rough, but it's probably good enough for our purposes here. Also note that a truth-preservationist quantum logician who was willing to simply revise the classical truth tables instead of expanding them could presumably make the same sort of move to justify themselves.)

Now, this sort of move may seem plausible when it comes to Van McGee's counter-example to Modus Ponens or relevance logicians' arguments against the classical notions of implication and entailment and so on, but it seems to me that the quantum context is the one in which this move falls the most spectacularly flat.

Surely, our ordinary-language, intuitive notions of "and" and "or" predate the discovery of subatomic particles, superposition and the rest by....

...well, by pretty much all of human history. These notions were developed to talk about the ordinary properties of ordinary "mid-sized dry goods."

Now, taking the "quantum leap" in our understanding of the physical world surely poses challenges to many of our ordinary intuitions about things like location, causation at a distance and so on. It's plausible that it turns out to upset all sorts of metaphysical applecarts. Some of these could even have consequences for logic, if logic is understood in truth-preservationist terms and odd possibilities, not accounted for in logical schemes developed in a previous era, turn out to sometimes be true if this strange domain of physical reality.

But. Surely the one thing that we won't discover from our experiments about the behavior of previously-unknown subatomic particles is that we were mistaken about what "and" and "or" meant in our ordinary pre-philosophical, pre-scientific discourse.

If revisionary moves about logic are justified as a result of the deliverances of quantum physics, it looks like we're going to have to justify them in terms of the truth-preservationist conception of validity. And the prospects for making sense of that are looking pretty dim as well.

So, if Distribution is safe after all, does that mean that classical logic has been comfortably and securely squared with quantum weirdness, that there are no remaining problems or challenges there?

Well....

Stay tuned for next week!

In The Absence Of A Real Monday Post...

2010-09-20T09:45:00.000-07:00

....I'll just note that barking mad seems like a severe understatement when it comes to describing this.

To put things a bit more bluntly:

If you think that the proposals in question fall short of being near-psychotically racist and evil, you are a fucking idiot who should be shunned in all important contexts. Obviously.

We'll be back to quantum logic next time.

Quantum Logic, Part II of IV

2010-09-15T00:01:00.000-07:00

Assume (until we get to Part III) that validity is a matter of truth-preservation.

Now, for any proposal that we "change logics" (i.e. revise our current ideas about what the true laws of logic are) to succeed in doing more than just changing the subject, we need to make sure the terms we're using mean the same things after the revision that they did before. For example, if I suggest that "P & Q" is true iff P is true, Q is true, or both, whereas "P v Q" is true iff P and Q are jointly true, my proposal amounts to reversing the usual notation for conjunction and disjunction, not changing our ideas about them. By contrast, if someone denies that the inference from any and every premise to any instance of the disjunction "P v ~P" is truth-preserving, and justifies it with some philosophical story about the unevenness with which vague predicates map onto the world, they really are rejecting classical logic.

Quine, in a famous passage in Philosophy of Logic, seems to suggest that all revisionary proposals about logic fall into the first category, that "the deviant logician" always succeeds merely in changing the subject. He argues, in particular, that when the dialetheist says that some contradictions are true and not everything follows from them, they have stopped using "not" and "~" in the usual ways.

For a variety of reasons, I don't this is one of Quine's better or more plausible moments. For my tastes, Quine is at his best in Two Dogmas of Empiricism, where he suggests that perhaps quantum physics will one day force us to abandon classic logic, and provides a holistic framework for thinking about questions of confirmation and belief revision within which that makes sense.

Without getting into a longer critique of Quine here, it's worth noting, as at least one major test of sameness of meaning, that the dialetheist, who takes some contradictions to be true and denies that anything and everything follows from any given contradiction--because she denies that Disjunctive Syllogism is universally truth-preserving--actually affirms all of the relevant classical truth tables. She just adds some extra lines. Where the classical logician says that ~P is true iff P is false and ~P is false iff P is true, the dialetheist agrees, and, given that they think that P can be both true or false, take the step they are required to take by the assumptions they share with the classical logician and say that ~P can be both true and false as well. Where the classical logician says that a disjunction is true iff at least one of the disjuncts is true--indeed, given that the disjunction symbol refers to "inclusive disjunction" (P or Q or both), "P v Q" and "at least one of these things is true: P, Q" seem to mean precisely the same thing--the dialetheist grants this, assents to all the lines of the classical truth table (where "F" is read as "just false") and adds some extra ones, getting the result that "at least one of these things is true: P, Q" can be true even if P and Q are both false and Q fails to be true, if P is both true and false. Once all that is in place, it's a straightforward, principled consequence of the view--with the truth tables for the two classical truth values (just true and just false) intact and the meaning securely unchanged--that, given the assumption that truth and falsity can overlap, Disjunctive Syllogism is not universally truth-preserving.

A similar story could be told about the inferences denied by deviant logicians who do things like deny the Law of the Excluded Middle, but the example of the dialetheist makes the point:

Even very radical proposals for logic change--and, in some ways, the claim that there are true contradictions seems much more radical than the quantum logician's rejection of Distribution--can pass the "not just changing the subject" test with flying colors. Whatever one thinks of their claim to have discovered new logical possibilities ignored by orthodoxy, their views about how the classical truth-values interact are thoroughly orthodox. Crucially, they respect the fact that "P or Q" in the sense captured by "P v Q" seems to mean the same thing as "at least one of these things is true: P, Q."

The quantum logician, on the other hand, seems to fail the test.

For the premise of some instance of Distribution--i.e. [P & (Q v R)]--to be true, surely P must be true. (If a conjunction can be true even though one of its conjuncts is not, the "change of meaning" charge starts to sound pretty convincing.) For the same reason, "Q v R" must be true. How, then, can the conclusion--[(P & Q) v (P & R)] be false? Well, we've already said that P is true. And if Q and R were both false, then (Q v R) would be false, and hence the premise as a whole would be false. Moreover, if one of them were true, the conclusion would be true!

In order for the premise to be true and the conclusion to have some other status, then, both Q and R must have some other truth-value. What other truth-value, though, would do the trick? If Q and R were both both true and false, the conclusion would be true. If Q and R were both neither true nor false, then how could "at least one of these things is true: P, Q" possibly be true? If Q and R were both somehow undecided or unsettled between truth and falsity, or on the vague borderline between them or something, why wouldn't their disjunction similarly be undecided or unsettled between truth and falsity, or on the vague borderline between them or something?

If someone has a proposal for what truth-value Q and R could have that would make the premise of non-truth-preserving instances of Distribution true without the conclusion being true, I'd love to hear about it in the comments, but right now it looks like nothing fits the bill.

But wait! Maybe we went wrong in our initial assumption that validity is truth-preservation. Perhaps, once we switch over to one of the other theories of validity, the quantum logician's move will start making more sense.

To which all I can say is...

Stay tuned!

Quantum Logic, Part I of IV

2010-09-13T00:01:00.000-07:00

So when I asked for reader requests last week, a couple of people asked for a post about quantum logic. The phrase "quantum logic" means a lot of different things these days, as indeed the word "logic" can mean a lot of different things. (See, for example, inductive "logic", computer-programming "logics" and so on.) So when many people talk about "quantum logic", they're talking about various formal or even mathematical constructions that model certain kinds of experimental results, the kind of thing for which no deeper philosophical justification is either offered nor required.

My interest in logic, however, veers towards what we can think of as 'logical metaphyics,' questions like, 'Do the inferences that we classically take to be universally truth-preserving really universally preserve truth? Are the claims we take to be logical truths really true?' (Hence my central research focus is in the semantic paradoxes, the question of whether there are any true contradictions, whether the Law of the Excluded Middle always holds and so on.) Because of that--and because I'm a Quineish confirmational holist--my interest in quantum logic is specifically on the question of whether the best explanation of the relevant physics might involve rejecting some of our current ideas about logical truth.

Historically, the most common proposal along these lines has been that, in response to quantum phenomena, we should reject Distribution, the principle that [P & (Q v R)] entails [(P & Q) v (P & R)]. Unless otherwise indicated, when I talk about "quantum logicians" or "the quantum logician", I'll be talking about the Distribution-rejecting quantum logician. In the new few posts, I'll argue that the prospects for their proposal are fairly bleak.

If we understand entailment in terms of truth-preservation (as I think we should), then, no matter how creative we get about adding in extra truth-values, there doesn't seem to be any plausible way to (a) get the result that the usual inferences about conjunction and disjunction that the quantum logician doesn't want to revise away remain valid, while (b) getting the result that Distribution is invalid. Without (a)--if, for example, it turns out that the truth of P isn't enough to guarantee the truth of (P v Q) in some truth-functional quantum logic--the 'change of meaning' charges often lobbed against quantum logic start seeming pretty hard to refute.

Of course, one could take all of this (in combination with whatever empirical case one thinks there is for quantum logic) as a good reason to reject truth-preservationism in favor of switching over to an inferentialist account of logical consequence, where primitive inference rules are taken to be "meaning-constituting" for logical connectives. Unfortunately, on closer inspection, things look even worse for the quantum logician here. It seems terribly implausible that experimental results about esoteric sub-atomic phenomena should show us that we were mistaken about the meaning of the terms "and" and "or."

At the level of description I'm giving here, it might seem like these are generic criticisms that would apply to *any* proposal to revise logic--"aren't classical logicians always accusing people with heterodox views of these sorts of things?"--but this isn't the case. Comparisons to other revisionary proposals will be instructive. After all, as we'll see, paracomplete theorists who reject instances of the Excluded Middle and Disjunctive Syllogism-rejecting dialetheists both pass the tests which (I argue) the Distribution-rejecting quantum logician fails.

Finally, though, I'll argue that even if the proposal that we reject Distribution to make sense of quantum phenomena isn't particularly plausible, that doesn't let classical orthodoxy 'off the hook.' Given the experiments that establish superposition and the rest, there are well-grounded worries that some sort of logically revisionary solution may be needed, even if rejecting Distribution doesn't fit the bill. I'll conclude with some tentative thoughts about that.

Meanwhile, though, I have to prep for teaching Philosophy of Art to some Koreans, so this post will have to remain nothing more than a preview for coming attractions. Stay tuned for Wednesday!

Ethical Quasi-Realism and Logical Truth

2010-09-08T00:01:00.000-07:00

Moral realists think that there are mind-independent facts of the matter that ground the truth-values of claims like "killing small children for sport is wrong" and "playing Wii Golf is wrong." Whether one is the more austere kind of minimalist about truth or the most extreme, early-Wittgenstein-style correspondence theorist, or anything in between, if one is a moral realist, one will--at least on the most basic level of description--tell pretty much the same story about why "killing small children for sport is wrong" is true and "playing Wii Golf is wrong" is false. In each case, the type of event being referred to either is the way it is being described as being (in which case it's true) or it isn't (in which case it's false). For moral error theorists, again, regardless of their precise position on truth as long as it's within the usual range, the story about how to evaluate such statements is precisely the same as for the moral realist. The only difference is that, according to the error theorist, *both* of the statements just mentioned are false, since neither type of event has the property in question. According to the error theorist, after all, *no* type of event has that property, since there is no such property as (moral) wrongness.

(An important sidenote is that people sometimes speak as if the moral error theorist had to give up on *all* evaluative language, but that's absurd. The usual J.L. Mackie-style arguments against moral properties are obviously inapplicable to other sorts of evaluative properties--e.g. ones pertaining to epistemic matters--that can be reduced to ordinary, non-"queer" properties far more easily and less problematically than *morally* evaluative terms can. Anyone who thinks that the moral error theorist is saying something false-according-to-error-theory when she says "you shouldn't be a moral realist, since there's no evidence for the existence of moral properties" hasn't thought very hard about the variety of different things that the word "should" can do in different contexts. I'm not an error theorist, but I do take it a bit more seriously than all that.)

Moral "quasi-realists" argue that this story about truth only works for one kind of truth--"descriptive truth," and that it fails for another kind of truth--"evaluative truth." (Everyone else, of course, thinks that the very idea of a "non-descriptive" form of truth is deeply confused.) When it comes to "evaluative truths", the statement is true iff it (depending on one's preferred flavor of quasi-realism) expresses the speaker's moral attitudes, or the moral attitudes the speaker would have under certain sorts of idealized circumstances, or something like that.

Now, generally speaking, quasi-realists are realistic enough about human psychology to grant that there's no particular reason to believe that, even under idealized circumstances, we'd all have precisely the same moral attitudes, and of course, no one claims that we all have the same moral attitudes right now. As such, any form of quasi-realism about morality automatically adds up to a sort of relativism--not relativism about moral properties, mind you, but relativism about truth. If you and I have different deep moral attitudes, still would under idealized circumstances, etc., it's that from my perspective, killing small children for sport is right-for-me and wrong-for-you, but that, from my perspective, my claim that it's right is true and your claim that it's wrong is false, and the opposite is true from your perspective. Although few would like to put it quite like this, the quasi-realist avoids right-for-you and wrong-for-me at the expense of embracing true-for-you and false-for-me.

Now, relativism about truth might seem like an intuitively unappealing enough consequence on its own, but relativism about logical truth would take the quasi-realist to a really awkward place. After all, a big part of any story about "idealized circumstances" is presumably that, under these circumstances, one has, for example, chosen between moral claims and their negations in every case, made sure that all of their moral claims are consistent with each other, and so on. Logical matters are a key part of how quasi-realists can still make sense of criticizing claims that (according to them) aren't in the business of trying to correctly describe anything. You might not know what someone's deep attitudes are, but surely you can know that they're saying two things that are implicitly inconsistent with each other.

Moreover, they seem to be on solid ground here. It might seem like almost nothing is universally agreed on in the philosophy of logic. (After all, it's arguably the area of philosophy where the claims in dispute are the most basic.) Are contradictions ever true? Are instances of Excluded Middle always true, or should we sometimes simultaneously reject claims are their negations? What does it mean to say that one thing follows from another? Is there One True Logic, or should we be pluralists?

In all of this chaos, however, one of the few things that nearly everyone agrees on is what we could call the Universality of Logical Truth Thesis (ULTT). Dialetheists believe that some contradictions are absolutely, perspective-independently true no less than their orthodox opponents believe that all contradictions are are absolutely, perspective-independently false. Paracomplete theorist reject the view that, say, the instances of Excluded Middle that are relevant to the Liar Paradox are true for *anyone.* Even logical pluralists--e.g. Greg Restall and JC Beall, currently the best-known advocates of logical pluralism--generally focus on questions of validity. They argue that there are a plurality of genuine, legitimate logical consequence relationships, that that each one really does delineate a class of "valid" inferences is universally true and that these claims do not contradict each other. It's simply that some logical consequence relationships are appropriate for regulating our reasoning about some areas and that others are appropriate for regulating our reasoning about other areas. The truth of these appropriateness claims themselves will, again, be universal.

Of course, most philosophers believe in what we could call the UATT--the Universality of All Truth Thesis--and wouldn't bother thinking of the ULTT as a separate matter. The ULTT is important, though, because even quasi-realists, having rejected the UATT, still have good reason to cling tight to the ULTT. In fact, the ULTT looks like it's going to be absolutely central to their project, for cashing out what "idealized circumstances" look like, for making sense of why we should criticize people for having internally inconsistent moral stances, etc.

But there's a problem.

What about people who literally don't have *any* moral attitudes, and who are psychologically constituted in such a non-standard way that they're just totally incapable of forming any attitude of that type--they're morally color-blind? Parsing the existing scientific literature on people who have deeply non-standard psychological make-ups in ways that are morally relevant--say, psychopaths--raises all kinds of complicated conceptual and empirical problems that we don't need to get into here, but surely it's at least *possible* in principle for such people to exist.

From their perspective, no matter of expressing attitudes that they do have or under any circumstances would have could possibly decide between the truth of a moral claim and the truth of its negation. It seems like, given the overall story, the most natural thing to do would be to say that, from the perspective of the morally disengaged observer listening to a moral debate, either the person who says "killing small children for sport is wrong" and the person who says "killing small children for fun isn't wrong" are both saying true things, or neither of them is saying a true thing, or perhaps there's simply no fact of the matter about whether any such statements are true. All of these options, of course, get you logically heterodox general results.

And that seems to show that quasi-realists really aren't entitled to the full ULTT. The universality of logical truth at least can break down.

And, given the ULTT's apparently centrality to the quasi-realist's project, that seems like a problem.

*Remember, on a quasi-realist story, "killing children for fun" isn't a descriptive statement--it's not even a descriptive statement about the speaker's attitudes--but an expression of those attitudes. As such, "killing children for fun is always wrong" is *also* an expression of an attitude, so (at the very least) it's not entirely clear that one can take a classical way out here and claim that all negations of positive moral claims are true (for someone totally incapable of forming attitudes of the relevant type) according to the quasi-realist, the way that all such statements are true (for everyone) according to the error theorist.

Reader Requests?

2010-09-06T23:59:00.000-07:00

In recent weeks, I've done two four-part sets of posts, one about the Liar Paradox and one about set theory. Is there anything in particular that anyone would like to hear about in upcoming posts?

Obviously, my likelihood of actually taking a suggestion is correlated with the degree to which I have much of anything to say about the topic....e.g. last time I asked for requests, someone wanted a post about theories of truth-makers, but sadly, all I have to say about truth-makers is "I'm a bit leery about propositions for the usual reasons that people are often a bit leery of propositions or other kinds of abstract objects, and I'm somewhat agnostic about precisely what the primary bearers of truth actually are." Which wouldn't have made a very interesting post. That said, I'm open to suggestions about anything from Yablo's Paradox to the new Of Montreal album, quantum logic to the mid-term elections, Wittgenstein to whiskey, so ask away.

Bertrand Russell: Portrait Of The Philosopher As A Young Man

2010-09-01T00:01:00.000-07:00

One of the classes I'm teaching in Korea this semester is called "Analytic Philosophy", and one of the textbooks I'm assigning is Bertrand Russell's book "My Philosophical Development."

In Chapter 3, "First Efforts," Russell records his first youthful doubts about conventional ideas.

"I began thinking about philosophical questions at the age of fifteen. From then until I went to Cambridge, three years later, my thinking was solitary and completely amateurish, since I read no philosophical books, before I read Mill's Logic in the last months before going to Trinity... I minded my theological doubts, not only because I had found comfort in religion, but also because I felt that these doubts, if I revealed them, would cause pain and bring ridicule, and I therefore became isolated and solitary. Just before and just after my sixteenth birthday, I wrote down my beliefs and un-beliefs, using Greek letters and phonetic spelling for purposes of concealment."

To which I have to say, uh, really?

The Greek letters thing is a nice, vivid, picturesque image, but you have to wonder if someone as smart as Russell obviously was, even at the age of 16, would have thought that this method would actually fool anyone in his household.

To review some relevant facts:

Russell's grandfather had been the Prime Minister in the 1840s and again in the 1860s. The family had been raised to peerage with the rise of the Tudor dynasty....i.e. a few centuries before Young Master Russell turned 16. It's safe to say that every male in the Russell clan since time immemorial would have received a good classical education. It doesn't seem like much of a stretch to say that they would have all been sufficiently used to reading Greek that any of them would have been able to tell that they were reading English words transliterated into the Greek alphabet if they'd just glanced at the page for long enough to read a sentence while shuffling around papers looking for a misplaced cup of tea.

(And that's just the boys. I also wonder if, in a family as progressive as the Rusells--keep in mind that Russell's godfather was John Stuart Mill--the education of the girls might have been considerably better than average as well.)

All of which makes me wonder: was young Bertie really particularly concerned about concealment, or did he just enjoy the romantic gesture of making a big elaborate show of concealment?

Moving on to the actual contents of the journal--which Russell faithfully reproduces in full in My Philosophical Development, while making embarrassed noises about the confused, undeveloped nature of a lot of the ideas therein--we find a lot of skepticism about traditional Christian dogmas, but Russell doesn't go quite so far at this point as to doubt the existence of God per se. When it comes to morality, we see a lot of the the sharp polemical humor you get in his later writings. For example, in one passage, he talks about his Presbyterian grandmother's view that, instead of using reason to tell right from wrong, one should follow the 'inner voice' of conscience, then a few paragraphs down he casually refers to "this inner voice, this God-given conscience which made Bloody Mary burn the Protestants..."

Everywhere, he protests in a fairly hyperbolic way about his dedication to rationality, e.g. "April 29. In all things, I have made the vow to follow reason, not the instincts inherited partly from my ancestors and gained gradually by selection and partly due to my education. How absurd it would be to follow these in the questions of right and wrong."

Keep that passage in mind while we go back and take a closer look at the bit about his grandmother:

"My rule of life which I guide my conduct by, and a departure from which I consider as a sin, is to act in the manner which I believe to be most likely to produce the greatest happiness considering both the intensity of the happiness and the number of people made happy. I know that my grandmother considers this an impractical rule of life and says that, since you can never know the thing which will produce the greatest happiness, you do much better in following the inner voice. The conscience, however, can easily be seen to depend mostly upon education (as, for example, common Irishmen do not consider lying wrong) which fact alone seems to be quite sufficient to disprove the divine nature of conscience."

Some thoughts about this:

(1) He considers his ideas about this subject to be shocking enough to go in his secret journal of forbidden thoughts, but he had at least one argument about it with grandma?

(2) The racism here is pretty awesome. It seems like a safe guess that the young English aristocrat writing this journal had never actually met a 'common Irishman', nor quite likely had he ever met anyone who had ever met one, so you have to wonder where exactly he got his information about The Irish And Their Propensity To Lie.

(3) He claims to have read no philosophy books at this time, and maybe he hadn't, but somehow or another he seems to have absorbed the utilitarian ideas of Bentham and Mill in full, complete with the precise characteristic turns of phrase and careful qualifications--"the greatest happiness considering both the intensity of the happiness and the number of people made happy."

(4) However that may have come about, it's awfully interesting that Russell's steadfast dedication to his sacred vow to follow reason alone in determining the difference between right and wrong, sweeping aside all the mental clutter derived from his ancestors and his education, led him to replicate, in a meticulously exact fashion, the precise moral opinions of his godfather, John Stuart Mill, and Mill's godfather, Jeremy Bentham.

Ed Feser And The Bomb

2010-08-30T00:01:00.000-07:00

Over at What's Wrong With The World, Ed Feser marked the anniversary of the bomb being dropped on Nagasaki with a post entitled "Happy Consequentialism Day!" He wrote:

"Certainly the image above – the aftermath of Fat Man’s explosion over Nagasaki – is a fitting symbol for consequentialism. Perhaps consequentialist ethicists should consider putting it on the covers of their books, or wear little mushroom cloud pins when they meet up at philosophical conferences. For one thing, since the consequentialist case for the bombings – that they would save more lives than an invasion of Japan would – carried the day with the Truman administration (and with defenders of the bombings ever since), it may be the most consequential piece of consequentialist reasoning ever formulated. For another, the bombings give a pretty good idea of what a world consistently run on consequentialist principles might look like."

Now, I personally find it fairly difficult to imagine a world consistently run on consequentialist principles. (Or even a rough approximation of it--say, a world where Peter Singer exerted as much influence as Henry Kissinger has exerted in the actual case.) I'd hazard a strong guess, though, that no nation-state in such a world would possess atomic weapons. Whatever. Disregarding that particular rhetorical fourish, Feser's point is clear. The bombings represent a grotesque reductio of consequentialist moral principles, right?

Well....

As anyone who knows anything about the relevant history knows, the consequentialist case for the bombings was always fantastically weak and unconvincing. And, indeed, Feser acknowledges--in the post, and in the comments--that most consequentialist moral philosophers are thoroughly unconvinced by it for obvious reasons, but:

"Whatever. What matters is that any consequentialist must allow that it is at least in principle legitimate intentionally to kill the innocent for the sake of a 'greater good.' And from the point of view of us reactionary, bigoted, unprogressive natural law theorists and Catholics, that is enough to make consequentialism a depraved doctrine. For it is never, never permissible to do what is intrinsically evil that good may come – not even if you’d feel much happier if you did it, not even if you’ve got some deeply ingrained tendency to want to do it, not even if it will shorten a war and save thousands of lives. Never."

Got that?

Whether or not the bombings are justifiable on consequentialist principles, Feser thinks that the bombings are a good reminder of why "consequentialism is a depraved doctrine." So, even if the consequentialist case for the bombings is unconvincing, consequentialist moral philosophers should still "wear little mushroom cloud pins when they meet up at philosophical conferences."

Why? Well, as far as I can tell, the whole of Feser's argument is that:

(1) Truman justified his actions by appealing to consequentialist considerations.
(2) Those who support the bombings typically do so by appealing to such considerations.
(3) In principle, if the bombings did lead to good enough consequences, they could be justified on consequentialist grounds.

Generalizing, we get the principle that it's appropriate to say that some group of moral philosophers should wear pins depicting Historically Important Action X iff (a) they subscribe to Moral View Y, (b) the perpetrators of Historical Important Action X appealed to Moral View Y to justify their action, and (c) according to Moral View Y, Historically Important Action X would, in principle, have been justifiable if only some relevant parameter had been met.

Keep that standard in mind.

Now, in the ensuing discussion in the comment box, someone made an obvious point:

As a devout Christian who believes that God is just, the Bible is divinely revealed, etc., isn't Feser committed to saying that it's sometimes acceptable to intentionally massacre innocent civilians after all? Isn't the Bible full of stories where the ancient Israelites do all sorts of nasty and even downright genocidal things--massacring "everything that breathes" in an enemy town, impaling babies on swords and so on--with God's enthusiastic approval? Doesn't this rather undermine all that self-congratulatory rhetoric about how "us reactionary, bigoted, unprogressive natural law theorists and Catholics" would never ever do things like bomb Hiroshima no matter what, and the fact that doing such things is OK "in principle" for consequentialists makes consequentialism a depraved doctrine?

Feser's response is that natural law morality, with its prohibitions on intentionally killing the innocent, is all about the natural ends of human beings, rather than God's supernatural ends. "So while we could never take it upon ourselves intentionally to kill the innocent, God can do so as part of His intention to realize for us a supernatural end."

Got that?

Human beings qua human beings aren't allowed to massacre towns full of innocent men, women and children, but if God wants them to and they are thus acting as human beings qua agents of God, everything's on the table! If God wants you to impale some Canaanite children, you should do it. If God wants you to slice up some women in a Philistine town in the fashion of Patrick Bateman in American Psycho, it would be wrong not to do it.

Now, in any given case where people think that God wants them to commit a massacre of the innocent, they can be wrong, and in terms of Feser's preferred framework of moral theology, it can be very clear that they're wrong, but their wrongness is contingent. In principle, someone who goes around in 2010 acting like one of those agents of God in the old Bible stories isn't doing something that's necessarily wrong. They're doing something that happens to be wrong in this particular context, because they're wrong about God's will.

Given all of this, I have a fairly specific idea about what image should adorn the pins worn to philosophy conferences by Feser and like-minded Christian ethicists.

Why I'm Not A Bayesian

2010-08-25T00:01:00.000-07:00

Every once in a while, I'll get into a bullshitting session with a grad school friend and they'll ask me what my Bayesian probability estimate would be of such-and-such claim turning out to be true. What do I think the probability of God existing is? Of the Singularity happening? Of alien life existing somewhere?

I always have to explain in buzz-kill-ish fashion almost worthy of Buzz Killington that I don't think that there are probabilities in the sense assumed by the question--to be more precise about this, I think the probability calculus is a marvelous mathematical tool for juggling frequencies, but I strongly reject the claim that it can be used to model "degrees of belief" or justification or confirmation or rational belief revision or anything of the kind--and that, since I think the game people are playing when one says that the probability that God exists is .01% and the other says, "really, you think that high? I'd say .0001%" and so on is a deeply confused and silly game that doesn't really shed light on anything, I decline to play. I'm happy to say things like "given the overall evidence, it's irrational to think that God exists", or "given the overall evidence, my best guess is that the Singularity will not happen" or whatever, but that's where I'll leave it. There may be things such that I'd be much more surprised if they turned out to be true than I would be if other things turned out to be true, some things I'm more likely to constantly scrutinize new evidence and new arguments to make sure I'm right about than others, etc., but--it's good to repeat this, because I find that many people whose philosophical training has simply assumed Bayesianism end up being so shocked when you say that you reject it that they assume you must be saying something else--I don't think anything like the probability calculus is particularly relevant to the regulation of rational belief formation or rational belief revision.

So, why do I think this strange thing?

I've covered a lot of this ground here before, so most of this will be linking and summing up.

For one thing, I think that one of the most obvious claims in all of epistemology is that if you're rationally entitled to believe all of the premises of a valid deductive argument, and you know that the argument is valid, you're rationally entitled to believe the conclusion on that basis. Of course, it could be that you *were* rationally entitled to believe all of the premises until you reached the conclusion, but that the absurd conclusion makes continued acceptance of the premises irrational. That's fine. The relationship works in both directions. As we constantly teach our introductory logic students, if you're confronted with an apparently valid deductive argument connecting premises you accept with a conclusion you reject, you can't just say, "oh, well, it doesn't matter. Even if my beliefs do entail that other thing, I still believe what I believe and not that." You only have three choices. You can re-examine and ultimately reject a premise, you can find a flaw in the reasoning connecting the premises to the conclusion or you can go ahead and accept the conclusion after all. There's a lot more to rational belief-revision than that--a lore more--but that's the core.

Not obvious enough for you? OK, how about the following, which I think is an even more basic and obvious epistemic principle. If you know that something absolutely can't be true, you shouldn't believe it.

Epistemic principles really don't get a lot more intuitively compelling than that, do they? Well, cases like the Lottery Paradox and the Preface Paradox show that the two principles just laid out are in direct conflict with Bayesianism. See here for a more detailed explanation.

Another problem is that it seems deeply, crazily irrational to me to think that we can be absolutely certain that our initial best guesses about logic must be right. Whether one thinks (as I do) that logical laws are a matter of universal truth preservation, and thus that logical truth supervenes on all other kinds of truth (facts about protons and electrons, tables and chairs, dogs and cats) and is thus vulnerable to possible revision in light of new developments elsewhere in overall theory of the world, or one thinks (as many others do) that logical laws encode certain "rules of use" implicit in our "language" or some such thing, neither of those stories (nor any other remotely plausible view about logic) gives us any reason to think that we can be absolutely sure about it. What about linguistic or psychological "rules of use" makes you think that we can be absolutely infallible in our epistemic access to them, that we're incapable of making mistakes about them? Still more so, if our current beliefs about which logical laws there are encode our best theory--relative to the level of generality and abstraction at which formal systems operate--of how Absolutely Everything is, it seems beyond foolish to think that we can be utterly and infallibly certain about *that.*

When Frege and Russell, a bit over a century ago, codified the system we know think of as "classical logic", they were doing exactly what their non-classical opponents have done since then, which is attempting to capture a bunch of intuitions. Since then, "classical logic" has been challenged on the basis of a bunch of other intuitions--about referring to non-existents (free logics), about what it takes for one claim to really "follow" from another (relevance logics) and so on--and they've tried to capture these intuitions in formal systems of their own. Some reasons to doubt some very central assumptions built into not only classical logic but also into the older kind of syllogistic logic inherited from ancient Greece have been around since ancient Greece--the "sea battle" problem about future contingents, the Liar Paradox, problems about vagueness, etc.--and there still isn't any clear consensus about what to make of them.

Now, I find myself in the orthodox camp here--I think Frege and Russell's best guesses are still pretty much our best guesses--but the idea of thinking that we're rationally entitled to be absolutely certain, that there's no room for doubt, that various objections to the classical view don't deserve at least some serious epistemic weight and consideration, seems utterly indefensible to me.

Why do I stress this so much?

Well, once again, this obvious-seeming view is utterly incompatible with Bayesianism.

A Reply To Same Objections Made During the Liar Paradox Series

2010-08-23T00:01:00.000-07:00

In the series of posts I did over the course of the last couple weeks, I laid out my preferred solution to the Liar Paradox. In Part I, I argued on the basis of disquotationalist considerations about truth that sentences that like the Liar ("this sentence is false") and the Truth-Teller ("this sentence is true"), while they may seem meaningful, are actually meaningless. Attributions of truth to a sentence mean nothing above and beyond the meaning of whatever sentence they attribute truth to, and attributions of falsehood to a sentence mean nothing above and beyond the meaning of the negation of the sentence they attribute falsehood to. Given a string of sentences like:

Sentence 1: Sentence 2 is true.
Sentence 2: Sentence 3 is true.
Sentence 3: Snow is white.

....the rest of the sentences in the series inherit their meaning from Sentence 3. This view can be summed up by saying that meaningful truth talk reduces to truth-free talk. Sentences like the Liar and the Truth-Teller, which lack any 'true'-free sentences from which they can inherit their meanings, are therefore literally meaningless. Although this is a straightforward consequence of an independently-motivated, principled view rather than a desperate attempt to save consistency or retain classical logic, it does have the happy consequence that, if we accept it (as opposed to many standard approaches to the Liar) we are entitled to hold onto full logical orthodoxy.

In Part II, I argued against the view that competent speakers are infallible on questions of meaningfulness. Rather, I argued, it's quite possible for a sentence to be meaningless even if most people's initial intuition is that it is meaningful, or vice versa. I also provided some specific reasons why this might be a case where people are particularly likely to make mistakes about meaningfulness. Part of that error theory, which I should make more explicit now, is that, in certain contexts, the same combination of words does add up to a meaningful sentence. If someone has written "the Normans conquered England in 1066" on a chalkboard and I point to it and say "this sentence is true", I have said something meaningful, since (given my preferred story about truth) what I have said can be accurately translated into "the Normans conquered England in 1066", a meaningful sentence. It's only if the "this" is intended to refer to the sentence being spoken--or some other bit of truth talk that doesn't succeed in inheriting its meaning from a 'true'-free source, or of course some other sort of meaningless sentence, like "colorless green ideas sleep furiously"--that "this sentence is true" is meaningless.

Similarly, "this is true" is surely often a meaningful utterance, as in the following snatch of conversation:

Tom: "The Normans conquered England in 1066."
Jerry: "This is true."

If, on the other hand, Jerry points to a rock and says "this is true," and subsequent questioning shows that he intends for the "this" to refer to the rock, "this is true" is meaningless. Rocks simply aren't the sorts of things to which truth talk meaningfully applies. Neither are bits of irreducible truth talk.

In Part III, I took on various revenge paradoxes like the one posed by "this sentence is either false or meaningless" and in Part IV I argued that the solution I'd been arguing for had the benefit of providing an absolutely unified account of all of the standard semantic paradoxes, and that this was a considerable advantage, given the embarrassingly disunified account of the Liar and Curry you get with the leading non-classical approaches.

There were a couple of interesting objections raised in the comment thread on Part II that I didn't really have time to address earlier, so, rather than going back to a two-week-old comment thread occasional readers are hardly likely to be haunting for further developments, I thought I'd say something about them here:

Jonas asked whether I'd characterize bits of self-referential truth talk that intuitively seemed clearly true or clearly false--one of his examples was "this sentence is obviously false"--as meaningless. (As I understand it, since it's not obvious one way or the other whether "this sentence is obviously false" is false, it is false, but not obviously so, making it unproblematically and unparadoxically false.) I think an even cleaner example of an unproblematically-false looking or unproblematically-true-looking Liar-like sentence would be "this sentence is either true or false." (We can call that the Disjunctive Liar/Truth-Teller.) And my answer would be that yes, it's meaningless. It's possible to engineer bits of irreducible truth talk that seem to be simply true or false (like the examples just given) or that seem to be both true and false (like standard Liars) or that seem to show that everything is true (like Curry) or whose truth-value seems to be totally and permanently mysterious (like the Truth-Teller). In all cases, though, if my arguments in Parts I and II go through, we both have a good, plausible reason to suppose that such sentences are meaningless and we have a good explanation of why they might initially seem meaningful.

(On a somewhat related point, in the comments on Part IV, Emil asked me what view of truth and meaning I was working from. On truth, my view is radically disquotationalist--the meaning of a sentence that attributes truth simply is the meaning of the sentence it attributes truth to--and I think the best argument for that is that it's the simplest, cleanest theory of truth talk that there is. Talking about elaborate structural correspondence or coherence with other sentences in an overall framework adopted at the end of inquiry or whatever--or even, as in more 'substantive' versions of disquotationalism like JC Beall's, talking about obeying certain inference rules--strikes me as introducing complications that are unnecessary to an explanation of the phenomenon at hand. In terms of meaning, I think it's a virtue of this view is that it doesn't require any controversial assumptions about meaning, above and beyond the view--which, as I argued in Part II, we have plenty of independent reasons to accept--that competent speakers are fallible on questions of meaningfulness. To make a synonymy claim, or even to simultaneously assert a whole type of synonymy claims--e.g. "a simple, literally-intended German sentence that does nothing more than refer to some stuff on the ground and say that it 'est schnee' doesn't mean anything different than a simple, literally-intended English sentence that does nothing more than to refer to the same stuff and say that it 'is snow'--isn't to commit yourself to any particular theory of meaning.)

Also in the comment thread on Part II, Jason Streitfeld made a comment that's interesting enough to be worth breaking down in detail:

#

I'm not sure why "This sentence is false" (P) is meaningless when said of "colorless green ideas sleep furiously." What if somebody really believed that idea made sense and was false? They might be wrong, but that doesn't make their assertion meaningless.

I think you are making a mistake in your argument. You over-generalize from Quine's disquotationalism. Quine's point is about adding "is true" to sentence. That does not mean that the meaning of all "X is true" sentences mean the same as X. For example, "What she says is true" does not simply mean "What she says." "What she says" is not a well-constructed sentence. Similarly, "This sentence is true" does not mean "This sentence," which also is not a well-constructed sentence.

As you agreed earlier, we can use "this sentence is true" in clearly meaningful ways. I think you want to distinguish these meaningful cases with the Liar Paradox by claiming that "this" in P lacks content (in the relevant contexts). But I don't see how you are establishing that.

#

Let's take this one piece at a time.

"I'm not sure why 'This sentence is false' (P) is meaningless when said of 'colorless green ideas sleep furiously.' What if somebody really believed that idea made sense and was false? They might be wrong, but that doesn't make their assertion meaningless."

Well, someone might be wrong about whether it was meaningful, and thus falsely say "'colorless green ideas sleep furiously' is a meaningful utterance", and in that instance, the fact that they were wrong certainly wouldn't make their assertion meaningless. (Their statement would be false, not meaningless.) If, on the other hand, they thought thought it was meaningful and true, and thus they asserted it (that being a natural thing to do when one takes a statement to be true), simply saying "colorless green ideas sleep furiously", then, of course, given that our starting point was that this is a meaningless combination of words, their statement would be meaningless. If they expressed their belief that it was true in a different way, by saying "'colorless green ideas sleep furiously' is true", would that be any more meaningful? How about it if they pointed at "colorless green ideas", written on a chalkboard, and said, "this sentence is true"? From my point of view, all three sentences--the raw assertion of the original sentence, the quotation of the original sentence with 'is true' appended to the quotation, and the reference to it as true, necessarily mean the same thing...which is to say that none of them mean anything at all.

If you, dear reader, do take this series of utterances--going from "colorless green ideas sleep furiously" to "'colorless green ideas sleep furiously' is true" to "this sentence is true"--where the "this sentence" is meant to refer to "colorless green ideas sleep furiously"--at some point becomes meaningful, I'd be interested in hearing in the comments section about where the cutoff point is. Which of these formulations is meaningful, and what's the difference between it and the meaningless formulation(s)?

Similarly, if someone believes--correctly!--that "snow is white" is meaningful and in fact true, and expresses this belief by simply saying "snow is white", or they express it in a slightly more elaborate way by saying "'snow is white' is true", or they express it a slightly more indirect way by pointing at "snow is white" written on a chalkboard and saying "this sentence is true", it seems to me that all three sentences have precisely the same meaning. None of them do anything but ascribe whiteness to snow. Again, if anyone has a different take on this, I'd be interested to hear it--if one of these sentences means something else, exactly what does it mean?

I suspect that the real work is being done in this paragraph by the phrase "really believed," which seems to sneak back in the idea that competent speakers are infallible about at least the meaningfulness or meaninglessness of their own utterances. This seems to me to be falsified by the "bored dinner companion" example I gave in Part II, and also by the simpler case where someone really believed that "colorless green ideas sleep furiously" was meaning, and who really believed that it was true, and therefore asserted it.

Similarly, I think that to assert that a sentence is false is simply to assert its negation--it has no additional content above and beyond this--so "this sentence is false" (where "this sentence" refers to the sentence "snow is white") means the same thing as "snow is not white." Similarly, "this sentence is false", where "this sentence" is supposed to refer to "colorless green ideas sleep furiously", means the same thing as "colorless green ideas don't sleep furiously", which is to say that it means nothing at all.

Moving on to the next paragraph:

"I think you are making a mistake in your argument. You over-generalize from Quine's disquotationalism. Quine's point is about adding 'is true' to sentence. That does not mean that the meaning of all 'X is true' sentences mean the same as X. For example, 'What she says is true' does not simply mean 'What she says.' 'What she says' is not a well-constructed sentence. Similarly, 'This sentence is true' does not mean "This sentence," which also is not a well-constructed sentence."

I want to start with a slightly nit-picky but I think potentially important point, which is that, as I've argued here on multiple occasions before, being "well-constructed" or not has nothing whatever to do with being meaningful or not. Or, to be more precise, "well-constructedness" is neither necessary nor sufficient for meaningfulness. It is, at best, a very weak and defeasible indicator. The usual linguist's and philosopher's stock example of a meaningless sentence--"colorless green ideas sleep furiously" is grammatically "well-constructed." A great many meaningful statements in ordinary conversations, blog posts, quick exchanges with cashiers at coffee shops, objections raised at philosophy talks and so on, are not. The widespread idea that "well-constructedness" (or, more typically, "well-formedness") has its roots, I suspect, in an exaggerated and idealized analogy between the grammatical rules of real languages and the formation rules of formal logical systems.

A perfect example of the obvious possibility of meaningless but not "well-formed" utterances comes from the paragraph I just quoted! "What she said" isn't in itself a well-constructed sentence, but in many contexts is a perfectly meaningful and instantly understood utterance. In fact, in ordinary conversational English as practiced by people of a certain age range, it's a fairly common one, e.g. this snatch of conversation people might have after putting up a roof.

Jack: "What do you think, Jill? Is that good enough."
Jill: "No way. It's going to collapse within a day, two days max."
Jack: "Huh. How 'bout you, John, what do you think?"
John: "What she said."

(In my view, actually, conversational short-cuts like "what she said" actually function quite a bit like truth-attributions.)

Moving on to more substantive issues with the quoted paragraph, to re-cap some of the discussion above, I'm curious about what the difference is supposed to be in content between, say, Sentences 2 and 3 in this series:

Sentence 1: Snow is white.
Sentence 2: 'Snow is white' is true.
Sentence 3: Sentence 1 is true.

My view is that all three sentences mean precisely the same thing. Of course, one could hold an odd sort of combined view where one analyzed Sentence 2 disquotationally and thus argued that it meant the same thing as Sentence 1, but switched to a substantive view of truth when analyzing Sentence 3, thus arguing that it meant, say, "there is a complicated structural correspondence relationship between Sentence 1 and Sentence 3." This hybrid view--that 'truth' means one thing in cases where one bothers to quote the sentence and another thing when one uses short hands like 'Sentence 1' or 'this sentence'--seems like an odd and implausibly complicated position, and I certainly see no reason to attribute it to Quine.

To be clear, my position is not that we accurately capture the meaning of every sentence with the words "is true" in it by chopping off those words from the end of the sentence. It's that all meaningful attributions of truth inherit their meanings from the sentences to which truth is being attributed. So, for example, in the following exchange:

Jill: "The Normans conquered England in 1066."
John: "What she says is true."

....I'd argue that John's statement simply means "the Normans conquered England in 1066."

Similarly, in a more complicated (and, granted, fairly artificial case) like this one:

Jill: "The Normans conquered England in 1066."
John: "What she says is true."
Jack: "The sentence just uttered by John is true."
Jake: "What Jack says is true."
Jose: "What Jake said? Yeah, that's true."
Juan: "My brother Jose just said a true thing."

....the content of every sentence in that series is "The Normans conquered England in 1066." Juan's sentence inherits its meaning from Jose's, Jose's from Jake's, Jake's from Jack's, Jack's from John's and finally John's from Jill's.

My view, then, isn't that "this sentence is true" means "this sentence", but that any instance of the combination of words "this sentence is true" means whatever the sentence being referred to with the phrase "this sentence" means. If "this sentence" is intended to refer to itself, "this sentence is true" means nothing at all.

Moving on to the final paragraph:

"As you agreed earlier, we can use 'this sentence is true' in clearly meaningful ways. I think you want to distinguish these meaningful cases with the Liar Paradox by claiming that 'this' in P lacks content (in the relevant contexts). But I don't see how you are establishing that."

We agree that the words "this sentence is true" can be used in meaningful ways. If one accepts Quine's point about "'snow is white' is true"--and doesn't hold the odd view that truth-attributions work in a fundamentally different way depending on whether one bothers to quote the whole sentence to which one is attributing truth or just uses a short-hand device like 'that sentence' or 'Sentence X' or whatever--an obvious consequence of Quine's view is that there are obviously, unambiguously meaningless instances of the combination of words "this sentence is true", which are just the instances in which "this sentence" refers to a meaningless sentence. Given that the existence of both categories of instances of "this sentence is true"--meaningful ones where the 'this sentence' refers to a meaningful sentence and meaningless ones where the 'this sentence' refers to a meaningless sentence--the interesting question is that of which category "this sentence is true" falls into where the "this sentence" is intended to refer to the sentence in which it appears. My view is that truth-attributions don't start working in a fundamentally different way in self-references cases. In all cases, truth-attributions inherit their meaning from the sentences to which they attribute truth. An infinite series of sentences like:

Sentence 1: Sentence 2 is true.
Sentence 2: Sentence 3 is true.
Sentence 3: Sentence 4 is true.
....and so on into infinity....

....never reaches a non-truth-attributing sentence from which all the truth-attributing sentences in their series can inherit their meaning, so they are meaningless. The self-referential "this sentence is true" has the same problem, and is similarly meaningless. Adding the word "not" to a meaningful sentence doesn't make it meaningful, and the relationship between the self-referential "this sentence is true" and the self-referential "this sentence is not true" (or, equivalently, "this sentence is false") is no exception.

My Take on the Liar Paradox (Part IV of IV)

2010-08-18T00:01:00.000-07:00

In Part I, argued on the basis of disquotationalist considerations about truth that ascriptions of truth (or falsehood) necessarily inherit their meanings from the meanings of the statements they ascribe truth to (or, in the case of falsehood, the negations of those statements), and that, as such, sentences like the Liar and the Truth-Teller, which have no ‘true’-free sentence from which they can inherit their meaning are, despite appearances , meaningless.

In Part II, I responded to the objection that meaningless sentences can’t appear to be meaningful to otherwise competent speakers of the language. On the contrary, I argued, such mistakes are quite possible (and, in areas less controversial than this one) even common, and we can provide a plausible error theory to explain why mistakes about meaningfulness are so common when it comes to these particular sentences.

In Part III, I responded to various “revenge paradoxes” that might seem to arise for the view I defend. One of the important points brought out in that discussion was that strings of words that have the grammatical structure of compound statements—like disjunctions, and, crucially, conditionals and that have meaningless strings of words as ‘disjuncts’ or ‘antecedents’ or whatever—are themselves meaningless, and that we have excellent reasons to think so quite apart from paradox-avoidance. Given this point, and the sort of disquotationalist story that I told to motivate the claim that the Liar was meaningless, an obvious consequence is that Curry sentences are also meaningless. The point I ended on, and which I want to draw out now as the capstone of the series of posts, is that this amounts to a massive advantage of the classical approach to the paradoxes that I’m arguing for over any of the leading non-classical approaches currently ‘on the market’: a completely unified solution to the Liar and Curry Paradoxes.

Here’s one way to think about all of this. The Liar Paradox amounts to a sort of prima facie sound argument for trivialism, consisting of two parts—standard liar reasoning (which gets us a contradiction) and the explosion proof (which gets us from that contradiction to triviality). The dialetheist solution blocks all of this by accepting the inference to contradiction, but by rejecting the inference from that contradiction to triviality. The paracomplete solution blocks it by denying the Excluded Middle instances you need to get the argument to a contradiction through standard liar reasoning off the ground. The Curry Paradox amounts to a simpler argument from triviality—a direct train from the truth conditions of the paradoxical, self-referential statement to triviality rather than one where you have to take one train from those truth conditions to contradiction, then transfer to a different train to get from contradiction to triviality. Neither the paracomplete theorist’s methods for derailing the train from truth conditions to contradiction nor the dialetheist’s methods for derailing the train from contradiction to triviality have the slightest effect on the progress of the direct train from truth conditions to triviality. It doesn’t matter whether Curry is false or neither true nor false or the sort of thing about which there is no fact of the matter about whether it is true or false or whatever, because the mere statement of its truth conditions generate triviality. Moreover, neither dialetheists nor paracomplete theorists are particularly eager to deny that those apparent truth conditions are what they appear to be, since, if they found that a plausible move in the first place, they could have used it to solve the Liar Paradox. If one’s willing to simply give up the Liar’s apparent T-Schema truth conditions, one doesn’t have to give up on Disjunctive Syllogism or the Law of the Excluded Middle in order to block triviality.

As such, Priest and Beall on the dialetheist side and Field on the paracomplete side all resort to solving the Curry Paradox in a way that has absolutely nothing to do with their solutions to the Liar—by weakening the inferential power of their conditionals. This should be a massive embarrassment for proponents of both approaches.

To see why, consider how much emphasis Priest places on the virtues of the dialetheist’s unified solution to the Liar and Russell’s Paradox, which we took a look at in the previous series. Surely, the Liar and Curry—as paradoxical sentences whose paradoxicality relies on their self-referential truth-talk—have vastly more in common with each other than either has in common with any paradox about set theory.

Priest takes the relevant feature binding the Liar and Russell’s Paradox to be the Inclosure Schema, and grants that, if expressed in terms of the old, classical, un-weakened conditionals, Curry is a Schema paradox, but argues that, once the conditionals have been suitably weakened, it does not. I have trouble seeing how different this would be from someone like Gil Harman, who argues that the lesson we should draw from the Liar is that the T-Schema has exceptions, saying that, once we deny the Liar’s truth conditions, it isn’t a Schema paradox like Russell’s Paradox, so it’s OK to solve the two paradoxes in different ways. However this may be, I think that given the obvious kinship between the two semantic paradoxes of self-reference under consideration, I’d say that if Curry isn’t a Schema paradox, so much the worse for the Schema as an intuitively plausible tool for separating paradoxes into relevant types demanding unified solutions.

Think of it this way:

Earlier, I said that one way to think about the Liar Paradox is as a prima facie sound argument for trivialism and that all solutions to it—classical, paracomplete, dialetheist or whatever—amount to different strategies for blocking triviality in the light of the existence of such sentences. After all, in classical logic, “Sentence X is false” and “if Sentence X is true, everything is true” are logically equivalent to each other. Thus, if some false sentence can be shown to also be true, the floodgates open and reasoning collapses into rubble. The argument for triviality, then, goes like this.

1. The Liar must be either true or false.
2. If it’s true, it’s false.
3. If it’s false, it’s true.
4. Given 1-3, it’s both true and false.
5. Given that it’s both true and false, everything’s true.

The truth-value gap theorist tries to escape triviality by denying (1). The obvious revenge paradox for them, is, of course, the Strengthened Liar:

“This sentence is not true.”

If it’s neither true nor false, it’s not true, so the inference to triviality continues to go through. If one “solved” the Simple Liar by denying Bivalence but tried to solve the Strengthened Liar in some totally different, disunified way, everyone would see this as a huge problem. Similarly, when people like me deny that the Liar is meaningful, we have to say something plausible about ‘revenge’ variants like...

“This sentence is either false or meaningful.”

If the gap theorist acknowledged that their solution to the Simple Liar didn’t work for the Strengthened Liar and tried to solve that by meddling with unrelated logical machinery, or if I acknowledged that my solution to standard Liars didn’t work for the sentence quoted above and tried to solve it in a way that had nothing to do with my approach to standard Liars, everyone would see this as a massive disadvantage. Whether or not one goes as far as Priest and insists that all structurally similar paradoxes be solved in the same way, a standard assumption made by everyone is that any plausible solution to the Liar Paradox must be one that extends in a natural way to various “revenge paradoxes” formulated using the terms of the solution. When it comes to paradox-solving, nearly everything is controversial, but that principle isn’t.

Now, just as the gap theorist tries to escape triviality by rejecting step 1 of the argument laid out above, the dialetheist tries to escape triviality by rejecting step 5. The dialetheist, in other words, escapes triviality by rejecting the classical equivalence between “Sentence X is false” and “if Sentence X is true, everything is true.” As such, using the standard revenger’s technique for formulating a revenge paradox using the precise move made by the paradox-solver against them, the obvious revenge paradox for the dialetheist would be...

“If this sentence is true, everything is true.”

….and, in fact, I’m quite sure that, if the Curry Paradox hadn’t already existed in the literature before the emergence of dialetheism, someone would have formulated it as a revenge paradox for the dialetheist. Like any revenge paradox, it efficiently fiddles with exactly the variable the paradox-solution in question focuses on in order to restore the conclusion the paradox-solver was trying to avoid.
Moreover, (a) “this sentence is false” has far more in common with (b) “if this sentence is true, everything is true” than it does with (c) “this sentence is either false or meaningless.” After all, in the classical context in which the problem arises in the first place, (b) and (c) are equivalent.

In other words, (a) and (c) have precisely as much in common with each other as (a) has with (d) “this sentence is not true.” Given the orthodox assumption of Bivalence, (a) and (d) are equivalent to each other, just as, given the orthodox assumption that contradictions imply everything, (a) and (c) are equivalent to each other. Of course, in both cases, if one rejects the relevant orthodox assumption in an attempt to get around the paradox, the equivalence disappears, but, in both cases, the revenge paradox decisively demonstrates that rejecting the equivalence accomplished nothing. The paradox remains in place.

My Take on the Liar Paradox (Part III of IV)

2010-08-16T00:01:00.000-07:00

In Part I, I used a disquotationalist picture of truth to motivate the claim that sentences like the Liar and the Truth-Teller, although they may seem meaningful, are in fact quite literally meaningless. If “’snow is white’ is true” means nothing above and beyond what “snow is white” means, if the former inherits 100% of its meaning from the latter, then a sentence like the Truth-Teller...

“This sentence is true.”

...means nothing at all. Ascriptions of truth must inherit their entire meaning from whatever sentence truth is being ascribed to--they have no ‘independent’ meaning—and, as an orphan with no ‘true’-free sentence from which it can inherit its meaning, the Truth-Teller means nothing at all. And, of course, as Carnap liked to point out, adding the word ‘not’ to a meaningless sentence does not magically convert it into a meaningful one, so what we’ve said about the Truth-Teller applies equally to the Liar.

In Part II, I argued against those who dismiss this sort of move as absurd on the basis of the “obvious” meaningfulness of such sentences, and argued, first, that ordinarily competent speakers of the language are in fact quite capable of mistakenly taking meaningless sentences to be meaningful and vice versa, and that, in this particular case, a plausible error theory is available to explain the widespread intuition that sentences like the Liar and the Truth-Teller are meaningful.

I’ve heard the objection, though, that even if we did have good reason to suppose such sentences to be meaningless, it wouldn’t help with the Liar Paradox.

After all, meaningless sentences aren’t true, so, if one takes Liars to be meaningless, the Strengthened Liar….

“This sentence is not true.”

...still ends up being both true and untrue as a result of standard liar reasoning.
That’s wrong. Ascriptions of truth to meaningless sentences (or to bits of melting candle wax) are themselves meaningless, and, as such, so are their negations. One is surely making a mistake of some sort when one points at a salt shaker and says “that’s true,” but it’s a nonsensical category mistake (which renders the utterance “that’s true” meaningless), not a substantive factual mistake (which renders the utterance “that’s true” false). After all, one is surely also making a mistake of some sort if one points to the same salt shaker and says “that’s false.” Given Bivalence and the claim that both mistakes are factual, we have a trivially easy argument for true contradictions, and dialetheists don’t need to get into anything as esoteric as semantic paradoxes in order to make their case. For every single item in existence other than meaningful declarative sentences (e.g. salt shakers, meaningless sentences, cats, dogs, stars, galaxies…..) there is a true contradiction about the meaning of that sentence!

I’ve also heard the objection, that while applications of the truth predicate to meaningless jumbles of words—e.g. “it’s true that colorless green ideas sleep furiously”—are meaningless, applications of the truth operator to an entire quoted jumble –e.g. “‘colorless green ideas sleep furiously’ is true” are just false. Pretty clearly, given the discussion in the last paragraph, no one who wants to maintain Bivalence without inheriting vast mountains of true contradictions can make this distinction. Moreover, recall that the disquotationalist “nothing above and beyond” principle is, in its original formulation, about truth as an operator rather than truth as a predicate. Hence the metaphor of “disquotation”, that the effect of ascribing truth to a quoted sentence is simply to cancel out the quotation marks.

Certainly, at any rate, no one who accepts the “nothing above and beyond” principle can accept that the negation of the ascription of truth to a meaningless sentence is itself true. If an ascription of truth to a sentence means nothing above and beyond what the sentence means, the ascription of truth to a meaningless sentence can’t be meaningful, and, again, the negation of nonsense is nonsense. Bivalence—the logical formula that tells us that for every claim P, either Tr(P) or F(P)—but meaningless sentences, salt shakers and the rest simply aren’t the kind of thing we can symbolize, write down in a truth table, perform logical operations on and so forth without committing a nonsensical category mistake.

Of course, some readers who are particularly skilled at finding clever new ways to formulate revenge paradoxes for consistent solutions to the paradox might think they’ve found one already, in a phrase I used in the third sentence of the paragraph before last. What, after all, can we say about the following sentence, which we can call The Mistake-Maker?

“One would be making a mistake of some sort if one said that this sentence was true.”

Hold that thought. First, let’s deal with an easier case (although an interesting important one), which we can call The Babbler.

“This sentence is meaningless.”

As I mentioned in Part II, one way of pushing the intuition that sentences like the Liar and the Truth-Teller are meaningless is by means of conversations like this:

“This statement is false.”

“Wait, what statement is false?”

“That one, that I just made.”

“OK, but what were you saying was false?”

...and so on. One could, of course, have the same conversation about the alleged meaninglessness of the Babbler, but, of course, for obvious reasons, if it is meaningless, it is true, and, by definition, all true statements are meaningful ones, so if it is meaningless, it is meaningful, and we have a contradiction.

The answer is that, although the same vague niggling intuitive sense that meaninglessness may be afoot might arise with both the Liar and the Babbler, there is a principled way of differentiating between the cases. In the case of the Liar, we can tell a well-motivated conceptual story to back up and justify our suspicion that the sentence may be meaningless, whereas nothing similar is available for anyone who may wish to argue that the Babbler is meaningless. The story we told about the Liar was specific to the notion of truth, and doesn’t automatically generalize to all other semantic notions. It may go for some—I’ve already indicated, in Part IV of the Russell’s Paradox series, that I think precisely the same deflationary story should be told for the meaning of the phrase “applies to itself”—but this sort of thing has to be decided on a case by case basis, and in the case of the meaningfulness predicate, we have multiple excellent reasons to rule out any similar story being told. For one thing, “‘snow is green’ is meaningful” pretty much has to mean something different than “snow is green”, since the two statements have opposite truth-values! Moreover, the whole linguistic function of the meaningfulness predicate is to differentiate meaningful statements from meaningless ones. It wouldn’t be worth anything if we couldn’t meaningfully apply it to meaningless sentences, and thus say things like “’colorless green ideas sleep furiously’ is a meaningless sentence.” Thus, for any sentence, whether it is meaningful or meaningless, the claim that it is meaningful (or the negation of that claim) is itself meaningful. As such, we have a good, principled, non-ad hoc reason to suppose that the Babbler is simply false. No contradiction follows.

All this leads nicely to the standard, throw-away revenge paradox that is standardly thought to sink meaninglessness solutions. We can call it the Strengthened Babbler.

“This sentence is either false or meaningless.”

If it’s true, then either its true and false or it’s true and meaningless, and either way a contradiction follows. If it’s false, it’s true. If it’s meaningless, and it’s true, and hence meaningful. All roads lead to contradiction.

Right?

Well, actually, I’d argue, no, no they don’t. It’s meaningless.

Wait! Doesn’t that mean that it’s true?

Nope.

Think of another Disjunctive Babbler:

“Either glork blork de glork or this sentence is meaningless.”

Or, for that matter, think of cases like...

“Either glork blork de glork or the Normans conquered England in 1066.”

No one would look at that sentence and say “oh, that’s a true disjunction, since the second disjunction is true” nor should they. Putting “or” and then a combination of words that, on their own, would be a true sentence, at the end of a meaningless string of nonsense doesn’t render the whole thing meaningful, much less true. We know what it means to say that the Normans conquered England in 1066, but we don’t know what it means to say that ‘either that happened or glork blork de glork.’ No one knows what the latter combination of words means, for the simple reason that there is nothing there to know. It’s meaningless.

If someone points to a rock and says “either that, or Japan bombed Pearl Harbor in 1941”, and subsequent investigation shows that he really means for the word “that” to refer to the rock itself rather than any claim about it, the pointer has not succeeded in making a true statement. Rocks aren’t the kinds of things that can be meaningfully symbolized, that we can perform logical operations on, that are candidates to be “disjuncts” of meaningful statements and so on. The fact that the “second disjunct” would, if isolated, be a true sentence, has absolutely nothing to do with whether the “disjunction” succeeds in being meaningful, much less true. The same goes for the Disjunctive Babbler. Meaningless statements are no more candidates for being disjuncts of meaningful statements than are rocks. The fact that we can carefully formulate the Disjunctive Babbler in terms like this...

Sentence DB: “Either sentence DB is false or sentence DB is meaningless.”

...such that the words that come after that “or”, if they were carved off into a separate sentence, would constitute a true sentence of their own…..

...simply does not make the original meaningless claim meaningful, much less true.

Moreover, the sentence we called the Mistake-Maker above, which initially seemed so worrying, is on closer inspection just a disguised form of the Disjunctive Babbler. Or, to be more careful about this, the relationship between the Mistake-Maker and the Disjunctive Babbler is more or less the same as the relationship between the Simple Liar and the Strengethened Liar.

So far, so good, but some readers might note that I’ve used some version of the phrase “if such-and-such sentence is true, it must be meaningful” and the like on several occasions, and wonder if this sort of thing couldn’t be used to bring the Strengthened Liar back into the equation as a problem for the solution I’m defending. After all, the following statement is obviously true:

“If some sentence is true, then that sentence is meaningful.”

Now, I’ve argued that sentence like, for example, the Strengthened Liar, are not meaningful. By Modus Tollens (one of those laws of classical logic I’m so keen to defend with this solution to the Liar Paradox), doesn’t it follow that the Strengthened Liar is not true? And from there, doesn’t it follow by standard liar reasoning that the Strengthened Liar is both true and untrue?

This all sounds pretty worrying, until you realize that there’s an absolutely missing step in that little proof, which is the inference from...

“If some sentence is true, then that sentence is meaningful.”

...to...

“If the Strengthened Liar sentence is true, then the Strengthened Liar sentence is meaningful.”

....a string of words that, according to my solution, obviously comes out as just as meaningless as:

“If blork glork de blork, then the Normans conquered England in 1166.”
Assuming that any string of words with the “antecedent” “if the Strengthened Liar is true” is meaningful, much less true, utterly and transparently begs the question against the question against meaninglessness solutions to the paradox. One may find the suggestion that such ‘conditionals’ are meaningless implausible, but surely it is no more implausible than the suggestion that Liar sentences are themselves meaningless. No additional objection to the view can be launched on the basis of one’s intuition that this obvious consequence of that classification is implausible.

‘If P, then Q’ and ‘either Q or ~P’ entail each other. As such, given classical logic—and, again, it begs the question against my defense of classical logic to assume that the classical equivalencies don’t hold—and the principle that the ‘negation’ of a meaningless string of words is itself meaningless, whether strings of words with the grammatical form of disjunctions and a meaningless string of words as one ‘disjunct’ are themselves meaningful is not a separate question from whether strings of words with the grammatical form of conditionals and meaningless strings of words as ‘antecedents’ are not separate questions.

Moreover, careful consideration of this last point should lead to one clear virtue of the classical approach to the paradoxes that I’m championing here over any of the leading non-classical approaches to those paradoxes on the market today. Both the dialetheist solution propounded by Priest, Beall and others, and the Excluded Middle-denying solution offered by ‘paracomplete’ theorists like Hartry Field, are disunified in their treatment of the Liar and Curry Paradoxes. We’ll pick up on that point in Part IV.

My Take on the Liar Paradox (Part II of IV)

2010-08-11T00:01:00.000-07:00

In Part I, I provided what I take to be a principled, non-ad hoc motivation for what, in In Contradiction, Graham Priest refers to as the “heroic solution” to the Liar Paradox. I claim that sentences of the type “this sentence is true”, “this sentence is false” and so on are meaningless.

Many people are so initially confused by this proposal that they try to charitably interpret it away. Early last year, while I was discussing my dissertation with some faculty members in my PhD program who were not on my committee, one junior professor wrinkled his forehead and said, “wait, you don’t think that the Liar is meaningless the way that ‘colourless green ideas sleep furiously’ is meaningless, do you?”

In the same spirit, in his book Saving Truth From Paradox, Hartry Field (who, a couple of years after that book came out, was on my dissertation committee!) says that those who claim that sentences like the Liar are meaningless must be using the term meaningless “in some special technical sense” that’s distinct from ordinary use of the term, and that, as such, such talk probably amounts to a confusing way of formulating something like his own “paracomplete” approach to the paradoxes.

So, to be clear, no, I don’t mean ‘meaningless’ in some non-standard way. I take sentences which (a) seem to say of themselves or other sentences that they are ‘true’ (or ‘false’ or whatever) but which (b) can’t be paraphrased into some set of base-language sentences which don’t use ‘true’ or its negation, as (c) literally meaningless, in precisely the same way that ‘colourless green ideas sleep furiously’ is meaningless. “‘Snow is white’ is true” doesn’t mean anything above and beyond what “snow is white” means, and “this sentence is true” doesn’t mean anything at all.

A major factor that seems to drive the incredulous stares often directed at meaninglessness solutions to the paradoxes is the notion that competent speakers of natural languages are infallible about questions of meaningfulness. (We could call this the Strong Principle.)

Now, one could claim that the underlying principle behind the incredulous stares is not the Strong Principle, but just the notion that it’s very unlikely that ordinary competent speakers could be wrong about issues of meaninglessness, and that we should have a strong presumption in favor of their initial intuitions. (We can call this the Weak Principle.)

Now, I actually agree with the Weak Principle, but I don’t think it explains the strength of the incredulous stare reaction meaninglessness solutions often receive.

Anecdotally, it is often the case that people react with incredulity to the claim that Liar-like sentences could be meaningless despite not being sure where to object to the reasoning that delivered that conclusion, or even having originally assented to that reasoning. If it’s very un-common for Croatians to drink single malt whiskey, and our strong default assumption for any given Croatian is that they don’t do so, but you notice a half-empty bottle of Laphroaig on Emil’s kitchen table and you smell peat on Emil’s breath, the strong statistically-based assumption becomes irrelevant. If one acknowledges the possibility of error about meaningfulness by competent speakers but finds it unlikely and subscribes to a strong default assumption against it, and then one is confronted by a good argument that a certain category of initially meaningful-seeming sentences are meaningless, then the probability becomes irrelevant.

The Strong Principle, on the other hand, seems to be trivially easy to falsify. The philosophers of the Vienna Circle were surely all competent speakers of the German language, but they mistakenly held many perfectly meaningful German sentences about various metaphysical topics to be literally meaningless, nonsense, “like music,” etc. Linguistically competent Wittgensteinians have incorrectly held contradictions to be meaningless rather than false. I flatter myself to think that I speak, read and write reasonably passable English, but, if the majority opinion on this issue is correct, I’m mistaken about the meaningfulness of Liar-like sentences.

In fact, the fun part of all of this is that, for all of the debates just mentioned, whichever side turns out to be right, given the disagreements, some otherwise competent speaker must be making a mistake about meaningfulness!

Maybe, however, the Strong Principle is implausibly strong, but something stronger than the Weak Principle is still plausible. Given standard, orthodox views on all the subjects just mentioned, one could propose the Still Fairly Strong Principle, that, while false negatives of meaningfulness are possible, cases where ordinarily competent speakers mistakenly take meaningful statements to be meaningless, false positives are still impossible.

I’d argue that this proposal is falsified by what I think of as “bored dinner guest”-type examples. Imagine that two people, Jack and Jill, are at dinner. Jill goes on at length about subjects that Jack is bored to tears by, and, after a while, Jack completely tunes out. He contributes nothing to the conversation except for the occasional “yeah,” “I agree” or “that’s true,” dictated by the rising and falling of
Jill’s voice and the appropriate pauses. After a while, Jill catches on. To test him, she starts emitting a string of nonsense syllables like “glork bork de glork”, but she keeps her tone normal, and makes sure her voice rises and falls in the normal way. At the appropriate pause, Jack says “oh yeah, that’s true.”

Now, Jack has no idea whether he just “that’s true”-ed a true statement or a false one, so he doesn’t know whether his statement was true or false. He does, however, surely assume that it was one or the other, that he at least said something meaningful.

But, to paraphrase our previous President, if “it’s true that ‘glork bork de glork’” isn’t meaningless, then meaninglessness has no meaning. Jack is an ordinarily competent speaker of English who is mistaken about the meaningfulness of *his own* utterance. False positives are indeed possible.

(Moreover, remember, especially on standard deflationary stories about truth, such “blind endorsements” constitute one of the most important linguistic purposes for which the truth predicate exists.)

Now, one could water down the Still Fairly Strong Principle to the Not Terribly Weak Principle:

False negatives are possible across the board, but false positives are only possible when the competent speaker in question isn’t aware of the content of the sentence in question.

Now, to canvass a broad range of approaches here, I’ve left that word “content” intentionally ambiguous. If by “content”, one means something merely syntactic, such that “knows the content of a sentence” just means “is aware of which words appear in the sentence”, then the Still Not Terribly Weak Principle looks awfully implausible.

For one thing, I think an awful lot of linguistically-but-not-epistemically-competent people could be suckered into believing that “colorless green ideas sleep furiously” was not only meaningful but true…for example, Deepak Chopra could tell them that quantum physics had proved that colorless green ideas did that.
In response, one could interpret “knows the content” in a stronger, semantic way, where “knows the content” means “grasps the content.” It may be possible to fool people in various ways into thinking that “colorless green ideas means something” but it’s impossible to fool them into actually knowing what it means, because there’s nothing there to know.

Fair enough, but at this point deploying the (now entirely trivial) principle which we are still referring to as the Still Not Terribly Weak Principle against meaninglessness solutions to the Liar Paradox would utterly and transparently beg the question. No one would deny that, of course, if competent speakers (or, for that matter, even normally wildly incompetent speakers!) grasp the meaning of a sentence they take to be meaningful, their belief that it’s meaningful can’t be wrong. To base on objection to meaninglessness solutions on this, however, is to simply assume the precise bone of contention, which is whether Liar-like sentences have a content to grasp.

Perhaps, somewhere along this spectrum, there’s some remotely plausible principle that’s relevantly stronger than the Weak Principle, and as such justifies the “well, that’s just obviously ridiculous” unargued brush-off that some people use to refute meaninglessness solutions. If anyone has any candidates they’d like the propose in the comments section, I think that might be interesting.

Meanwhile, of course, showing that it’s possible for competent speakers of English to be mistaken about the meaningfulness of syntactically innocuous sentences formed out of English words is quite a different thing from showing that, in the case of the particular category of sentences under consideration, they actually are mistaken. In Part I, I sketched out an argument, but we need more than that. We also need an error theory to explain why many people have such a strong mistaken intuition. This obligation becomes more pressing as the number of mistaken people rises. If one ordinarily mathematically competent reasoner gets the wrong result for a simple algebra problem, then “he didn’t have enough coffee that morning” may be a plausible error theory. If, on the other hand, out of the thousands of ordinarily mathematically competent reasoners to have tried their hand at the problem, all but one have gotten the wrong answer, we need something a bit more robust to explain this away.

Now, many people make claims from the armchair that most speakers have the intuition that Liar-like sentences are meaningful, but I’ve never seen any concrete empirical evidence to back this up. (This has always sounded like a job for x-phi to me.) In fact, anecdotally, my impression is that, on first contact with the Liar and its ilk, some non-philosophers will respond in a way that indicates what may be precisely the opposite intuitions, having conversations like this one….

“This statement is false.”

“Wait, what statement is false?”

“Well, the one I just made.”

“Yes, but what did you say?”

“I said that what I said was false.”

“Yes, but….”

….and so on. None of this, of course, adds up to a good reason to suppose that Liars are meaningless, but it might reflect a suspicion that this is the case.

Certainly, though, whatever the proportions might turn out to be at some point in the future when we’ve collected some empirical evidence, there’s no denying that plenty people, philosophically trained and otherwise, have the meaningfulness intuition, certainly enough that “Priest and Field don’t drink enough coffee”-type explanations won’t cut it.

I think one piece of the puzzle (although, to be clear, only a secondary one) is a matter of training. Intuitions are formed, changed and molded by one’s educational experiences. The people whose intuitions about the Liar we know the most about are professional philosophers. The Liar is an ancient and venerable philosophical difficulty, and even professional philosophers who have never thought in depth about it at least have years under their belt of being vaguely aware of it in a context in which one of the fundamental background assumptions to get the difficulty off the ground is that it is meaningful. (Moreover, as good rhetoricians have known since time immemorial, it’s often easier to influence people with the opinions you never get around to explicitly stating. Think about Mark Antony’s funeral speech in Shakespeare’s “Julius Caeser.”) So people’s intuitions are shaped by the shared assumptions of the people around them. Think about the way that, in the other direction from the case at hand (non-existence solutions to Russell’s Paradox being as popular as they are unpopular when it comes to the Liar), people who work on set theory, and who initially have ‘naïve’ intuitions, often claim to have new intuitions about what sorts of things sets are and which might exist, formed as a result of the experience of exclusively working in the cumulative hierarchy of ZFC or some similar system. So that’s surely part of the story, and I think it accounts for some of the strength and self-assured fervor of meaningfulness intuitions among professional philosophers.

For many reasons, however, this can only be a small part of the overall story. After all, how do we explain how meaningfulness assumptions initially came into the picture? Besides, it’s surely plausible that a great many people have the meaningfulness intuition on first contact with the Liar “in the wild,” even in eccentric circumstances where it isn’t presented to them as a famous and interesting puzzle.

I think that the first thing to notice is that odd and unusual sentences involving self-reference are usually meaningful—indeed, they’re usually obviously and unproblematically true or obviously and unproblematically false. Consider cases like...

“This sentence has seven words in it.”

When it comes, however, to sentences whose truth-value is utterly and stubbornly inaccessible to us, we usually have no reason to doubt that they are meaningful and that they thus have truth-values. Consider “Alexander the Great’s maternal grandmother’s paternal grandfather once had a splinter in his thumb at some point during the month after his sixth birthday.” Given, among other problems, the absence of time travel, no one has the slightest idea of how to find out whether this is true or false, but this doesn’t (and shouldn’t!) make us question whether it has a semantic content and that this content either lines up with the facts or fails to.

Finally (although it would certainly be possible to go on) the Liar is a syntactically “well-formed” sentence. Now, it’s easy to show with examples that being “well-formed” is neither necessary nor sufficient for being meaningful, and I’m somewhat inclined to think that the use of the phrase “well-formed” to describe natural language sentences relies on an exaggerated, idealized and un-helpful analogy between natural language and formal logical “languages,” but let’s put that to one side and acknowledge that sentences that are “well-formed” (i.e. composed entirely of normal natural language words, arranged in a way that conforms to grammatical rules, etc.) are at least less likely to turn out to be meaningless jumbles of words than other sorts of utterances, and that most of the grammatically innocuous assertion-style combinations of words we have cause to run into in the ordinary course of things are meaningful.

At this point, we can see that the Liar and its ilk sit at the intersection of several categories of sentences such that the overwhelming majority of members of each category are meaningful. It is, then, fantastically unsurprising that most people’s initial intuitive reaction is that it is meaningful. (If, indeed, this turns out to be the case.) Moreover, even when subjecting that intuition to critical scrutiny, it’s perfectly rational to apply a bit of simple probabilistic inference here and argue that, all else being equal, it’s reasonable to infer that a sentence that participates in a lot of categories of typically-meaningful sentences is itself meaningful.

All else is, in this particular case, simply not meaningful, as established by the argument from the disquotationalist principle that quotation marks and ‘is true’ don’t jointly add anything to the content of a sentence to the conclusion that a sentence that attempts to apply truth talk to itself won’t be meaningful. This is a substantive discovery in the course of reasoning about difficult problems, not something that one would expect to be immediately pre-reflectively obvious.

Coming up next time:

Revenge Paradoxes!

My Take on the Liar Paradox (Part I of IV)

2010-08-09T11:04:00.000-07:00

I've certainly spent enough time in the last few years taking pot shots at other people's approaches to the Liar. Now that I've finished the degree and all that, it's probably time to present my own in detail. Devastating objections go in the comment box. Have fun!

JC Beall calls his view of truth “transparent disquotationalism.” His idea is that the truth predicate is a “transparent device” introduced into our language in order to make certain sorts of generalizations, blind endorsements and so on. (For example, “everything the Pope says is true.”) The device is exhaustively defined by the two rules that govern its behavior—Capture (the inference from ‘P’ to ‘Tr(P)’) and Release (the inference from ‘Tr(P)’ to ‘P’). It is ‘transparent’ because it allows us to see through to the ‘base language’ facts we use it to describe. (The ‘base language’ here is the fragment of the language in which the truth predicate is not used. It may, of course, be mentioned, as it is in the base-language sentence which came just before this one.) In Beall’s story, despite the truth predicate being introduced into the language as a way of talking about base language facts, this isn’t the only thing it can be used to do. After all, while normal sentences in which ‘true’ is used can be converted into sentences in which it is not used by applications of Release, no such procedure is available to convert a Liar sentence like “this sentence is false” (or its Truth-Telling brother “this sentence is true”) into a base-language sentence.

Thus, Beall’s use of the phrase “transparent disquotationalism” is a bit misleading. For Beall, as we’ve just seen, transparency is a merely contingent feature of typical uses of ‘true,’ not something necessary or universal. There are failures of transparency, applications of the truth predicate that don’t function to let us see through to some base-language fact. He’d be better off calling his view “mostly transparent disquotationalism.”

This creates tricky problems for Beall when it comes to how to think about the truth or falsity of non-transparent sentences. In Spandrels of Truth, he rejects the possibility of truth-value gaps, which leaves him with three values—true, false, and both. Non-triviality dictates that Curry sentences like “if this sentence is true, everything is true” get the value “false”, and as a dialetheist, Beall takes standard Liar reasoning to guarantee that (non-Curry) paradoxical sentences get the value “both”, but what to do with all the non-paradoxical and also non-transparent sentences? Beall decrees that they’re all “both” as well, but gives no indication whatsoever of any sort of principled motivation for this above and beyond the fact that it’s simpler to assume that it’s the case.

In fact, it seems terribly difficult to see what sort of principled reason anyone could ever give for either asserting or denying that any such sentence is true. “’Snow is white’ is true” is true and “’snow is green’ is true” is false. Both of these facts are functions of the actual color of snow. Clearly, however, no base-language fact will come to the rescue of a “transparent disquotationalist” trying to figure out the truth-value of the sentence “this sentence is true,” nor can they use the sort of reasoning they use to assign “both” to Liar sentences. Moreover, throwing up one’s hands in agnostic confusion wouldn’t be a principled move here, since the issue here clearly has nothing to do with some limit of our epistemic access to any relevant information. What ingredient do we not have, such that if we had it, the truth-value of such sentences would become clear?
I think that where Beall goes wrong is in assuming truth to be merely typically transparent. Let’s think about a classic statement of disquotationalism, Quine’s “nothing above and beyond” principle. In In Pursuit of Truth, he says this:

“To ascribe truth to the sentence [‘Snow is white.’] is to ascribe whiteness to snow…Ascription of truth just cancels out the quotation marks. Truth is disquotation.”

So, for any base-language sentence P, “’P’ is true” and “P” have, literally, exactly the same meaning. No additional content whatsoever is added to P by the application of the truth predicate.

My proposal, which, for contrast to Beall’s formulation, we can call “reductive disquotationalism,” is that the truth predicate does not suddenly become more substantive or play by different rules when we construct a sentence like “this sentence is true.” If the application of the truth predicate to a sentence doesn’t add any content to a sentence, a sentence with no content other than the use of the truth predicate has no content at all. Consider Sentences 1-3:

Sentence 1: Sentence 2 is true.
Sentence 2: Sentence 3 is true.
Sentence 3: Snow is white.

Now, imagine an infinite chain of sentences where each sentence ascribes truth to the next one in the series:

Sentence 4: Sentence 5 is true.
Sentence 5: Sentence 6 is true.
Sentence 6: Sentence 7 is true.
…and so on into eternity.

If, by the Quinean principle just mentioned, the meaning of Sentence 1 is entirely inherited from the meaning of Sentence 3, with nothing left over, where does Sentence 4 get its meaning?

My answer would be that it doesn’t get one. There is no base-language sentence from which it could inherit its meaning, so it is literally meaningless. Similarly for a simple case like “this sentence is true.” And, of course, adding the word ‘not’ into the mix changes nothing. As the philosophers of the Vienna Circle were so fond of pointing out, the negation of nonsense is nonsense.

We can summarize reductive disquotationalism as the claim that all meaningful sentences in which the truth predicate (or, equivalently, the truth operator) is used can, in principle, be accurately paraphrased without the use of the truth predicate (or operator). That “in principle” is important, because, for example, we might not have enough information to supply the paraphrase (as in “what John just said is true” where we don’t remember what John said), or the paraphrase could be infinitely long. If one finds the “in principle” suspect, we can express the principle by saying that, for every meaningful sentence in which ‘true’ is used, there is a set of sentences such that (a) none of them use ‘true’ and (b) they collectively mean exactly the same thing as the ‘true’-using sentence which they paraphrase. They capture its entire content.

Thus, sentences like “this sentence is true” and “this sentence is false,” although the may seem meaningful, are actually meaningless. Nor is this some desperate ad hoc maneuver to save consistency. It’s a fairly-clear cut application of a principle (“nothing above and beyond”) which is widely held for independent reasons, which has obvious advantages in terms of simplicity and so on, and which is closely linked to principles held by some significant figures on the other side of the dialetheism debate (e.g. Beall). Happily, though, it allows us to save not only consistency but classical logic. No rules of inference are being sacrificed, no additional values are being added to truth-tables, nothing revisionary, in fact, is going at all, except that those with the initial intuition that such sentences are meaningful are shown, for non-question-begging reasons, to be mistaken.

Once, after sketching out this solution to a friendly acquaintance who works in the same area, he politely told me that it was intuitively appealing but he’d have to “wait to see the formal details.” This is very funny to me, as an illustration of how deeply entrenched the notion that solutions to the Liar need to be logically revisionary has become in the era of Kripke, Field and the rest. A solution that saves classical logic, non-regimented natural language, the ‘naïve theory of truth’ complete with Capture and Release, etc., is one that can’t have ‘formal details.’ There’s no new syntactic machinery to show off, to prove the consistency of, to play with and test the limits of and so on. Everything is being done on the semantic, ‘informal’ side. Which is not to say, of course, that there isn’t plenty of tricky detail-level work to be done, as we’ll see when we address the various revenge paradoxes…it’s just that this work is all happening on the ‘informal,’ semantic side.

Now, dear reader, you may be getting pretty frustrated at this point, impatiently wondering how I could be so confused as to think that it’s even possible for such sentences to be literally meaningless when ordinary, competent speakers of the language take themselves to grasp their meaning. I must be either talking nonsense or using ‘meaningless’ in some strange non-standard sense of the term, right? And anyway, surely meaningless sentences aren’t true, so how am I getting around the Strengethened Liar? And even if I had some way around that, surely the solution must collapse on contact with sentences like “this sentence is either false or meaningless”?

And even if some plausible way can be found around that, then, surely, the Universal Law Of Liar Paradox Solutions (ULOLPS) still dictates that, for any claimed consistent solution to the paradoxes, a ‘revenge paradox’ can be formulated using the clever new terminology of the solution. Perhaps, these days, Hartry Field has convinced a few people that the ULOLPS breaks down in the case of a few fantastically formally sophisticated solutions, much the same way that normal rules about velocity and simultaneity break down as we approach the speed of light, but even those who grant this remain largely convinced that the mighty ULOLPS remains in full force down here at the lower reaches where a few fools still try to reason about the paradoxes with nothing more than simple juggling of intuitions and counter-examples.

If you have any of these worries all I can say is, well, stay tuned for the next few posts!

Russell's Paradox as a Paradox About Properties (The Russell's Paradox Series, Part IV of IV)

2010-08-04T00:01:00.000-07:00

Some dialetheists take Russell's Paradox to form the basis of a sound argument for the existence of true contradictions. In Part I of this series, I argued that, given the obvious limitations of our epistemic access to the realm of mathematical objects (if it exists), we should be extremely cautious and conservative in coming to conclusions about its logical structure. In Part II, I argued that, given that our best reason for believing in the existence of any sets comes from Quinean indispensibility considerations, and that we don't need anything even close to the over-populated universe of naive set theory to reconstruct all the mathematics which can't be eliminated from our best science, belief in unrestricted comprehension axiom of naive set theory (and, hence, belief in the existence of the Russell Set) is unmotivated. In Part III, I responded to the best obvious objection--that I was solving Russell's Paradox by appealing to considerations about the epistemology of mathematics that would be obviously inapplicable to the Liar Paradox, and thus solving paradoxes that are 'of a type' in an unacceptable disunified way. I argued that, in fact, given plausible and widely held views about mathematical objects, and Priest's own view about puzzles like "the Barber Paradox," solving Russell's Paradox in a way highly distinct from how one solves the Liar Paradox is actually entirely reasonable.

All of this, however, leaves a large remaining question about Russell's Paradox unanswered. All of my arguments about the rationality of rejecting belief in the Russell Set only solve the paradox if we assume that Russell's Paradox is fundamentally a paradox about sets, i.e. abstract mathematical objects that have other objects as members. Recently, however, Hartry Field and JC Beall, in Saving Truth From Paradox and Spandrels of Truth respectively, have argued that there are two versions of Russell's Paradox--one about sets and the other about properties--and that, even if the former is solvable in straightforwardly orthodox ways, the latter requires some sort of non-classical solution.

In Part I, I said the following:

"JC Beall, if I understand him correctly, follows Hartry Field in separating the paradox into two versions, the traditional set-theoretic one, and a paradox about the property 'does not apply to itself'--Beall accepts classical orthodoxy about the former and handles the latter dialethetically, just as Field accepts classical orthodoxy about the former and handles the latter by denying the relevant instances of the Excluded Middle. To me, the Field/Beall position on the taxonomy of the paradoxes seems bizarre. Russell's Paradox is about sets, and the 'does not apply to itself' paradox is about properties. They have similar structures, but what of it? Lots of paradoxes have Russell's-Paradox-ish structures, and saying that this means that the 'does not apply to itself' paradox is a version of Russell's Paradox strikes me as making no more sense than describing the Liar as 'the sentence version of' Russell's Paradox. (In fact, I'm inclined to think that, in so far as 'applies to' can be paraphrased as something like 'is said of itself in a true sentence', the 'does not apply to itself' paradox has a lot more in common with the Liar than it does with Russell's Paradox.) In both cases, one can talk that way if one wishes, but it doesn't strike me as shedding much light on anything. I think that things are kept clearer by regarding the 'does not apply to itself' paradox as an interesting puzzle in its own right that doesn't have much of anything to do with Russell's Paradox."

Whether or not this view about how to classify the property paradox is correct, since I've been blogging about Russell's Paradox, and some smart big-name philosophers regard the puzzle about properties as a version of Russell's Paradox, I should say something about it here.

My preferred solution would be to say that the predicate "applies to itself" (and, thus, its negation) is only meaningful parasitically. That is to say, for example, the true statement, "the property of 'being a property' applies to itself" means nothing above and beyond "the property of 'being a property' is a property." (Similarly, the false statement, "the property of 'being a property that no one has ever mentioned' applies to itself" means nothing above and beyond "the property of 'being a property that no one has ever mentioned' is a property that no one has ever mentioned.) The phrase 'applies to itself' (or 'does not apply to itself') does not name a distinct property. It is, rather, a convenient linguistic device for referring to other properties. What, after all, could be the extra content of "the property of 'being a property that no one has ever mentioned' applies to itself," above and beyond what's captured by the paraphrase "the property of 'being a property that no one has ever mentioned' is a property that no one has ever mentioned?"

If, however, the meaning of the predicate "applies to itself" is entirely inherited from the meaning of whatever phrase it is appended to, and it has no 'independent' meaning whatsoever, a straightforward consequence of this fact is that the phrase "the property 'does not apply to itself' applies to itself" seems to mean something but is actually meaningless. Ungrounded uses of the predicate "applies to itself" (or, of course, its negation) are as devoid of semantic content as "colorless green ideas sleep furiously."

"Wait a damn second," you might be thinking, "if it's meaningless, how can so many competent speakers of the language think they grasp its meaning? You must mean 'meaningless' in some special technical way that's not how the term is ordinarily used. Also, are you going to try to pull something like this as your solution to the Liar Paradox? If so, aren't you going to have a hell of a hard time getting around revenge paradoxes? What about 'this sentence is either false or meaningless'?"

....to all of which all I can say is "stay tuned for next Monday's post on the Liar Paradox!"

Russell's Paradox, the Liar, the Barber and the Inclosure Schema (The Russell's Paradox Series, Part III of IV)

2010-08-02T00:01:00.000-07:00

In Parts I and II, I responded to the bullet-biting approach to Russell's Paradox--that it shows that there really are true contradictions--by appealing to some general epistemic considerations about how and when we might be justified in coming to believe in the existence of mathematical objections with inconsistent properties. I argued that, given our notorious lack of direct epistemic access to mathematical objects, if there are any such things (think of Benaceraff-type arguments here), we should be extremely cautious and conservative in our theory of them. (The default assumption should be that they exhibit the same logical structure as the parts of reality we do have direct access to.) Moreover, I argued that the best argument we have for supposing any mathematical objects to exist is one from Quinean indispensibility-to-our-best-science, I pointed out that we hardly need all of the sets in the crowded universe of naive set theory to reconstruct the fragment of mathematics we need for our best current science, and ended by leaving a fallibilistic back door open for a way that hypothetical future science could force inconsistent mathematical objects on us after all.

All well and good, someone might argue, but these sorts of considerations clearly won't help us with the Liar Paradox. Any epistemic access we might have to sets is indirect and holistic, fair enough, but sentences are clearly a different matter. Thus, any plausible consistent solution to the Liar Paradox must surely be different from the solution I've offered here for Russell's Paradox. Given the deep structural similarity between the Liar and Russell's Paradox, though, aren't they both 'of a type'? And doesn't that mean that the right solution to the two paradoxes must be 'unified'?

Graham Priest has argued in exactly this way, in his article "The Structure of the Paradoxes of Self-Reference" and many subsequent articles. He argues that the Liar Paradox, Russell's Paradox, and many other paradoxes besides, are all instances of a common pattern, the Inclosure Schema, and that the right solution to any given Inclosure Schema paradox must apply to the rest as well. The technical details of the Schema (diagnolizing functions, etc.) don't really concern us at the level of detail we're going to operate on here, but the important point is that there are three elements: Existence, Closure and Transcendence.

(He often only talks about Closure and Transcendence in the articles, but existence is obviously a necessary component--there's nothing paradoxical about saying "if there was such-and-such an object, it would have strange paradoxical properties--and in Priest's book Beyond The Limits Of Thought, which includes his longest discussion of the Schema, he emphasizes Existence as much as the other elements.)

Priest's approach is to bite the bullet of inconsistency, so in each case, he accepts that there is something (existence), that it falls into a category (closure) and falls outside of it as well (transcendence). Any consistent solution to Russell's Paradox, like mine, that starts from considerations about the limits of our epistemic access to the realm of abstract mathematical objects, considerations that will obviously be irrelevant to consideration of the semantic paradoxes, must be much less unified than this, right?

Well, first of all, not necessarily, no. If one takes Liar sentences to be meaningless, or to be meaningful but to fail to express propositions, or to be expressing different propositions than they seem to express, or anything that's remotely in that family of approaches to the Liar, then one is denying the Existence condition. The solution to Russell's Paradox that I argued for in the last two posts also amounts to denying Existence. (Indeed, almost all consistent solutions to Russell's Paradox amount to denying Existence. In fact, Penelope Maddy's "set-membership gaps" approach--which amounts to denying both Closure and Transcendence--is the only exception that I can think of right now.) Thus, if you solve the Liar Paradox in any of the ways just gestured at, and pair that solution with my view about Russell's Paradox, then, at the level of abstraction the Inclosure Schema operates at, your solution will be exactly as unified as Priest's dialetheist soltuion to all of the Schema paradoxes. In all cases, Priest affirms Existence and Closure and Transcendence. In all cases, you deny Existence. There's simply no gap whatsoever in the uniformity of your respective solutions.

Now, since I find myself in the happy position of having a view of the Liar that falls within the camp of Existence-denying solutions, I'm in the clear here. What, however, if you don't find Existence-denying solutions to the Liar plausible? For example, what if you accept Barwise & Etchemendy's solution from The Liar: An Essay On Truth and Circularity, which (by denying that the Liar's truth-value depends on itself in the obvious way) arguably amounts to denying both Transcendence and Closure?

(I actually think that, as is the case with a great many solutions to the Liar Paradox, which element of the Schema is being denied is interestingly ambiguous. Whether this fact exposes a diagnostic flaw in the usefulness of the Schema is an interesting question I'll leave for another time. For now, let's say that the Barwise/Etchemendy solution--which proceeds from a complicated version of the correspondence theory of truth and some set-theoretic modeling to the surprising conclusion that Liar sentences are (just) false, since its truth-value is not included in the situation that determines that truth-value--amounts to denying Closure and Transcendence but not Existence.)

So, if one solves Russell's Paradox in my preferred Quine-inspired way, and solves the Liar in the Barwise/Etchemendy way (thus picking different aspects of the Schema to go after in each case), is one solving the paradoxes in a less uniform way than Priest is, and thus losing plausibility points?

I don't think so. Here's why not:

Imagine that someone tells you and a group of your friends that they heard that, in the village of Alcala, the barber shaves all the men in the village who don't shave themselves. (Subsequent questioning reveals that the barber is a man and that he lives in the village.) Inevitably, someone asks whether the barber shaves himself, and everyone quickly realizes that, given the information you have so far, he must both shave himself and not shave himself.

No one will say, "huh, I guess there are true contradictions about the observable physical world. Wow!" Even if your friends are all philosophy professors who accept dialetheism and believe that the Liar Paradox and Russell's Paradox must be solved in the same way because they are both instances of the Inclosure Schema, none of them will respond in this way. The response of any normal, reasonably bright human being over the age of six would, rather, be to demand evidence of the existence of such a barber. For a variety of reasons, different philosophers may have different views about whether any possible evidence would ever justify belief in the existence of such a barber, but it would take a lot to satisfy even the most zealous fallibilist.

Pretty clearly, evidence of the existence of a barber who's stated policy is to shave everyone who doesn't shave themselves won't cut it. Nor, given the enormity of the claim, would eyewitness evidence be sufficient. Perhaps, a la the set-of the short story Sylvan's Box, live streaming video of the barber simultaneously shaving and failing to shave himself, might satisfy such a demand. Certainly, one couldn't imagine much less than that convincing someone like Graham Priest or JC Beall--much less those of us who don't start out by believing in true contradictions--of the barber's existence.

But wait! Structurally, the Barber Paradox surely has at least as much in common with Russell's Paradox as the Liar Paradox does. In fact, no less an authority on Russell's Paradox than Bertrand Russell introduced the Barber Paradox as an illustration of the structure of Russell's Paradox. If the Barber Paradox turned out that it didn't fit the Inclosure Schema, this would be fairly damning evidence that the Schema had failed to capture the pattern of Russell's Paradox. Isn't someone who solves the Barber Paradox by denying existence and solves Russell's Paradox by embracing Existence, Closure and Transcendence violating the principle that all paradoxes of the same structural 'type' should be solved in precisely the same way?

In Graham Priest's article The Import of Inclosure: Some Comments on Grattan-Guiness (from the October 1998 issue of Mind and available on Jstor), Priest acknowledges that "the Barber certainly can be put into the form of the Schema" and that he (like everyone else) denies that there ever has been or ever will be a barber of the relevant kind, and that he couldn't be convinced otherwise by anything so straightforward as, for example, the testimony of people who claim this about their village barber.

It's important to note that, although he doesn't say so, in the case of a non-paradoxical claim like "the barber in our town shaves all the men in the town except for himself," Priest (like any other normal human being operating with normal, everyday epistemic standards) would probably take a whole lot less than the combined testimony of every single resident of the town before he accepted that the claim was probably true. This is despite the fact that he starts from the assumption that true contradictions are possible. Still, reasonably enough, he finds the suggestion of contradictions in the observable physical world to be surprising and counter-intuitive enough to require more-than-usual amounts of evidence. One might think that, by analogy, those of us who don't even start out believing in the existence of true contradictions should be well within our rights to demand greater-than-usual evidence of the existence of the Russell Set once we realize that, if it existed, it would have inconsistent properties. It's not clear, then, why (even on the assumption that naive comprehension is correct and that there are true contradictions) Priest thinks that we're rationally obligated to set the epistemic bar for what should count as evidence of the existence of the Russell Set as low as the mere fact that "for every description we can come up with, there's a set with all and only the objects satisfying that description as members" is an intuitively simple comprehension axiom.

Let's put that to one side for now, and go back to the original question. Given that the Barber Paradox fits the Schema, and that Priest solves it by denying Existence rather than by embracing Existence, Transcendence and Closure (as he does for Russell's Paradox), isn't Priest guilty of disunified solution?

He argues that "there is more to the Schema than its form...An Inclosure Argument is valid. But one needs more than this for a paradoxical argument: the premises must also be true, at least prima facie--or no one would suppose the situation paradoxical. It is this fact that rules out the Barber Paradox and its ilk as inclosure paradoxes. We have no good reason to suppose that there is a barber of the required kind." (p. 386)

Now, in this passage, Priest is teetering dangerously close to abandoning the claim that the Schema has any role at all as a tool for showing us which paradoxes are 'of a type' and hence require uniform solution. Premises, after all, can fail to be true for all sorts of different reasons in different cases, and its not clear what's added by the phrase "at least prima facie." Now, I have enormous amounts of respect for Priest's philosophical abilities--if I didn't think the man was making good arguments, I wouldn't waste so much time and energy worrying about how to respond to them!) And, certainly, it doesn't seem terribly charitable to read him as claiming here that, whenever the premises of some Barber-like puzzle initially seem true, we acquire some permanent duty to solve it in precisely the same way we solve Russell's Paradox, but that we don't have such a duty if we see through the false premise right away. That said, since it's unclear what significance he does attach to the phrase, let's just move on.

Given that, if we have a good enough reason to doubt that "the premises are true," it's OK to solve a paradox with the form of the Inclosure Schema in a way that's different from the way you solve the rest of the Schema paradoxes, is there a way still available for Priest to insist that the Liar and Russell's Paradox be solved in the same way?

He thinks there is. He goes on to say that, to be a Schema Paradox in good standing to which one's uniform solution must apply, it's important "not just that the premises are prima facie true, but that they are a priori so. For exactly this is true of the standard paradoxes of self-reference: Transcendence and Closure would appear to be a priori certified. The contradictions that these give rise to seem to be inherent in thought itself, intrinsic to our conceptual structures... The premises in the case of the Barber and its ilk, were they ever to be true, could only ever be so a postereori." (p. 386)

With that firmly in mind, let's remember Stephen Yablo's summary of the recent history of philosophy, which I quoted in my last post.

"About fifty years ago, Quine convinced almost everyone who cared that the argument for abstract objects, if there was going to be one, would have to be a posteriori in nature. And it would have to be an a posteriori argument of a particular sort: an indispensability argument representing numbers, to use that example, as entities that ‘total science’ cannot do without."

In other words, at this point most people are convinced that they only good reason we could ever have to believe in the existence of any sets or any other mathematical objects--surely, a crucial premise of the dialetheist's argument from Russell's Paradox--is, in its own way, as a posteriori as any reason we could ever have to believe in the existence of the self-shaving-and-non-self-shaving barber.

Now, I don't, of course, want to engage in an appeal to popularity here. Priest may well have a knock-down argument against the prevailing consensus up his sleeve. Failing that, though, given Priest's own standards about which sorts of Schema-fitting arguments need to solved with one's favored 'uniform solution,' anyone who finds themselves in Yablo's "almost everyone" should feel free to solve Russell's Paradox and the Liar Paradox in entirely different ways.

Russell's Paradox, Quine's Argument and the Empirical Refutation of the Law of Non-Contradiction (The Russell's Paradox Series, Part II of IV)

2010-07-29T05:22:00.000-07:00

In my last post, I said the following to defuse the idea that any particular scheme for determining which sets exist, whether the unrestricted comprehension axiom of naive set theory or the more conservative axioms of various later theories, is something we can be particularly confident about:

"The realm of sets, if it exists, is notoriously epistemically inaccessible to us. (In fact, as Benaceraff famously pointed out, if we assume that it exists, it still seems to be the case that, if every set in that realm disappeared tomorrow, we'd never know.) Even if we assume that there's a compelling case for set-theoretic realism--i.e. for the conclusion that at least some sets exist--the question of which sets exist is still very much open."

OK, so, given that, what reason do we have (if any) to suppose that at least some sets exist? Here's what Stephen Yablo says, in his article "A Paradox of Existence":

"About fifty years ago, Quine convinced almost everyone who cared that the argument for abstract objects, if there was going to be one, would have to be a posteriori in nature. And it would have to be an a posteriori argument of a particular sort: an indispensability argument representing numbers, to use that example, as entities that ‘total science’ cannot do without."

If that's one's reason for granting the at least some existence of sets, though, we clearly don't need all the sets you get from the unrestricted comprehension axiom. For example, the sets in the ZFC cumulative hierarchy should be (way, way) more than enough to reconstruct the fragment of mathematics you need for our best current scientific theories.

Remember that the original context of this series of posts is Graham Priest's argument for dialetheism on the basis of the antinomies of naive set theory. It seems to me, though, that we have to start by having a pretty damn good reason to accept unrestricted comprehension if we're going to be willing to sacrifice Non-Contradiction for its sake. Of course, if we're going to engage in the project of arguing with dialetheists (which is, after all, what I'm doing here), "your premises must be wrong because I don't like your conclusion" isn't much of a counter-argument. Fortunately, in this case, we can do better.

After all, to review:

(1) The most plausible argument for thinking that any sets exist at all is the indispensibility argument,
&
(2) To set-theoretically reconstruct the fragment of mathematics that's actually indispensible to our best-supported scientific theories, you don't need as many sets as unrestricted comprehension gets you,
&
(3) As I argued last time, if we lack evidence one way or the other about what exists in some realm of reality to which we have no direct access, it makes sense to conservatively assume that its logical structure doesn't radically depart from the logical structure of the parts of reality we are familiar with. This seems like a good general principle that everyone can accept--e.g. dialetheists who (like Priest) don't think that there are any counter-examples to the Law of the Excluded Middle have every reason to reject evidentially unmotivated claims that such counter-examples exist in some strange, epistemically inaccessible domain of reality.

Putting (1)-(3) together, it looks like, unless we've already been convinced of dialetheism by some other argument, the paradoxes of naive set theory give us no reason to believe in the existence of sets with inconsistent properties. Considered as an independent argument for dialetheism (which is how Priest seems to view it), we have excellent, non-question-begging reasons to reject one of the crucial premises.

So far, so good. Does this mean, though, that no argument for independent argument for dialetheism could be mounted on the basis of the existence of inconsistent mathematical objects, given our limited epistemic access to such objects, the conservative principle (3) above, and so on?

Actually, no. Indispensibility cuts both ways. Here's what I say about that in my dissertation:

#

Now, this suggestion--that our best reason to believe that the total universe of sets fails to contain the Russell Set is that we are only justified in being realists about the sorts of mathematical entities that our best-confirmed scientific theories are ontologically committed to--has a consequence that many of my fellow monaletheists[*] might find extremely unattractive. So as not to look like I’m skirting the issue, it’s worth pausing for a moment to spell it out, and to explain why I regard it not as a bug but as a feature.

While Priest usually likes to portray the intellectual adjustment involved in admitting the possibility of true contradictions as a smaller change than it might appear, one without sweeping consequences for our practices of reasoning--we’ll come back to this point with a vengeance in Chapter Seven, when we consider his “classical re-capture”--in other, more enthusiastically optimistic moods, he likes to speculate about the wide-ranging consequences that the jump to the ‘realm of the transconsistent’ might have.

"In modern science, the inferentially sophisticated part is nearly always mathematical. An appropriate mathematical theory is found, and its theorems are applied. Hence, a likely way for an inconsistent theory to arise now in science is via the application of an inconsistent mathematical theory. Though the construction of inconsistent mathematical theories (based on adjunctive paraconsistent logics) is relatively new, there are already a number of inconsistent number theories, linear algebras, category theories; and it is clear that there is much more scope in this area. The theories have not been developed with an eye to their applicability in science—just as classical group theory was not. But once the paraconsistent revolution has been digested, it is by no means implausible to suggest that these theories, or ones like them, may find physical application—just as classical group theory did. For example, we might determine that certain physical magnitudes appear to be governed by the laws of some inconsistent arithmetic, where, for example, if n and m are magnitudes no smaller than some physical constant k, n + m = k (as well as its being the case that n+ m ≠k). There are, after all, plenty of episodes in the history of science where we came to accept that certain physical magnitudes had somewhat surprising mathematical properties (being imaginary, non-commuting, etc.). Why not inconsistency?"

The suggestion may seem absurd, but consider the analogy of physical geometry. For thousands of years, pretty much everyone was what we might think of as a “Euclidean monist,” meaning that they took it as obvious that there could only be one shortest path between any two points, and that they understood this not just as a claim about the particular hypothetical realm of one axiomatic system but as a (a priori knowable) claim about physical reality as well. Even when non-Euclidean geometries started to be developed, they were largely regarded as esoteric curiosities that couldn’t possibly model anything in the real world. If a student went to Immanuel Kant’s table at his tavern in Konninsberg and suggested to the great man that future science might plausibly one day falsify Euclidean monism and allow for multiple non-equivalent shortest paths between two physically real points, the young man would have presumably been laughed out of the place. Yet, history has rendered the opposite verdict; Einstein has shown that the true geometry of space-time is non-Euclidean, and so much the worse for our spatial intuitions.

Of course, for the student to demand without special evidence or argument that Herr Professor Kant accept that this was possible (particularly in any richer sense than epistemically possible, where ‘epistemically possible’ is taken to mean something like ‘having a status such that one should keep an open mind about it’) would be to beg the question against Euclidean monism, and the good professor would have been fully rationally entitled to decline to do so. Similarly in this case, if we take Priest’s rhetorical question as a demand that we (monaletheists) take his wild speculation about inconsistent magnitudes seriously as a possibility. The problem, of course, as the Einsteinian case shows, is that the lesson of the history of science seems to be that even intuitively well-grounded claims about impossibility can be falsified, and that we shouldn’t be dogmatically closed off to the (epistemic) possibility that we’re wrong about (logical and mathematical) possibility. Priest lays out a fanciful sort of scenario where monaletheist assumptions about logical and mathematical possibility are in fact falsified by empirical research.

"Let us suppose that come to predict a collision between an enormous star and a huge planet. Using a standard technique, we compute their masses as x1 and y1 respectively. Since masses of this kind are, to within experimental error, the sum of the masses of the baryons (protons and neutrons) in them, it will be convenient to take a unit of measurement according to which a baryon has mass 1. In effect, therefore, these figures measure the number of baryons in the masses. After the collision, we measure the mass of the resulting (fused) body, and obtain the figure z, where z is much less than x1 + y1. Naturally, our results are subject to experimental error. But the difference is so large that it cannot possibly be explained by this. We check our instruments, suspecting a fault, but cannot find one; we check our computations for error, but cannot find one. We have a puzzle. Some days later, we have a chance to record another collision. We record the masses before the collision. This time they are x2 and y2. Again, after the collision, the mass appears to be z (the same as before), less than x2 + y2. The first result was no aberration. We have an anomaly.

"We investigate various ways of solving the anomaly. We might revise the theories on which our measuring devices depend, but there is no obvious way of doing this. We could say that some baryons disappeared in the collision; alternatively, we could suppose that under certain circumstances the mass of a baryon decreases. But either of these options seems to amount to a rejection of the law of conservation of mass (-energy), which would seem to be a rather unattractive course of action.

"Then someone, call them Einquine, fixes on the fact that the resultant masses of the two collisions were the same in both cases, z. This is odd. If mass has gone missing, why should this produce the same result in both cases? An idea occurs to Einquine. Maybe our arithmetic for counting baryons is wrong. Maybe the appropriate arithmetic is one where z is the least inconsistent number, and p (the period of the cycle) = 1. For in such an arithmetic x1 + y1 = x2 + y2 = z, and our observations are assumed without having to assume that the mass of baryons has changed, or that any are lost in the collisions! Einquine hypothesizes that z is a fundamental constant of the universe, just like the speed of light, or Planck’s constant."

The story goes on, but the general idea should be clear enough. We have a localized change of the arithmetic assumed by our best science, one that doesn’t require us to change the way we count match-sticks, any more than the non-Euclidean curve of space-time stops us from making Euclidean assumptions about how to make geometrically complicated bank shots when we play pool, but which is just as deadly to monaletheism as Einstein’s discovery was to Euclidean monism. Of course, the details of this story could doubtless be nit-picked in many ways, but I very much doubt that someone in the late eighteenth century trying to imagine a way that science could falsify Euclidean monism could do any better. It’s in the nature of scientific revolutions that, before they come along, they’re not only unpredictable but often almost completely unimaginable. Similarly, at the level of detail that’s Priest’s given us, there’s no way to be sure whether (even in the extreme, strange, counterpossible hypothetical situation being considered) revising the underlying arithmetic of our theory from a classical to a paraconsistent one would be the most rational response. As per our discussion at the end of the introduction, recalcitrant evidence can always be taken as evidence against many different parts of our overall package of beliefs, and any given proposal for belief-revision has to be reasoned out on the specifics of the case, specifics that are unknowable in consideration of these sorts of extreme hypotheticals. So we can’t be too confident in advance that, even in this specific scenario, the best scientific move would be the one Priest hypothesizes.

That said, if Priest’s scenario came true, and it turned out that the best, most reasonable response was to formulate a scientific theory to which inconsistent mathematical entities were indispensible, then, by my lights, we should be realists about such entities, just as we have a good reason to be realists about as much set theory as we need to make sense of our best current science. Thus we have the potentially unattractive consequence mentioned earlier: my solution to Russell’s Paradox leaves the empirical/mathematical back door open for dialetheism. If the empirical results were to go in certain extremely unexpected directions, then inconsistent mathematical results would give us an excellent reason to abandon monaletheism.

As I said, this strikes me as not a bug but a feature of my solution. A belief that no evidence would ever dislodge is a dogma, and one doesn’t need to subscribe to all the details of Karl Popper’s epistemically impoverished theory of justification to agree with him that one of the great lessons of the last few hundred years of scientific progress is that unfalsifiability is a mark of theories that don’t deserve to be taken seriously. If monaletheism is intuitively compelling, there are no good arguments for dialetheism, and the consequences of accepting dialetheism are unappetizing, then we’re within our rational rights to retain the monaletheist structure of our beliefs, but we should always keep the door open for the world to push back against those beliefs and show us that we’re wrong.

Of course, we don’t have a right to even this limited conclusion until we’ve satisfactorily blocked the single strongest argument for dialetheism, which is the argument from the Liar Paradox. It is to this, then, which we now turn.

[*"Monaletheism" is the view that contradictions are never true. When being careful, I like to talk about "monaletheism" rather than "the LNC" as the view being challenged by dialetheists because, technically speaking--if by "the LNC" one just means the logical formula that tells us that for any conjunction of a claim and its negation, the negation of that conjunction is true--then dialetheists don't generally reject the LNC. That formula continues to be a logical truth in systems like Priest's favored logic LP. It's just that dialetheists who accept it also think it has true--and false--counter-examples.]

#

But wait! Doesn't this amount to solving Russell's Paradox and the Liar Paradox in different ways, despite their deep structural similarity? And doesn't that violate the Principle Of Uniform Solution?

Stay tuned for next Monday's post to find out!

Russell's Paradox and Logical Conservativism (The Russell's Paradox Series, Part I of IV)

2010-07-26T02:18:00.000-07:00

Graham Priest often treats Russell's Paradox as the basis of a positive argument for dialetheism. He has even said that in certain respects it is a more invulnerable dialetheist argument than the argument from the semantic paradoxes, since there's no issue of meaningfulness--since it concerns sets rather than sentences, one can't get around it by denying that some sentence is meaningful.

This is, of course, not the only dialetheist position on the paradox. JC Beall, if I understand him correctly, follows Hartry Field in separating the paradox into two versions, the traditional set-theoretic one, and a paradox about the property 'does not apply to itself'--Beall accepts classical orthodoxy about the former and handles the latter dialethetically, just as Field accepts classical orthodoxy about the former and handles the latter by denying the relevant instances of the Excluded Middle. To me, the Field/Beall position on the taxonomy of the paradoxes seems bizarre. Russell's Paradox is about sets, and the 'does not apply to itself' paradox is about properties. They have similar structures, but what of it? Lots of paradoxes have Russell's-Paradox-ish structures, and saying that this means that the 'does not apply to itself' paradox is a version of Russell's Paradox strikes me as making no more sense than describing the Liar as 'the sentence version of' Russell's Paradox. (In fact, I'm inclined to think that, in so far as "applies to" can be paraphrased as something like "is said of itself in a true sentence", the "does not apply to itself" paradox has a lot more in common with the Liar than it does with Russell's Paradox.) In both cases, one can talk that way if one wishes, but it doesn't strike me as shedding much light on anything. I think that things are kept clearer by regarding the 'does not apply to itself' paradox as an interesting puzzle in its own right that doesn't have much of anything to do with Russell's Paradox.

In any case, a third possible dialetheist position on Russell's Paradox might be to take naive set-theoretic realism and the associated contradictions as something that one might as well accept once one has accepted dialetheism, but something which the non-dialetheist has no particular reason to be bothered about unless they are confronted with some separate compelling argument for dialetheism. Someone who held this position might think that, e.g. the Liar Paradox formed the basis for a compelling argument for dialetheism, and consistent solutions to the Liar are severely wanting, so, given the existence of Liar sentences, we should all be dialetheists (at which point we would have no particularly good remaining reason to reject the unrestricted comprehension axiom of naive set theory), but that, in the absence of other reasons to accept true contradictions, there's nothing particularly problematic about orthodox, consistent views about set theory.

Now, I'm not a dialetheist of any sort, and I think that the Liar Paradox can be plausibly solved within a consistent framework. (In fact, once I'm done with the Russell's Paradox series, I've been thinking about starting up a series on my preferred solution to the Liar.) That said, in this and in follow-up posts, I want to argue that dialetheists should accept Dialetheist Option #3.

To see why, let's start with what I'd like to think of as something like the Principle of Logical Conservativism (PLC), which is a special case of a general rule of ontological caution. Here's how I describe it in my dissertation:

#

Any time we expand our ontology, the following seems like an eminently reasonable principle of caution:

• Until we have concrete evidence to the contrary, we should take whatever principles we previously took to apply to absolutely everything whatsoever to apply to the newly discovered objects as well.

By analogy, when we discover the existence of a distant galaxy that we have not been yet able to send probes to photograph, or SETI messages to search for alien life in, the fact that we can imagine or describe perpetual motion machines gives us no reason to think that aliens living on one of the planets of the newly discovered galaxy have developed such a machine. As far as we know, such machines are impossible. If, however, when we send the probes and the SETI messages, we make contact with aliens living in that galaxy, and they show us what seem to be perpetual motion machines, after careful consideration of their evidence, we should be prepared to revise our old ideas about physical possibility. Just so for sets and logical possibility.

#

To expand on the thought a bit:

At one point in In Contradiction, Graham Priest says that his view that the cumulative hierarchy of sets postulated by ZFC and similarly orthodox set-theoretic options is an interesting mathematical structure that mathematicians may have their own reasons to choose to concentrate on, but that the whole universe of naive set theory--including sets with contradictory properties, such as the Russell Set and the set of all ordinal numbers--also exists, is "quite compatible" with a view often expressed by more conservative theorists who are agnostic about the existence of any sets outside the cumulative hierarchy. To me, this exactly misses the point--assuming that one has a reason to believe that at least the sets in the cumulative hierarchy exist, a reasonable stance might be (i) confidence that no sets with inconsistent properties, like the Russell Set and the set of all ordinal numbers don't exist, grounded in a general confidence that no objects have inconsistent properties, combined with (ii) agnosticism about the existence of sets that are outside of the cumulative hierarchy but whose existence would be compatible with what we know about the logical structure of reality in general.

The realm of sets, if it exists, is notoriously epistemically inaccessible to us. (In fact, as Benaceraff famously pointed out, if we assume that it exists, it still seems to be the case that, if every set in that realm disappeared tomorrow, we'd never know.) Even if we assume that there's a compelling case for set-theoretic realism--i.e. for the conclusion that at least some sets exist--the question of which sets exist is still very much open.

Different set theories have different comprehension axioms that give us different results about that question, and we don't have any direct evidence that settles the question of which one gets us the right result. The unrestricted comprehension axiom of naive set theory gives us a particularly simple and clear-cut formula for deciding which sets exist, but one which, as far as we know, can't be the right one, since, as far as we know, there are no true contradictions. If we had evidence of the existence of sets with inconsistent properties, then unrestricted comprehension might be the best option, but that's not the situation that we're in.

Now, a fair-minded critic might start to worry at this point that I'm throwing in this conditional--and the "concrete evidence to the contrary" clause in the principle of caution, above--to make my stance on all this sound more fallibilistic and open-minded than it really is. After all, given the Benacerraf-style worries just mentioned, what would ever count as concrete evidence of the existence of inconsistent sets?

All I can say is that, if you share this concern, you should stay tuned for the next installment!

Nolan & Co. On Moral Fictionalism (Pt. 2)

2010-07-21T00:01:00.000-07:00

(Read Pt. 1 here.)

Daniel Nolan, Greg Restall and Caroline West have an interesting article called Moral Fictionalism versus the Rest available in its longer form here and in a shorter form in the Australasian Journal of Philosophy, 83 (2005), 307–330. (Thanks to Greg Restall for popping up in the comments last time to direct people to the longer version available online, after someone noted that they couldn't access the actual AJP article from where they are. For the sake of convenience, when I refer to page numbers in what follows, they'll be for that online version.

I was initially interested to see the article mostly because meta-ethics is a strong secondary interest for me at this point, and I was delighted to see some of my favorite figures in my primary area of interest (phil of logic) tackling it. In the last post, I started out by briefly going through other meta-ethical options, setting the whole thing up in terms of how these other options make sense of our intuition that it is somehow correct to say that "torturing children for fun is wrong" and somehow incorrect or mistaken to say that "playing Wii Golf is wrong."

I went on to say that the "distinction surely has to do with our moral intuitions, and for shaved apes like us to 'have moral intuitions' is surely ultimately for us to have a certain sort of neurological event, one resulting from genetics, environment or some eccentric combination of the two. If moral properties exist, and they are non-physical (as they certainly seem like they would be), no remotely plausible causal story relates them to humans having moral intuitions about them.

"At this point, of course, one may go in any number of directions. For example, one can try to make sense of moral properties so they aren't non-physical. Volumes can be (and have been) written about this proposal in all of its myriad flavors--synthetic reductionisms, sentimentality theories and so on--but for the moment I'll just say that, while I've argued for this kind of approach in the recent past, at the moment I find the prospects for this sort of project fairly bleak."

If you're interested in why I tend to find those prospects fairly bleak, see my subsequent post on why I think typical, initially plausible-sounding naturalistic stories about moral realism tend to collapse into relativism. Of course, some sort of frankly relativistic realism of one sort or another might well be the right option (keeping in mind that not all forms of relativism are as silly as the simple variety that introductory ethics instructors the world over spend the first day of class demolishing), but, for now, let's put that to one side.

If not non-naturalistic moral realism, and not naturalistic moral realism, then what? Well, there's always outright error theory, but if we're interested in capturing our intuition that it's somehow correct to say that "torturing children for fun is wrong" and somehow mistaken to say that "playing Wii Golf is wrong," then standard forms of error theory aren't real options.

An additional choice on the market today is Blackburn/Timmons/Gibbard-style "quasi-realism." I find it unattractive mostly because I'm a minimalist about truth, and, while every figure mentioned in the last sentence claims the same thing, I think that their view amounts to a theory of truth no less "substantive" than coherence, young-Wittgenstein-style correspondence and so on. (Note that this is not a criticism that Nolan & Co. make, although I think they should.) See below for an explanation of all that.

Meanwhile, at this point, it might start to look like all the possible options are wrong, and we're in serious trouble. Indeed, it's precisely because this is how things look that I find fictionalism so interesting. It lets have our "some moral statements seem correct and others seem incorrect" cake and not only eat it too but wash it down with both a shot of Respectably Naturalistic Picture Of The World and a Plausible View About Truth chaser. What's not to like?

Before going much further, it's worth fleshing out for a moment what "fictionalism" means. Any fictionalist view about moral statements is one according to which moral discourse is like fictional discourse. What semantic status an apparently correct moral statement like "torturing children for fun is wrong" has according to any version of fictionalism, then, is a matter of what status the fictionalist in question takes apparently correct fictional statements like "Sherlock Holmes is a detective, not a carpenter" to have. There are, at this point, at least two options (actually, there are definitely a lot more, but, for the time being, let's stick with just these two, since one of them is the Restall/Nolan/West-preferred and the other helps us bring out some interesting things about it):

(1) What we can call "truth-fictionalism," which stems from the view that the natural language sentence "Sherlock Holmes is a detective" is true, although its surface grammar has to be re-interpreted a bit. When we say "Sherlock Holmes is a detective, not a carpenter," we really mean "In fiction, Sherlock Holmes is a detective, not a carpenter." Similarly, with whatever the meta-ethical version of a fiction operator may be implicitly affixed the beginning of the sentence, "torturing children for fun is wrong" is true as well, despite the fact that we're neither postulating a really existing property of wrongness nor playing quasi-realist games with the notion of truth.

(2) What we can call "falsehood-fictionalism," which stems from the view that the natural language sentence "Sherlock Holmes is a detective" is just as false as "Sherlock Holmes is a carpenter," but, while discussing fiction, it's correct to assert the former and incorrect to assert the latter. If we take the surface grammar seriously, and we take reference failures to guarantee falsehood in the typical Russellian sort of way, the falsehood of any statement one makes about Sherlock Holmes one way or the other would seem to be straightforward enough. The appropriateness or inappropriateness of asserting any such (false) sentence, however, is a function not just of their truth or falsehood but of the purpose of fictional discourse, and similarly, even if "torturing children for fun is wrong" is, strictly speaking, just as false as "playing Wii Golf is wrong", it's still correct to assert the former and incorrect to assert the latter. This difference in correctness is not a matter of the way that the world is (since, on this view, the world lacks properties like "wrongness") but simply a matter of the function of moral language.

Before we go any further, it's worth addressing one bit of taxonomy--why not call (1) a version of moral realism and (2) a version of error theory? This seems to me to be largely a matter of decision and convenience. If one describes any view according to which moral statements are true as realist and any view according to which they are false as error-theoretic, we can talk about realist fictionalism and error-theoretic fictionalism, but I think that the shape of things is clearer if we reserve the word "realism" for views according to which the world contains really existing moral properties and the word "error theory" for views according to which there's no sense in which some moral statements are more correct than others.

A more interesting question, to my mind, is the one of what the difference is between (2) and quasi-realism. Both "falsehood-fictionalism" and quasi-realism agree that (a) the world isn't obliging enough to provide us with metaphysical facts to make true statements like "torturing children for fun is wrong", but (b) the nature of our moral discourse still makes some such statements (but not others) correctly assertible. Where they come apart is on the question of whether (c) is therefore true.

My gloss on this would be that the main difference is that falsehood-fictionalism is a view that you get when you start out from quasi-realist-like assumptions, but you take minimalism about truth seriously. (I'm certainly not claiming that you have to be a minimalist to be a (2)-style fictionalist. Indeed, Nolan/Restall/West, while advocating (2)-style fictionalism in the linked article, don't claim to be minimalists, and indeed seem quite prepared to cede the term to the quasi-realist. Rather, my claim is that minimalists who agree with the starting points of the quasi-realist story necessarily have to become (2)-style fictionalists in order to remain consistent.) If truth is nothing more than T-Schema instances--or, arguably equivalently, "'P' is true" never means anything above and beyond what P means--then, if I ascribe a non-existent property to something, my statement is straightforwardly false. (If, say, we call the fictional color in H.P. Lovecraft's classic horror story "The Colour Out Of Space" 'glack,' our statement "snow is glack" is obviously and straightforwardly false.) It doesn't matter how our discourse about that property works, what useful function might be served by that bit of our language, or what the conditions might or might not be for in any sense 'correctly' or 'appropriately' asserting that something has that property. None of those things can enter into our assessment of the truth or falsehood of the statement, because there's nothing more to truth than the instances of the T-Schema. The quasi-realist's move is, inevitably, to build things about assertion into their account of truth, and at this point, I can't for the life of me see how their view about truth is any less "robust" or "substantive" than, say, the picture-theory version of correspondence. The picture-theorist, after all, accepts all instances of the T-Schema, they just tell a substantive, definitional story about what all those instances have in common.

(Of course, if the quasi-realist wants to frankly admit that they have a substantive theory of truth and frankly argue for that theory and against any kind of minimalist or deflationary option, that's their right, and any such proposal needs to be carefully considered on its merits. What's annoying is that quasi-realists tend to wrap themselves up in the banner--and inherited intuitive appeal--of minimalism, when their implicit theory of truth is really anything but.)

(As a further sidenote to all this, depending on how a (2)-style fictionalist fills out her preferred story about moral language, (2)-style fictionalism might amount to a version of expressivism, if we use that term to refer not just to quasi-realism but to a whole family of related views, like Ayer's non-cognitivism and Hume's projectivism. Hume, remember, took moral statements to be false. When we paint the world in the colors of our moral reactions to it, we are misrepresenting it, not doing something non-descriptive.)

In any case, before saying anything more about (2)--of which the Nolan/West/Restall position is a variation--we should take a harder look at option (1), "truth-fictionalism." What's wrong with that?

One objection might be that, even if the "implicit fiction operator" story is a plausible reconstruction of what ordinary speakers are getting at when they make positive assertions about Sherlock Holmes--if you say "Sherlock Holmes lived on 221b Baker Street" and I offer to pour through the records of the place to prove that no such person ever occupied it, the obvious response would be "you know that's not what I mean"--it still isn't a particularly plausible reconstruction of what ordinary speakers mean when they talk about wrongness. I'm not sure how devastating this objection is, given cases of fictional discourse whose fictionalness isn't universally agreed on. For example, if Robin is an agnostic who doubts the historicity of the Exodus, and Jane is a devout Christian fundamentalist who takes the Bible as the inerrant Word of God, and one of them thinks that Aaron was Moses' brother and the other of us was sure that Aaron was Moses' son, Jane and Jill don't seem to talking past each other. The (1)-style fictionalist could perhaps make sense of this by saying that both our statements have an implicit "according to the Bible..." operator B(...), and our disagreement about whether B(P) universally entails P is beside the point.

On the other hand, the (2)-style fictionalist might claim to have a more straightforward account of this case...and thus, by analogy, a more straightforward account of what's going on when Mark the Robust Non-Naturalistic Realist and Ryan the Fictionalist disagree about whether abortion is wrong. The (2)-style fictionalist could gloss the Jane and Robin case by saying that the statements they are asserting directly contradict each other, neither being prefaced by any sort of implicit operator, and that their disagreement about whether whatever the correct answer is is assertible because, say, they know it to be true, or because, despite its falsity, the function of fictional discourse entitles us to assert it, is quite irrelevant to the case. Similarly, in the Mark and Ryan case, the claims "abortion is wrong" and "abortion is not wrong" directly contradict each other, and their disagreement about whether the correct answer is assertible because it captures the moral facts or whether it's a matter of the function of moral language is quite beside the point.

This leads directly into another advantage for (2)-style fictionalism over (1)-style fictionalism. "According to morality, abortion is wrong" and "according to morality, abortion is not wrong" don't contradict each other. (They might implicitly contradict each other if we make the background assumption that our moral fiction is internally consistent, but that needs to be separately argued for. Famously, ordinary fictions are often quite inconsistent. "In the Sherlock Holmes stories, Watson's war wound is on his right shoulder" and "in the Sherlock Holmes stories, Waston's war wound is not on his right shoulder" are both true. There are desperate ways of interpreting away such inconsistencies, but they fail when we come to deliberately inconsistent works of fiction like Graham Priest's playful short story "Sylvan's Box.") (1)-style fictionalism, in other words, faces a Frege-Geach problem about whether the logic of moral discourse is going to end up being revisionary. The (2)-style fictionalist, on the other hand, would seem to have no such problem. Even if one is a truth-preservationist about validity, since moral statements are just ordinary statements, some moral arguments are valid and others are invalid (although, of course, none are sound) and there will be nothing revisionary about any of this. (Similarly, the inference from "snow is glack" and "if snow is glack, grass is glack" to "grass is glack" is valid.) So far, so good.

Indeed, West, Nolan & Restall put quite a bit of emphasis on Frege-Geach, and the attractiveness of their avoidance of the problem. (See, for example, p. 25 of the linked article.) There are, however, some tricky issues about how thoroughly they've escaped Frege-Geach, and I'll end on a quick look at that point.

To start to get a handle on it, we can ask a basic question:

What's the value of moral discourse, given that it doesn't have the value of "getting at the truth"?

In the case of fictional discourse with the rules of assertion that go with it (according to the (2)-style fictionalist), the whole story might simply be that the pleasure of conversationally recreating the details of our favorite novels and short stories is an extension of the original pleasure of reading them, and that might be all there is to it. Moral discourse, however, seems to serve richer human purposes. It seems to be intimately linked to all sorts of things that definitely do exist. Even if there's no such thing as wrongness, and as such it's not true that "torturing children for fun is wrong", it's still true, in a typical discussion about the matter, regardless of their meta-ethical views, all participants prefer that no one tortures children for fun, are upset by the idea, disapprove of other people doing so and plan not to do so themselves. Presumably, even if we take truth out of the equation, the remaining purpose and importance of continuing to engage in moral argumentation has something to do with all of this.

In their discussion of the advantages of moral fictionalism over standard error-theoretic accounts, Nolan & Co. write:

"A fourth advantage of moral fictionalism over eliminitavism has to do with its capacity to salvage the important role moral discourse is widely thought to play in coordinating attitudes and regulating interpersonal conflict in cases where people disagree about what they are to do, especially where collective action is needed or the proposed actions of different people interfere with each other." (p. 21)

In filling in the details of how one sort of plausible fictionalist story about this might work, they go on to say that one "reply might be to connect non-moral preferences and what is true in fiction via internalist bridge laws (though care must be taken in stating these). If the fiction is set up in such a way that it is guaranteed that what is good-in-the-story that the people engaged in the story have certain non-cognitive attitudes towards it, then coming to realize that some course of action does have certain moral properties according to the story should prompt the realization that the action is one that the agent has certain attitudes towards.

"Or the fictionalist could tell an externalist story: it might be the case that, by and large, people contingently want to bring about situations which are true according to the fiction. According to the externalist, no mere cognitive belief alone will affect people's preferences, but that does not mean that people may not alter their preferences to reflect what is true in the fiction." (p. 22)

The last point, it seems to me, could be more happily paraphrased as something like "people may have an over-arching preference for their behavioral preferences to line up with what, in the fiction, is 'right.'" (Indeed, given Motivational Humeanism--a view, which, anecdotally, it seems to me is accepted by an awful lot of philosophers with realist views about morality--there's a certain sense in which the fictionalist is in no worse boat than a robust moral realist. Even if there are moral facts, and even if we have access to them, people will only be motivated by them to the extent that they happen to have a desire to be moral.) In any case, all of this talk of regulating preferences brings us back to Frege-Geach.

It seems obvious that it's at least possible to have inconsistent approvals, preferences or plans. I can prefer for two mutually inconsistent things to happen, I can approve of them both, and I can plan to do both. My preferences and approvals will never be fully satisfied and my plans will never be fully carried out, but this can be true without inconsistency rearing its ugly head. (For example, I might plan to always carry out every promise I ever make to anyone about anything, but be psychologically incapable of making good on this.) Conversely, I might not have any preference, approval or plan one way or the other about some issue.

It might seem, though, that part of the value of moral discourse is that it helps reduce both the inconsistency and the incompleteness of my attitudes. Therein, indeed, might lie much of the usefulness of careful, rational discussion of moral issues. So far, so good.

But, given the first option sketched above (internalist bridge laws), it seems like the incompleteness and inconsistency of my plans will tend to infiltrate into the moral fiction and this useful purpose will be undermined. It could be that, even in idealized circumstances, I would still equally prefer for P and ~P to come true, or have no preferences one way or the other. If the moral story we've all been telling to each other seems humans started to develop ideas about morality is "designed" so as to line up with our preferences or plans (or even hypothetical preferences or plans formed under idealized circumstances), then won't it reflect the inconsistency and incompleteness of our (actual or hypothetical) plans and preferences?

OK, but what about the externalist option? Going with that, it could be that the fictional world of rights, duties and the rest described by our moral discourse is both complete and consistent, and that "by and large" people happen to have the happy trait of over-archingly preferring to line up their behavioral preferences with the imaginary properties that exist in that world. Outstanding! Even here, though, problems of consistency and completeness arise. Without infallibly internalist bridge laws, the fictional world of moral obligations might be as complete and consistent as external reality--there might be a "Moral Law of the Excluded Middle" and a "Moral Law of Non-Contradiction"--but why suppose that things have worked out that way? If morality is a human construction--like the world of the Sherlock Holmes stories, it's something we made up--why should it be any more complete or consistent that the Holmes stories are?

Take a typical case of reflective equilibrium, where a conflict between immediate moral intuitions is used to help us to regiment them into a consistent system, in precisely the sort of way that's important to the function of moral discourse described above (to help regulate preferences and coordinate collective action): the argument against consequentialism based on the second version of the Trolley Problem. We start by pointing out that in all morally relevant ways, the switch-pushing case and the fat-man-pushing case seem to be equivalent, and thus reason that "if it's OK to push the switch, it's OK to push the fat man." We then assert that it's not OK to push the fat man, and conclude that it's not OK to push the lever. So far, so good, and all precisely the sort of thing the fictionalist is supposed to be able to make sense of, despite the fact that they take all of the statements with the moral predicate "OK" in them to be, strictly speaking, false.

Why, however, should we still take this to be a valid argument, given that the moral fiction, being a human construction, might well contain inconsistencies? Surely, what it's correct to assert about rightness and wrongness is a function of which things are right and wrong according to the story, and, in the story, (a) if it's OK to push the switch, it's OK to push the fat man" and (b) "it's not OK to push the fat man" could both be true in the story without (c) "it's not OK to push the switch" being true in the story. After all, (d) it's OK to push the fat man" might, in principle, also be true in the world of the story if we have no guarantee that it's an entirely internally consistent story.

Of course, the fictionalist could stick to their guns and say that validity isn't a matter of correct-assertibility-preservation, and certainly not of truth-according-to-some-fictional-story-preservation, but a matter of truth-preservation in the actual case. This is fair enough, but I think that at the end of the day, the fictionalist is faced with an unpleasant choice:

(1) They could accept a correct-assertibility-preservation account of logical validity at the price of accepting revisionary conclusions about the logic of moral discourse, which is to say, re-introducing the Frege-Geach Problem, the avoidance of which was supposed to be a major selling point of the account*,

or

(2) They could accept a truth-preservation account of logical validity at the price of destroying the value of moral argumentation. That is to say, they could continue to say that some logical arguments are valid and others are invalid and that the rules for deciding which are which are the same as they are for arguments about anything else, at the expense of logical validity being an important virtue for moral arguments. If, after all, moral statements are always false (so the point of moral arguments can't be to convince us that the conclusions of those arguments are true), and logically valid arguments can take us from moral premises which it is correct to assert to moral conclusions that it is not correct to assert, then why should moral fictionalists care whether or not some argument is valid?

*Restall, as a logical pluralist, might have the resources to accept this with a shrug--perhaps the logical consequence relation appropriate for moral argumentation is inappropriate for other contexts--but Nolan, who has argued extensively (in his work on counterpossible conditionals) in defense of a monist view about logic, should presumably be more troubled. I'm not sure what West's position is. In any case, in the article, they jointly present fictionalism's alleged avoidance of Frege-Geach-style problems as a major selling point.

Follow-Up On Free Will and the "Logical" Problem of Evil (The Hitler Sperm Post)

2010-07-19T19:04:00.000-07:00

In my last post, I argued that Plantinga's version of the free will defense (whereby the bare logical possibility of e.g. demons causing earthquakes is supposed to solve the "logical problem of evil") is a non-starter. If one is concerned about (a) the logical compatibility of the appearance of extreme suffering by innocents with the existence of all all-PKG deity, that's easily enough demonstrated, without any need to bring out to the sort of conceptual technology about actualizing possible worlds that Plantinga deploys. If, on the other hand, one is concerned with the considerably more interesting question of (b) the logical compatibility of the existence of an all-PKG deity with the real extreme suffering of really existing innocents caused in the way that we know it to be really caused, then the demonic free will maneuver clearly fails. What the maneuver demonstrates, at best (i.e. if there's a viable free will defense against the problem of evil in general, which I deny) is (c) the logical compatibility of the existence of an all-PKG God with the real extreme suffering of really existing innocents. And, I argued, (c) only sounds interesting because, if one isn't paying close attention, it looks like (b). We know that earthquakes are caused by the autonomous operation of natural processes in pretty much the same way that we know that the Haitian children who died in last year's earthquake weren't Chalmersian zombies or holograms.

That seems clear enough, but an interesting further question remains--whether some sort of free-will-based solution can at least solve a severely restricted form of the "logical problem of evil" where we confine our attention to human evil. (If that much turned out to be true, some entirely separate defense would still be needed in order to demonstrate the logical compatibility of the existence of all-PKG deity with the existence of natural evil.) I suggested that:

(i) If one is a compatibilist about free will (as I think one should be), there's quite obviously no case to made that God couldn't actualize a possible world with free will and no evil. As far as I know, Plantinga's published arguments against compatibilism tend to be a matter of foot-stomping and table-banging.

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(ii) Even if compatibilism is wrong, this hardly establishes the logical possibility that libertarianism is right. There are tricky issues here, but it's at least prima facie unclear that the notion that our decisions could be neither random nor causally inevitable is an internally conceptually coherent one. (Think about, for example, the rollback argument.)

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(iii) Even if some form of libertarianism is both internally incoherent and the account that best captures our intuitive concept of free will, it hardly follows that the libertarian's proposed conditions for free will are ever met in the actual case. This might seem irrelevant, since we're supposed to be talking purely about bare logical possibility, but if libertarianism is incompatible with what we know about the concrete world as a result of the deliverances of the empirical sciences, then--a la the point above about demons and earthquakes--an obvious argument can be made that no form of the free will defense gets around any interesting form of the logical problem of (even human) evil.

So far, so good, but I ended with a promissory note for something better. I said that I'd go on to give an argument that even if (i)-(iii) were all wrong, no form of the free will defense succeeds in demonstrating the possible co-existence of an all-PKG deity and, for example, the Holocaust. I also promised I'd use the phrase "Hitler sperm" in my explanation.

To get the second part out of the way:

Hitler sperm.

To go into greater detail:

There are many strange things about free will defenses in general. One is that they tend to rely on the assumption that not only would a just God want human beings to be the sort of creatures that are generally capable of making free decisions, but that He would unrestrictedly allow them to exercise that capacity and to carry out their freely-decided plans...except when some other, non-divine agent stops them from doing so, the normal operation of some divinely-established natural process stops them, etc. This is a significant point: Consider that proponents of the free will defense take the importance of free will to explain (or possibly explain, or whatever) why a just God wouldn't stop Hitler. Consider too that no one takes the generic statement "humans have the capacity to freely decide between various courses of action and act on those decisions" to be falsified by the fact that human police agencies sometimes catch murderers and rapists before they carry out their freely-decided courses of action and lock them up in confined places where they will be unable to do so (or, even more efficiently, execute them so they are too dead to do so). Why, then, should we suppose that the same generic statement would have been falsified by, for example, God supernaturally intervening (in a way that, obviously, no one would have ever known about) to move the briefcase bomb that almost killed Hitler in 1944 a few feet closer to the Führer, or arranging for there to be one hole in one fence somewhere so positioned as to have saved even one single one of Hitler's millions victims? How can one coherently believe that (1) "a just God would allow his creatures free will" explains God's failure to stop even one person on one occasion from carrying out their freely-decided plans, but also that (2) free will continues to exist in a world where people are constantly stopped from implementing their plans by everything from the intervention of other agents to heart attacks, freak accidents and so on?

Recall that Plantinga-style invocations of free will rely on the radical claim that it would be impossible for even all omnipotent being--presumably "omnipotent" in the sense of "being constrained only by the boundaries of logical possibility"--to actualize a version of the world in which human beings had free will and even one fewer Jew died in the Holocaust. Now, a theist could grant that the generic statement "humans have free will" would still have been true even if God had saved an additional one or two of the six million, but argue that a just God would have to allow not just the existence of free will in general but "at least exactly as much free will as humans actually have in the actual world." That is to say, a just God could allow people's plans to be foiled by the actions of other people, by the normal operation of divinely-established natural processes and so on, but he wouldn't act directly to stop any person from making or carrying out any decision about anything. We can think of this as a bit like a strange cosmic counterpart to the constitutional prohibition on Congress making a bill of attainder.

This is, to say the least, a lot less intuitive than the simpler version of the free will defense. The burden would certainly be on the theist to explain why we should take this modified version seriously. Beyond that, though, there's one last important point to be made about this:

Even if we accept for the sake of argument that the actual world is not deterministic, that compatibilism is not a viable theory of free will but that libertarianism is, and that the actual world conforms to the libertarian picture of free will....

....and we accept for the sake of argument that, for some reason, a just God would always allow every person in existence to carry out their plans unless stopped in some other way than by direct divine intervention....

....it still doesn't follow that an all-PKG God couldn't have prevented Adolf Hitler from ordering the extermination of European Jews. Standard glosses on omniscience (the "K" part of "all-PKG") have it that God knows everything that will ever happen as well as everything that has ever happened. (Whether to cash this out in terms of "foreknowledge" or atemporal knowledge of all of time is irrelevant for our present purposes.) This is, happily, quite compatible with libertarianism. (If one goes with "foreknowledge" rather than "atemporal knowledge," there are some complications, but if one is willing to grant the possibility of backward causation, it all works out well enough.) Even if some entity knows which radically self-caused free decisions I will make at some point in the future, I can still be the cause of that decision, fulfilling whatever one's favorite libertarian requirements for free will might be.

Given that, an all-PKG deity would know everything Hitler would do in his life, every radically self-caused free decision that Hitler would (if allowed to come into existence) would make before He allowed the particular sperm and the particular egg that became Hitler to come into contact with each other. As such, even given the extreme, strange, ad hoc claim that we assented to above (that, for whatever reason, an all-good God would never intervene to stop anyone from exercising their free will in any way that they were not constrained from doing by the normal rules of His governance of the natural world or by the decisions of other agents), unless one ascribes free will to sperm, there is absolutely no reason why a just God couldn't or wouldn't actualize a possible world in which libertarian free will existed but the Holocaust did not.

(BTW, thanks to my good friend Ryan Lake for an extremely informative discussion about this a while back. Like most of what I say about free will, the clever bits are mostly due to his influence.)

Plantinga and the Problem(s) of Evil

2010-07-14T09:57:00.000-07:00

In Monday's post about the atheist bus campaign, I suggested that, given the powerful anti-theistic arguments available to us, "God probably doesn't exist" dramatically under-states the case. Even allowing for general fallibilism, a more accurate formulation would be "we can be as sure of God's non-existence as we can be of anything." While most of the post was spent taking apart the stranger bits of conventional wisdom often spouted by people with "middle of the road" positions on God--"science and religion can't conflict because they concern different subjects," "you can't prove a negative"--I did mention the Problem of Evil in passing, saying that (a) there were purely logical problems with making sense of the notion of an "all-powerful" being existing, that (b) "we have powerful empirical evidence against the existence of God in the form of the Problem of Evil," and that (c), given that we have no evidence for the existence of God, even if we didn't have any evidence against the existence of God--i.e. even if (a) and (b) were both false--then, in that "all else being equal" hypothetical, atheism would be rationally mandatory purely as a matter of ontological simplicity.

In the comments, Simon Bunckenburg asked:

"Hi, re your treatment of the problem of evil. I'm suprised you do not mention Platinga's free will response? seems to make sense to me."

To quickly review for readers who may not be familiar with this, it's customary to separate out "the logical problem of evil" from "the evidential problem of evil." A standard way to explain the distinction is that the former problem is about whether the existence of evil makes it logically impossible that God exists, whereas the latter problem is about whether the existence of evil makes it merely extremely improbable that God exists. Plantinga's version of the Free Will Defense isn't intended to address the "evidential problem of evil," but merely to show that the existence of evil doesn't make the existence of God logically impossible. While normal Free Will Defenses at best only account for human evil, Plantinga's is notable because he extends it to natural evil by pointing out that, for example, it is at least logically possible that earthquakes could be caused by demons exercising their free will.

So, the obvious, standard way to reply to Simon would be to point out that my only claim about the Problem of Evil in my post was that the existence of evil constitutes "powerful empirical evidence against the existence of God", and that Plantinga's response doesn't touch that claim, and that indeed it isn't intended to touch it. The reasons why I'm devoting an entire post to this, instead of just giving that one-sentence response in the comments, are that:

(1) For a variety of reasons, I'm not entirely satisfied with the standard way of chopping up the Problem of Evil into "logical" and "evidential" forms,
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(2) In any case, even if I was completely comfortable with standard procedure here, I'd strongly reject the claim that Plantinga has refuted the "logical" form of the problem. I think that his version of the Free Will Defense (like all versions of it) fails for all sorts of interesting reasons, worth mentioning.

(Note that if you want to skip all the epistemic stuff and just see what I have to say about Plantinga, I mark the break between my discussions of (1) and (2) with a ************.)

On (1), I'd start by noting that, while I'm in a distinct minority here on the contemporary philosophical scene, I reject the whole notion of epistemic probability. (I've posted about this before, and may again, but for now I'll just note that I agree with the conclusions Simon Evnine comes to in his discussion of the Lottery and Preface Paradoxes in his book Epistemic dimensions of personhood. Briefly: the right lesson to draw from the paradoxes is that high probability is neither universally necessary nor universally sufficient for epistemic justification.) A second, related point, is that I'm a confirmational holist. (I think all of rational inference is a matter of coming up with the best total explanation of the evidence, where "total explanation" includes logical and metaphysical components as well as more obviously "empirical" ones, and it's all intertwined.) Putting the two together, let's go back to something I said about the refutation of theories in my last post:

"If a consequence of Astronomical Theory A is that such-and-such planet will be at such-and-such position at a certain time, and at the relevant time, we observe the relevant position and the planet isn't there, that's evidence against Astronomical Theory A. (Similarly, an obvious consequence of theism is that unnecessary suffering shouldn't exist, but it does exist, in great quantities, and the theist has no convincing way to explain it away. This is evidence against theism.) Without this sort of negative evidence, the process of doing science would be unrecognizable."

Let's try to be a bit more precise about all of this:

Is the refutation of Astronomical Theory A "logical"? Well, in the example, the conclusion that Astronomical Theory A is false comes at the end of a valid and sound logical argument (an instance of Modus Tollens), and I don't accept that probability has anything to do with it. In that sense, it's certainly logical. On the other hand, there are in any such cases any number of creative ad hoc maneuvers one could go through in order to deny the first premise of that argument ("if the theory were true, such-and-such planet would be observed at such-and-such position at a certain time"). If we stick to our guns and continue to assert that first premise and thus the conclusion as well, it's because we've examined these explanations and decided that our best overall theory of the world is a simpler one where we don't try to explain away the evidence in these complicated ways. Note, however, that I'd say precisely the same thing if someone tried to get around the conclusion by denying the validity of Modus Tollens (for example, by postulating true contradictions and pointing out that, given that assumption, it follows that MT isn't universally truth-preserving, since 'if P, then Q' 'P', 'Q' and '~Q' would all be jointly true). My attitude wouldn't be "either the probability of the conclusion being false is 0, which licenses me to dogmatically ignore all dissenters, or it's over 0, in which case I have to prefix key parts of the argument with the word 'probably'", but rather that everything's on the table and has to be evaluated case-by-case.

Given my Quinean epistemic picture, then, I'd argue that there's just one Problem of Evil. Given the existence of all sorts of apparently gratuitous suffering and evil, should we believe in God?

********************

On (2), I'd note first that we need to be careful about delineating exactly what Plantinga is trying to do. Just showing that the appearance of the existence of extreme suffering by innocent people is logically compatible with theism doesn't require anything nearly so complicated as Plantinga's manuevers. If you're friendly to qualia, there's Ryan's excellent zombie solution, and with a little creativity one could come up with something similar that doesn't involve qualia to show that the appearance of innocent people suffering in an extreme ways is some sort of illusion. This is all a lot more simple and elegant than banging on about free will and God's ability to actualize certain possible worlds. It isn't sufficient for Plantinga, though, because he seems to aspire to a slightly more interesting project: he wants to show that even if we take the appearance of extreme suffering by innocent people seriously, and postulate that it is just as it appears, this assumption is compatible with the claim that God exists.

That project is considerably more interesting, and I think he pisses it away in a spectacularly uninteresting way when he starts talking about demons exercising their free will by causing earthquakes. If successful (which I don't take it to be....see below) that technically shows that the real existence of extreme suffering is compatible with the existence of God, but it doesn't show that the real existence of extreme suffering caused in the way that it is really caused is compatible with the existence of God. The project "show that God's existence is logically compatible with something we all know to be true (even if there's some sense in which we can't absolutely rule out extreme fantastical scenarios on which it would be false)" becomes totally uninteresting and pointless if the price of the compatibility is that you have to continue to admit that the existence of God is logically incompatible with something else that we equally well all know to be true (even if there's some sense in which we can't absolutely rule out extreme fantastical scenarios on which it would be false). We know damn well that earthquakes aren't caused by demons, just as we know damn well that the inmates at Auschwitz weren't Chalmersian zombies. The two claims have precisely the same epistemic status. Showing that "the existence of earthquakes that really kill and maim really existing and really conscious innocent children" is logically consistent with the existence of God has no value if you aren't also showing that "the existence of earthquakes really caused in the way we know them to be caused that really kill and maim really existing and really conscious innocent children."

So, even if Plantinga's maneuver were successful in his project, there would still be an interesting "logical problem of evil" that his solution wouldn't touch, and, in fact, I would argue that that the version addressed by his solution only ever looked interesting because, if you squint, it looks like the filled-out version spelled out at the end of the last paragraph.

Still, Plantinga's solution might at least defeat the interesting logical problem of evil when it comes to human evil, right? Like, a separate solution is needed to show that God's existence is logically compatible with natural evil, but showing it for human evil would still be worthwhile.

Well...

The first problem is that, given compatibilism about free will, it's obviously not the case that God couldn't actualize deterministic possible worlds where everyone fulfilled the conditions for being free and there was no moral evil. Arguing for compatibilism is a much bigger project--there's a vast literature there--but for now I'll just report that (a) I'm a compatibilist, and (b) that I'm fairly unimpressed with Plantinga's dismissals of compatibilism, which he doesn't tend to take seriously enough to provide much of anything resembling an argument against.

The second, related, problem is that even if you accept that libertarians are right and compatibilists are wrong about the conditions for free will--which I don't--Plantinga has only demonstrated the logical possibility of the co-existence of God with human evil given the significant further assumption that it's logically possible for the libertarian's conditions for free will could be fulfilled. (After all, one could take e.g. Naomi Arpaly's position, eloquently argued for in her book Merit, Meaning and Human Bondage that we do desire free will, but that it's quite possible to wish for deeply impossible things.) I actually think that there are considerable reasons to doubt that this would be logically possible. Depending on exactly how one understands the details, free will as conceived by the libertarian might seem to require a form of causation that is neither deterministic nor random, and it's just not obvious that this notion can be cashed out in a logically coherent way.

Finally, even given the joint assumptions that (a) the libertarian is right about the conditions for freedom, and (b) it's at least possible for those conditions to be fulfilled (plus, of course, some additional controversial assumptions, like, "free will is more good than genocide is bad"), it's not clear that any of this establishes the possible co-existence of God with the specific sorts of extreme human evil that actually exist.

If you're the sort of theistic libertarian who finds that last claim intriguingly strange--given both libertarianism about free will, and the assumption that a just God would allow his creatures free will, how could all that not at least add up to a solution to the logical Problem of Evil?--then stay tuned for next week, because that will be the subject of Monday's post. As an extra feature, I promise to use the phrase "Hitler semen" in my answer.

Science, Religion And Two Very Silly Claims Often Made By Agnostics

2010-07-12T00:01:00.000-07:00

If the people running the atheist bus campaign had asked me, I would have balked at the word 'probably' and pushed for something like, "We Can Be As Sure Of The Non-Existence Of God As We Can Of Anything, So Stop Worrying And Enjoy Your Life." Then they would have pointed out that my version is a lot clunkier than theirs, and that it would be harder to fit on the side of a bus in letters big enough for people to read as it rolled down the street, and I would have said, OK, despite my deep commitment to fallibilism about absolutely everything (and, yes, that includes fallibilism, and no, you can't run a self-refutation argument against fallibilism on that point, unless you want to beg the question by simply assuming that certainty is a requirement for knowledge), I'm slightly uncomfortable with the word "definitely", but I'll accept it as a necessary simplification for the sake of space.

As it happens, Richard Dawkins, being prominently involved in the project, actually did argue for the phrasing "almost certainly." Whether or not this is equivalent to my "as sure as we can be about anything" formulation presumably depends on whether Professor Dawkins thinks that we can be absolutely certain about anything. (I'm not sure.) My position, at any rate, is that (a) the Stone Paradox is just as much of a logical problem for the claim that an omnipotent being exists as Russell's Paradox is for naive set theory, (b) we have powerful empirical evidence against the existence of God in the form of the Problem of Evil, and that, (c) given the total absence of anything resembling evidence for the existence of God, even if there were no evidence against the existence of God, and theism were a logically coherent position, and we were in an "all else being equal" situation, atheism would still win hands down as a matter of sheer ontological simplicity.

(If I tell you that an invisible three-inch-tall elf is jumping around on my right hand as I type, tap-dancing and singing show tunes at a frequency that no human or piece of recording equipment can detect, your response will obviously and correctly be active disbelief. No rational person would respond with, "well, it's probably wiser for us all to be humble and admit that no one really knows whether or not there's an elf....")

Despite all of this, a lot of nice enlightened tolerant liberal agnostic types balk at the claim that we can be confident about the non-existence of God, citing the deeply silly and confused claim that "you can't prove a negative." A related and equally silly objection comes from people who, often in the context of tut-tutting at the involvement of folks like Dawkins in things like the atheist bus campaign, say (often in a tone that indicates that they take themselves to be delivering a great insight) that "there can't be a conflict between science and religion because they concern different subjects."

(Note that, while I have heard one or two people who do philosophy for a living express similar sentiments--even the best of us have days where we haven't had enough coffee in the morning to think straight--I'm mostly talking about "the folk" here, or rather a certain very recognizable subsection of the folk who read the New York Times, listen to National Public Radio and studiously avoid having "extreme" or "strident" opinions about anything.)

The force of that second objection, of course, relies on playing with an ambiguity about the word "science." If by "science", you mean to narrowly refer to chemistry, physics, biology and so on, then, sure, it's true that there's no direct logical conflict between theism itself and the deliverances of these fields. Of course, many religions do make specific claims that can and do conflict with the findings of physics, chemistry, biology and so on--creationism being only the most obvious example--but other religious views are specifically designed to avoid such conflicts, and certainly the claim "an all-powerful, all-knowing, all-good being exists" doesn't by itself directly conflict with the findings of any of those fields.

If, however, by "science", one means to refer more broadly to the overall project of trying to understand the world in a rational, evidence-driven way--with chemistry, physics, biology and the rest being important elements of that project, but not the whole of it--then, yes, "science" in that sense does indeed conflict with religion, in so far as the Problem of Evil gives us an extremely convincing empirical, evidential argument against theism, there's no evidence for the existence of God, and the sorts of considerations of simplicity, non-adhocness and so on that necessarily drive theory selection in every part of the overall project should lead us to favor atheism.

The first objection is even stranger. Of course you can prove a negative. I'm constantly amazed by the number of otherwise bright, college-educated people who have somehow gotten it into their heads that it's impossible to do so. If by "prove," one means "show by means of evidence," then, remember, Sir Karl Popper thought that the whole of science was a matter of disconfirming theories--conjectures and refutations--and never a matter of confirming them. Even if one isn't willing to go quite that far, it's still indisputably true that we often have evidence that shows us that certain things are not the case. If a consequence of Astronomical Theory A is that such-and-such planet will be at such-and-such position at a certain time, and at the relevant time, we observe the relevant position and the planet isn't there, that's evidence against Astronomical Theory A. (Similarly, an obvious consequence of theism is that unnecessary suffering shouldn't exist, but it does exist, in great quantities, and the theist has no convincing way to explain it away. This is evidence against theism.) Without this sort of negative evidence, the process of doing science would be unrecognizable.

(Also, of course, given the logical law of double negation, any evidential confirmation of any claim P is also a confirmation of the negation of ~P.)

If by "proof", one does not mean "demonstration by evidence" but "proof" in the strict sense used by mathematicians or logicians, then the claim that "you can't prove a negative" becomes, if possible, even sillier and stranger than it was when we understood "proof" evidentially. There are simple, elegant and decisive proofs of the fact that there's no largest prime number, that there's no set of all ordinal numbers, and so on. So, whether one means "prove" in this narrow sense or in a broad enough sense to include evidential confirmation, the claim that one can't prove a negative is bizarre.

Still, can we specifically prove this negative (that God doesn't exist), in the strict, narrow sense of "prove" just discussed? I think so, yeah.

One of the most famous negative proofs in the history of philosophy and mathematics is Bertrand Russell's refutation of naive set theory, which goes like this:

Start with the claim that, for every description you can come up with, there's a set of all and only and only the objects that match that description. Question: What about the set of all sets that are not members of themselves? Is it a member of itself? If it is, it isn't, and it isn't, it is. Either way, we've got a contradiction.

Here's the equivalent for God:

Start with the claim that a being exists who can do anything. Question: What about creating a stone so heavy that He Himself cannot lift it? If He can create it, then there's something the being that can do anything can't do (create the stone), and if He can't, then there's something the being that can do anything can't do (lift it). Either way, we've got a contradiction.

"But, wait," you say, "can't we just be more careful and re-define omnipotence to avoid these problems? Let's just say that God can do pretty much anything, but there are some limits imposed by logical possibility."

Sure, you can say that. Similarly, the naive set theorist could respond to Russell's Paradox by being more careful about how to express their unrestricted comprehension axiom. They could just say that for pretty much any description we can come up with, there's a set of all and only the objects that meet the description, but that there are some restrictions on this imposed by logical possibility. Or they could reject one of the logical laws used to derive the contradiction. Or they could pull a Modus Ponens where everyone else opts for Modus Tollens, and take Russell's Paradox as an argument for the existence of true contradictions. (Graham Priest does, in fact, take precisely this line, and people who've read this blog before know that, while I disagree with Priest, I take his argument very seriously.) There's a general lesson here: given a conflict among one's beliefs, there are always many different ways of changing those beliefs to resolve the conflict. The process of rational belief revision is all about carefully weighing the options and choosing the best and most plausible alternative. Considerations of simplicty, non-adhocness and so on will come into play, and no solution can be absolutely ruled out in advance.

....which brings us back to where we started. Can we be absolutely certain that God doesn't exist? Nah. But we can be as sure about that as we can about anything, and the bus-campaigner's adjective "probably" severely understates the case.

So, I Have A Full-Time Job For Next Year

2010-07-07T00:06:00.001-07:00

It's a one-year (but renewable for up to two more) non-tenure track Assistant Professor position in the Philosophy Department at the University of Ulsan. 3/3 teaching load, mostly Intro but with some opportunity to teach advanced undergraduate classes.

And, yes, that's the Ulsan in South Korea. (The teaching, fortunately, is entirely English-language.) And the school year starts on September 1st.

So, unless some unexpected last minute glitch comes up with getting a visa or some such, I'll be moving to Korea in mid-August.

A Quick Thought About Meta-Ethics

2010-07-05T17:29:00.000-07:00

Think about statements (1)-(3)....

(1) Bob seems tall to me.
(2) Bob is a tall guy.
(3) Bob is 6'5.

....and compare them to statement (4):

(4) Murder is wrong.

Of (1)-(3), which one is most like (4)?

If your answer is (1), then you're endorsing a sort of extreme moral relativism whereby the only properties tracked by moral statements are individual preferences. If I think abortion is OK, and you think that it's wrong, we aren't really disagreeing. It's just wrong for you, but not for me. (As Kang says in the classic Simpsons Halloween segment, "abortions for some and miniature American flags for others!")

If your answer is (2), then we've moved to a more moderate and interesting form of relativism. It is, after all, possible to be mistaken about tallness and to be shown your mistake with evidence and arguments. (E.g. "Wow, that guy's tall!" "No, he isn't, man. He just looks tall because of the angle that photo was taken at. Plus, look at his shoes...") Still, exactly what (or, more to the point, "vaguely what") "tall" means surely varies depending on the time, place and context of utterance. Tall-in-12th-century-China and tall-in-21st-century-America are not the same.

The view that moral statements like (4) should be thought of on the model of (3)--combined with the claim that at least some such statements are true is, of course, objectivist moral realism. This is an attractive view for a variety of reasons, but, at least from a naturalistic point of view, it's often hard to make sense of. The easiest, most natural way of cashing out objectivist moral realism--that moral facts exist eternally outside of space and time--is immediately confronted with obvious epistemic difficulties, more or less equivalent to Benacerraf-style worries about numbers-as-abtract-objects. If there were such non-naturalistic moral facts, how would we ever know about them? What reason would we ever have to think that our moral intuitions, refined by some sort of process of reflective equilibrium, even roughly, non-coincidentally tracked them?

One sort of story one could tell to try to reconcile objectivist moral realism with some kind of respectably naturalistic metaphysical and epistemic framework would be to identify moral properties with whatever cluster of properties "out in the world" happen to provoke the right sort of intuitive responses in people under the right circumstances. When cashing out this sort of meta-ethical stories, it's fashionable to talk a lot about evolution, and to invoke analogies to Chomskyan "language engines." This provides at least a rough model, and lets the whole enterprise bask in the reflected glory of more scientifically rigorous disciplines.

Now, at this point, the obvious line of skeptical reply involves questioning the evidence, unfavorably comparing the rich body of rigorous empirical research backing up Chomskyan ideas about language with the paperclips and chewing gum that tend to be used to hold together any remotely plausible-sounding meta-ethical story about a "moral engine." The next interesting question is whether the end result of that line of attack is that the naturalistic objectivist moral realist needs to work harder, and that for now we should all be cautious agnostics about the whole thing, or that the "moral engine" research program is dead in the water....but I don't want to get into that now.

Instead, let's just assume for the sake of argument that the "moral engine" concept is right on the money. Evolution programs us to have certain instinctive reactions to certain things in reliable ways (perhaps excepting sociopaths and other genetic abnormalities, people whose cultural brainwashing has overwhelmed and overridden their instinctual moral reactions, etc.), and that's what our moral language tracks. Just to finish off the picture, we can throw in some Kripkean stuff about "rigidification" here to explain just how concepts like "wrongness" relates to various items of external reality. (Now, our view is basking in the combined reflective glow of Chomsky, Darwin and Kripke. It's unstoppable!) So far, so good.

Now, this means we can be mistaken about moral claims not just because we make logical mistakes in moral arguments, but because the core intuitions that lie at the justificatory base of the whole structure are flawed in some way (genetic abnormality, cultural brainewashing, etc.) One can argue flawlessly, be aware of every relevant counter-example, etc., have thought of everything, but can have intuitions that differ from the ones that fix our common moral language and thus be morally mistaken.

This is an important point, because without this, the whole project collapses into a kind of relativism.

Now, surely moral language existed two thousand years ago. (I.e. post-Plato, post-Aristotle, just before the rise of Christianity, etc.) That said, a lot of judgments of right and wrong that seem clear-cut now were nowhere to be seen then.

Slavery existed, but hardly anyone (if anyone) thought that slavery was universally morally wrong. Different people had different views about men, women and gender, but if anyone thought that men and women should have completely equal rights, then not a lot of them were writing this down. (The notion of "rights" itself doesn't seem to track much of anything in e.g. ancient Greek moral philosophy.)

Etc., etc., etc.

Now, all of this is plausibly explained by empirical progress in terms of common knowledge of relevant non-moral facts, by cultural blinders and prejudices that have been overcome, by bad moral reasoning that's been corrected, etc., and all of that's fair enough. Imagine, however, the following (admittedly silly and fantastical) discovery:

Some substance that was, up until a couple of thousand years ago, trapped deep beneath the surface of the earth, slowly started to leak upward at that time until it eventually made it into the world's supply of drinking water, and we've all been drinking it ever since. This substance causes a specific sort of brain disease that doesn't in any way impair or impede people's power of *reasoning* but does have one major consequence, but does slightly alter people's moral intuitions in a specific way: it makes them identify more with the plight of those who are different from them in various respects, in a way that makes a process of reflective equilibrium more likely to end up with them being sympathetic to the plight of slaves, women, religious minorities, etc.

Thus, given the sort of naturalistic objectivist view sketched out above, we should all (upon the discovery of this substance and its effect on our moral intuitions) realize that we were mistaken in coming to the conclusion that, for example, "slavery is morally wrong." Not because of any logical flaw in our reasoning, not because we weren't clever or imaginative enough in formulating examples and counter-examples and considerations, but because our deepest moral intuitions were defective, because they differed from the ones that the genetics would have predisposed us to if all else had been equal, the ones involved in the rigidification of moral language. Remember, earlier, our recognition of the possibility of these sorts of mistakes was essential. Without it, the view collapses into relativism.

So....thoughts?

Why Theistic Libertarians Should Believe In Future Facts and Backward Causation

2010-06-29T19:46:00.001-07:00

...at least if they believe in infallible divine foreknowledge.

I'm not aware of any empirical research to back this up, but my strong anecdotal impression is that theistic philosophers who take an interest in free will are, in the vast majority of cases, libertarians. Now, libertarians come in all sorts of different flavors, but the relevant features for our purposes right now are just:

(a) a belief that humans do indeed have free will &
(b) incompatibilism about free will and determinism

Now, historically, many smart people have worried about the compatibility of divine foreknowledge with free will, but (at least given the positions formed in response to the question of *determinism* and free will) it's not clear what the big deal is. As long as you believe in future facts, there's no problem with (a) believing that there are facts about what undetermined radically self-caused free decisions people will make in the future, and that(b) God, being omniscient, knows all such facts, without anything about that knowledge undermining the libertarian picture.

One problem, though, is that this would seem to commit anyone who believed both (a) and (b) to a form of backward causation. Free decisions I will make in 2020 are causing God's knowledge of them in 2010. Some people view the thought of any sort of causal sequence where the effect precedes the cause with extreme discomfort. Can theistic libertarians, then, make sense of things *without* postulating a process of backward causation whereby future facts cause present divine mental states?

Well, here are a couple of easy ways out, that allow the theistic libertarian to believe that God's knowledge includes knowledge of precisely which free decisions we'll make in 2020:

(i) Deny that this is *foreknowledge*, or
(ii) Deny that it is infallible

Some theologians argue that God is, in some sense, "outside of time." It's controversial whether there's even any coherent way to make sense of this idea, and I'm pretty skeptical, but that's a complicated and interesting subject for another time. For the moment, just note that it amounts to (i). God's atemporal knowledge of all of time wouldn't be foreknowledge, so this amounts to giving up on the project of making sense of divine foreknowledge without determinism or backward causation.

More creatively, one could argue that, even in an indeterministic universe, certain sorts of evidence about the present give one a very high degree of justification for one's beliefs about what people will do in the future, enough so that (if they are true) those beliefs count as knowledge. Surely, all evidentially-based human knowledge of the future is of this kind, whether or not we live in an indeterministic universe. After all, even in a deterministic universe, there's always the possibility that we're misinterpreting the evidence. Despite this, surely we at least sometimes know at least some things about what will happen in the future. Surely, in an indeterministic universe, God, with His complete and flawless knowledge of every aspect of the present, would have extremely well-justified beliefs about what will happen in the future, beliefs that, by parity of reasoning, would count as knowledge if those beliefs were true. So far, so good, but given indeterminism, surely some of God's beliefs about the future would be *false*, right?

Well....

If a clock is broken at 8 O'clock, it's possible that every time anyone ever happens to look at it, it will be either 8 AM or 8 PM. If a coin is fair, it is nevertheless just barely possible that it happens to come up heads every time it happens to be flipped. Perhaps, through a massive, bizarre coincidence, God's fallible knowledge of the future never happens to fail. To claim that this is how God has an entirely accurate and complete knowledge of the future would be massively ad hoc, but it's just barely in the logical space of possibilities.

Still, this doesn't seem to be what people typically mean by divine foreknowledge. It seems to me that, built into ordinary usage of the concept, is the notion that God's foreknowledge is infallible. It's not just that God's beliefs have always and will always luckily happen to be true, but that there's a deep sense in which it would impossible for God to make a mistake.

So, how about reconciling infallible divine foreknowledge with libertarianism without backward causation?

Many theists seem to think this can be done with "middle knowledge." God knows what you would do under certain circumstances, and God knows which circumstances will arise, therefore even without (a) determinism, or (b) God just having direct access to future facts, it is still the case that (c) God knows absolutely everything that will happen in the future.

Now, first notice that we're attributing to God not just knowledge of the "counterfactuals of freedom" of people who already exist, but also of people who will come into existence in the future. Thus, without God having direct epistemic access to the future, God knows the exact details of every free decision that will ever be made by every person who will come into existence 10,000 years in the future.

Now, for the theistic compatibilist (who accepts determinism), that's no problem--given determinism, everything about the character of future people, what kind of decisions they will make, etc., is a function of genetics, environment, etc., and is ultimately all built into the present physical state of the universe. If God knows everything about the present and everything about the laws of nature, He can extrapolate the total state of the universe 10,000 years in the future, including (given a compatibilist understanding of freedom) every free decision made by those future people.

But....for a theistic libertarian, just how does this work? For the sake of simplicity, let's stick with the (presently non-existent) children of present people. Given His knowledge of present-tense facts about you and the person who you will one day have kids with, combined with His knowledge of facts about which external circumstances will arise, God can have foreknowledge of which people will come into existence. How, however, does He know (without direct access to future facts) what decisions those (presently non-existent) children will make under various circumstances, unless that's simply a function of their genetics and environment? Given a view of free will that says that a decision can't be simultaneously determined and free, middle knowledge of presently non-existent people would seem to obviously, trivially rule out free will.

It gets worse. Even for God to have short-term infallible foreknowledge of the future actions of presently existing people without having direct access to future facts, determinism creeps back in. After all, if, given present facts about the agent's character, unavoidable external circumstances, etc., their eventual decision is unavoidable--which is what it amounts to to say that a being with infallible knowledge of all those present facts would therefore have infallible extrapolative knowledge of their future decisions--then those decisions are causally determined. The agent can't do otherwise in the sense of "can do otherwise" that differentiates libertarians from compatibilists.

Of course, there are "softer" versions of libertarianism whereby it's OK for most decisions to be causally determined given previous facts about character, etc., provided that certain key "character-forming decisions" are radically free from deterministic chains of cause-and-effect. Even those kinds of moderate libertarian views won't help here, though, given that God would not have had even short-term infallible foreknowledge of those key character-forming decisions.

One way or the other, without backward causation (i.e. direct divine access to future facts), either you have to give up on across-the-board infallible divine foreknowledge or you have to give up on indeterminism. It looks to me like anyone who wants to combine classical theism (complete with infallible divine foreknowledge) with a libertarian view of free will would be well-advised to get over any squeamishness they might have about future facts and backward causation.

Thoughts?