Monday, August 30, 2010

Ed Feser And The Bomb

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?


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.

Wednesday, August 25, 2010

Why I'm Not A Bayesian

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.

Monday, August 23, 2010

A Reply To Same Objections Made During the Liar Paradox Series

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.

Wednesday, August 18, 2010

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

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.

Monday, August 16, 2010

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

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.


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

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


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.”

“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.

Wednesday, August 11, 2010

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

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!

Monday, August 9, 2010

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

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!

Wednesday, August 4, 2010

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

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

Monday, August 2, 2010

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

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.