A popular view about the semantic function of the truth predicate is that "truth is disquotation." A classic explanation of this comes from Quine, who claims in a famous passage in "Philosophy of Logic" that the semantic effect of appending "is true" to a quoted sentence is to cancel out the quotation marks. Thus, when we say "'Snow is white' is true", we're not talking about the sentence and attributing some property to it--e.g. correspondence or coherence or whatever. Instead, we're talking about snow and attributing whiteness to it. (Of course, in that example, asserting that snow is white in this indirect way is just redundant, but there are plenty of familiar contexts--e.g. "blind endorsements" like Jack saying "everything Jill just said is true" when he can't necessarily remember every detail of what she said--in which having the truth predicate in our language does allow us to say things we couldn't say without it.)
A lot of people who call themselves disquotationalists don't spell out exactly what they mean by 'disquotationalism', preferring to leave it at a rough intuitive level, and I'm pretty sure quite a few pretty much what Quine did, but a lot of others (like, e.g. JC Beall) make it clear that they mean something slightly different. They start with something like inferentialism about logical connectives--the view that the meaning of the connectives is defined by the logical rules governing them--and apply this to the truth predicate, such that the rules allowing one to derive "'P' is true" from "P" and vice versa are taken to somehow "define" the term "true."
We can think of the first version as "meaning-equivalence disquotationalism" and the second version as "inferentialist disquotationalism." I argued in my dissertation that meaning-equivalence disquotationism gives us a good, principled non-ad-hoc reason to think that sentences like "This sentence is true" and, crucially, its paradoxical cousin "This sentence is not true" are meaningless. After all, if "is true" doesn't add any semantic content to a sentence--and a sentence like "this sentence is true" has nowhere else to gets its content--it ends up just not having any.
But what about inferentialist disquotationalism?
A familiar objection to inferentialism in general is the "tonk problem." What if I introduce a connective called "tonk" with the same introduction rules as disjuncton--so, "P tonk Q" follows from "P"--and the same elimination rules as conjunction, such that "Q" follows from "P tonk Q"?
This problem has been around forever, and while some people--and I'm one of them--take it to render inferentialism painfully implausible, clearly, lots of inferentialists are willing to live with it. "Living with it" clearly means imposing some sort of restrictions on which sorts of rules can create meanings. The phrase "well-defined" is often muttered at this juncture.
Of course, tonk is entirely well-defined--it simply combines the well-defined introduction rules of disjunction with the well-defined elimination rules of conjunction--unless part of what you mean by well-defined is something like "consistent" or "non-trivial" or some other way of saying "fits with our basic logical assumptions."
....which is fine. (Again, not my preferred way of going about things--especially since making those sorts of moves makes it impossible to have a non-question-begging argument with someone who questions those basic logical assumptions--but, for present purposes, let's say that it's fine.) But, if we can just see the logical implications of allowing some connective and declare it logically impossible that it's meaningful, why can't we do the same thing with sentences? If inferentialism works, it's because we can impose a logical-possibility constraint on meaningfulness, and if we can do that, it looks to me like we have an even quicker way of getting to the meaninglessness conclusion about paradoxical sentences than we did with meaning-equivalence disquotationalism.
Wednesday, August 8, 2012
Monday, August 6, 2012
New Developments
I just got a one-year full-time Lectureship in the Philosophy Department at the University of Miami. Among other things, the new job means that I'll have a lot more time to work on papers, and, as always, this blog is a useful way of kicking around the initial half-baked versions of paper ideas and getting people's thoughts about them.
I've already written and scheduled a post for Wednesday called "643 Words About Inferentialism, Disquotationalism, the Tonk Problem and the Liar Paradox."
(It's been a while since I've posted here, so I thought I shouldn't skimp on the content when I started back...)
Meanwhile, here's a picture of the job-offer paperwork I just signed....
Wednesday, August 31, 2011
Beall Against Pinocchio
It's a fun paradox, and Peter Eldridge-Smith argues (convincingly, by my lights) that it creates problems for the claim that 'semantic' contradictions can be true, but not 'metaphysically' substantive ones. JC Beall's half of the exchange is available for free here.
(Eldridge Smith's half is available for free too, but if you're reading this on a computer at an institution with an online subscription to Analysis.)
One problem I have with Beall's response is that it's far from clear what sort of "impossibility" he has the resources to assign to base-language contradictions. It's one thing to say that the actual world lacks them--it certainly seems to!--but Beall, of course, can hardly claim that non-trivial worlds containing base-language contradictions are *logically* impossible. As he himself convincingly argues in Spandrels of Truth, dialetheists can hardly go around claiming that some falsehoods are 'more false' than others, such that contradictions involving them really would be explosive, since one could always construct a paradoxical 'spandrel' which attributed precisely this sort of extra-special-super-falseness to itself. If the claim is that they're metaphysically-but-not-logically impossible, I think that requires considerable fleshing out. *Why* would they be metaphysically impossible?
Someone with orthodox views would say that they're metaphysically impossible *because* they're logically impossible. Once we've blocked off that route by accepting (even "purely semantic") true contradictions, an alternative explanation is required.
(Eldridge Smith's half is available for free too, but if you're reading this on a computer at an institution with an online subscription to Analysis.)
One problem I have with Beall's response is that it's far from clear what sort of "impossibility" he has the resources to assign to base-language contradictions. It's one thing to say that the actual world lacks them--it certainly seems to!--but Beall, of course, can hardly claim that non-trivial worlds containing base-language contradictions are *logically* impossible. As he himself convincingly argues in Spandrels of Truth, dialetheists can hardly go around claiming that some falsehoods are 'more false' than others, such that contradictions involving them really would be explosive, since one could always construct a paradoxical 'spandrel' which attributed precisely this sort of extra-special-super-falseness to itself. If the claim is that they're metaphysically-but-not-logically impossible, I think that requires considerable fleshing out. *Why* would they be metaphysically impossible?
Someone with orthodox views would say that they're metaphysically impossible *because* they're logically impossible. Once we've blocked off that route by accepting (even "purely semantic") true contradictions, an alternative explanation is required.
Monday, August 29, 2011
Wednesday, May 11, 2011
Analysis
I just had a paper, entitled "Paracompleteness and Revenge," accepted for publication at Analysis.
It's about revenge problems for "paracomplete" solutions to the Liar Paradox, a topic which of course I've discussed in this space before.
It's about revenge problems for "paracomplete" solutions to the Liar Paradox, a topic which of course I've discussed in this space before.
Wednesday, April 27, 2011
Singapore
I just gave a talk at the National University of Singapore, entitled "Liar Paradox II: Revenge of the Liar Paradox." Singapore, by the way, is lovely, and I'm going to be sorry to leave on Thursday. On the surface it feels enough like Miami to make me feel nostalgic--humid, windy, full of palm trees and outdoor bars and a mishmash of different languages and cultures--while having a lot of appealing un-Miami-ish traits, like being chock-full of restaurants serving delicious Indian food. (The strangest thing I've seen here to date has been Haw Par Villa, a "moral instruction" theme park based on Confucianism and Chinese mythology put up by some early-twentieth-century Chinese millionaires who'd made a killing in the tiger balm trade. The main attraction is the Ten Courts of Hell, where lurid statues and signs depict the punishments sinners are sentenced to by the Emperors of Hell. For example, "cheating on examinations" gets you your intestines ripped out by demons.) While I've been in Singapore, I've been staying with NUS prof Neil Sinhababu, of Possible Girls fame, who will henceforth always have a special place in my heart for saying, when I came in on Saturday night, "I made sure to save some Laphroaig for your visit."
As far as the talk itself, here's the abstract:
Dialetheists like Graham Priest and JC Beall conclude from the Liar Paradox that sentences like “This sentence is not true” are fact both true and untrue, and that we must therefore revise our logic to accommodate the existence of true contradictions. Similarly, “paracomplete” theorists like Hartry Field avoid the contradiction posed by the Liar Paradox by rejecting one of the central elements of classical logic, the Law of the Excluded Middle. A more conservative solution starts from the claim that sentences that attempt to attribute truth or untruth to themselves are meaningless, and therefore simply not the kinds of things we can logically symbolize or apply truth talk to without committing a nonsensical category mistake. The most common objections to this move are (1) that the “meaninglessness solution” is refuted by the existence of “revenge paradoxes” like the one revolving around the sentence “This sentence is either false or meaningless”, and that (2) the sentences involved are so obviously meaningful that it’s just not possible to take seriously the claim that they’re literally meaningless in any ordinary sense, like “Blorks geblork” or “Colorless green ideas sleep furiously,” whereas the dialetheist and paracomplete approaches have the advantages that they (1*) make room for the perfectly obvious fact that, in any language with normal expressive resources, we can construct perfectly meaningful sentences that attribute untruth to themselves, and (2*) are immune to refutation by means of “revenge paradoxes.” I will argue that (1), (2), (1*) and (2*) are all completely wrong.
In terms of the talk itself, I'm never 100% sure what to think about the ethics of blogging in-person discussions, given that I'm sure I wouldn't want to be represented by someone else's half-clear recollection of what I said on the spur of the moment, so I'll pretty well stick to representing what I said myself, with one exception (one hopes, a benign one): In the talk, I spent a few minutes hammering the standard Priest/Beall/Field sort line on Curry's Paradox. In the Q&A, NUS prof Ben Blumson took issue with some of that, and later in the day I ended up spending a couple of hours in his office genially arguing about Curry and related issues, and if I'm still not utterly convinced, I will definitely say that he did a better job of presenting a fairly plausible defense of the approach to Curry I was criticizing than any other defense of that approach I've seen or read before, and in future I will be scaling back at least some of my initial objections in light of some of the points he made.*
In any case, Curry aside, a lot of the ground I covered should be familiar to regular readers here. While I briefly presented my disquotationalist argument for the claim that ungrounded truth talk is literally meaningless, including the Greenness Paradox as a way of defusing the worry that, since we are able to reason about the Liar, we know what follows from it, thus what it means and thus that it means something, my main focus was on revenge paradoxes. (A paper I'm working on making some of these points is tentatively entitled 'Who Among Us Is Safest From The Liar's Revenge?') Conventional wisdom says that the dialetheist and paracomplete approaches to the Liar, given their willingness to engage in radical surgery to our basic logical notions, gain immunity from the revenge paradoxes that typically plague classical solutions to the paradoxes, of which a particularly clear case is supposed to be the problem posed by (1) for those of us who take these sentences to be meaningless:
(1) Sentence (1) is either false or meaningless.
Since paracompletists don't assert anything about the semantic status of such sentences, but rather reject the relevant instances of Excluded Middle, reject the negation of those instances and so on 'all the way down the line,' they seem to be immune from danger from sentences that attribute to themselves the status paracompletists attribute to such sentences. Even more so, dialetheists seem to be immune from any revenge problems, because any 'revenge' liar would at worst just generate yet another true contradiction, and true contradictions don't generate triviality, given the dialetheist's claim that Disjunctive Syllogism isn't universally truth-preserving.
My claim is that (1) is not a problem for the meaninglessness solution at all. A meaningless sentence does not become meaningful once we attach the word "or" to it and paste (what would otherwise be) a meaningful sentence to its tail. Just because a meaningless sentence has the syntactic form of a disjunction and a true second disjunct does not mean that it's meaningful, much less true.
On the other hand, I argue that the paracompletist has a real problem about sentence (2) and that the dialetheist has a real problem about sentence (3).
(2) An ideally rational being who did not lack any relevant information would not accept sentence (2).
(3) It is not the case that sentence (3) is related to truth.
Regular readers will recognize that (2) is the latest form of a revenge paradox for paracompletism I've been tinkering with for some time. The problem, as I see it, is that, if (2) is true, we have the starkly counter-intuitive result that an ideally rational being would not accept a sentence it knew to be true, if (2) is false, we have the equally counter-intuitive result that an ideally rational being would accept a sentence it knew to be false, and if (2) is one of the sentences about which the most rational option is to 'go paracomplete' and reject both the sentence and its negation, then its a sentence that any ideally rational being would not accept (it would reject the sentence instead of accepting it!) and the sentence is true, and, once again, we have the conclusion that an ideally rational being would fail to accept a sentence it knew to be true.
(3) is a familiar problem, but as I argued here, the exact nature of the biggest problem it poses doesn't seem to be widely realized. If being-related-to-truth and not-being-related-to-truth overlap, just as being-related-to-truth and being-related-to-falsehood overlap, then when the dialetheist shows that (given the assumption of dialetheism) there are cases in which all the premises of Disjunctive Syllogism are related to truth and the conclusion is not, they have no more shown that DS is not universally truth-preserving than they would if they'd 'just' showed that all the premises of DS were true and the conclusion was false. No one thinks that "all the premises of argument A are true and the conclusion is false" is a dialetheistically-acceptable way of establishing a failure of truth-preservation. Why should "all the premises of A are related to truth and the conclusion is not" be even a little bit different? Without a better answer to that question, the dialetheist claim that Liars can be both true and false without triviality following simply doesn't hold up.
*On a similar note, I should include a quick shout-out to Brandon Watson for giving me a hard time recently about my error theory about the mistakenness of ordinary competent speakers who take Liars to be meaningful. My preferred way of putting the point now, which I used in the talk, goes about like this: Self-referential sentences are often meaningful--e.g. 'This sentence has seven words in it'--and sentences with precisely the same wording as the Liar sentence are meaningful in other context--'This sentence is false' said while pointing at a sentence about some substantive subject written on a chalkboard. To realize that it's meaningless when the intended reference of the 'this' is that very sentence is a conclusion that takes careful philosophical argumentation. Given those two facts, its quite natural that most people don't realize that it's true. By analogy, if an object is far away and looks a certain (misleading) way from a great distance, and out of a whole crowd of people watching the object, only Bob has a telescope, it's utterly unsurprising that most competent-users-of-functioning-human-eyes end up having a mistaken impression about the object. All Bob needs to do by way of explanation of the disconnect is to say, "yeah, you don't have a telescope. But, hey, look through it yourselves and you'll see where you went wrong here."
As far as the talk itself, here's the abstract:
Dialetheists like Graham Priest and JC Beall conclude from the Liar Paradox that sentences like “This sentence is not true” are fact both true and untrue, and that we must therefore revise our logic to accommodate the existence of true contradictions. Similarly, “paracomplete” theorists like Hartry Field avoid the contradiction posed by the Liar Paradox by rejecting one of the central elements of classical logic, the Law of the Excluded Middle. A more conservative solution starts from the claim that sentences that attempt to attribute truth or untruth to themselves are meaningless, and therefore simply not the kinds of things we can logically symbolize or apply truth talk to without committing a nonsensical category mistake. The most common objections to this move are (1) that the “meaninglessness solution” is refuted by the existence of “revenge paradoxes” like the one revolving around the sentence “This sentence is either false or meaningless”, and that (2) the sentences involved are so obviously meaningful that it’s just not possible to take seriously the claim that they’re literally meaningless in any ordinary sense, like “Blorks geblork” or “Colorless green ideas sleep furiously,” whereas the dialetheist and paracomplete approaches have the advantages that they (1*) make room for the perfectly obvious fact that, in any language with normal expressive resources, we can construct perfectly meaningful sentences that attribute untruth to themselves, and (2*) are immune to refutation by means of “revenge paradoxes.” I will argue that (1), (2), (1*) and (2*) are all completely wrong.
In terms of the talk itself, I'm never 100% sure what to think about the ethics of blogging in-person discussions, given that I'm sure I wouldn't want to be represented by someone else's half-clear recollection of what I said on the spur of the moment, so I'll pretty well stick to representing what I said myself, with one exception (one hopes, a benign one): In the talk, I spent a few minutes hammering the standard Priest/Beall/Field sort line on Curry's Paradox. In the Q&A, NUS prof Ben Blumson took issue with some of that, and later in the day I ended up spending a couple of hours in his office genially arguing about Curry and related issues, and if I'm still not utterly convinced, I will definitely say that he did a better job of presenting a fairly plausible defense of the approach to Curry I was criticizing than any other defense of that approach I've seen or read before, and in future I will be scaling back at least some of my initial objections in light of some of the points he made.*
In any case, Curry aside, a lot of the ground I covered should be familiar to regular readers here. While I briefly presented my disquotationalist argument for the claim that ungrounded truth talk is literally meaningless, including the Greenness Paradox as a way of defusing the worry that, since we are able to reason about the Liar, we know what follows from it, thus what it means and thus that it means something, my main focus was on revenge paradoxes. (A paper I'm working on making some of these points is tentatively entitled 'Who Among Us Is Safest From The Liar's Revenge?') Conventional wisdom says that the dialetheist and paracomplete approaches to the Liar, given their willingness to engage in radical surgery to our basic logical notions, gain immunity from the revenge paradoxes that typically plague classical solutions to the paradoxes, of which a particularly clear case is supposed to be the problem posed by (1) for those of us who take these sentences to be meaningless:
(1) Sentence (1) is either false or meaningless.
Since paracompletists don't assert anything about the semantic status of such sentences, but rather reject the relevant instances of Excluded Middle, reject the negation of those instances and so on 'all the way down the line,' they seem to be immune from danger from sentences that attribute to themselves the status paracompletists attribute to such sentences. Even more so, dialetheists seem to be immune from any revenge problems, because any 'revenge' liar would at worst just generate yet another true contradiction, and true contradictions don't generate triviality, given the dialetheist's claim that Disjunctive Syllogism isn't universally truth-preserving.
My claim is that (1) is not a problem for the meaninglessness solution at all. A meaningless sentence does not become meaningful once we attach the word "or" to it and paste (what would otherwise be) a meaningful sentence to its tail. Just because a meaningless sentence has the syntactic form of a disjunction and a true second disjunct does not mean that it's meaningful, much less true.
On the other hand, I argue that the paracompletist has a real problem about sentence (2) and that the dialetheist has a real problem about sentence (3).
(2) An ideally rational being who did not lack any relevant information would not accept sentence (2).
(3) It is not the case that sentence (3) is related to truth.
Regular readers will recognize that (2) is the latest form of a revenge paradox for paracompletism I've been tinkering with for some time. The problem, as I see it, is that, if (2) is true, we have the starkly counter-intuitive result that an ideally rational being would not accept a sentence it knew to be true, if (2) is false, we have the equally counter-intuitive result that an ideally rational being would accept a sentence it knew to be false, and if (2) is one of the sentences about which the most rational option is to 'go paracomplete' and reject both the sentence and its negation, then its a sentence that any ideally rational being would not accept (it would reject the sentence instead of accepting it!) and the sentence is true, and, once again, we have the conclusion that an ideally rational being would fail to accept a sentence it knew to be true.
(3) is a familiar problem, but as I argued here, the exact nature of the biggest problem it poses doesn't seem to be widely realized. If being-related-to-truth and not-being-related-to-truth overlap, just as being-related-to-truth and being-related-to-falsehood overlap, then when the dialetheist shows that (given the assumption of dialetheism) there are cases in which all the premises of Disjunctive Syllogism are related to truth and the conclusion is not, they have no more shown that DS is not universally truth-preserving than they would if they'd 'just' showed that all the premises of DS were true and the conclusion was false. No one thinks that "all the premises of argument A are true and the conclusion is false" is a dialetheistically-acceptable way of establishing a failure of truth-preservation. Why should "all the premises of A are related to truth and the conclusion is not" be even a little bit different? Without a better answer to that question, the dialetheist claim that Liars can be both true and false without triviality following simply doesn't hold up.
*On a similar note, I should include a quick shout-out to Brandon Watson for giving me a hard time recently about my error theory about the mistakenness of ordinary competent speakers who take Liars to be meaningful. My preferred way of putting the point now, which I used in the talk, goes about like this: Self-referential sentences are often meaningful--e.g. 'This sentence has seven words in it'--and sentences with precisely the same wording as the Liar sentence are meaningful in other context--'This sentence is false' said while pointing at a sentence about some substantive subject written on a chalkboard. To realize that it's meaningless when the intended reference of the 'this' is that very sentence is a conclusion that takes careful philosophical argumentation. Given those two facts, its quite natural that most people don't realize that it's true. By analogy, if an object is far away and looks a certain (misleading) way from a great distance, and out of a whole crowd of people watching the object, only Bob has a telescope, it's utterly unsurprising that most competent-users-of-functioning-human-eyes end up having a mistaken impression about the object. All Bob needs to do by way of explanation of the disconnect is to say, "yeah, you don't have a telescope. But, hey, look through it yourselves and you'll see where you went wrong here."
Sunday, April 24, 2011
Some Objections to the Meaninglessness Solution to the Liar Paradox, Part IV of IV
(Is it Wednesday already? Oh well, better late than never....)
I've argued that, if (as I think) the truth predicate/operator is just a device used to assert things (just as the falsehood predicate/operator is just a device used to assert their negations), it can pretty clearly only be meaningfully applied when there is something there to be asserted--thus, it can only be meaningfully applied to claims about something other than truth. Thus, for example, in a Yablo-like series of sentences where each sentence ascribes truth to the next sentence in the series,
T1: T2 is true.
T2: T3 is true.
T3: T4 is true.
....and so on forever, all the sentences in the infinite series are literally as devoid of meaning as strings of nonsense syllables, or 'Colorless green ideas sleep furiously.' If, on the other hand, sentence T1000000 is "Snow is white," the rest of the sentences inherit their meanings (and, thus, truth-values) from that.
A semantic property pretty clearly *unlike* truth in this respect is meaningfulness itself. If the meaningfulness predicate only applied to meaningful sentences, it wouldn't fulfill its sole communicative function of separating out the meaningful sentences from the meaningless ones. This is important when we consider (18), which, by analogy to the Liar, we can call The Babbler:
(18) Sentence (18) is meaningless.
If (18) is true, it's both true and meaningless, therefore both meaningful and meaningless, and, of course, if it's meaningless, it's both true and meaningless, therefore both meaningful and meaningless. As such, on pain of contradiction, (18) had better just be false.
Fortunately, in light of the above, we have a good principled reason to think that this is indeed the case. If the function of the meaningfulness predicate is to separate out the meaningful from the meaningless sentences, it has to apply to all sentences. Therefore, it's meaningful to say of any sentence that it's meaningful or meaningless, regardless of the nature of the sentence we're talking about. As such, if all a sentence does is assert a view about the meaningfulness of some sentence, even itself, there's no reason for it not to be meaningful. Thus, (18) is false and (19):
(19) Sentence (19) is meaningful.
...is true.
One important principle, underlying the whole business of revenge-paradoxology, is worth calling attention to here, since I've been implicitly using it a lot. Given these sorts of examples, or, better yet, cased like (20) and (21):
(20) This sentence has seven words in it.
(21) This sentence has twenty words in it.
....where it would be clearly absurd to assert about sentence (20), for example, that is seven words long, without granting that sentence (21) is true, we have what we can call the Meaningfulness of Self-Reference Principle: "If Sentence X has property Y, and Sentence X *states* that Sentence X has property Y, then Sentence X is true (and thus, of course, meaningful)."
With all that in mind, and the demonstrations in Parts II and III that it clearly is possible to engage in apparent reasoning about even the most clearly meaningless sentences--meaning that it's not a problem for meaninglessness solutions to the Liar that it's "clearly possible to reason about it, and we all know what does and doesn't follow from it"--let's turn to the apparently troubling revenge paradox for my view that I ended with last time:
(17) Sentence (17) is one that one would have to ultimately label as "false" if one treated it as being meaningful and went through the motions of "reasoning" about it without making the sort of mistake we would regard in normal contexts as a mistake in reasoning.
So, playing along with the game of treating it as meaningful for a moment, an obvious first question is this:
Does (17) take a stand on the question of its own meaningfulness? In other words, does it (a) say of itself that it's meaningful, (b) say of itself that it's meaningless, or (c) remain neutral on that topic?
The wording strongly suggests that (a) would be the wrong gloss--talk of treating it 'as meaningful' and 'going through the motions' strongly suggests that the point is to, at the very least, keep open the possibility that it's meaningless, if not to actively assert it. That said, if (a) is right--it's taking a stand on its own meaningfulness in the directing of asserting it--then to say that, if one went through the motions of reasoning about it, one would make something we would regard in other contexts--i.e. really reasoning about meaningful things--as a mistake, is to say that, if one reasoned about it and failed to come to the conclusion that it was false, one would be making a real, full-fledged mistake in reasoning--a factual mistake, landing us with the wrong answer. In other words, given (a), (17) is just a normal if un-usually phrased Liar sentence, the normal meaninglesness solution applies to it, and the Principle of the Meaningfulness of Self-Reference is not violated if we simultanneously say of it that, although meaningless, going through the motions seems to get us the result that it's false (and true), given that what it's saying is that this isn't a matter of going through the motions in an empty context, because it really is false. (It can be neutral about its own falsehood, given that it asserts its own meaningfulness and thus converts the neutral-sounding language about apparent mistakes into, in effect, the positive claim that one would be making a substantive mistake and getting the wrong result.)
If (b) is the case, then we have a disguised conjunction of two claims: (i) a claim about its alleged meaninglessness, and (ii) a claim about whether any possible analysis of it that (1) took it as meaningful and (2) failed to include any mistakes unrelated to the meaningfulness question would therefore (3) diagnose (17) as false. There's a lot to untangle here, but suffice to say that if it is meaningless, then the true 'first conjunct' asserting as much doesn't make the whole thing meaningful, for reasons examined when we looked at (2), above, and if it's not meaningless, the falsity of the first conjunct guarantees the falsity of the whole thing without fear of contradiction. Really, though, I think the most natural reading is (c), and that's where the real problem seems to be.
If (c) is, then, the case, as should be clear by now, (17) really amounts to a disguised disjunction between the claim that (i*) reasoning about (17) and failing to come to the conclusion that it's false would be a *factual* mistake, and (ii*) that 'reasoning' about (17) leads us to the apparent conclusion that it is false, but only because we're indulging a nonsensical category mistake. In other words, given (c), what we end up with is a disguised version of sentence (2), above:
(2) The sentence marked (2) is either false or meaningless.
...which we already dealt with in Part I. Since I enjoy the circularity of ending by directing back to the first post in the series, I think I'll just leave off there and throw open the floor to questions, comments and devastating objections.
I've argued that, if (as I think) the truth predicate/operator is just a device used to assert things (just as the falsehood predicate/operator is just a device used to assert their negations), it can pretty clearly only be meaningfully applied when there is something there to be asserted--thus, it can only be meaningfully applied to claims about something other than truth. Thus, for example, in a Yablo-like series of sentences where each sentence ascribes truth to the next sentence in the series,
T1: T2 is true.
T2: T3 is true.
T3: T4 is true.
....and so on forever, all the sentences in the infinite series are literally as devoid of meaning as strings of nonsense syllables, or 'Colorless green ideas sleep furiously.' If, on the other hand, sentence T1000000 is "Snow is white," the rest of the sentences inherit their meanings (and, thus, truth-values) from that.
A semantic property pretty clearly *unlike* truth in this respect is meaningfulness itself. If the meaningfulness predicate only applied to meaningful sentences, it wouldn't fulfill its sole communicative function of separating out the meaningful sentences from the meaningless ones. This is important when we consider (18), which, by analogy to the Liar, we can call The Babbler:
(18) Sentence (18) is meaningless.
If (18) is true, it's both true and meaningless, therefore both meaningful and meaningless, and, of course, if it's meaningless, it's both true and meaningless, therefore both meaningful and meaningless. As such, on pain of contradiction, (18) had better just be false.
Fortunately, in light of the above, we have a good principled reason to think that this is indeed the case. If the function of the meaningfulness predicate is to separate out the meaningful from the meaningless sentences, it has to apply to all sentences. Therefore, it's meaningful to say of any sentence that it's meaningful or meaningless, regardless of the nature of the sentence we're talking about. As such, if all a sentence does is assert a view about the meaningfulness of some sentence, even itself, there's no reason for it not to be meaningful. Thus, (18) is false and (19):
(19) Sentence (19) is meaningful.
...is true.
One important principle, underlying the whole business of revenge-paradoxology, is worth calling attention to here, since I've been implicitly using it a lot. Given these sorts of examples, or, better yet, cased like (20) and (21):
(20) This sentence has seven words in it.
(21) This sentence has twenty words in it.
....where it would be clearly absurd to assert about sentence (20), for example, that is seven words long, without granting that sentence (21) is true, we have what we can call the Meaningfulness of Self-Reference Principle: "If Sentence X has property Y, and Sentence X *states* that Sentence X has property Y, then Sentence X is true (and thus, of course, meaningful)."
With all that in mind, and the demonstrations in Parts II and III that it clearly is possible to engage in apparent reasoning about even the most clearly meaningless sentences--meaning that it's not a problem for meaninglessness solutions to the Liar that it's "clearly possible to reason about it, and we all know what does and doesn't follow from it"--let's turn to the apparently troubling revenge paradox for my view that I ended with last time:
(17) Sentence (17) is one that one would have to ultimately label as "false" if one treated it as being meaningful and went through the motions of "reasoning" about it without making the sort of mistake we would regard in normal contexts as a mistake in reasoning.
So, playing along with the game of treating it as meaningful for a moment, an obvious first question is this:
Does (17) take a stand on the question of its own meaningfulness? In other words, does it (a) say of itself that it's meaningful, (b) say of itself that it's meaningless, or (c) remain neutral on that topic?
The wording strongly suggests that (a) would be the wrong gloss--talk of treating it 'as meaningful' and 'going through the motions' strongly suggests that the point is to, at the very least, keep open the possibility that it's meaningless, if not to actively assert it. That said, if (a) is right--it's taking a stand on its own meaningfulness in the directing of asserting it--then to say that, if one went through the motions of reasoning about it, one would make something we would regard in other contexts--i.e. really reasoning about meaningful things--as a mistake, is to say that, if one reasoned about it and failed to come to the conclusion that it was false, one would be making a real, full-fledged mistake in reasoning--a factual mistake, landing us with the wrong answer. In other words, given (a), (17) is just a normal if un-usually phrased Liar sentence, the normal meaninglesness solution applies to it, and the Principle of the Meaningfulness of Self-Reference is not violated if we simultanneously say of it that, although meaningless, going through the motions seems to get us the result that it's false (and true), given that what it's saying is that this isn't a matter of going through the motions in an empty context, because it really is false. (It can be neutral about its own falsehood, given that it asserts its own meaningfulness and thus converts the neutral-sounding language about apparent mistakes into, in effect, the positive claim that one would be making a substantive mistake and getting the wrong result.)
If (b) is the case, then we have a disguised conjunction of two claims: (i) a claim about its alleged meaninglessness, and (ii) a claim about whether any possible analysis of it that (1) took it as meaningful and (2) failed to include any mistakes unrelated to the meaningfulness question would therefore (3) diagnose (17) as false. There's a lot to untangle here, but suffice to say that if it is meaningless, then the true 'first conjunct' asserting as much doesn't make the whole thing meaningful, for reasons examined when we looked at (2), above, and if it's not meaningless, the falsity of the first conjunct guarantees the falsity of the whole thing without fear of contradiction. Really, though, I think the most natural reading is (c), and that's where the real problem seems to be.
If (c) is, then, the case, as should be clear by now, (17) really amounts to a disguised disjunction between the claim that (i*) reasoning about (17) and failing to come to the conclusion that it's false would be a *factual* mistake, and (ii*) that 'reasoning' about (17) leads us to the apparent conclusion that it is false, but only because we're indulging a nonsensical category mistake. In other words, given (c), what we end up with is a disguised version of sentence (2), above:
(2) The sentence marked (2) is either false or meaningless.
...which we already dealt with in Part I. Since I enjoy the circularity of ending by directing back to the first post in the series, I think I'll just leave off there and throw open the floor to questions, comments and devastating objections.
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