So when we left off our story, Barwise and Etchemendy were modelling Russellian propositions with Aczelian hypersets, and the Liar Sentence came out false, but this fact wasn't included in "the world," because if it was, that would violate the "coherence condition" preventing a set-theoretic proposition-object and its "dual" from both being present in a model.
They describe this consequence as "counter-intuitive." I'd describe it as absolutely incoherent.
In any case, when they turn to Austinian propositions, one of the big advantages of the shift was supposed to be that this "counter-intuitive" consequence can be avoided. Remember, Austinian propositions include the situations they are about--so, e.g. if you are at a card game and you mistake the player holding the 3 of Hearts for Jill and claim that "Jill has the 3 of Hearts," the proposition comes out false even if by coincidence Jill actually is playing cards across town and she does happen to have the 4 of Hearts.
As such, the proposition being expressed by any given use of the LS is about some specific situation, since all propositions are on the "Austinian" model. How does this help?
Well, as in the Russellian case, the LS always comes out false. We can see why by analogy to the behavior of the Truth-Teller ("This sentence is true.") In situations that don't include any semantic facts, the Truth-Teller is false, since there is nothing to make it true. In situations that include semantic facts, the Truth-Teller is sometimes true and sometimes false, depending on what those semantic facts are...i.e. in a situation that includes the semantic fact that the Truth-Teller of that situation is true, it's true, and in a situation that includes the semantic fact that the Truth-Teller of that situation is false, it's false. Where these "semantic facts" about the Truth-Teller are supposed to come from, since they obviously don't supervene on the non-semantic facts, is a bit of a mystery to me, but whatever. Let's move on.
The Liar behaves in exactly the same way, with one major difference. The relevant semantic facts are always excluded from the situation used to determine its truth value. Why? Well, if we allowed them to included, that would give us the result that each Liar was both true and false, and that would violate the coherence rule. So it's in "the world," but not the part of it used to evaluate the truth-value of the proposition.
So the Liar of a situation that doesn't have any semantic facts in it is false--just false and not true, which is a neat trick--and the semantic fact that "the Liar of Situation 1 is false" is included not in Situation 1 but in Situation 2. Sure, Situation 2 has a Liar of its own, which is made false because there is nothing in Situation 2 to make it true (after all, the semantic fact that the Liar of Situation 1 is false is about the Liar of Situation 1, not this new Liar, so it is irrelevant), and its falsehood is a fact not of Situation 2 but of Situation 3. On and on forever.
Two comments seem to be in order about this picture.
The first is that, at the beginning of the book, Barwise and Etchemendy rejected Tarski's hierarchy-of-artificial-languages solution because it was arbitrary, didn't really solve anything on the intuitive level, etc. I agree with them on that. It is interesting, though, given that, that the situation-theoretic hierarchy they posit seems to be structurally exactly parallel to Tarski's languages model.
Secondly, and more importantly, it seems to me that the one fact being excluded from the situations is the only one that cannot be plausibly excluded from them. It is, in fact, the ONLY thing that belongs in them. If the Liar Sentence really does express a proposition, then the one and only thing its about is its own truth value. That semantic fact is the one and only fact of any kind that belongs in its Austinian "situation," so a Liar that really was 'about' a situation without semantic facts, or even without this one relevant semantic fact, is simply an incoherent non-possibility.
As such, taken as a defense of the Law of Non-Contradiction--and, as an attempt to explain why the Liar does not have its prima facie property of being true if false and false if true, that's what it amounts to, even without a dialetheist who actually asserts that it is both true and false as the conscious target target--it boils down to the following argument.
"There are no counter-examples to the LNC, therefore the Liar is not a counter-example to the LNC."
That's a perfectly valid instance of universal instantiation, but those of us who want to defend the LNC in the context of an actual argument are going to have to do better than that.
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6 comments:
Actually Prior's solution of the "Liar" seems pretty sound. An assertive sentence already possesses an implicit truth claim: when Sollie shreiks, "there are bats in the lavatory!" He obviously means that "it's True that there are bats in the lavatory". If he said it "It is true and it is false that there are bats in the lavatory," he contradicts himself (or, as gangstas say, he's talkin' sheet). But there is no paradox.
SO the supposed paradox of the liar becomes "this sentence is true and it is false," which is just contradiction, not a paradox (I think this can be applied to other situtations as well). Kripke said something similar, it appears (see Wiki).
The ZFC stuff (and Russ. paradox) is another, more difficult matter (I have yet to read--even from a set theory guru like Suppes, a detailed proof of Zermelo's "separation" axiom), but I think with most set theorists (even Frege and Lord Bertie) they start from platonic assumptions (about sets, universals, number, infinity, functions, etc.) which are themselves questionable. In the "game" of set theory, OK, that's understandable, but some of the peeps often seem like they are discussing some abstract, platonic forms ("Set world") which have little to do with how mathematics functions in the real world
Prior's solution, as you describe it doesn't work.
Of course, even if it worked, I would still be reluctant to accept it, simply because I find it extremely implausible that every ordinary proposition its own subject. The subject of the claim there are bats in the lavatory is the bats, not the proposition that there are bats in the lavatory. It strikes me that an infinite regress problem looms here, since if 'there are bats in the lavatory' really means "it is true that 'there are bats in the lavatory,' shouldn't the embedded token also mean "it is true that 'there are bats in the lavatory'?" If so, "there are bats in the lavatory" really means 'it is true that it is true that it is true that....[on and on into infinity before we get to the original atomic proposition that] there are bats in the lavatory." There's an open question about whether this kind of infinite series of embeddings would even be coherent. At any rate, it certainly isn't plausible.
Fortunately, all this is irrelevant, because's Prior's suggestion as you describe it simply doesn't work. Here's why not:
The analogue for the Liar Sentence of reading "there are bats in the lavatory" as "it's true that there are bat's in the lavatory" is not "this sentence is true and it is false," but "it is true that this sentence is false," which is just as paradoxical as the original sentence.
Thus, this suggestion gets us nowhere.
One more thing about this: if instead of reading the Liar by analogy with the Bat sentence as 'it is true that this sentence is false,' you want to read it as the conjunction of the Liar and the Truth-Teller , you have to go back and read the bat sentence too as a hidden conjunction of the actual bat sentence and the Truth-Teller sentence (the content-free "This sentence is false").
Since, even for those who think that the Liar Sentence is by the relevant semantic rules necessarily both true and false, it's horribly unclear what the truth value of the Truth-Teller is--Barwise and Etchemendy say that it is "up for grabs"--we'd have to do some hard and strange re-thinking of all of our existing truth-value assignments if we did see every atomic proposition as 'really' a conjunction of the atomic proposition and the Truth-Teller. (Even Graham Priest regards the Truth-Teller as a good prima facie candidate for a truth-value gap, as opposed to the Liar, which he regards as a clear truth value glut.) Fortunately, as you describe Prior's thought, it doesn't sound like Prior is suggesting this manuever. Unfortunately, what he is saying doesn't sound like it would help.
Reading back to this, I think my first reply was clear, but I'm not 100% sure, so let's try it this way:
Sentence A: Hitler won WWII.
Sentence B: "Sentence A is false."
Sentence C: "It is true that Sentence A is false."
Sentence D: "Sentence A is both true and false."
A is false, B is true, C is true and only D is a contradiction. Whatever B's truth-status was, C would necessarily have the same one. Same goes for the Liar Sentence and the version of it starting with "it is true that..."
When Sollie says "There are bats in the lavatory!", he makes an existence claim (which, arguably, the truth function hinges upon). So he says something like "it is true that there are bats in the lavatory" (which is to say, if serious, he suggests that his proposition points to some real state of affairs).
The liar sentence as stated does not really point to any real state of affairs: it's something like "it is true that this sentence is false". So regardless of Prior's solution (which seems fairly sound), what event/state of affairs does the sentence refer to? You can flip these pseudo-statements around (like the cards example), but if one grants that a assertive sentence makes an implicit existence claim about, like, objects in the world (for lack of a better term) the paradox sort of vanishes.
Russell's paradox should not be considered as part of the Liar; tho' some seem to suggest that. RP concerns sets (and the set of sets that are not members of themselves): my own view is that RP followed from Cantor's view of infinity (if not platonic accounts of mathematical objects) which are untenable.
The Paradox still can arise (say with lists, or data bases, perhaps), yet there are ways around it (including even Russell's solution via theory of types), mostly by limiting self-referentiality in ways. It's not as crucial as some in the philosophy biz make it out to be, at least for nominalists and those who question the platonic account of mathematical objects.
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