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.