[The following are excerpts from the portion of the rough draft of my dissertation where I discuss the Curry Paradox. Note that by "Liar-based dialetheists," I just mean any dialethest who takes Liar sentences to be both true and false. The numbering system is retained from the original.]
...one could argue that it’s impossible to beg a question against a trivialist, because the trivialist accepts everything, necessarily including the wrongness of trivialism. This would, however, be entirely too hasty. The epistemic reasons that begging the question is barred from rational debate don’t cease to apply in the strange case of our opponents cheerfully granting whatever is throw at them. Priest himself has some very sensible things to say about precisely this subject in Doubt Truth To Be A Liar, and when looking at how he fares on these grounds, it will be sufficient to compare his actual performance with the standards he himself lays out in that book. What he asks us to do in that book is to imagine, when arguing against the trivialist, that we are not trying to convince the trivialist (who, after all, is already convinced of everything) but that we are trying to convince a neutral arbitrator who has cautiously not yet made up her mind about the issue of trivialism, and who will accept or reject arguments and evidence brought up one way or the other, according to normal rules applicable to any inquiry about anything. Convincing the trivialist that trivialism is wrong is a trivial matter. Convincing the imaginary arbitrator to rule in our favor is not. It is the latter, harder standard to which we should set ourselves.
"Convincing the arbitrator is…not as easy as it might first appear… For example, an obvious argument to use is that the trivialist’s position is inconsistent, and so ought not to be believed. (Naturally, a dialetheist is not going to make this objection; but a classical logician might.) It is clear that this argument fails, though, since it begs the question. The trivialist affirms…all contradictions. Simply to claim that these cannot be true is to take for granted part of what is at issue. "
Things are not so hopeless as they may appear, however. In fact, there is excellent reason to be confident that a nontrivialist with normal beliefs about the world could make a lot of good, convincing anti-trivialist arguments of the kind that the hypothetical neutral arbitrator would allow.
"The trivialist believes many strange things. They believe, for example, that you are a scrambled egg. This is objectionable since there is just no evidence to justify it. The rational person should not believe anything for which there is no reason. The arbitrator must agree. And if the trivialist argues that there is a reason, namely that it follows from trivialism, then it is they who now beg the question. The trivialist will also claim that there is independent evidence. (They will claim everything.) But claiming does not of course make it so. The evidence needs to be produced for the arbitrator…."
This should be enough to get across Priest’s general idea about how to argue against trivialism, and it is an idea that seems absolutely correct to me. Moreover, the anti-trivialist argument just quoted seems to be a good one, and many similarly good arguments can be made. Before concluding that we have a good, principled reason to reject trivialism, however, we must not just consider the case against trivialism but the case for it as well.
Just as the best argument for dialetheism is the Liar Paradox, the best argument for trivialism is the Curry Paradox. Liars say of themselves that they are false, or untrue, or (just) untrue, or something else of the same general kind. Curry sentences, by contrast, say of themselves that if they are true, some α is true as well, where α can be any arbitrarily chosen sentence that you like. For example, take sentence (45).
(45) If the sentence marked as (45) is true, then Graham Priest is a scrambled egg.
Symbolize (45) as α. The relevant instance of the biconditional truth schema for (45) is “‘If the sentence marked as (45) is true, then Graham Priest is a scrambled egg’ iff if the sentence marked as (45) is true, then Graham Priest is a scrambled egg.” If we name the sentence α, and the claim that Graham Priest is a scrambled egg β, then we can symbolize this as:
1. Tr<α> ↔ (Tr<α> →β)
Plugging in the definition of ↔, we get:
2. [Tr<α> → (Tr<α> → β)] & [(Tr<α> → β) → Tr<α>]
By Conjunction-Elimination, we get:
3. Tr<α> → (Tr<α> → β)
Now, from here, an easy conditional proof, using only Modus Ponens (MP), will get us to Tr<α> → β.
4. Tr<α> A for CP
5. Tr<α> → β 3, 4, MP
6. β 4, 5, MP
7. Tr<α> → β 4-6, CP
Now, applying Conjunction-Elimination once again to 2, we get:
8. (Tr<α> → β) → Tr<α>
From 7, 8 and Modus Ponens, we get:
Now, from 7, 9 and Modus Ponens, we get:
Graham Priest is indeed a scrambled egg. In fact, since Curry sentences can be constructed with any arbitrary consequent that you care to pick, we can use this argument to prove all of the trivialist’s claims without exception, or just cut to the chase by having the consequent be “trivialism is true.”
This seems to be as good an argument for trivialism as the argument from the Liar Paradox is for dialetheism. In the latter case, in Chapter Three, we considered Gilbert Harman’s attempt to block the derivation of inconsistency by a rejection of the biconditional truth schema and rejected it as question-begging. Priest’s hypothetical neutral arbitrator would surely rule out a parallel rejection of the relevant schema instance in order to block the derivation of triviality for precisely the same reason. Similarly with any otherwise unmotivated tinkering with the formal machinery we used to derive triviality from Curry.
[....In the skipped section, I basically note that any consistent solution to the Liar worth its salt can be re-purposed as a non-trivial solution to Curry. Both Liar and Curry sentences, after all, purport to discuss their own truth-values. The two paradoxes are of a type.]
By contrast, the Liar-based dialetheist is in a tough spot here. They are clearly committed to the claim that self-referential truth talk is meaningful, and, in fact, truth-evaluable. Curry sentences must, then, be meaningful declarative sentences, as would the relevant instances of the biconditional truth schema, and no one who admits this can have a good, principled, nonadhoc and non-question-begging reason to avoid granting that these schema instances give the correct truth conditions for Curry sentences. Once that point has been granted, all we need is Modus Ponens and conditionalization (the rule that we can use conditional proofs to prove the truth of conditionals). Rejecting Modus Ponens is extremely implausible. In fact, no one is more adamant about this point than Priest, who says that,
"Any conditional worth its salt, →, should satisfy the modus ponens principle…This is, indeed, analytically part of what implication is."
Now, whatever one thinks of the analytic/synthetic distinction, or the complicated issue of the relationship between the meaning of logical constants and the rules governing their behavior that we looked at in Chapter One, Priest’s basic point in this passage is a good one, which we can re-phrase in weaker terms that are neutral on those other debates by saying that if someone denies that α, α→β ┠ β is a truth-preserving inference, this counts as pretty good evidence that they don’t mean the same thing by → that the rest of us do. We can extend (this charitably weakened version of) Priest’s point by saying that if someone denied conditionalization, that would be equally good evidence that they didn’t mean the same thing by → as the rest of us. The two rules are conceptually parallel to each other, a fact recognized by introductory logic texts that introduce conditionalization as “→-introduction” and Modus Ponens as “→-elimination.” In one case, the fact that α implies β is enough to guarantee that it entails it, and in the other case, the fact that α entails β is enough to guarantee that it implies it. This mutual relationship between entailment and implication seems obvious enough that no one thinks much of anything of talking about the “rule form” and the “conditional form” of various logical rules, and it’s not at all clear to me why a denial of one half of this relationship should count as better evidence than the denial of the other for the claim that the denier doesn’t mean what the rest of us mean by →.
Denying either of these rules in order to escape the inference from Curry sentences seems like a desperate, adhoc maneuver. This is, however, precisely the route that Priest takes. In In Contradiction, he rejects ‘absorption’ (also called ‘contraction’)—the inference from [α → (α → β)] to (α → β)—on the grounds that “entailment must not fall foul of Curry paradoxes.” Absorption, however, just is conditionalization plus Modus Ponens, so, given Priest’s endorsement of Modus Ponens, his rejection of absorption boils down to a rejection of conditionalization.
He just barely gets to keep Modus Ponens itself (in its rule form), but his anti-Curry reasoning has the further unattractive consequence that he must reject the conditional form of Modus Ponens--[α ∧ (α → β)] → β], sometimes called ‘assertion’--which, like absorption, is the product of accepting the validity of both Modus Ponens and conditionalization. Priest lays out a view about implication that avoid all of this, but the important point for the present discussion is that he justifies this theory by the need to bring about these counter-intuitive consequences, and that he justifies that with the need to get around Curry.
"It must be admitted that the assertion principle looks acceptable enough, but…a criterion of adequacy for a solution to the problem of formulating an account of implication is that it must not validate assertion (or absorption)."
Priest, in other words, is tinkering with otherwise intuitively plausible rules of inference in order to avoid being committed to triviality. If this is the best he can do when the trivialist brings her argument from the Curry Paradox to the neutral arbitrator, then the arbitrator will have no choice but to rule out this objection on the grounds of flagrant question-begging and, ultimately, rule in favor of the trivialist against the nontrivialist dialetheist.