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In the SEP article for dialetheism, it is said that

A dialetheist, though, cannot simply accept that the Curry sentence is both true and false, because if it is true then ⊥ follows. Dialetheists need a different treatment of Curry.

I've always thought that this is true. However, in Priest's Logic of Paradox (LP):

  1. Semantic entailment Σ ⊨ A is defined as there is no v s.t. v(A) = f and for all B ∈ Σ, v(B) = t or p.
  2. The truth table for the conditional is:
    LP conditional truth table

So modus ponens fails. Particularly, if A is p and A → B is p, then B could still be f. This blocks the Curry proof (the proof needs modus ponens and contraction) of ⊥, but also allow for the curry sentence (and its truth) to be both true and false.

Additional questions:

  1. Perhaps all of the above is accurate, but we can still charge Priest with solving the Curry in a different manner than he solves the Liar, since modus ponens fails?
  2. I'm ignorant of other conditionals that could be used in LP. Perhaps under such conditionals, the curry sentence is not both true and false? What would one such conditional be? (SEP suggests a noncontractive conditional, but I'm not sure if that renders the curry sentence both true and false)
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  • From what I understand, if they accept it as a true contradiction they cannot block inferences to triviality by their usual means, and have to resort to artificial limiting devices that they criticize non-dialetheists for, see e.g. Burgis:"[Priest] mentions the paradox to justify the rejection of Contraction in his favored logic LP... One obvious problem with this move is that it does not seem to have anything to do with paraconsistency per se. Contraction plays no role in the inference from contradictions to triviality."
    – Conifold
    Commented Apr 12 at 0:23
  • @Conifold yeah that is correct, although they don't accept the Curry sentence as a true contradiction under certain conditions (see my answer). Commented Apr 12 at 14:44

1 Answer 1

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I found the following information in JC Beall (a dialetheist)'s Spandrels of Truth (chapter 2).

Beall notes that if we interpret the Curry sentence's conditional as LP's barebones conditional (the truth table I gave above), then since the conditional is defined as:
A → B ≡ ¬A ∨ B
The Curry sentence is just a disjunctive Liar:
C = ¬T(⌜C⌝) ∨ ⊥

And hence it is straightforward to say that it is both true and false. However, the dialetheist still needs a suitable conditional, one that is detachable (modus ponens is valid). Under his suitable conditional (and Priest's as well, although their semantics for LP are distinct?), modus ponens succeeds:
A, A → B ⊨ B
But both absorption (contraction) and pseudo modus ponens fails:
A → (A → B) ⊭ A → B
⊭ (A ∧ (A → B)) → B

Interpreting the Curry sentence with the suitable conditional, it turns out that the Curry sentence is just false. So Beall's reply is that either the Curry sentence is both true and false, or just false, depending on which conditional is at play.

Given a detachable conditional, it is not rational to accept that such Curry sentences are gluts (both true and false), since such sentences imply triviality. On my account, Curry sentences are false; I reject that they’re true.
[...] I noted my openness to an asymmetric treatment of such sentences [like the Liar] (e.g., treating some truth-ineliminable sentences as gluts, some classically [as just false]), but officially embraced the simple approach according to which all such sentences are gluts—transparently true with transparently true negations. This position remains, but only for the conditional-free fragment [as in free of the suitable conditional].

I don't know yet if it is possible for a conditional that is (1) detachable, (2) noncontractive, and (3) renders the Curry sentence both true and false. As far as I can tell, Beall doesn't conjecture either way. But for now this answer is sufficient.

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  • Burgis discusses Beall's version too:"Beall seems to accept that the distinction between rationally rejecting one element of a set of possible conclusions and thus rationally accepting the other and rationally rejecting one disjunct of a disjunction and thus rationally accepting the other does not make a difference... That is just Principle R... every bit as unacceptable to a dialetheist as Disjunctive Syllogism." After a discussion of rational dilemmas, he calls them "fatal to Beall’s attempt to use rejection talk." I am guessing "suitable conditional" does not exist.
    – Conifold
    Commented Apr 12 at 22:21
  • That sounds right, although I haven't read it closely. I was interested in this question because I found preist's principle of uniform solution reasonable, and so if he deals with the Liar and the Curry differently, then it would pose a problem to his view. Of course, according to his Inclosure Schema, the Liar and Curry are different type of paradoxes, but I think there is a better schema where both paradoxes fall under the same category. Commented Apr 14 at 6:12

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