I am reading Graham Priest's In Contradiction (P.12), where he is asserting that English satisfies the Tarski condition (a variant of semantic closure) and thus contains true contradiction.

He mentioned that a possible rebuttal involves appealing to truth gap to reject 'the reductio principle of condition 3'. I am not sure what this section is about.

I don't understand what the reductio principle of condition 3 is; reductio usually refers to assuming the contrary of the conclusion, then by proving a contradiction one would have proved that the conclusion is valid.

If Priest is trying to prove that English does satisfy cond. 3 by reductio, he would assume that ¬(α Λ ¬α) (ie. negation of the conclusion) and then try to prove a contradiction. And if the opponent is to reject this line of argument, as I believe this is what he is talking about, then the opponent would have to show that the proof of contradiction does not work by appealing to truth value gap. But I don't understand how a truth value gap would do this.

In particular, I don't at all understand what he is talking about regarding conditional. Gap in/gap out? Valueless spread to the whole? I just haven't the slightest clue what these mean. Likewise, the bit about 'reductio scheme is equivalent to the law of excluded middle' is equally baffling.

Could anyone help please?

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    Quick comment first, before trying an answer - I think "The reductio principle" of 3 is simply the inference rule it describes, construed as implying that there is a contradiction following from this expression of the liar sentence. If, however, a 3-valued "gappy" logic does not permit the inference of the contradiction from the conditional premise then being able to say "L if and only if ¬L" for some L does not itself mean that the language is inherently contradictory; L might simply just have no truth value. Does that help at all? – Paul Ross Jun 14 '20 at 13:23
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    A iff ¬A is the "typical" expression of a paradoxical statement, like e.g. Russell's one. The above rule amounts to assuming that every paradoxical statement implies a contradiction. You have to recall that Priest is a supporter of dialetheism and thus Excluded Middle, Priciple of Explosion and similar are not part of the "basic" logic. – Mauro ALLEGRANZA Jun 14 '20 at 13:31
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    His inference rule is: α ↔ ¬α entails α Λ ¬α. He then explains how this fails assuming truth value gaps. If α is a gap the conjunction isn't true, but both ¬α → α and α → ¬α, and hence the biconditional, may be true. This is because the implication may not be truth functional, i.e. it may not return a gap whenever any or all of its inputs are gaps (what he calls "gap in/gap out"). He further says that his rule is even equivalent to the law of excluded middle ("under very weak conditions"), i.e. to the absence of truth value gaps. – Conifold Jun 15 '20 at 7:02
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    Maybe useful: Dialetheism – Mauro ALLEGRANZA Jun 15 '20 at 8:20
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    The first part is right. But it doesn't block opponents, they are happy to reject LEM as well. At the end he promises to show that dialetheism still can not be avoided by some further argument. – Conifold Jun 17 '20 at 14:55

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