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?