This may be catastrophically simple-minded and lazy to boot, but I'm just not sure that dialetheism (the view that there are true contradictions) should be thought of as leading to logical revision. [Before reading any more, I realize that I'm way out of date on all of this, and that there is an extant literature surrounding "classical recapture" that is relevant and that I haven't addressed in what follows.]
The argument that dialetheism entails revising classical inference simply notes that in normal classical logic a contradiction entails everything (ex falso quodlibet, P, ~P |- Q). This is actually a really hard principle to give up because any logic that entails disjunctive syllogism entails ex falso quodlibet [Assume you've got a contradiction P and ~P. From P by disjunction introduction you've got (P v Q) and ~P. But then by disjunctive syllogism you can conclude Q. So from any P and ~P, any Q follows.] So if I want to say with Graham Priest that paradoxical sentences such as "I am not true" are both true and false, and I don't want to be committed to the truth of everything, I better adopt a paraconsistent logic that does not license ex falso quodlibet.
But couldn't a true contradiction work as a logical/pragmatic stop-sign? The thought would be that if you derive a contradiction within something assumed for further discharge such as negation introduction or disjunction elimination you go about your business and continue to use classical logic, but if you derive it outside of these contexts it tells you to stop using the logic.
This works really well for true contradictions that are generated outside of paradoxes involving self-reference (I don't know any other philosophers besides me, in my essay in The Law of Non-Contradiction, who have argued for such). Say that our only notion of ethical truth is truth-relative-to-a-moral-tradition such as in MacIntyre's historicist days (and that the T-Schema is valid for each relativized truth predicate). These traditions will each be independently plausible, but inconsistent with one another. Given the T schema you can generate contradictions from the traditions' joint inconsistency.
And people are pretty good at bouncing back and forth between different moral theories, combining them when it's safe. But sometimes we mess up and apply both in situations where they contradict one another (maybe think of Kantian and Millian ethics on whether a given lie is morally permissible). If the best we can do at the limit of reflective equilibrium is getting a set of good mutually inconsistent (but each internally consistent) theories (something I think holds for ethical, philosophical, and aesthetic theories generally), then a true contradiction is just a place where two of the theories at the limit of reflective equilibrium disagree.
But at least that kind of true contradiction should not lead you to abandon classical logic. In subvaluational "glut semantics" a sentence is supertrue in a model if there is at least one valuation in that model that makes it true. True contradictions happen when a sentence is made true on one valuation and false on another.
On the account of entailment where you say gamma entails alpha if and only if alpha is subtrue when all of gamma are you get a horrifically messy non-classical entailment relation (in Williamson's book on vagueness this is called "wide entailment," albeit there in respect to gap supervaluational semantics for vagueness, where a sentence is true in a model if it is true at *all* relevant interpretations). But such semantics also give you the option of just keeping classical entailment, gamma entails alpha if and only if there is no valuation (remember that models are sets of such valuations) that makes all of gamma true and alpha false. Williamson argues that this form of narrow entailment should not the relevant one for vagueness for the supervaluationist, but I remember finding his arguments here less than persuasive. One would need to examine them in this context.
But the model is compelling. Each one of the interpretations making up the glut semantics model is analogous to a correct theory in the limit of investigation after reflective equilibrium is considered. And narrow entailment can be understood as legitimizing classical logic as long as we in some sense stay within one of the correct theories in the limit of investigation after reflective equilibrium is considered. Since we're in the limit of investigation all a true contradiction tells you is that your bringing together of two of the theories into the same context has run into trouble. It just tells you not to do that in this context.
The big if is whether such a view can be argued to apply to self referential paradoxes. I'm not at all sure that it can. Consider the paradox of the stone. What would the analog to the two interpretations be? I'm going to look at Priest's brilliant 1994 Mind paper "The Structure of the Paradoxes of Self-Reference" (still brilliant even if one agrees with the conclusion of Badici's "The Liar Paradox and the Inclosure Schema," Australasian, 86, 2008) really carefully over the next two weeks and see if there is any hope at all for the idea true contradictions generated from paradoxes can be argued to be consistent with classical logic.
I'm not that optimistic, because you'd have to say something like a new subvaluational interpretation is generated every time Priest's Transcendence is applied. To the extent that it might work, one could probably piggy back on extant attempted solutions to the paradoxes that involve indexing in various ways. But that literature is a mess and one of the primary benefits of dialetheism is that you were supposed to be able to circumvent it (albeit at the cost of a messy logic).
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