Inconsistent Thoughts

In his Relevant Logic: a Philosophical Interpretation, Mares discusses several famous examples of arguments for explosion, the principle which says that a contradiction entails everything. What I am about to discuss is, I think, a minor part of Mares’ overall strategy in arguing against explosion, but it struck me as funny so I wanted to say something about it.

C.I. Lewis defended explosion. He also criticized the material conditional. In a critical article, Lewis frames a dialogue between himself and a fictional interlocutor. Lewis remarks to his interlocutor at one point that “If you grant the paradoxer his assumption…” then his conclusions materially follow. By taking seriously the possible truth of certain assumptions he argues that material implication runs into greatly counter-intuitive results. Mares tries to turn the tactic against Lewis:

Suppose that we ‘grant some paradoxer his assumption’ that is true. We then have to take seriously what would happen in a context in which a contradiction is true. (p.9)

Once we have swallowed this much, we can see that Lewis defense of explosion was wrong. If we take seriously the possibility of true contradictions, then there must be situations in which both of and are true. From this alone it would not follow that other arbitrary propositions are true, so it is not the case that a contradiction entails everything. He repeats the strategy later in the book: “in an implication we have to take the antecedent seriously.” (p.84) I think it is fair to say that Mares here has in mind someone like Priest who advocates dialetheism, and Mares is claiming that we have to take the dialetheist seriously just as Lewis took the advocate of material implication seriously.

This strikes me as a really terrible way to argue against the validity of explosion. After all, what prevents generalizing this argument to absurdity? Take any formal schema at all, for examples consider . Formulae of this form are almost universally taken to be logical truths, but imagine someone who asserts that such propositions can be false. According to Mares we have to take this seriously, so we have to imagine possible situations in which some proposition does not imply itself. Since other propositions would be true in this situation, does not follow vacuously from anything, and so it is not a logical truth.

In other words, we can’t just grant anyone whatever assumptions they want in Mares sense. This grants too much. Mares seems be saying that the only way to be fair to our interlocutors in disagreements over logic is to build into our logical framework possibilities in which our interlocutors assumptions hold. But taken generally, this would force us to accept that any proposition is non-trivially possibly true and non-trivially possible false, so nothing follows from anything.