Is the Liar Sentence Both True and False?

There are many reasons why one might be tempted to reject certain instances of the law of excluded middle. And it is initially natural to take ‘reject’ to mean ‘deny’, that is, ‘assert the negation of’. But if we assert the negation of a disjunction, we certainly ought to assert the negation of each disjunct (since the disjunction is weaker1 than the disjuncts). So asserting ¬(A ∨ ¬A) should lead us to assert both ¬A and ¬¬A. But to assert both a sentence (¬A) and its negation is, in at least one sense of the phrase, to assert a contradiction. Accepting contradictions would be intolerable if contradictions implied everything: we would be logically committed to every imaginable absurdity. But there are "paraconsistent logics" where contradictions in the above sense (pairs consisting of B and ¬B, for some B) don’t entail everything. It is not especially controversial that paraconsistent logics might be useful for certain purposes, e.g. analyzing certain notions of "relevant implication" and/or "what a possibly inconsistent theory should be taken as directly committed to". But I’m interested in the issue of a particular kind of use, the one motivated above: a use of paraconsistent logic to license the simultaneous literal belief in both B and ¬B, in full knowledge that we believe both, and where such knowledge gives no pressure to revise one of the beliefs. In short, where the beliefs, though "contradicting" each other, are not in any serious sense in conflict. I will adapt Graham Priest’s term ‘dialetheism’ for the doctrine that we should fully accept certain sentences while also accepting their negations. This is not quite Priest’s usage, as we’ll see. Nonetheless, Priest is an advocate of dialetheism in this sense; in fact, its most prominent advocate. The argument with which I began shows that if we want to disbelieve instances of excluded middle (in the sense of, believe their negations) then we should be dialetheists (not merely that we should accept paraconsistent logics for some purposes). And as Priest has often urged (e.g. [12]), the most familiar arguments against the coherence of dialetheism are seriously faulty, a result of a refusal to take the doctrine seriously.