Absolute Contradiction, dialetheism, and Revenge

Is there a notion of contradiction—let us call it, for dramatic effect, “absolute”— making all contradictions, so understood, unacceptable also for dialetheists? It is argued in this paper that there is, and that spelling it out brings some theoretical benefits. First it gives us a foothold on undisputed ground in the methodologically difficult debate on dialetheism. Second, we can use it to express, without begging questions, the disagreement between dialetheists and their rivals on the nature of truth. Third, dialetheism has an operator allowing it, against the opinion of many critics, to rule things out and manifest disagreement: for unlike other proposed exclusion-expressing-devices (for instance, the entailment of triviality), the operator used to formulate the notion of absolute contradiction appears to be immune both from crippling expressive limitations and from revenge paradoxes—pending a rigorous nontriviality proof for a formal dialetheic theory including it. Nothing is, and nothing could be, literally both true and false. [. . . ] That may seem dogmatic. And it is: I am affirming the very thesis that [the dialetheists] have called into question and—contrary to the rules of debate—I decline to defend it. Further, I concede that it is indefensible against their challenge. They have called so much into question that I have no foothold on undisputed ground. So much the worse for the demand that philosophers always must be ready to defend their theses under the rules of debate. – David Lewis, Logic for Equivocators §1. Debating dialetheism. In his 1969 criticism of Hegel and Marx’s “dialectical logic”, Popper observed that arguing against someone who accepts contradictions is methodologically puzzling. Let T = {A1, . . . , An} be a theory or belief set. One criticizes the T-theorist, or T-believer, by inferring from premises in {A1, . . . , An}, via rules of inference he accepts, some B he rejects. A standard reductio move takes B = ∼Ai , 1 ≤ i ≤ n. But the Marxist, who accepts “dialectical contradictions” in reality, can be unyielding: he can maintain his T without releasing Ai , and take ∼Ai on board too. One who finds contradictions acceptable may not revise beliefs on pain of contradiction. Popper saw in this the death of criticism, freedom and democracy. Long before the cold war, Aristotle had stated in his Metaphysics (1005b 25-6) that, when someone claims “For some A, A and not-A are both true” (the villain was, in that Received: October 20, 2012. c © Association for Symbolic Logic, 2014 1 doi:10.1017/S175502031400001X