On Inconsistent Arithmetics: A Reply to Denyer

In "Is Arithmetic Consistent?" (1994a-hereafter, IAC) I drew attention to the fact that there are inconsistent but non-trivial theories that contain all the sentences true in the standard model of arithmetic, N. The theories are not, of course, classical theories, but paraconsistent ones. I also argued that it is not as obvious that N is the correct arithmetic as one might suppose, and that there are reasons for taking one of the inconsistent arithmetics, M, with least inconsistent number, m, to be the correct one. Two reasons were given. The first was a direct one; the second an indirect one, to the effect that M avoids most of the limitative theorems of classical metatheory. In "Priest's Paraconsistent Arithmetic" (1995hereafter PPA), Nicholas Denyer gives a critique of the paper. The first four sections attack the indirect argument, the fifth the direct argument, and the sixth, and final, section is an ad hominem attack. The paper is a mixture of insightful criticism, over-swift argument and misreading. The purpose of this note is to point out which parts are which.