Jc Beall

In his ‘On t and u and what they can do’, Greg Restall presents an apparent problem for a handful of well-known non-classical solutions to paradoxes like the liar. In this article, we argue that there is a problem only if classical logic – or classical-enough logic – is presupposed. 1. Background Many have thought that invoking non-classical logic – in particular, a paracomplete or paraconsistent logic – is the correct response to the liar and related paradoxes. At the most basic level, the target non-classical idea is that some expressions, like ‘all and only the true propositions’, do not behave as we would expect from classical logic. Non-classical theorists argue that the class of all and only the truths is either incomplete or inconsistent: when you truly speak of all and only truths, you're either leaving some truths out, or you're letting some untruths in. Truth, in a slogan, is either gappy or glutty. Non-classicality is not a glib or easy-way-out response to the paradoxes. Innocuous-seeming notions can turn out to be philosophically substantial. Moreover, apparently correct forms of reasoning can turn out to be incorrect. To take a example, a glut theorist must hold that the following argument form is not in general valid. Either p is true or q is true. But p is untrue. So q is true. This argument form may strike our ears as acceptable. But if p is a truth value glut, then the inference fails to preserve truth. According to glut theorists, the inference breaks down in inconsistent contexts: if the subject matter involves gluts, then the inference is to be rejected. And the liar paradox, according to such theorists, shows that truth is exactly the kind of subject matter that yields inconsistency. In the …

I believe that, for reasons elaborated elsewhere (Beall, 2009; Priest, 2006a, 2006b), the logic LP (Asenjo, 1966; Asenjo & Tamburino, 1975; Priest, 1979) is roughly right as far as logic goes.1 But logic cannot go everywhere; we need to provide nonlogical axioms to specify our (axiomatic) theories. This is uncontroversial, but it has also been the source of discomfort for LP-based theorists, particularly with respect to true mathematical theories which we take to be consistent. My example, throughout, is arithmetic; but the more general case is also considered.