I’m on a roll with the writing: I’ve managed to get another paper converted from slides-from-a-talk to real, live, words on a page and actual proofs. The paper I’ve worked on over the last couple of days is from my presentation from WCP4 on non-classical theories like non-classical theories of numbers, classes, or truth. I’m very happy with this one, as it’s an application of my recent work – on proof theory – to a topic I’ve been writing about for some time, non-classical theories like naïve theories of classes or of truth. Here’s the abstract.
In this paper I urge friends of truth-value gaps and truth-value gluts – proponents of paracomplete and paraconsistent logics – to consider theories not merely as sets of sentences, but as pairs of sets of sentences, or what I call ‘bitheories,’ which keep track not only of what holds according to the theory, but also what fails to hold according to the theory. I explain the connection between bitheories, sequents, and the speech acts of assertion and denial. I illustrate the usefulness of bitheories by showing how they make available a technique for characterising different theories while abstracting away from logical vocabulary such as connectives or quantifiers, thereby making theoretical commitments independent of the choice of this or that particular non-classical logic.
Examples discussed include theories of numbers, classes and truth. In the latter two cases, the bitheoretical perspective brings to light some heretofore unconsidered puzzles for friends of naïve theories of classes and truth.
This paper is a turnabout for me. While I’ve been making a slow turn for some time, from being a friend of naïve set theory in paraconsistent logics to a friend of truth-value gaps over gluts to a friendly critic of non-classical approaches to things like Curry’s paradox. In this paper, I present what I take to be much more problematic paradoxes around Frege’s Law (V), for which connectives like negation and the conditional do not feature at all, so questions of the design of the non-classical logic of negation or of the conditional are beside the point when it comes to the paradoxes.