Late last month, my little manuscript on Proofs and Models in Philosophical Logic was published by Cambridge University Press. This, like all entries in the new Cambridge Elements series, is a tiny little manscript, with an aim to give students, and researchers in allied fields, a quick, accessible introduction to a research topic and current methods. My mansucript is a breezy 84 pages, and it tries to introduce the role of proofs and models in contemporary philosophical logic, with a focus on work on the paradoxes.
You can download the book for free on the Cambridge University Press website until April 13. After that, you’ll need to either buy a hardcopy or an electronic copy, or access it with some institutional subscription.
Here are some thoughts about Proofs and Models, and why I put the book together in the way I did. Hopefully, it helps put the work in context.
Feature 1: The guiding theme in the book is the even-handed treatment of proofs and models. (Hence, the title.) It’s written from the perspective that logical notions like validity, consistency, etc., are best viewed from both proof-first and model-first perspectives, and that both angles can provide their own unique insight, not only into technical results, but into how logical concepts connect with how we represent things, and how we use logical vocabulary.
Feature 2 is a corollary of Feature 1: With the relationship between proof theory and model theory in focus, I’m able to draw connections to recent work in philosophy on the interface between inferentialism and representationalism (cf. Brandom), and the related difference between semantic realism and semantic anti-realism or expressivism (cf. Dummett in metaphysics, or Gibbard in metaethics), and current discussions about “easy ontology” or thin/thick realism (cf. Amie Thomason). You can think of this little book as a guide to understanding recent work in logic in ways that don’t simply take one side or other in those debates at the outset.
Feature 3 is the book’s centre of attention: Recent work on the paradoxes. There’s been a lot of ink spilled on the paradoxes of self-reference (like the liar paradox) & paradoxes of vagueness (like the sorites), so there’s a lot of ground to cover. I’ve tried to do this in a gentle way, with an aim to elucidate the issues at the heart of these paradoxes. Since the Liar Paradox and Curry’s paradox are so similar, the lesson shouldn’t just be some revisionary view of negation: after all, Curry’s paradox is negation-free. I also give a general treatment of the Kripke-Brady-Woodruff-Gilmore fixed point construction that doesn’t fixate on how the model at the end is to be interpreted, since the one and the same model works for K3 (truth-value gaps), LP (gluts) or ST (classical but without Cut). The aim, as ever, is to introduce these tools and techniques on their own merits, with the benefit of all the hindsight we have gained in the early 2020s.