*Proofs and Models in Philosophical Logic* (Elements in Philosophy and Logic). Cambridge University Press, 2022.

This is a short book, in the Cambridge Elements series in Philosophical Logic.

This is a general introduction to recent work in proof theory and model theory of non-classical logics, with a focus on the application of non-classical logic to the semantic paradoxes and (to a lesser extent), the sorites paradox. After a short introduction motivating general notions of proof and of models, I introduce and motivate a simple natural deduction system, and present the structure of the liar paradoxical argument (concerning truth) and Curry’s paradox (concerning class membership). I introduce and motivate the notion of a structural rule, in both natural deduction and the sequent calculus and I compare and contrast the different approaches to substructural treatments of the paradox, contrasting the roles that *contraction*, *cut* and *identity* play in the derivations of the paradoxes.

In the next section, I introduce model theoretic treatments of the paradoxes, introducing supervaluations, and three-valued treatments of vagueness, and of the semantic paradoxes. I explain the fixed-point model construction that shows how to construct three-valued models for theories of truth, which can be used to then give models for different logics: K3 (with truth-value gaps), LP (with truth-value gluts) and ST (which supports all of classical logic, at the cost of invalidating the *cut* rule). I compare and contrast these approaches, and introduce the Routley—Meyer ternary relational semantics one way to model logics without the structural rules of contraction, or of weakening.

In the final section, I explore the relationship between proofs and models, explaining how the soundness and completeness theorems are proved for classical logic and for ST. I then discuss the issue of whether proofs or models play the primary *semantic* role, relating this question to the broader context of inferentialist and representationalist theories of meaning. This section closes with a discussion of the different resources that proof-first and model-first accounts of semantics have for answering the general question of how a language might be generally resistant to triviality due to paradox.

It is available from Cambridge University Press.

I’m *Greg Restall*, and this is my *personal website*. ¶ I am the Shelby Cullom Davis Professor of Philosophy at the University of St Andrews, and the Director of the Arché Philosophical Research Centre for Logic, Language, Metaphysics and Epistemology ¶ I like thinking about – and helping other people think about – logic and philosophy and the many different ways they can inform each other.

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