Will the Real Higher-Order Logic Please Stand Up?

Abstract: In this work-in-progress talk, I will attempt to clarify some of the issues surrounding second- and higher-order logics. How should we understand the relationship between the proof-theoretic inferentialist presentations of such logics and their model-theoretic truth conditional semantics? I will argue that in an important sense, the so-called full semantics for second- and higher-order logics is too strong, and that a wider class of models is not only necessary, but is mathematically and philosophically desirable. The good news: such a class of models already exists and is well understood. (And no, it’s just not the Henkin semantics.) The slides for the talk can be downloaded here. The talk is a presentation in the Arché Metaphysics and Logic Seminar, at the University of St Andrews.

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