Abstract: In this informal talk, I will revisit some longstanding issues in philosophical logic in the light of some contemporary developments.
The longstanding issues? (1) Michael Dummett’s challenge in The Logical Basis of Metaphysics to the effect that to get anywhere in fundamental issues of metaphysics we would do well to attend to the fundamental commitments of our theory of meaning—and that those concerns lead to the conclusion that we can find common ground in intuitionistic logic, not classical logic. (2) The issue of pluralism (or monism) about logical consequence. Contemporary work in logic is filled with a range of different (and seemingly opposed) accounts of what follows from what. Many different kinds of logical pluralism have arisen to attempt to make sense of the diversity of logical analyses, and just as many defences of logical monism have been offered.
The contemporary developments? The rise of dependent type theory in computer science and the consequent rise of proof assistants in the formalisation of mathematics. Different proof assistants (Agda, Idris, Lean, Isabelle, Rocq/Coq) make different choices in the formal representation of mathematical reasoning, but the predominant choice of these proof assistants is to represent proofs constructively, in what amounts to intuitionistic logic and not classical logic. Proof assistants are steadily gaining ground in the mathematical community—mathematicians use these tools as conversation partners in the development of mathematical proofs, and many of these conversation partners have intuitionistic scruples.
I will reexamine Dummett’s appeal to revisit fundamental commitments in our theory of meaning, and the challenge of logical pluralism, given the example of the use of proof assistants in mathematical reasoning. We will see that paying attention to different aspects of our reasoning practices can clarify what is at stake in debates over the use of this or that logic, or about the propriety of some semantic principle. In the end, I hope to show that—as Dummett argued—it is worthwhile to spend some time attending to the concepts we use in our theorising, just as the astronomer must take care of her telescopes if she wants to see clearly and see far, and understand the power and limits of her tools. However, I will also show that taking Dummett’s advice does not mean that we must accept his restrictive conclusion about the impropriety of classical reasoning. Classical reasoning can be vindicated, but at some cost.
This talk is a presentation at the Arché Metaphysics and Logic Seminar.
You can download the slides here.
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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