Modal Homotopy Type Theory

Abstract

In[KF1] 1914, in an essay entitled ‘Logic as the Essence of Philosophy’, Bertrand Russell promised to revolutionize philosophy by introducing there the ‘new logic’ of Frege and Peano: “The old logic put thought in fetters, while the new logic gives it wings.” A century later, this book proposes a comparable revolution with a newly emerging logic, modal homotopy type theory. Russell’s prediction turned out to be accurate. Frege’s first-order logic, along with its extension to modal logic, is to be found throughout anglophone analytic philosophy. This book provides a considerable array of evidence for the claim that philosophers working in metaphysics, as well as those treating language, logic or mathematics, would be much better served with the new ‘new logic’. It offers an introduction to this new logic, thoroughly motivated by intuitive explanations of the need for all of its component parts—the discipline of a type theory, the flexibility of type dependency, the more refined homotopic notion of identity and a powerful range of modalities. Innovative applications of the calculus are given, including analysis of the distinction between objects and events, an intrinsic treatment of structure and a conception of modality both as a form of general variation and as allowing constructions in modern geometry. In this way, we see how varied are the applications of this powerful new language—modal homotopy type theory.