Reference : Mechanizing Principia Logico-Metaphysica in Functional Type Theory
Scientific journals : Article
Engineering, computing & technology : Computer science
Arts & humanities : Philosophy & ethics
Computational Sciences
Mechanizing Principia Logico-Metaphysica in Functional Type Theory
Kirchner, Daniel [Freie Universität Berlin > Mathematics]
Benzmüller, Christoph mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Computer Science and Communications Research Unit (CSC)]
Zalta, Edward N. [Stanford University > Center for the Study of Language and Information (CSLI)]
Review of Symbolic Logic
Cambridge University Press
[en] own ; Higher Order Logic ; Higher Order Modal Logic Universal Reasoning ; Computational Metaphysics
[en] Principia Logico-Metaphysica contains a foundational logical theory for metaphysics, mathematics, and the sciences. It includes a canonical development of Abstract Object Theory [AOT], a metaphysical theory (inspired by ideas of Ernst Mally, formalized by Zalta) that distinguishes between ordinary and abstract objects.

This article reports on recent work in which AOT has been successfully represented and partly automated in the proof assistant system Isabelle/HOL. Initial experiments within this framework reveal a crucial but overlooked fact: a deeply-rooted and known paradox is reintroduced in AOT when the logic of complex terms is simply adjoined to AOT’s specially formulated comprehension principle for relations. This result constitutes a new and important paradox, given how much expressive and analytic power is contributed by having the two kinds of complex terms in the system. Its discovery is the highlight of our joint project and provides strong evidence for a new kind of scientific practice in philosophy, namely, computational metaphysics.

Our results were made technically possible by a suitable adaptation of Benzmüller’s metalogical approach to universal reasoning by semantically embedding theories in classical higher-order logic. This approach enables one to reuse state-of-the-art higher-order proof assistants, such as Isabelle/HOL, for mechanizing and experimentally exploring challenging logics and theories such as AOT. Our results also provide a fresh perspective on the question of whether relational type theory or functional type theory better serves as a foundation for logic and metaphysics.

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