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MECHANIZING PRINCIPIA LOGICO-METAPHYSICA IN FUNCTIONAL TYPE-THEORY

Published online by Cambridge University Press:  12 July 2019

DANIEL KIRCHNER ,
CHRISTOPH BENZMÜLLER  and
EDWARD N. ZALTA
DANIEL KIRCHNER*
Affiliation:
Fachbereich Mathematik und Informatik, Freie Universität Berlin
CHRISTOPH BENZMÜLLER*
Affiliation:
Fachbereich Mathematik und Informatik, Freie Universität Berlin and Computer Science and Communications, University of Luxembourg
EDWARD N. ZALTA*
Affiliation:
Center for the Study of Language and Information, Stanford University
*
*FACHBEREICH MATHEMATIK UND INFORMATIK FREIE UNIVERSITÄT BERLIN ARNIMALLEE 14, 14195 BERLIN, GERMANY E-mail: daniel@ekpyron.org
†FACHBEREICH MATHEMATIK UND INFORMATIK FREIE UNIVERSITÄT BERLIN ARNIMALLEE 14, 14195 BERLIN, GERMANY E-mail: c.benzmueller@fu-berlin.de and COMPUTER SCIENCE AND COMMUNICATIONS UNIVERSITY OF LUXEMBOURG 2, AVENUE DE L’UNIVERSITÉ L-4365 ESCH-SUR-ALZETTE, LUXEMBOURG E-mail: christoph.benzmueller@uni.lu
‡CENTER FOR THE STUDY OF LANGUAGE AND INFORMATION STANFORD UNIVERSITY CORDURA HALL, 210 PANAMA STREET STANFORD, CA 94305-4115, USA E-mail: zalta@stanford.edu
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Abstract

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.

Keywords

03Axx03B1503B4503B6003B8068T1568T2768T30computational metaphysicsAbstract Object Theoryshallow semantical embeddinghigher-order logictheorem provinguniversal reasoning
Type
Research Article
Information
The Review of Symbolic Logic , Volume 13 , Issue 1 , March 2020 , pp. 206 - 218
Copyright
Copyright © Association for Symbolic Logic 2019 

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References

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