Mechanizing Principia Logico-Metaphysica in Functional Type Theory

Principia Logico-Metaphysica proposes a foundational logical theory for metaphysics, mathematics, and the sciences. It contains a canonical development of Abstract Object Theory [AOT], a metaphysical theory (inspired by ideas of Ernst Mally, formalized by Zalta) that differentiates 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 the fruitful reuse of state-of-the-art higherorder 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. §1. Abstract Summary. Principia Logico-Metaphysica (PLM) [13] aims at a foundational logical theory for metaphysics, mathematics and the sciences. It contains a canonical presentation of Abstract Object Theory (AOT) [14, 15], which distinguishes between abstract and ordinary objects, in the tradition of the work of Mally [6]. The theory, outlined in §2, systematizes two fundamental kinds of predication: classical exemplification for ordinary and abstract objects, and encoding for abstract objects. The latter is a new kind of predication that provides AOT with expressive power beyond that of quantified second-order modal logic, and this enables elegant formalizations of various metaphysical objects, including the objects presupposed by mathematics and the sciences. More generally, the system offers a universal logical theory that is capable of accurately representing the contents of human thought. Independently, the use of shallow semantical embeddings (SSEs) of complex logical systems in classical higher-order logic (HOL) has shown great potential as a metalogical approach towards universal logical reasoning [1]. The SSE approach aims to unify logical reasoning by using HOL as a universal metalogic. Only the distinctive primitives of a target logic are represented in the metalogic