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See detailMechanizing Principia Logico-Metaphysica in Functional Type Theory
Kirchner, Daniel; Benzmüller, Christoph UL; Zalta, Edward N.

in Review of Symbolic Logic (2019)

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 ... [more ▼]

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. [less ▲]

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See detailCompleteness of Aqvist's systems E and F
Parent, Xavier UL

in Review of Symbolic Logic (2015), 8(1), 164-177

This paper tackles an open problem posed by Åqvist. It is the problem of whether his dyadic deontic systems E and F are complete with respect to their intended Hanssonian preference- based semantics. It ... [more ▼]

This paper tackles an open problem posed by Åqvist. It is the problem of whether his dyadic deontic systems E and F are complete with respect to their intended Hanssonian preference- based semantics. It is known that there are two different ways of interpreting what it means for a world to be best or top-ranked among alternatives. This can be understood as saying that it is optimal among them, or maximal among them. First, it is established that, under either the maximality rule or the optimality rule, E is sound and complete with respect to the class of all preference models, the class of those in which the betterness relation is reflexive, and the class of those in which it is total. Next, an analogous result is shown to hold for F. That is, it is established that, under either rule, F is sound and complete with respect to the class of preference models in which the betterness relation is limited, the class of those in which it is limited and reflexive, and the class of those in which it is limited and total. [less ▲]

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