Automating Emendations of the Ontological Argument in Intensional Higher-Order Modal Logic

Abstract

Abstract. A shallow semantic embedding of an intensional higher-order modal logic (IHOML) in Isabelle/HOL is presented. IHOML draws on Montague/Gallin intensional logics and has been introduced by Melvin Fitting in his textbook Types, Tableaus and Gödel's God in order to discuss his emendation of Gödel's ontological argument for the existence of God. Utilizing IHOML, the most interesting parts of Fitting's textbook are formalized, automated and verified in the Isabelle/HOL proof assistant. A particular focus thereby is on three variants of the ontological argument which avoid the modal collapse, which is a strongly criticized side-e↵ect in Gödel's resp. Scott's original work.

Computer-supported Analysis of Positive Properties, Ultrafilters and Modal Collapse in Variants of Gödel's Ontological Argument

“…Further work (see the references in [24,4]) contributed a range of similar studies on variants of the modal ontological argument that have been proposed by Anderson [1], Anderson and Gettings [2], Hájek [21,22,23], Fitting [16], and Lowe [26]. Particularly relevant for this article is some prior formalization work by the authors that has been presented in [18,17]. The use of ultrafilters to study the distinction between extensional and intensional positive properties in the variants of Scott, Anderson and Fitting has first been mentioned in invited keynotes presented at the AISSQ-2018 [6] and the FMSPh-2019 [3] conferences.…”

“…The formal analysis in this article takes Fitting's book [16] as a starting point; see also [18,17]. Fitting suggests to carefully distinguish between intensions and extensions of positive properties in the context of Gödel's argument, and, in order to do so within a single framework, he introduces a sufficiently expressive higher-order modal logic enhanced with means for the explicit representation of intensional terms and their extensions, which we have termed intensional higher-order modal logic (IHOML) in previous work [17]. The SSE of IHOML in HOL, that we utilize in the remainder of this article, is presented in §3.1.…”

“…This approach enables the reuse of existing, interactive and automated, theorem proving technology for HOL to mechanize also non-classical higher-order reasoning. Some of the findings reported in this article have, at an abstract level, already been summarized in the literature before [24,6,17], but they have not been published in full detail yet (for example, the notions of "modal" ultrafilters, as employed in our analysis, have not been made precise in these papers). This is the contribution of this article.…”

Universal (meta-)logical reasoning: Recent successes

“…The heterogeneous group compositions, the 24/7 feedback from the Isabelle/HOL environment, and the motivating project topics were prime reasons, as the author believes, for the very good overall results of the course. A selection of project results has meanwhile been presented at conferences or published as book chapters or journal articles [98,66,74,73,99,100,67,101]. Several students picked up follow-up topics and turned them into BSc or MSc thesis projects [102,103,104,105,106].…”

Recent Successes with a Meta-Logical Approach to Universal Logical Reasoning (Extended Abstract)

“…Student projects originating from this course have led to impressive new contributions (cf. [1,25,27]; further papers are submitted), including Kirchner's already mentioned embedding of the PLM in HOL, a computer-assisted reconstruction of an ontological argument by Leibniz and a verification of (main parts of) prominent textbooks by Fitting [23] and Boolos [21]. A key factor in the successful implementation of the course has been, that a single methodology and overall technique (the SSE approach) was used throughout, enabling the students to quickly adopt a wide range of different logic variants in short time within a single proof assistant (Isabelle/HOL).…”