Computer Science and Metaphysics: A Cross-Fertilization

Abstract

Computational philosophy is the use of mechanized computational techniques to unearth philosophical insights that are either difficult or impossible to find using traditional philosophical methods. Computational metaphysics is computational philosophy with a focus on metaphysics. In this paper, we (a) develop results in modal metaphysics whose discovery was computer assisted, and (b) conclude that these results work not only to the obvious benefit of philosophy but also, less obviously, to the benefit of computer science, since the new computational techniques that led to these results may be more broadly applicable within computer science. The paper includes a description of our background methodology and how it evolved, and a discussion of our new results.

MECHANIZING PRINCIPIA LOGICO-METAPHYSICA IN FUNCTIONAL TYPE-THEORY

“…7 In the very latest versions of PLM, AOT's free logic has been extended to cover λexpressions, and so [λx ∃F (xF & ¬F x)] is now treated as well formed but non-denoting. This latest development is briefly discussed in [9]. 8 Note that ∀x∃F (xF & ¬F x) is a well-formed formula of the system, but in fact false.…”

“…8 But, as we've seen, if [λx ∃F(xF & ¬Fx)] were a term subject to β-conversion, AOT would yield a contradiction. 9 Thus, it is not trivial to devise a semantical embedding that supports AOT's distinction between formulas and propositional formulas, but at the same time preserves a general theory of quantification. Another challenge has been to accurately represent the hyperintensionality of AOT: while relations in AOT are hyperintensional (i.e., necessarily equivalent relations may be distinct), functions (and relations) in HOL are fully extensional, and can not be used to represent the relations of AOT directly.…”

“…However, the negation of this formula is true, and our question is how to interpret the embedded quantifier in functional type theory. 9 Readers familiar with Isabelle/HOL might find this notation confusing. In AOT, the symbol φ is a metavariable that ranges over formulas which may contain free occurrences of x that can be bound by a binding operator.…”

“…Now, logical existence is (a) defined using exemplification predication instead of using identity for individual terms and (b) also defined for relation terms, this time using encoding predication. These new developments of AOT and its embedding are briefly discussed in [9]. §6.…”

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

“…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.…”