Universal (meta-)logical reasoning: Recent successes

Abstract

Classical higher-order logic, when utilized as a meta-logic in which various other (classical and non-classical) logics can be shallowly embedded, is suitable as a foundation for the development of a universal logical reasoning engine. Such an engine may be employed, as already envisioned by Leibniz, to support the rigorous formalisation and deep logical analysis of rational arguments on the computer. A respective universal logical reasoning framework is described in this article and a range of successful first applications in philosophy, artificial intelligence and mathematics are surveyed.

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

“…This section is intended to make the article sufficiently self-contained and to give references to related prior work. The presentation in this section is taken and adapted from a recently published related article [24, §1.1]; readers already familiar with the SSE approach may simply skip it, and those who need further details may consult further related documents [7,4].…”

“…Earlier papers, cf. [4] and the references therein, focused on the development of SSEs. These papers show that the standard translation from propositional modal logic to first-order logic can be concisely modeled (i.e., embedded) within higher-order theorem provers, so that the modal operator 2, for example, can be explicitly defined by the λ-term λϕ.λw.∀v.…”

“…Utilizing these notions, and extending Fitting's analysis, the modifications as introduced by Anderson and Fitting to Gödel's concept of positive properties are formally studied and compared. Our computersupported analysis, which is carried out in the proof assistant system Isabelle/HOL [27], is technically enabled by the universal logical reasoning approach [4], which exploits shallow semantical embeddings (SSEs) of various logics of interest -such as intensional higher-order modal logics (IHOML) in the present article -in Church's simple type theory [5], aka. classical higher-order logic (HOL).…”

A Computational-Hermeneutic Approach for Conceptual Explicitation

“…Hence the principle builds upon the possibility of intersubjective agreement about external facts among speaker and interpreter. The principle of charity can be invoked to make sense of a speaker's ambiguous utterances and, in our case, to presume 12 As will be described in section 4.1, the semantical embeddings approach [10,4] allows us to embed different non-classical logics (modal, deontic, intuitionistic, etc.) in higher-order logic (as meta-language), and to combine them dynamically by adding and removing axioms.…”

Universal (meta-)logical reasoning: The Wise Men Puzzle (Isabelle/HOL dataset)

“… Data source location Berlin, Germany, Freie Universität Berlin, Institute of Computer Science Data accessibility Data is with this article Related research article C. Benzmüller, Universal (meta-)logical reasoning: Recent successes, Science of Computer Programming 172:48-62 (2019). https://doi.org/10.1016/j.scico.2018.10.008 [1] …”