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

Christoph Benzmüller
2017 Lecture Notes in Computer Science  
The quest for a most general framework supporting universal reasoning is very prominently represented in the works of Leibniz. He envisioned a scientia generalis founded on a characteristica universalis, that is, a most universal formal language in which all knowledge about the world and the sciences can be encoded. A quick study of the survey literature on logical formalisms suggests that quite the opposite to Leibniz' dream has become reality. Instead of a characteristica universalis, we are
more » ... oday facing a very rich and heterogenous zoo of different logical systems, and instead of converging towards a single superior logic, this logic zoo is further expanding, eventually even at accelerated pace. As a consequence, the unified vision of Leibniz seems farther away than ever before. However, there are also some promising initiatives to counteract these diverging developments. Attempts at unifying approaches to logic include categorial logic algebraic logic and coalgebraic logic. My own research draws on another alternative at universal logical reasoning: the shallow semantical embeddings (SSE) approach. This approach has a very pragmatic motivation, foremost reuse of tools, simplicity and elegance. It utilises classical higherorder logic [22] as a unifying meta-logic in which the syntax and semantics of varying other logics can be explicitly modeled and flexibly combined (cf. [6] and the references therein). Off-the-shelf higher-order interactive and automated theorem provers [7] can then be employed to reason about and within the shallowly embedded logics. Respective experiments have e.g. been conducted in metaphysics. An initial focus thereby has been on computer-supported assessments of modern variants of the ontological argument for the existence of God, where the SSE approach has been utilised in particular for automating variants of higher-order (multi-)modal logics [9] . In the course of these experiments (cf. [17,15,16,18,19,14] for details), my prover LEO-II [10] detected an previously unnoticed inconsistency in Kurt Gödel's [26] prominent variant of the ontological argument, while the slightly modified variant by Dana Scott [32] was verified in the interactive proof assistants Isabelle/HOL [30] and Coq [20] . Further modern variants of the argument have subsequently been studied with the approach, and theorem provers have even contributed to the clarification of an unsettled philosophical dispute [13] . Another, more ambitious study has focused on Ed Zalta's Principia Logico-Metaphysica (PLM) [38] , which aims at a foundational logical theory for all of metaphysics and the sciences. This includes mathematics, and in this sense it is more ambitious than Russel's Principia Mathematica. The semantical embedding of PLM in HOL has been very challenging, since in addition to its size, its foundational theory is complicated: the PLM is based on hyperintensional higher-order modal logic S5 defined on top of a relational (as opposed to a functional) type theory that comes with restricted comprehension principles (the use of full comprehension in the PLM has been known to cause para-
doi:10.1007/978-3-319-70848-5_2 fatcat:5dhzxs57mzbi7ju2h2nrpkejoa