Combining and automating classical and non-classical logics in classical higher-order logics

Abstract

Numerous classical and non-classical logics can be elegantly embedded in Church's simple type theory, also known as classical higher-order logic. Examples include propositional and quantified multimodal logics, intuitionistic logics, logics for security, and logics for spatial reasoning. Furthermore, simple type theory is sufficiently expressive to model combinations of embedded logics and it has a well understood semantics. Off-the-shelf reasoning systems for simple type theory exist that can be uniformly employed for reasoning within and about embedded logics and logics combinations. In this article we focus on combinations of (quantified) epistemic and doxastic logics and study their application for modeling and automating the reasoning of rational agents. We present illustrating example problems and report on experiments with off-the-shelf higher-order automated theorem provers.

Higher-order aspects and context in SUMO

“…These examples together with their performance results are available in the TPTP THF library (cf. [75] for further details) and they provide first evidence for the practicality of our proposed solution.…”

“…Most importantly, encodings based on modal logics and first-order logic can be elegantly combined in classical higher-order logics. It is this representational power which we have exploited in our recent work [75,79,74,86,87] and which we do also employ here. Moreover, deeper semantic issues can be clarified when taking a solid higher-order perspective.…”

“…A recent case study by Otten and Raths on automating quantified monomodal logics [71] actually confirms that higherorder automated theorem provers such as Satallax and LEO-II can in fact compete with the above mentioned specialist reasoners for reasoning in quantified monomodal logics when using our semantic embeddings based approach. Moreover, a number of alternative examples requiring combinations of modalities have been studied and automated [75]. These examples together with their performance results are available in the TPTP THF library (cf.…”

“…Moreover, there is already some evidence that our automation approach scales to at least some reasonable number of combinations and nestings of modal operators [75]. Since there is currently no practical system in the direct or first-order approach available that supports flexible combinations and nestings of modalities, a solid comparison with alternative approaches is not feasible at this stage.…”

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

“…For example, the knowledge and belief of intelligent agents can be modelled with epistemic and doxastic logics, which are directly amenable to the SSE approach, since they are just particular modal logics. To demonstrate this, prominent AI puzzles about knowledge and belief, including the well known wise men puzzle, have been successfully automated [3,36]. Moreover, the semantic web description logic ALC is just a reinvention of basic multi-modal logic K and, hence, the SSE approach is immediately applicable to it.…”