The Power of Generalized Clemens Semantics

2024-12-28

Electronic Proceedings in Theoretical Computer Science

AI Summary

This study addresses the semantic interpretation and reasoning challenges posed by contradictory information in many-valued logics—specifically LP and K3—within multi-agent epistemic and philosophical epistemological contexts. Method: We introduce, for the first time, an *n*-ary ordered-pair semantics, generalizing Clemens’s original ordered-pair semantics to *n*-tuples and quantified languages, and endow it with a multi-agent epistemic interpretation. Concurrently, we reconstruct LP/K3 semantics over classical logic to enable quantified modeling of vague predicates and characterize mixed consequence relations. Contributions: (1) We establish the first unified *n*-ary Clemens semantics system; (2) we forge cross-domain bridges among many-valued logic, epistemic philosophy, and vagueness theory; and (3) we provide an interpretable, classically anchored pathway for “informational contradictions,” systematically addressing semantic challenges raised by Egré, Cobreros, and others.

Technology Category

• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Multiagent Systems: Agreement, Argumentation & Negotiation

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

In this paper, we elaborate on the ordered-pair semantics originally presented by Matthew Clemens for LP (Priest's Logic of Paradox). For this purpose, we build on a generalization of Clemens semantics to the case of n-tuple semantics, for every n. More concretely, i) we deal with the case of a language with quantifiers, and ii) we consider philosophical implications of the semantics. The latter includes, first, a reading of the semantics in epistemic terms, involving multiple agents. Furthermore, we discuss the proper understanding of many-valued logics, namely LP and K3 (Kleene strong 3-valued logic), from the perspective of classical logic, along the lines suggested by Susan Haack. We will also discuss some applications of the semantics to issues related to informative contradictions, i.e. contradictions involving quantification over different respects a vague predicate may have, as advanced by Paul 'Egr'e, and also to the mixed consequence relations, promoted by Pablo Cobreros, Paul 'Egr'e, David Ripley and Robert van Rooij.

Problem

• Multi-valued Logic
• Semantic Understanding
• Epistemology

Innovation

• Extended Semantic Pairs
• Multi-valued Logics
• Contradictory Information Handling