Tableaux and Restricted Quantification for Systems Related to Weak Kleene Logic

Abstract

Logic-driven applications like knowledge representation typically operate with the tools of classical, first-order logic. In these applications’ standard, extensional domains—e.g., knowledge bases representing product features—these deductive tools are suitable. However, there remain many domains for which these tools seem overly strong. If, e.g., an artificial conversational agent maintains a knowledge base cataloging e.g. an interlocutor’s beliefs or goals, it is unlikely that the model’s contents are closed under Boolean logic. There exist propositional deductive systems whose notions of validity and equivalence more closely align with legitimate inferences over such intentional contexts. E.g., philosophers like Kit Fine and Stephen Yablo have made compelling cases that Richard Angell’s $\mathsf {AC}$ characterizes synonymy, under which such intentional contexts should be closed. In this paper, we adapt several of these systems by introducing sufficient quantification theory to support e.g. subsumption reasoning. Given the close relationship between these systems and weak Kleene logic, we initially define a novel theory of restricted quantifiers for weak Kleene logic and describe a sound and complete tableau proof theory. We extend the account of quantification and tableau calculi to two related systems: Angell’s $\mathsf {AC}$ and Charles Daniel’s $\mathsf {S}^{\star }_{\mathtt {fde}}$, providing new tools for modeling and reasoning about agents’ mental states.