Logic and Topology for Knowledge, Knowability, and Belief

TL;DR

This paper develops a topological framework to analyze the relationship between knowledge, knowability, and belief, refining existing logical models and providing sound, complete axiomatizations for these concepts.

Contribution

It introduces a topological subset space semantics for a trimodal logic of knowledge, knowability, and belief, and offers new axiomatizations including for irreducible belief.

Findings

Provided a sound and complete axiomatization for the logic of knowledge, knowability, and belief.

Developed novel topological semantics for irreducible notions of belief.

Demonstrated that belief can be defined in terms of knowledge and knowability within this framework.

Abstract

In recent work, Stalnaker proposes a logical framework in which belief is realized as a weakened form of knowledge. Building on Stalnaker's core insights, and using frameworks developed in previous work by Bjorndahl and Baltag et al., we employ topological tools to refine and, we argue, improve on this analysis. The structure of topological subset spaces allows for a natural distinction between what is known and (roughly speaking) what is knowable; we argue that the foundational axioms of Stalnaker's system rely intuitively on both of these notions. More precisely, we argue that the plausibility of the principles Stalnaker proposes relating knowledge and belief relies on a subtle equivocation between an "evidence-in-hand" conception of knowledge and a weaker "evidence-out-there" notion of what could come to be known. Our analysis leads to a trimodal logic of knowledge, knowability, and belief interpreted in topological subset spaces in which belief is definable in terms of knowledge and knowability. We provide a sound and complete axiomatization for this logic as well as its uni-modal belief fragment. We then consider weaker logics that preserve suitable translations of Stalnaker's postulates, yet do not allow for any reduction of belief. We propose novel topological semantics for these irreducible notions of belief, generalizing our previous semantics, and provide sound and complete axiomatizations for the corresponding logics.