Logic and Topology for Knowledge, Knowability, and Belief - Extended Abstract

Abstract

In recent work, Stalnaker proposes a logical framework in which belief is realized as a weakened form of knowledge [33]. Building on Stalnaker's core insights, and using frameworks developed in [12] and [4], 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.

A Simplicial Model for $KB4_n$: Epistemic Logic with Agents that May Die

“…It would be interesting to relate our work with the topological approaches to epistemic logic, that have proven so useful to study the nature knowledge and belief, as well as topological semantics beginning with McKinsey and Tarski [23] and the subset space semantics introduced by Moss and Parikh [27], see e.g. [3,29] for additional references.…”

Uncertainty About Evidence

“…It is natural to wish to extend the framework we have developed to include a representation not only for knowledge but also belief. Defining this extension is relatively straightforward, as it parallels a similar construction from previous work [12]. The interest here arises not in the definition itself, but from the subsequent investigation into the interplay between belief and uncertainty about the interpretation of evidence.…”

“…Thus, it may be easier to think of (E, ⊕, I) as the analog of a basis for X , rather than a full topology. We can also define a kind of generalized interior operator in evidence interactions models, and use it to articulate a notion of measurability corresponding to what the agent could come to know after taking a sufficiently good measurement or otherwise obtaining a sufficiently strong piece of evidence (see [11,12,10]). Given an evidence interaction model (X , E, ⊕, I, v) and an evidence scenario (x, e), we interpret the propositional variables, Boolean connectives, K, and E as before, and for we define…”

“…Consider a public announcement style update mechanic in which knowability plays the role of the precondition of the corresponding announcement, as in [11]. The effect of an announcement is then manifested as a transition from the initial evidence state to a more informative one, without requiring global changes in the given model, as in logics of information dynamics interpreted on subset space models [28,12,16,6,3]. The enriched structure owing to the evidence states and their variable interpretations raises the question of whether such a dynamic logic can be reduced to a weaker, static logic, as is often the case in similar settings.…”

“…And a variety of logical models for evidence and belief update assume that each piece of evidence corresponds to a set of possible worlds U and entails exactly those propositions ϕ such that U ⊆ [[ϕ]] [24,14,26,9,7,8]. Of special note are those models in which the collection of evidence is assumed to take the structure of a topology [6,12,11,2,25]; the framework we propose can be viewed as a generalization of this paradigm.…”