Logic and Topology for Knowledge, Knowability, and Belief

Abstract

In recent work, Stalnaker proposes a logical framework in which belief is realized as a weakened form of knowledge [30]. Building on Stalnaker's core insights, and using frameworks developed in [11] 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.

The Epistemology of Nondeterminism

“…Topological subset models have experienced renewed interest in recent years [3,8,9,10], beginning with the work in [3] studying public announcements in the topological setting. Standard semantics for public announcement logic take the precondition of an announcement of ϕ to be the truth of ϕ; in the topological setting, this precondition is strengthened to the knowability of ϕ.…”

“…, x n+1 ) through X such that, for each i, x i R πi x i+1 . 10 Networks through M constitute the points of the dynamic topological model we are building: X = {α : α is a network through M . }.…”

Arbitrary Public Announcement Logic with Memory

“…In the memory-free context, it was shown in [19] that the natural and apparently 'obvious' translation of [G] ϕ into GAL is in fact invalid. 14 Moreover, no recursive axiomatization is known for CAL, or for any of its extensions. In contrast, in this paper we show that, in our memory-enhanced framework, the obvious translation is valid: the analogue of the coalition announcement modality is definable in GALM.…”

“…This reveals deep connections between apparently very different formalisms. Moreover, this alternative semantics is of independent interest, giving us a more general setting for modeling knowability and learnability (see, e.g., [10,13,14]). Various combinations of PAL or APAL with subset space semantics have been investigated in the literature [9-11, 13, 38, 39, 44, 45], including a version of SSL with backward looking public announcement operators that refer to what was true before a public announcement [8].…”

“…As a consequence, all the evidence gathered in between the initial and the current state can be compressed into one set E, called "the evidence" (rather than keeping a growing tail-sequence of all past evidence sets). Similarly, in our logic, all the past communication is compressed in its end-result, namely in the set W of current possibilities, which plays the same role as the evidence set E in Bayesian models 14. But it is not known if another, valid translation exists 15.…”