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LOGIC AND TOPOLOGY FOR KNOWLEDGE, KNOWABILITY, AND BELIEF

Published online by Cambridge University Press:  09 October 2019

ADAM BJORNDAHL  and
AYBÜKE ÖZGÜN
ADAM BJORNDAHL*
Affiliation:
Carnegie Mellon University
AYBÜKE ÖZGÜN*
Affiliation:
ILLC, University of Amsterdam; and Arché, University of St. Andrews
*
*DEPARTMENT OF PHILOSOPHY CARNEGIE MELLON UNIVERSITY PITTSBURGH, PA, USA E-mail: abjorn@cmu.edu
†ILLC, UNIVERSITY OF AMSTERDAM AMSTERDAM, THE NETHERLANDS and ARCHÉ, UNIVERSITY OF ST. ANDREWS ST. ANDREWS, SCOTLAND E-mail: a.ozgun@uva.nl
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Abstract

In recent work, Stalnaker proposes a logical framework in which belief is realized as a weakened form of knowledge 35. Building on Stalnaker’s core insights, and using frameworks developed in 11 and 3, 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.

Keywords

03B42epistemic/doxastic logictopological semanticsevidence
Type
Research Article
Information
The Review of Symbolic Logic , Volume 13 , Issue 4 , December 2020 , pp. 748 - 775
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Copyright
Copyright © Association for Symbolic Logic 2019 

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