Logic and Logical Philosophy 29 (3):381-413 (2020)

Given the frequency of human error, it seems rational to believe that some of our own rational beliefs are false. This is the axiom of epistemic modesty. Unfortunately, using standard propositional quantification, and the usual relational semantics, this axiom is semantically inconsistent with a common logic for rational belief, namely KD45. Here we explore two alternative semantics for KD45 and the axiom of epistemic modesty. The first uses the usual relational semantics and bisimulation quantifiers. The second uses a topological semantics and standard propositional quantification. We show the two different semantics validate many of the same formulas, though we do not know whether they validate exactly the same formulas. Along the way we address various philosophical concerns.
Keywords belief  topology  bisimulation quantifiers  derived set  modesty  humility  KD45  Formal Epistemology  derivative  Gettier
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On Logics of Knowledge and Belief.Robert Stalnaker - 2006 - Philosophical Studies 128 (1):169-199.
Propositional Quantifiers in Modal Logic.Kit Fine - 1970 - Theoria 36 (3):336-346.
The Algebra of Topology.J. C. C. Mckinsey & Alfred Tarski - 1944 - Annals of Mathematics, Second Series 45:141-191.

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