Logics of False Belief and Radical Ignorance

TL;DR

This paper addresses the open problem of axiomatizing the transitive logic of false belief, introducing new axioms and a canonical relation to unify completeness proofs, and extending results to radical ignorance.

Contribution

It provides the first axiomatization of the transitive logic of false belief and introduces a canonical relation for completeness proofs, also extending to radical ignorance.

Findings

Almost definability schema guides core axioms.

Canonical relation handles completeness uniformly.

Results extend to logic of radical ignorance.

Abstract

In the literature, the question about how to axiomatize the transitive logic of false belief is thought of as hard and left as an open problem. In this paper, among other contributions, we deal with this problem. In more details, although the standard doxastic operator is undefinable with the operator of false belief, the former is {\em almost definable} with the latter. On one hand, the involved almost definability schema guides us to find the desired core axioms for the transitive logic and the Euclidean logic of false belief. On the other hand, inspired by the schema and other considerations, we propose a suitable canonical relation, which can uniformly handle the completeness proof of various logics of false belief, including the transitive logic. We also extend the results to the logic of radical ignorance, due to the interdefinability of the operators of false belief and radical ignorance.