Fitch's Paradox in Fusions of Epistemic and Alethic Logics

Abstract

In this paper, we analyze Fitch’s paradox of knowability in the framework of fusions of epistemic and alethic modal logics. The paradox arises from accepting the knowability principle, which states that all truths are knowable. However, this leads to the unacceptable conclusion that all truths are known. We introduce a logical system that incorporates all assumptions used by Fitch in his original reasoning, including the knowability principle. We present a natural semantics for this logic, proving the soundness and completeness theorem. Additionally, we present a new semantic proof of the knowability paradox, demonstrating that the problematic conclusion can be derived independently of Fitch’s original proof and showing that the knowability principle itself is the source of the paradox. Using the formal tools introduced, we conduct a semantic analysis of the paradox, which allows us to identify the root cause of its occurrence. Finally, we propose a weakened version of the knowability principle that avoids paradoxical conclusions.