A Constructive Epistemic Logic with Public Announcement (Non-Predetermined Possibilities)

TL;DR

This paper introduces a constructive epistemic logic that accounts for non-predetermined possibilities, extending classic epistemic logic with public announcement operators and applying it to resolve the Surprise Exam Paradox.

Contribution

It develops a new constructive epistemic logic incorporating non-predetermined worlds and demonstrates its soundness, completeness, and application to the Surprise Exam Paradox.

Findings

The logic accounts for non-predetermined possibilities ignored in classic logic.

Public announcement operator extension enhances the logic's expressiveness.

The paradox is resolved by recognizing the non-predetermined nature of exam days.

Abstract

We argue that the notion of epistemic \emph{possible worlds} in constructivism (intuitionism) is not as the same as it is in classic view, and there are possibilities, called non-predetermined worlds, which are ignored in (classic) Epistemic Logic. Regarding non-predetermined possibilities, we propose a constructive epistemic logic and prove soundness and completeness theorems for it. We extend the proposed logic by adding a public announcement operator. To declare the significance of our work, we formulate the well-known Surprise Exam Paradox, $\mathbf{SEP}$, via the proposed constructive epistemic logic and then put forward a solution for the paradox. We clarify that the puzzle in the $\mathbf{SEP}$ is because of students'(wrong) assumption that the day of the exam is necessarily predetermined.