Knowing-Value Logic with Successor Arithmetic

Abstract

In their prior work, Wang and Fan proposed conditional knowing-value logic and provided a complete axiomatization. However, in natural language scenarios and logic puzzles, knowing-value reasoning often appears together with arithmetic operations, which motivates us to enrich knowing-value logic with arithmetic function symbols. In this paper, we extend the language of conditional knowing-value logic with equality and the successor function. Due to the failure of compactness over the class of standard models, we additionally introduce non-standard models to facilitate the technical analysis. Our main results establish the finite model property and provide an axiomatization that is strongly complete with respect to the class of non-standard models and weakly complete with respect to the class of standard models. Furthermore, we extend our logic with public announcement operators and use the resulting system to formalize and solve the "Consecutive Numbers" puzzle. This work provides a novel framework for integrating epistemic logic with arithmetic.