"Knowing value" logic as a normal modal logic

TL;DR

This paper demonstrates that the 'knowing value' logic can be reformulated as a normal modal logic using a simplified Kripke semantics with a ternary relation, leading to clearer understanding and easier analysis.

Contribution

It shows that the 'knowing value' logic is a disguised normal modal logic, simplifying its semantics and providing a transparent modal system.

Findings

The original first-order semantics can be simplified with a ternary relation in Kripke models.

The logic based on these models is equivalent to Wang and Fan's earlier work.

A binary generalization of the 'knowing value' diamond does not increase expressive power.

Abstract

Recent years witness a growing interest in nonstandard epistemic logics of "knowing whether", "knowing what", "knowing how", and so on. These logics are usually not normal, i.e., the standard axioms and reasoning rules for modal logic may be invalid. In this paper, we show that the conditional "knowing value" logic proposed by Wang and Fan \cite{WF13} can be viewed as a disguised normal modal logic by treating the negation of the Kv operator as a special diamond. Under this perspective, it turns out that the original first-order Kripke semantics can be greatly simplified by introducing a ternary relation $R_i^c$ in standard Kripke models, which associates one world with two $i$-accessible worlds that do not agree on the value of constant $c$. Under intuitive constraints, the modal logic based on such Kripke models is exactly the one studied by Wang and Fan (2013,2014}. Moreover, there is a very natural binary generalization of the "knowing value" diamond, which, surprisingly, does not increase the expressive power of the logic. The resulting logic with the binary diamond has a transparent normal modal system, which sharpens our understanding of the "knowing value" logic and simplifies some previously hard problems.