"Knowing value" logic as a normal modal logic
This work provides a foundational simplification for researchers in epistemic logic, though it is incremental as it builds on existing 'knowing value' logic.
The paper tackles the problem of interpreting 'knowing value' logic as a normal modal logic by redefining its operator, resulting in a simplified Kripke semantics and a transparent normal modal system that does not increase expressive power.
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.