Conditionally Knowing What

Classic epistemic logic focuses on propositional knowledge expressed by “knowing that” operators. However, there are various types of knowledge used in natural language, in terms of “knowing how”, “knowing whether”, “knowing what”, and so on. In [10], Plaza proposed an intuitive know-what operator which was generalized in [16] by introducing a condition. The latter know-what operator can express natural conditional knowledge such as “I know what your password is, if it is 4-digits”, which is not simply a material implication. Essentially this know-what operator packages a first-order quantifier and an S5-modality together in a non-trivial way, thus making it hard to axiomatize. In [16] an axiomatization is given for the single-agent epistemic logic with both know-that and know-what operators, while leaving axiomatizing the multi-agent case open due to various technical difficulties. In this paper, we solve this open problem. The completeness proof is highly non-trivial, compared to the singleagent case, which requires different techniques inspired by first-order intensional logic.