Reasoning about knowledge
Gentzen Style Systems for K45 and K45D
Proceedings of the Symposium on Logical Foundations of Computer Science: Logic at Botik '89
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We investigate, for several modal logics but concentrating on KT, KD45, S4 and S5, the set of formulas B for which $${\square B}$$ is provably equivalent to $${\square A}$$ for a selected formula A (such as p, a sentence letter). In the exceptional case in which a modal logic is closed under the (`cancellation') rule taking us from $${\square C \leftrightarrow \square D}$$ to $${C \leftrightarrow D}$$, there is only one formula B, to within equivalence, in this inverse image, as we shall call it, of $${\square A}$$ (relative to the logic concerned); for logics for which the intended reading of "$${\square}$$ " is epistemic or doxastic, failure to be closed under this rule indicates that from the proposition expressed by a knowledge- or belief-attribution, the propositional object of the attitude in question cannot be recovered: arguably, a somewhat disconcerting situation. More generally, the inverse image of $${\square A}$$ may comprise a range of non-equivalent formulas, all those provably implied by one fixed formula and provably implying another--though we shall see that for several choices of logic and of the formula A, there is not even such an `interval characterization' of the inverse image (of $${\square A}$$) to be found.