Inverse Images of Box Formulas in Modal Logic

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.