On Modal Logics with an Intuitionistic Base

(Abstract) In [6] Prior proposed a modal calculus, M IP C, which is an extension of the intuitionist propositional calculus. This system, which I call S 5 IC, turns out to be the intuitionist analogue of Lewis's S 5 in the sense that 1) S 5 IC plus A ∨ ¬A is equivalent to S 5 , 2) collapsing the modal operators yields Heyting's calculus. The question as to whether S 5 IC is to be considered analogous to S 5 is not completely answered by 1) and 2) in so far that they are not sufficient for singling out the " correct " S 5-analogue. Bull however showed in [1] that S 5 IC has another characteristic which definitely settles the question of analogy, i.e. 3) there is a translation map T , from S 5 IC formulae into Intuitionistic predicate calculus (IP C) formulae with one variable for which S 5 IC A iff IP C T A. It is well known that 3) holds if (in both systems) we replace IC by the ordinary classical calculus. This third condition suggests that there might exist a general rule by which one could find, in correspondence to a classical modal system, its intuitionist counterpart. My purpose then is to define a general criterion for the concept of " intuitionist modal analogue " in the form of another theorem preserving translation which will be seen to be a natural generalization of Gödel's translation from IC to S 4. The following considerations will be helpful in formulating my proposal. Let S 5 IC be the above modal extension of the intuitionist proposi-tional calculus with the usual connectives and operators