Lewis meets Brouwer: constructive strict implication

C. I. Lewis invented modern modal logic as a theory of “strict implication” J. Over the classical propositional calculus one can as well work with the unary box connective, which became the modal primitive ever since. Over the intuitionistic propositional calculus, however, the strict implication has greater expressive power than 2 and allows to make distinctions invisible in the ordinary syntax. In particular, the logic determined by the most popular semantics of intuitionistic K becomes a proper extension of the minimal normal logic of the binary connective. Furthermore, even an extension of this minimal logic with the “strength” axiom, which classically would trivialize a normal modality to a propositional constant, preserves the distinction between the binary and the unary setting. In fact, this distinction and the strong constructive strict implication itself has been discovered by the functional programming community in their study of “arrows” as contrasted with “idioms”. Our particular focus is on arithmetical interpretations of intuitionistic J in terms of preservativity in extensions of HA, i.e., Heyting’s Arithmetic.