Lewis meets Brouwer: constructive strict implication

TL;DR

This paper explores the expressive power of constructive strict implication in intuitionistic logic, its semantics, and its arithmetical interpretations, revealing distinctions from classical modal logic and connections to functional programming concepts.

Contribution

It introduces a new perspective on constructive strict implication, analyzing its semantics, logical extensions, and arithmetical interpretations, highlighting differences from classical modal logic.

Findings

Constructive strict implication has greater expressive power than the box in intuitionistic logic.

The logic based on this implication extends minimal normal logic and preserves distinctions under certain axioms.

Connections are drawn between constructive strict implication and concepts in functional programming, such as arrows.

Abstract

C. I. Lewis invented modern modal logic as a theory of "strict implication". Over the classical propositional calculus one can as well work with the unary box connective. Intuitionistically, however, the strict implication has greater expressive power than the box 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. Even an extension of this minimal logic with the "strength" axiom, classically near-trivial, preserves the distinction between the binary and the unary setting. In fact, this distinction and the strong constructive strict implication itself has been also discovered by the functional programming community in their study of "arrows" as contrasted with "idioms". Our particular focus is on arithmetical interpretations of the intuitionistic strict implication in terms of preservativity in extensions of Heyting's Arithmetic.