Carnap's problem for intuitionistic propositional logic

TL;DR

This paper proves that intuitionistic propositional logic has a unique interpretation of its connectives across various semantics, establishing its Carnap categoricity and reinforcing its foundational consistency.

Contribution

It demonstrates that intuitionistic propositional logic is Carnap categorical across multiple well-known semantics, highlighting its unique interpretative stability.

Findings

Intuitionistic logic is Carnap categorical under Kripke, Beth, Dragalin, and topological semantics.

Categoricity also holds for algebraic semantics, but in a different sense.

The result emphasizes the interpretative uniqueness of intuitionistic propositional logic.

Abstract

We show that intuitionistic propositional logic is \emph{Carnap categorical}: the only interpretation of the connectives consistent with the intuitionistic consequence relation is the standard interpretation. This holds relative to the most well-known semantics with respect to which intuitionistic logic is sound and complete; among them Kripke semantics, Beth semantics, Dragalin semantics, and topological semantics. It also holds for algebraic semantics, although categoricity in that case is different in kind from categoricity relative to possible worlds style semantics.