Self-referentiality of Brouwer–Heyting–Kolmogorov semantics

Abstract

The Gödel–Artemov framework offered a formalization of the Brouwer–Heyting–Kolmogorov (BHK) semantics of intuitionistic logic via classical proofs. In this framework, the intuitionistic propositional logic IPC is embedded in the modal logic S4, S4 is realized in the Logic of Proofs LP, and LP has a provability interpretation in Peano Arithmetic. Self-referential LP-formulas of the type ‘t is a proof of a formula ϕ containing t itself’ are permitted in the realization of S4 in LP, and if such formulas are indeed involved, it is then necessary to use fixed-point arithmetical methods to explain intuitionistic logic via provability. The natural question of whether self-referentiality can be avoided in realization of S4 was answered negatively by Kuznets who provided an S4-theorem that cannot be realized without using directly self-referential LP-formulas. This paper studies the question of whether IPC can be embedded in S4 and then realized in LP without using self-referential formulas. We consider a general class of Gödel-style modal embeddings of IPC in S4 and by extending Kuznetsʼ method, show that there are IPC-theorems such that, under each such embedding, are mapped to S4-theorems that cannot be realized in LP without using directly self-referential formulas. Interestingly, all double-negations of tautologies that are not IPC-theorems, like ¬¬(¬¬p→p), are shown to require direct self-referentiality. Another example is found in IPC→, the purely implicational fragment of IPC. This suggests that the BHK semantics of intuitionistic logic (even of intuitionistic implication) is intrinsically self-referential.This paper is an extended version of [26].