Self-referentiality in the Brouwer–Heyting–Kolmogorov Semantics of Intuitionistic Logic

Abstract

The intended provability semantics of Intuitionistic Propositional Logic IPC (also called Brouwer–Heyting–Kolmogorov semantics) has been formalized within Gödel-Artemov’s framework. According to this approach, IPC is embedded in modal logic S4 by Gödel embedding and S4 is realized in Artemov’s Logic of Proofs LP which has a provability interpretation in Peano Arithmetic. Artemov’s realization of S4 in LP uses self-referential LP-formulas of the form $t\!\!:\!\!\phi(t)$, namely, ‘t is a proof of a formula φ containing t itself.’ Kuznets showed that this is not avoidable by offering S4-theorems that cannot be realized without using self-referential LP-formulas. This paper extends Kuznets’ method to find IPC-theorems that call for direct self-referentiality in LP. Roughly speaking, examples include double-negations of tautologies that are not IPC-theorems, e.g., $\neg\neg(\neg\neg p\!\rightarrow\! p)$, and there are also examples in the purely implicational fragment IPC _→. This suggests that the Brouwer–Heyting–Kolmogorov semantics of intuitionistic logic is intrinsically self-referential.