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.
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Yu, J. (2013). Self-referentiality in the Brouwer–Heyting–Kolmogorov Semantics of Intuitionistic Logic. In: Artemov, S., Nerode, A. (eds) Logical Foundations of Computer Science. LFCS 2013. Lecture Notes in Computer Science, vol 7734. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35722-0_29
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DOI: https://doi.org/10.1007/978-3-642-35722-0_29
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