Negation-as-Failure in the Base-extension Semantics for Intuitionistic Propositional Logic

10/11/2022

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by   Alexander V. Gheorghiu, et al.

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Proof-theoretic semantics (P-tS) is the paradigm of semantics in which meaning in logic is based on proof (as opposed to truth). A particular instance of P-tS for intuitionistic propositional logic (IPL) is its base-extension semantics (B-eS). In this paper, we prove completeness of IPL with respect to the B-eS through logic programming (LP). This reveals that inherent to P-tS is proof-search and the negation-as-failure protocol. Traditionally, the denial of a proposition is understood as the assertion of its negation, which is indeed the case in the model-theoretic semantics for IPL, but the connection to LP reveals that in P-tS the denial of a proposition refers to the failure of finding a proof of it. In this way, assertion and denial are both prime concepts in P-tS.