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Embedding classical in minimal implicational logic

Hajime Ishihara & Helmut Schwichtenberg

Mathematical Logic Quarterly 62 (1-2):94-101 (2016)

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The Jacobson Radical of a Propositional Theory.Giulio Fellin, Peter Schuster & Daniel Wessel - 2022 - Bulletin of Symbolic Logic 28 (2):163-181.details Alongside the analogy between maximal ideals and complete theories, the Jacobson radical carries over from ideals of commutative rings to theories of propositional calculi. This prompts a variant of Lindenbaum’s Lemma that relates classical validity and intuitionistic provability, and the syntactical counterpart of which is Glivenko’s Theorem. The Jacobson radical in fact turns out to coincide with the classical deductive closure. As a by-product we obtain a possible interpretation in logic of the axioms-as-rules conservation criterion for a multi-conclusion Scott-style entailment (...) Download Export citation Bookmark
(1 other version)Proof Compression and NP Versus PSPACE II.Lew Gordeev & Edward Hermann Haeusler - 2020 - Bulletin of the Section of Logic 49 (3):213-230.details We upgrade [3] to a complete proof of the conjecture NP = PSPACE that is known as one of the fundamental open problems in the mathematical theory of computational complexity; this proof is based on [2]. Since minimal propositional logic is known to be PSPACE complete, while PSPACE to include NP, it suﬃces to show that every valid purely implicational formula ρ has a proof whose weight and time complexity of the provability involved are both polynomial in the weight of (...) Download Export citation Bookmark 1 citation
Classifying material implications over minimal logic.Hannes Diener & Maarten McKubre-Jordens - 2020 - Archive for Mathematical Logic 59 (7):905-924.details The so-called paradoxes of material implication have motivated the development of many non-classical logics over the years, such as relevance logics, paraconsistent logics, fuzzy logics and so on. In this note, we investigate some of these paradoxes and classify them, over minimal logic. We provide proofs of equivalence and semantic models separating the paradoxes where appropriate. A number of equivalent groups arise, all of which collapse with unrestricted use of double negation elimination. Interestingly, the principle ex falso quodlibet, and several (...) Download Export citation Bookmark

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