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On The Role of The Polynomial (X → Y) → Y in Some Implicative Algebras

Mathematical Logic Quarterly 34 (2):117-122 (1988)

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Decomposability of free Łukasiewicz implication algebras.Jose Patricio Díaz Varela & Antoni Torrens Torrell - 2006 - Archive for Mathematical Logic 45 (8):1011-1020.details Łukasiewicz implication algebras are {→,1}-subreducts of Wajsberg algebras (MV-algebras). They are the algebraic counterpart of Super-Łukasiewicz Implicational logics investigated in Komori, Nogoya Math J 72:127–133, 1978. The aim of this paper is to study the direct decomposability of free Łukasiewicz implication algebras. We show that freely generated algebras are directly indecomposable. We also study the direct decomposability in free algebras of all its proper subvarieties and show that infinitely freely generated algebras are indecomposable, while finitely free generated algebras can be (...) Download Export citation Bookmark 1 citation
(1 other version)Glivenko like theorems in natural expansions of BCK‐logic.Roberto Cignoli & Antoni Torrens Torrell - 2004 - Mathematical Logic Quarterly 50 (2):111-125.details The classical Glivenko theorem asserts that a propositional formula admits a classical proof if and only if its double negation admits an intuitionistic proof. By a natural expansion of the BCK-logic with negation we understand an algebraizable logic whose language is an expansion of the language of BCK-logic with negation by a family of connectives implicitly defined by equations and compatible with BCK-congruences. Many of the logics in the current literature are natural expansions of BCK-logic with negation. The validity of (...) Download Export citation Bookmark 13 citations
(1 other version)Glivenko like theorems in natural expansions of BCK‐logic.Roberto Cignoli & Antoni Torrens Torrell - 2004 - Mathematical Logic Quarterly 50 (2):111-125.details The classical Glivenko theorem asserts that a propositional formula admits a classical proof if and only if its double negation admits an intuitionistic proof. By a natural expansion of the BCK‐logic with negation we understand an algebraizable logic whose language is an expansion of the language of BCK‐logic with negation by a family of connectives implicitly defined by equations and compatible with BCK‐congruences. Many of the logics in the current literature are natural expansions of BCK‐logic with negation. The validity of (...) Download Export citation Bookmark 14 citations
For Want of an ‘And’: A Puzzle about Non-Conservative Extension.Lloyd Humberstone - 2005 - History and Philosophy of Logic 26 (3):229-266.details Section 1 recalls a point noted by A. N. Prior forty years ago: that a certain formula in the language of a purely implicational intermediate logic investigated by R. A. Bull is unprovable in that logic but provable in the extension of the logic by the usual axioms for conjunction, once this connective is added to the language. Section 2 reminds us that every formula is interdeducible with (i.e. added to intuitionistic logic, yields the same intermediate logic as) some conjunction-free (...) Download Export citation Bookmark 2 citations

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