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On two fragments with negation and without implication of the logic of residuated lattices

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Abstract

The logic of (commutative integral bounded) residuated lattices is known under different names in the literature: monoidal logic [26], intuitionistic logic without contraction [1], H BCK [36] (nowadays called by Ono), etc. In this paper we study the -fragment and the -fragment of the logical systems associated with residuated lattices, both from the perspective of Gentzen systems and from that of deductive systems. We stress that our notion of fragment considers the full consequence relation admitting hypotheses. It results that this notion of fragment is axiomatized by the rules of the sequent calculus for the connectives involved. We also prove that these deductive systems are non-protoalgebraic, while the Gentzen systems are algebraizable with equivalent algebraic semantics the varieties of pseudocomplemented (commutative integral bounded) semilatticed and latticed monoids, respectively. All the logical systems considered are decidable.

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Authors and Affiliations

  1. School of Information Science, Japan Advanced Institute of Science and Technology, Japan

    Félix Bou

  2. Institute of Investigation in Artificial Intelligence, CSIC, Spain

    Àngel García-Cerdaña

  3. Department of Philosophy, Autonomous University of Barcelona, Spain

    Àngel García-Cerdaña

  4. Department of Logic, History and Philosophy of Science, University of Barcelona, Spain

    Ventura Verdú

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  1. Félix Bou
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  2. Àngel García-Cerdaña
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  3. Ventura Verdú
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Correspondence to Àngel García-Cerdaña.

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Bou, F., García-Cerdaña, À. & Verdú, V. On two fragments with negation and without implication of the logic of residuated lattices. Arch. Math. Logic 45, 615–647 (2006). https://doi.org/10.1007/s00153-005-0324-9

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