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Implicational (semilinear) logics I: a new hierarchy

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Abstract

In abstract algebraic logic, the general study of propositional non-classical logics has been traditionally based on the abstraction of the Lindenbaum-Tarski process. In this process one considers the Leibniz relation of indiscernible formulae. Such approach has resulted in a classification of logics partly based on generalizations of equivalence connectives: the Leibniz hierarchy. This paper performs an analogous abstract study of non-classical logics based on the kind of generalized implication connectives they possess. It yields a new classification of logics expanding Leibniz hierarchy: the hierarchy of implicational logics. In this framework the notion of implicational semilinear logic can be naturally introduced as a property of the implication, namely a logic L is an implicational semilinear logic iff it has an implication such that L is complete w.r.t. the matrices where the implication induces a linear order, a property which is typically satisfied by well-known systems of fuzzy logic. The hierarchy of implicational logics is then restricted to the semilinear case obtaining a classification of implicational semilinear logics that encompasses almost all the known examples of fuzzy logics and suggests new directions for research in the field.

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Authors and Affiliations

  1. Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod vodárenskou věží 2, 182 07, Prague, Czech Republic

    Petr Cintula

  2. Department of Mathematics and Computer Science, University of Siena, Pian dei Mantellini 44, 53100, Siena, Italy

    Carles Noguera

Authors
  1. Petr Cintula
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  2. Carles Noguera
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Correspondence to Carles Noguera.

Additional information

Petr Cintula was partly supported by grant ICC/08/E018 of the Grant Agency of the Czech Republic (a part of ESF Eurocores-LogICCC project FP006) and partly by Institutional Research Plan AVOZ10300504. Carles Noguera acknowledges partial support from the grant 2006-BP-A-10043 of the Departament d’Educació i Universitats of the Generalitat de Catalunya and the Spanish project MULOG2 (TIN2007-68005-C04) and ESF Eurocores-LogICCC / MICINN project (FFI2008-03126-E/FILO).

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Cintula, P., Noguera, C. Implicational (semilinear) logics I: a new hierarchy. Arch. Math. Logic 49, 417–446 (2010). https://doi.org/10.1007/s00153-010-0178-7

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  • DOI: https://doi.org/10.1007/s00153-010-0178-7

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