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Sufficient triangular norms in many-valued logics with standard negation

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Abstract

In many-valued logics with the unit interval as the set of truth values, from the standard negation and the product (or, more generally, from any strict Frank t-norm) all measurable logical functions can be derived, provided that also operations with countable arity are allowed. The question remained open whether there are other t-norms with this property or whether all strict t-norms possess this property. We give a full solution to this problem (in the case of strict t-norms), together with convenient sufficient conditions. We list several families of strict t-norms having this property and provide also counterexamples (the Hamacher product is one of them). Finally, we discuss the consequences of these results for the characterization of tribes based on strict t-norms.

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Authors and Affiliations

  1. Department of Mathematics, University of Haifa, 31 905, Haifa, Israel

    Dan Butnariu

  2. Department of Knowledge-Based Mathematical Systems, Johannes Kepler University, 4040, Linz, Austria

    Erich Peter Klement

  3. Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University of Technology, 81 368, Bratislava, Slovakia

    Radko Mesiar

  4. Center for Machine Perception, Department of Cybernetics, Faculty of Electrical Engineering, Czech Technical University, 166 27, Prague 6, Czech Republic

    Mirko Navara

  5. Institute of Information Theory and Automation, Czech Academy of Sciences, 182 08, Prague 8, Czech Republic

    Radko Mesiar

Authors
  1. Dan Butnariu
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  2. Erich Peter Klement
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  3. Radko Mesiar
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  4. Mirko Navara
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Correspondence to Dan Butnariu.

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Butnariu, D., Klement, E., Mesiar, R. et al. Sufficient triangular norms in many-valued logics with standard negation. Arch. Math. Logic 44, 829–849 (2005). https://doi.org/10.1007/s00153-004-0267-6

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  • DOI: https://doi.org/10.1007/s00153-004-0267-6

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