8 found
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  1.  41
    Product Ł Ukasiewicz Logic.Rostislav Horčík & Petr Cintula - 2004 - Archive for Mathematical Logic 43 (4):477-503.
    Łu logic plays a fundamental role among many-valued logics. However, the expressive power of this logic is restricted to piecewise linear functions. In this paper we enrich the language of Łu logic by adding a new connective which expresses multiplication. The resulting logic, PŁ, is defined, developed, and put into the context of other well-known many-valued logics. We also deal with several extensions of this propositional logic. A predicate version of PŁ logic is introduced and developed too.
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  2.  8
    Formal Systems of Fuzzy Logic and Their Fragments.Petr Cintula, Petr Hájek & Rostislav Horčík - 2007 - Annals of Pure and Applied Logic 150 (1):40-65.
    Formal systems of fuzzy logic are well-established logical systems and respected members of the broad family of the so-called substructural logics closely related to the famous logic BCK. The study of fragments of logical systems is an important issue of research in any class of non-classical logics. Here we study the fragments of nine prominent fuzzy logics to all sublanguages containing implication. However, the results achieved in the paper for those nine logics are usually corollaries of theorems with much wider (...)
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  3.  55
    Nonassociative Substructural Logics and Their Semilinear Extensions: Axiomatization and Completeness Properties: Nonassociative Substructural Logics.Petr Cintula, Rostislav Horčík & Carles Noguera - 2013 - Review of Symbolic Logic 6 (3):394-423.
    Substructural logics extending the full Lambek calculus FL have largely benefited from a systematical algebraic approach based on the study of their algebraic counterparts: residuated lattices. Recently, a nonassociative generalization of FL has been studied by Galatos and Ono as the logic of lattice-ordered residuated unital groupoids. This paper is based on an alternative Hilbert-style presentation for SL which is almost MP -based. This presentation is then used to obtain, in a uniform way applicable to most substructural logics, a form (...)
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  4.  18
    On N -Contractive Fuzzy Logics.Rostislav Horčík, Carles Noguera & Milan Petrík - 2007 - Mathematical Logic Quarterly 53 (3):268-288.
    It is well known that MTL satisfies the finite embeddability property. Thus MTL is complete w. r. t. the class of all finite MTL-chains. In order to reach a deeper understanding of the structure of this class, we consider the extensions of MTL by adding the generalized contraction since each finite MTL-chain satisfies a form of this generalized contraction. Simultaneously, we also consider extensions of MTL by the generalized excluded middle laws introduced in [9] and the axiom of weak cancellation (...)
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  5.  36
    Standard Completeness Theorem for ΠMTL.Rostislav Horĉík - 2004 - Archive for Mathematical Logic 44 (4):413-424.
    .ΠMTL is a schematic extension of the monoidal t-norm based logic by the characteristic axioms of product logic. In this paper we prove that ΠMTL satisfies the standard completeness theorem. From the algebraic point of view, we show that the class of ΠMTL-algebras in the real unit interval [0,1] generates the variety of all ΠMTL-algebras.
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  6.  34
    Archimedean Classes in Integral Commutative Residuated Chains.Rostislav Horčík & Franco Montagna - 2009 - Mathematical Logic Quarterly 55 (3):320-336.
    This paper investigates a quasi-variety of representable integral commutative residuated lattices axiomatized by the quasi-identity resulting from the well-known Wajsberg identity → q ≤ → p if it is written as a quasi-identity, i. e., → q ≈ 1 ⇒ → p ≈ 1. We prove that this quasi-identity is strictly weaker than the corresponding identity. On the other hand, we show that the resulting quasi-variety is in fact a variety and provide an axiomatization. The obtained results shed some light (...)
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  7.  22
    Minimal Varieties of Representable Commutative Residuated Lattices.Rostislav Horčík - 2012 - Studia Logica 100 (6):1063-1078.
    We solve several open problems on the cardinality of atoms in the subvariety lattice of residuated lattices and FL-algebras [4, Problems 17—19, pp. 437]. Namely, we prove that the subvariety lattice of residuated lattices contains continuum many 4-potent commutative representable atoms. Analogous results apply also to atoms in the subvariety lattice of FL i -algebras and FL o -algebras. On the other hand, we show that the subvariety lattice of residuated lattices contains only five 3-potent commutative representable atoms and two (...)
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  8.  10
    Full Lambek Calculus with Contraction is Undecidable.Karel Chvalovský & Rostislav Horčík - 2016 - Journal of Symbolic Logic 81 (2):524-540.