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  1.  33
    The Proof by Cases Property and its Variants in Structural Consequence Relations.Petr Cintula & Carles Noguera - 2013 - Studia Logica 101 (4):713-747.
    This paper is a contribution to the study of the rôle of disjunction inAlgebraic Logic. Several kinds of (generalized) disjunctions, usually defined using a suitable variant of the proof by cases property, were introduced and extensively studied in the literature mainly in the context of finitary logics. The goals of this paper are to extend these results to all logics, to systematize the multitude of notions of disjunction (both those already considered in the literature and those introduced in this paper), (...)
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  2.  21
    Distinguished Algebraic Semantics for T -Norm Based Fuzzy Logics: Methods and Algebraic Equivalencies.Petr Cintula, Francesc Esteva, Joan Gispert, Lluís Godo, Franco Montagna & Carles Noguera - 2009 - Annals of Pure and Applied Logic 160 (1):53-81.
    This paper is a contribution to Mathematical fuzzy logic, in particular to the algebraic study of t-norm based fuzzy logics. In the general framework of propositional core and Δ-core fuzzy logics we consider three properties of completeness with respect to any semantics of linearly ordered algebras. Useful algebraic characterizations of these completeness properties are obtained and their relations are studied. Moreover, we concentrate on five kinds of distinguished semantics for these logics–namely the class of algebras defined over the real unit (...)
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  3.  25
    Implicational (Semilinear) Logics I: A New Hierarchy. [REVIEW]Petr Cintula & Carles Noguera - 2010 - Archive for Mathematical Logic 49 (4):417-446.
    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 (...)
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  4.  13
    A New Hierarchy of Infinitary Logics in Abstract Algebraic Logic.Carles Noguera & Tomáš Lávička - 2017 - Studia Logica 105 (3):521-551.
    In this article we investigate infinitary propositional logics from the perspective of their completeness properties in abstract algebraic logic. It is well-known that every finitary logic is complete with respect to its relatively subdirectly irreducible models. We identify two syntactical notions formulated in terms of intersection-prime theories that follow from finitarity and are sufficient conditions for the aforementioned completeness properties. We construct all the necessary counterexamples to show that all these properties define pairwise different classes of logics. Consequently, we obtain (...)
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  5.  7
    A Note on Natural Extensions in Abstract Algebraic Logic.Petr Cintula & Carles Noguera - 2015 - Studia Logica 103 (4):815-823.
    Transfer theorems are central results in abstract algebraic logic that allow to generalize properties of the lattice of theories of a logic to any algebraic model and its lattice of filters. Their proofs sometimes require the existence of a natural extension of the logic to a bigger set of variables. Constructions of such extensions have been proposed in particular settings in the literature. In this paper we show that these constructions need not always work and propose a wider setting in (...)
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  6. Fraïssé Classes of Graded Relational Structures.Guillermo Badia & Carles Noguera - 2018 - Theoretical Computer Science 737:81–90.
    We study classes of graded structures satisfying the properties of amalgamation, joint embedding and hereditariness. Given appropriate conditions, we can build a graded analogue of the Fraïssé limit. Some examples such as the class of all finite weighted graphs or the class of all finite fuzzy orders (evaluated on a particular countable algebra) will be examined.
     
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  7.  37
    Perfect and Bipartite IMTL-Algebras and Disconnected Rotations of Prelinear Semihoops.Carles Noguera, Francesc Esteva & Joan Gispert - 2005 - Archive for Mathematical Logic 44 (7):869-886.
  8.  22
    Syntactic Characterizations of First-Order Structures in Mathematical Fuzzy Logic.Guillermo Badia, Pilar Dellunde, Vicent Costa & Carles Noguera - forthcoming - Soft Computing.
    This paper is a contribution to graded model theory, in the context of mathematical fuzzy logic. We study characterizations of classes of graded structures in terms of the syntactic form of their first-order axiomatization. We focus on classes given by universal and universal-existential sentences. In particular, we prove two amalgamation results using the technique of diagrams in the setting of structures valued on a finite MTL-algebra, from which analogues of the Łoś–Tarski and the Chang–Łoś–Suszko preservation theorems follow.
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  9.  8
    Implicational Logics III: Completeness Properties.Petr Cintula & Carles Noguera - 2018 - Archive for Mathematical Logic 57 (3-4):391-420.
    This paper presents an abstract study of completeness properties of non-classical logics with respect to matricial semantics. Given a class of reduced matrix models we define three completeness properties of increasing strength and characterize them in several useful ways. Some of these characterizations hold in absolute generality and others are for logics with generalized implication or disjunction connectives, as considered in the previous papers. Finally, we consider completeness with respect to matrices with a linear dense order and characterize it in (...)
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  10.  16
    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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  11.  19
    On Triangular Norm Based Axiomatic Extensions of the Weak Nilpotent Minimum Logic.Carles Noguera, Francesc Esteva & Joan Gispert - 2008 - Mathematical Logic Quarterly 54 (4):387-409.
    In this paper we carry out an algebraic investigation of the weak nilpotent minimum logic and its t-norm based axiomatic extensions. We consider the algebraic counterpart of WNM, the variety of WNM-algebras and prove that it is locally finite, so all its subvarieties are generated by finite chains. We give criteria to compare varieties generated by finite families of WNM-chains, in particular varieties generated by standard WNM-chains, or equivalently t-norm based axiomatic extensions of WNM, and we study their standard completeness (...)
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  12.  47
    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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  13.  11
    Implicational Logics II: Additional Connectives and Characterizations of Semilinearity.Petr Cintula & Carles Noguera - 2016 - Archive for Mathematical Logic 55 (3-4):353-372.
  14.  7
    Löwenheim–Skolem Theorems for Non-Classical First-Order Algebraizable Logics: Table 1.Pilar Dellunde, Àngel García-Cerdaña & Carles Noguera - 2016 - Logic Journal of the IGPL 24 (3):321-345.
  15.  7
    A Henkin-Style Proof of Completeness for First-Order Algebraizable Logics.Petr Cintula & Carles Noguera - 2015 - Journal of Symbolic Logic 80 (1):341-358.
  16.  5
    First-Order T-Norm Based Fuzzy Logics with Truth-Constants: Distinguished Semantics and Completeness Properties.Francesc Esteva, Lluís Godo & Carles Noguera - 2009 - Annals of Pure and Applied Logic 161 (2):185-202.
    This paper aims at being a systematic investigation of different completeness properties of first-order predicate logics with truth-constants based on a large class of left-continuous t-norms . We consider standard semantics over the real unit interval but also we explore alternative semantics based on the rational unit interval and on finite chains. We prove that expansions with truth-constants are conservative and we study their real, rational and finite chain completeness properties. Particularly interesting is the case of considering canonical real and (...)
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  17.  14
    A Logical Framework for Graded Predicates.Petr Cintula, Carles Noguera & Nicholas J. J. Smith - 2017 - In Alexandru Baltag, Jeremy Seligman & Tomoyuki Yamada (eds.), Logic, Rationality, and Interaction: LORI 2017. Berlin: Springer. pp. 3-16.
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  18.  5
    Extension Properties and Subdirect Representation in Abstract Algebraic Logic.Tomáš Lávička & Carles Noguera - 2018 - Studia Logica 106 (6):1065-1095.
    This paper continues the investigation, started in Lávička and Noguera : 521–551, 2017), of infinitary propositional logics from the perspective of their algebraic completeness and filter extension properties in abstract algebraic logic. If follows from the Lindenbaum Lemma used in standard proofs of algebraic completeness that, in every finitary logic, intersection-prime theories form a basis of the closure system of all theories. In this article we consider the open problem of whether these properties can be transferred to lattices of filters (...)
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  19. Handbook of Mathematical Fuzzy Logic - Volume 3.Petr Cintula, Christian Fermüller & Carles Noguera (eds.) - 2015 - College Publications.
     
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