31 found
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  1.  30
    Categorical Abstract Algebraic Logic: Equivalent Institutions.George Voutsadakis - 2003 - Studia Logica 74 (1-2):275 - 311.
    A category theoretic generalization of the theory of algebraizable deductive systems of Blok and Pigozzi is developed. The theory of institutions of Goguen and Burstall is used to provide the underlying framework which replaces and generalizes the universal algebraic framework based on the notion of a deductive system. The notion of a term -institution is introduced first. Then the notions of quasi-equivalence, strong quasi-equivalence and deductive equivalence are defined for -institutions. Necessary and sufficient conditions are given for the quasi-equivalence and (...)
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  2.  30
    Categorical Abstract Algebraic Logic: Models of Π-Institutions.George Voutsadakis - 2005 - Notre Dame Journal of Formal Logic 46 (4):439-460.
    An important part of the theory of algebraizable sentential logics consists of studying the algebraic semantics of these logics. As developed by Czelakowski, Blok, and Pigozzi and Font and Jansana, among others, it includes studying the properties of logical matrices serving as models of deductive systems and the properties of abstract logics serving as models of sentential logics. The present paper contributes to the development of the categorical theory by abstracting some of these model theoretic aspects and results from the (...)
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  3.  8
    Categorical Abstract Algebraic Logic: Equivalent Institutions.George Voutsadakis - 2003 - Studia Logica 74 (1):275-311.
    A category theoretic generalization of the theory of algebraizable deductive systems of Blok and Pigozzi is developed. The theory of institutions of Goguen and Burstall is used to provide the underlying framework which replaces and generalizes the universal algebraic framework based on the notion of a deductive system. The notion of a term π-institution is introduced first. Then the notions of quasi-equivalence, strong quasi-equivalence and deductive equivalence are defined for π-institutions. Necessary and sufficient conditions are given for the quasi-equivalence and (...)
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  4.  4
    Categorical Abstract Algebraic Logic: Equivalential Π-Institutions.George Voutsadakis - 2008 - Australasian Journal of Logic 6:1-24.
    The theory of equivalential deductive systems, as introduced by Prucnal and Wrónski and further developed by Czelakowski, is abstracted to cover the case of logical systems formalized as π-institutions. More precisely, the notion of an N-equivalence system for a given π-institution is introduced. A characterization theorem for N-equivalence systems, previously proven for N-parameterized equivalence systems, is revisited and a “transfer theorem” for N-equivalence systems is proven. For a π-institution I having an N-equivalence system, the maximum such system is singled out (...)
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  5.  37
    Categorical Abstract Algebraic Logic: Prealgebraicity and Protoalgebraicity.George Voutsadakis - 2007 - Studia Logica 85 (2):215-249.
    Two classes of π are studied whose properties are similar to those of the protoalgebraic deductive systems of Blok and Pigozzi. The first is the class of N-protoalgebraic π-institutions and the second is the wider class of N-prealgebraic π-institutions. Several characterizations are provided. For instance, N-prealgebraic π-institutions are exactly those π-institutions that satisfy monotonicity of the N-Leibniz operator on theory systems and N-protoalgebraic π-institutions those that satisfy monotonicity of the N-Leibniz operator on theory families. Analogs of the correspondence property of (...)
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  6.  18
    Categorical Abstract Algebraic Logic: More on Protoalgebraicity.George Voutsadakis - 2006 - Notre Dame Journal of Formal Logic 47 (4):487-514.
    Protoalgebraic logics are characterized by the monotonicity of the Leibniz operator on their theory lattices and are at the lower end of the Leibniz hierarchy of abstract algebraic logic. They have been shown to be the most primitive among those logics with a strong enough algebraic character to be amenable to algebraic study techniques. Protoalgebraic π-institutions were introduced recently as an analog of protoalgebraic sentential logics with the goal of extending the Leibniz hierarchy from the sentential framework to the π-institution (...)
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  7.  9
    Categorical Abstract Algebraic Logic Metalogical Properties.George Voutsadakis - 2003 - Studia Logica 74 (3):369-398.
    Metalogical properties that have traditionally been studied in the deductive system context and transferred later to the institution context [33], are here formulated in the π-institution context. Preservation under deductive equivalence of π-institutions is investigated. If a property is known to hold in all algebraic π-institutions and is preserved under deductive equivalence, then it follows that it holds in all algebraizable π-institutions in the sense of [36].
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  8.  10
    Categorical Abstract Algebraic Logic: Categorical Algebraization of Equational Logic.George Voutsadakis - 2004 - Logic Journal of the IGPL 12 (4):313-333.
    This paper deals with the algebraization of multi-signature equational logic in the context of the modern theory of categorical abstract algebraic logic. Two are the novelties compared to traditional treatments: First, interpretations between different algebraic types are handled in the object language rather than the metalanguage. Second, rather than constructing the type of the algebraizing class of algebras explicitly in an ad-hoc universal algebraic way, the whole clone is naturally constructed using categorical algebraic techniques.
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  9.  19
    Categorical Abstract Algebraic Logic Metalogical Properties.George Voutsadakis - 2003 - Studia Logica 74 (3):369 - 398.
    Metalogical properties that have traditionally been studied in the deductive system context (see, e.g., [21]) and transferred later to the institution context [33], are here formulated in the -institution context. Preservation under deductive equivalence of -institutions is investigated. If a property is known to hold in all algebraic -institutions and is preserved under deductive equivalence, then it follows that it holds in all algebraizable -institutions in the sense of [36].
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  10.  5
    Categorical Abstract Algebraic Logic: Bloom's Theorem for Rule-Based Π-Institutions.George Voutsadakis - 2008 - Logic Journal of the IGPL 16 (3):233-248.
    A syntactic machinery is developed for π-institutions based on the notion of a category of natural transformations on their sentence functors. Rules of inference, similar to the ones traditionally used in the sentential logic framework to define the best known sentential logics, are, then, introduced for π-institutions. A π-institution is said to be rule-based if its closure system is induced by a collection of rules of inference. A logical matrix-like semantics is introduced for rule-based π-institutions and a version of Bloom's (...)
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  11.  10
    Categorical Abstract Algebraic Logic: The Criterion for Deductive Equivalence.George Voutsadakis - 2003 - Mathematical Logic Quarterly 49 (4):347-352.
    Equivalent deductive systems were introduced in [4] with the goal of treating 1-deductive systems and algebraic 2-deductive systems in a uniform way. Results of [3], appropriately translated and strengthened, show that two deductive systems over the same language type are equivalent if and only if their lattices of theories are isomorphic via an isomorphism that commutes with substitutions. Deductive equivalence of π-institutions [14, 15] generalizes the notion of equivalence of deductive systems. In [15, Theorem 10.26] this criterion for the equivalence (...)
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  12.  47
    Categorical Abstract Algebraic Logic Categorical Algebraization of First-Order Logic Without Terms.George Voutsadakis - 2004 - Archive for Mathematical Logic 44 (4):473-491.
    An algebraization of multi-signature first-order logic without terms is presented. Rather than following the traditional method of choosing a type of algebras and constructing an appropriate variety, as is done in the case of cylindric and polyadic algebras, a new categorical algebraization method is used: The substitutions of formulas of one signature for relation symbols in another are treated in the object language. This enables the automatic generation via an adjunction of an algebraic theory. The algebras of this theory are (...)
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  13.  7
    Categorical Abstract Algebraic Logic: Structurality, Protoalgebraicity, and Correspondence.George Voutsadakis - 2009 - Mathematical Logic Quarterly 55 (1):51-67.
    The notion of an ℐ -matrix as a model of a given π -institution ℐ is introduced. The main difference from the approach followed so far in CategoricalAlgebraic Logic and the one adopted here is that an ℐ -matrix is considered modulo the entire class of morphisms from the underlying N -algebraic system of ℐ into its own underlying algebraic system, rather than modulo a single fixed -logical morphism. The motivation for introducing ℐ -matrices comes from a desire to formulate (...)
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  14.  12
    Categorical Abstract Algebraic Logic: Gentzen Π ‐Institutions and the Deduction‐Detachment Property.George Voutsadakis - 2005 - Mathematical Logic Quarterly 51 (6):570-578.
    Given a π -institution I , a hierarchy of π -institutions I is constructed, for n ≥ 1. We call I the n-th order counterpart of I . The second-order counterpart of a deductive π -institution is a Gentzen π -institution, i.e. a π -institution associated with a structural Gentzen system in a canonical way. So, by analogy, the second order counterpart I of I is also called the “Gentzenization” of I . In the main result of the paper, it (...)
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  15.  13
    Categorical Abstract Algebraic Logic: The Largest Theory System Included in a Theory Family.George Voutsadakis - 2006 - Mathematical Logic Quarterly 52 (3):288-294.
    In this note, it is shown that, given a π -institution ℐ = 〈Sign, SEN, C 〉, with N a category of natural transformations on SEN, every theory family T of ℐ includes a unique largest theory system equation image of ℐ. equation image satisfies the important property that its N -Leibniz congruence system always includes that of T . As a consequence, it is shown, on the one hand, that the relation ΩN = ΩN characterizes N -protoalgebraicity inside the (...)
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  16.  7
    Categorical Abstract Algebraic Logic: The Categorical Suszko Operator.George Voutsadakis - 2007 - Mathematical Logic Quarterly 53 (6):616-635.
    Czelakowski introduced the Suszko operator as a basis for the development of a hierarchy of non-protoalgebraic logics, paralleling the well-known abstract algebraic hierarchy of protoalgebraic logics based on the Leibniz operator of Blok and Pigozzi. The scope of the theory of the Leibniz operator was recently extended to cover the case of, the so-called, protoalgebraic π-institutions. In the present work, following the lead of Czelakowski, an attempt is made at lifting parts of the theory of the Suszko operator to the (...)
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  17. Categorical Abstract Algebraic Logic: Syntactically Algebraizable \-Institutions.George Voutsadakis - 2009 - Reports on Mathematical Logic:105-151.
    This paper has a two-fold purpose. On the one hand, it introduces the concept of a syntactically \-algebraizable \-institution, which generalizes in the context of categorical abstract algebraic logic the notion of an algebraizable logic of Blok and Pigozzi. On the other hand, it has the purpose of comparing this important notion with the weaker ones of an \-protoalgebraic and of a syntactically \-equivalential \-institution and with the stronger one of a regularly \-algebraizable \-institution. \-protoalgebraic \-institutions and syntactically \-equivalential \-institutions (...)
     
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  18. Categorical Abstract Algebraic Logic: Full Models, Frege Systems and Metalogical Properties.George Voutsadakis - 2006 - Reports on Mathematical Logic.
     
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  19.  13
    Categorical Abstract Algebraic Logic: Truth-Equational $Pi$-Institutions.George Voutsadakis - 2015 - Notre Dame Journal of Formal Logic 56 (2):351-378.
    Finitely algebraizable deductive systems were introduced by Blok and Pigozzi to capture the essential properties of those deductive systems that are very tightly connected to quasivarieties of universal algebras. They include the equivalential logics of Czelakowski. Based on Blok and Pigozzi’s work, Herrmann defined algebraizable deductive systems. These are the equivalential deductive systems that are also truth-equational, in the sense that the truth predicate of the class of their reduced matrix models is explicitly definable by some set of unary equations. (...)
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  20.  11
    Categorical Abstract Algebraic Logic: Behavioral Π-Institutions.George Voutsadakis - 2014 - Studia Logica 102 (3):617-646.
    Recently, Caleiro, Gon¸calves and Martins introduced the notion of behaviorally algebraizable logic. The main idea behind their work is to replace, in the traditional theory of algebraizability of Blok and Pigozzi, unsorted equational logic with multi-sorted behavioral logic. The new notion accommodates logics over many-sorted languages and with non-truth-functional connectives. Moreover, it treats logics that are not algebraizable in the traditional sense while, at the same time, shedding new light to the equivalent algebraic semantics of logics that are algebraizable according (...)
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  21.  13
    Categorical Abstract Algebraic Logic: Skywatching in Semilattice Systems.George Voutsadakis - 2016 - Logic Journal of the IGPL 24 (2):138-155.
  22.  4
    Categorical Abstract Algebraic Logic: Pseudo-Referential Matrix System Semantics.George Voutsadakis - 2018 - Bulletin of the Section of Logic 47 (2):69.
    This work adapts techniques and results first developed by Malinowski and by Marek in the context of referential semantics of sentential logics to the context of logics formalized as π-institutions. More precisely, the notion of a pseudoreferential matrix system is introduced and it is shown how this construct generalizes that of a referential matrix system. It is then shown that every π–institution has a pseudo-referential matrix system semantics. This contrasts with referential matrix system semantics which is only available for self-extensional (...)
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  23.  1
    Categorical Abstract Algebraic Logic: Compatibility Operators and Correspondence Theorems.George Voutsadakis - 2018 - In Janusz Czelakowski (ed.), Don Pigozzi on Abstract Algebraic Logic, Universal Algebra, and Computer Science. Springer Verlag.
    Very recently Albuquerque, Font and Jansana, based on preceding work of Czelakowski on compatibility operators, introduced coherent compatibility operators and used Galois connections, formed by these operators, to provide a unified framework for the study of the Leibniz, the Suszko and the Tarski operators of abstract algebraic logic. Based on this work, we present a unified treatment of the operator approach to the categorical abstract algebraic logic hierarchy of π-institutions. This approach encompasses previous work by the author in this area, (...)
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  24.  23
    Categorical Abstract Algebraic Logic: Referential Algebraic Semantics.George Voutsadakis - 2013 - Studia Logica 101 (4):849-899.
    Wójcicki has provided a characterization of selfextensional logics as those that can be endowed with a complete local referential semantics. His result was extended by Jansana and Palmigiano, who developed a duality between the category of reduced congruential atlases and that of reduced referential algebras over a fixed similarity type. This duality restricts to one between reduced atlas models and reduced referential algebra models of selfextensional logics. In this paper referential algebraic systems and congruential atlas systems are introduced, which abstract (...)
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  25.  16
    Categorical Abstract Algebraic Logic: Algebraic Semantics for (Documentclass{Article}Usepackage{Amssymb}Begin{Document}Pagestyle{Empty}$Bf{Pi }$End{Document})‐Institutions.George Voutsadakis - 2013 - Mathematical Logic Quarterly 59 (3):177-200.
  26.  8
    Categorical Abstract Algebraic Logic: The Diagram and the Reduction Operator Lemmas.George Voutsadakis - 2007 - Mathematical Logic Quarterly 53 (2):147-161.
    The study of structure systems, an abstraction of the concept of first-order structures, is continued. Structure systems have algebraic systems as their algebraic reducts and their relational component consists of a collection of relation systems on the underlying functors. An analog of the expansion of a first-order structure by constants is presented. Furthermore, analogs of the Diagram Lemma and the Reduction Operator Lemma from the theory of equality-free first-order structures are provided in the framework of structure systems. (© 2007 WILEY-VCH (...)
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  27. Categorical Abstract Algebraic Logic: The Criterion for Deductive Equivalence: The Criterion for Deductive Equivalence.George Voutsadakis - 2003 - Mathematical Logic Quarterly 49 (4):347.
     
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  28.  7
    Categorical Abstract Logic: Hidden Multi-Sorted Logics as Multi-Term Π-Institutions.George Voutsadakis - 2016 - Bulletin of the Section of Logic 45 (2).
    Babenyshev and Martins proved that two hidden multi-sorted deductive systems are deductively equivalent if and only if there exists an isomorphism between their corresponding lattices of theories that commutes with substitutions. We show that the π-institutions corresponding to the hidden multi-sorted deductive systems studied by Babenyshev and Martins satisfy the multi-term condition of Gil-F´erez. This provides a proof of the result of Babenyshev and Martins by appealing to the general result of Gil-F´erez pertaining to arbitrary multi-term π-institutions. The approach places (...)
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  29. Caal: Categorical Abstract Algebraic Logic: Coordinatization Is Algebraization.George Voutsadakis - 2012 - Reports on Mathematical Logic:125-145.
     
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  30.  11
    Corrigendum to “Categorical Abstract Algebraic Logic: The Criterion for Deductive Equivalence”.George Voutsadakis - 2005 - Mathematical Logic Quarterly 51 (6):644-644.
    We give a correction to the paper [2] mentioned in the title.
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  31. Secrecy Logic: Protoalgebraic S-Secrecy Logics.George Voutsadakis - 2012 - Reports on Mathematical Logic:3-28.
     
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