Switch to: References

Add citations

You must login to add citations.
  1. An Autobiography of Polyadic Algebras.Paul R. Halmos - 2000 - Logic Journal of the IGPL 8 (4):383-392.
    Download  
     
    Export citation  
     
    Bookmark   3 citations  
  • Lambek Calculus and its Relational Semantics: Completeness and Incompleteness. [REVIEW]Hajnal Andréka & Szabolcs Mikulás - 1994 - Journal of Logic, Language and Information 3 (1):1-37.
    The problem of whether Lambek Calculus is complete with respect to (w.r.t.) relational semantics, has been raised several times, cf. van Benthem (1989a) and van Benthem (1991). In this paper, we show that the answer is in the affirmative. More precisely, we will prove that that version of the Lambek Calculus which does not use the empty sequence is strongly complete w.r.t. those relational Kripke-models where the set of possible worlds,W, is a transitive binary relation, while that version of the (...)
    Download  
     
    Export citation  
     
    Bookmark   11 citations  
  • Directions in Generalized Quantifier Theory.Dag Westerståhl & J. F. A. K. van Benthem - 1995 - Studia Logica 55 (3):389-419.
    We give a condensed survey of recent research on generalized quantifiers in logic, linguistics and computer science, under the following headings: Logical definability and expressive power, Polyadic quantifiers and linguistic definability, Weak semantics and axiomatizability, Computational semantics, Quantifiers in dynamic settings, Quantifiers and modal logic, Proof theory of generalized quantifiers.
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  • A Finitely Axiomatized Formalization of Predicate Calculus with Equality.Norman D. Megill - 1995 - Notre Dame Journal of Formal Logic 36 (3):435-453.
    We present a formalization of first-order predicate calculus with equality which, unlike traditional systems with axiom schemata or substitution rules, is finitely axiomatized in the sense that each step in a formal proof admits only finitely many choices. This formalization is primarily based on the inference rule of condensed detachment of Meredith. The usual primitive notions of free variable and proper substitution are absent, making it easy to verify proofs in a machine-oriented application. Completeness results are presented. The example of (...)
    Download  
     
    Export citation  
     
    Bookmark  
  • Omitting Types for Finite Variable Fragments and Complete Representations of Algebras.Hajnal Andréka, István Németi & Tarek Sayed Ahmed - 2008 - Journal of Symbolic Logic 73 (1):65-89.
    We give a novel application of algebraic logic to first order logic. A new, flexible construction is presented for representable but not completely representable atomic relation and cylindric algebras of dimension n (for finite n > 2) with the additional property that they are one-generated and the set of all n by n atomic matrices forms a cylindric basis. We use this construction to show that the classical Henkin-Orey omitting types theorem fails for the finite variable fragments of first order (...)
    Download  
     
    Export citation  
     
    Bookmark   9 citations  
  • Algebraic Structuralism.Neil Dewar - 2019 - Philosophical Studies 176 (7):1831-1854.
    This essay is about how the notion of “structure” in ontic structuralism might be made precise. More specifically, my aim is to make precise the idea that the structure of the world is given by the relations inhering in the world, in such a way that the relations are ontologically prior to their relata. The central claim is the following: one can do so by giving due attention to the relationships that hold between those relations, by making use of certain (...)
    Download  
     
    Export citation  
     
    Bookmark  
  • A Representation Theorem for Measurable Relation Algebras.Steven Givant & Hajnal Andréka - 2018 - Annals of Pure and Applied Logic 169 (11):1117-1189.
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  • Omitting Types for Finite Variable Fragments of First Order Logic.T. Sayed Ahmed - 2003 - Bulletin of the Section of Logic 32 (3):103-107.
    Download  
     
    Export citation  
     
    Bookmark   3 citations  
  • Cylindric Modal Logic.Yde Venema - 1995 - Journal of Symbolic Logic 60 (2):591-623.
    Treating the existential quantification ∃ν i as a diamond $\diamond_i$ and the identity ν i = ν j as a constant δ ij , we study restricted versions of first order logic as if they were modal formalisms. This approach is closely related to algebraic logic, as the Kripke frames of our system have the type of the atom structures of cylindric algebras; the full cylindric set algebras are the complex algebras of the intended multidimensional frames called cubes. The main (...)
    Download  
     
    Export citation  
     
    Bookmark   24 citations  
  • Squares in Fork Arrow Logic.Renata P. de Freitas, Jorge P. Viana, Mario R. F. Benevides, Sheila R. M. Veloso & Paulo A. S. Veloso - 2003 - Journal of Philosophical Logic 32 (4):343-355.
    In this paper we show that the class of fork squares has a complete orthodox axiomatization in fork arrow logic (FAL). This result may be seen as an orthodox counterpart of Venema's non-orthodox axiomatization for the class of squares in arrow logic. FAL is the modal logic of fork algebras (FAs) just as arrow logic is the modal logic of relation algebras (RAs). FAs extend RAs by a binary fork operator and are axiomatized by adding three equations to RAs equational (...)
    Download  
     
    Export citation  
     
    Bookmark  
  • Modal Languages and Bounded Fragments of Predicate Logic.Hajnal Andréka, István Németi & Johan van Benthem - 1998 - Journal of Philosophical Logic 27 (3):217 - 274.
    What precisely are fragments of classical first-order logic showing “modal” behaviour? Perhaps the most influential answer is that of Gabbay 1981, which identifies them with so-called “finite-variable fragments”, using only some fixed finite number of variables (free or bound). This view-point has been endorsed by many authors (cf. van Benthem 1991). We will investigate these fragments, and find that, illuminating and interesting though they are, they lack the required nice behaviour in our sense. (Several new negative results support this claim.) (...)
    Download  
     
    Export citation  
     
    Bookmark   70 citations  
  • On Amalgamation in Algebras of Logic.Tarek Sayed Ahmed - 2005 - Studia Logica 81 (1):61-77.
    We show that not all epimorphisms are surjective in certain classes of infinite dimensional cylindric algebras, Pinter's substitution algebras and Halmos' quasipolyadic algebras with and without equality. It follows that these classes fail to have the strong amalgamation property. This answers a question in [3] and a question of Pigozzi in his landmark paper on amalgamation [9]. The cylindric case was first proved by Judit Madarasz [7]. The proof presented herein is substantially different. By a result of Németi, our result (...)
    Download  
     
    Export citation  
     
    Bookmark   4 citations  
  • Sahlqvist's Theorem for Boolean Algebras with Operators with an Application to Cylindric Algebras.Maarten de Rijke & Yde Venema - 1995 - Studia Logica 54 (1):61-78.
    For an arbitrary similarity type of Boolean Algebras with Operators we define a class ofSahlqvist identities. Sahlqvist identities have two important properties. First, a Sahlqvist identity is valid in a complex algebra if and only if the underlying relational atom structure satisfies a first-order condition which can be effectively read off from the syntactic form of the identity. Second, and as a consequence of the first property, Sahlqvist identities arecanonical, that is, their validity is preserved under taking canonical embedding algebras. (...)
    Download  
     
    Export citation  
     
    Bookmark   12 citations  
  • On Fork Arrow Logic and its Expressive Power.Paulo A. S. Veloso, Renata P. de Freitas, Petrucio Viana, Mario Benevides & Sheila R. M. Veloso - 2007 - Journal of Philosophical Logic 36 (5):489 - 509.
    We compare fork arrow logic, an extension of arrow logic, and its natural first-order counterpart (the correspondence language) and show that both have the same expressive power. Arrow logic is a modal logic for reasoning about arrow structures, its expressive power is limited to a bounded fragment of first-order logic. Fork arrow logic is obtained by adding to arrow logic the fork modality (related to parallelism and synchronization). As a result, fork arrow logic attains the expressive power of its first-order (...)
    Download  
     
    Export citation  
     
    Bookmark  
  • Taming Logic.Maarten Marx, Szabolcs Mikul & István Németi - 1995 - Journal of Logic, Language and Information 4 (3):207-226.
    In this paper, we introduce a general technology, calledtaming, for finding well-behaved versions of well-investigated logics. Further, we state completeness, decidability, definability and interpolation results for a multimodal logic, calledarrow logic, with additional operators such as thedifference operator, andgraded modalities. Finally, we give a completeness proof for a strong version of arrow logic.
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  • On Complete Representations of Reducts of Polyadic Algebras.Tarek Sayed Ahmed - 2008 - Studia Logica 89 (3):325-332.
    Following research initiated by Tarski, Craig and Nemeti, and futher pursued by Sain and others, we show that for certain subsets G of $^\omega \omega $ , atomic countable G poiyadic algebras are completely representable. G polyadic algebras are obtained by restricting the similarity type and axiomatization of ω-dimensional polyadic algebras to finite quantifiers and substitutions in G. This contrasts the cases of cylindric and relation algebras.
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  • A Note on Neat Reducts.Tarek Sayed Ahmed - 2007 - Studia Logica 85 (2):139-151.
    SC, CA, QA and QEA denote the class of Pinter’s substitution algebras, Tarski’s cylindric algebras, Halmos’ quasi-polyadic and quasi-polyadic equality algebras, respectively. Let . and . We show that the class of n dimensional neat reducts of algebras in K m is not elementary. This solves a problem in [2]. Also our result generalizes results proved in [1] and [2].
    Download  
     
    Export citation  
     
    Bookmark   1 citation  
  • Derivation Rules as Anti-Axioms.Yde Venema - 1993 - Journal of Symbolic Logic 58:1003-1034.
    Download  
     
    Export citation  
     
    Bookmark   3 citations  
  • Interpolation in Algebraizable Logics Semantics for Non-Normal Multi-Modal Logic.Judit X. Madarász - 1998 - Journal of Applied Non-Classical Logics 8 (1):67-105.
    ABSTRACT The two main directions pursued in the present paper are the following. The first direction was started by Pigozzi in 1969. In [Mak 91] and [Mak 79] Maksimova proved that a normal modal logic has the Craig interpolation property iff the corresponding class of algebras has the superamalgamation property. In this paper we extend Maksimova's theorem to normal multi-modal logics with arbitrarily many, not necessarily unary modalities, and to not necessarily normal multi-modal logics with modalities of ranks smaller than (...)
    Download  
     
    Export citation  
     
    Bookmark   2 citations  
  • Derivation Rules as Anti-Axioms in Modal Logic.Yde Venema - 1993 - Journal of Symbolic Logic 58 (3):1003-1034.
    We discuss a `negative' way of defining frame classes in (multi)modal logic, and address the question of whether these classes can be axiomatized by derivation rules, the `non-ξ rules', styled after Gabbay's Irreflexivity Rule. The main result of this paper is a metatheorem on completeness, of the following kind: If Λ is a derivation system having a set of axioms that are special Sahlqvist formulas and Λ+ is the extension of Λ with a set of non-ξ rules, then Λ+ is (...)
    Download  
     
    Export citation  
     
    Bookmark   27 citations  
  • Decidability of Cylindric Set Algebras of Dimension Two and First-Order Logic with Two Variables.Maarten Marx & Szabolcs Mikulás - 1999 - Journal of Symbolic Logic 64 (4):1563-1572.
    The aim of this paper is to give a new proof for the decidability and finite model property of first-order logic with two variables (without function symbols), using a combinatorial theorem due to Herwig. The results are proved in the framework of polyadic equality set algebras of dimension two (Pse 2 ). The new proof also shows the known results that the universal theory of Pse 2 is decidable and that every finite Pse 2 can be represented on a finite (...)
    Download  
     
    Export citation  
     
    Bookmark  
  • Undecidable Relativizations of Algebras of Relations.Szabolcs Mikulás & Maarten Marx - 1999 - Journal of Symbolic Logic 64 (2):747-760.
    In this paper we show that relativized versions of relation set algebras and cylindric set algebras have undecidable equational theories if we include coordinatewise versions of the counting operations into the similarity type. We apply these results to the guarded fragment of first-order logic.
    Download  
     
    Export citation  
     
    Bookmark  
  • Complexity of Equational Theory of Relational Algebras with Projection Elements.Szabolcs Mikulás, Ildikó Sain & Andras Simon - 1992 - Bulletin of the Section of Logic 21 (3):103-111.
    The class \ of t rue p airing a lgebras is defined to be the class of relation algebras expanded with concrete set theoretical projection functions. The main results of the present paper is that neither the equational theory of \ nor the first order theory of \ are decidable. Moreover, we show that the set of all equations valid in \ is exactly on the \ level. We consider the class \ of the relation algebra reducts of \ ’s, (...)
    Download  
     
    Export citation  
     
    Bookmark   5 citations  
  • Step by Step – Building Representations in Algebraic Logic.Robin Hirsch & Ian Hodkinson - 1997 - Journal of Symbolic Logic 62 (1):225-279.
    We consider the problem of finding and classifying representations in algebraic logic. This is approached by letting two players build a representation using a game. Homogeneous and universal representations are characterized according to the outcome of certain games. The Lyndon conditions defining representable relation algebras (for the finite case) and a similar schema for cylindric algebras are derived. Finite relation algebras with homogeneous representations are characterized by first order formulas. Equivalence games are defined, and are used to establish whether an (...)
    Download  
     
    Export citation  
     
    Bookmark   10 citations  
  • Notions of Density That Imply Representability in Algebraic Logic.Hajnal Andréka, Steven Givant, Szabolcs Mikulás, István Németi & András Simon - 1998 - Annals of Pure and Applied Logic 91 (2-3):93-190.
    Henkin and Tarski proved that an atomic cylindric algebra in which every atom is a rectangle must be representable . This theorem and its analogues for quasi-polyadic algebras with and without equality are formulated in Henkin, Monk and Tarski [13]. We introduce a natural and more general notion of rectangular density that can be applied to arbitrary cylindric and quasi-polyadic algebras, not just atomic ones. We then show that every rectangularly dense cylindric algebra is representable, and we extend this result (...)
    Download  
     
    Export citation  
     
    Bookmark   7 citations  
  • On the Definition and the Representability of Quasi-Polyadic Equality Algebras.Miklós Ferenczi - 2016 - Mathematical Logic Quarterly 62 (1-2):9-15.
    Download  
     
    Export citation  
     
    Bookmark  
  • Monadic GMV-Algebras.Jiří Rachůnek & Dana Šalounová - 2008 - Archive for Mathematical Logic 47 (3):277-297.
    Monadic MV-algebras are an algebraic model of the predicate calculus of the Łukasiewicz infinite valued logic in which only a single individual variable occurs. GMV-algebras are a non-commutative generalization of MV-algebras and are an algebraic counterpart of the non-commutative Łukasiewicz infinite valued logic. We introduce monadic GMV-algebras and describe their connections to certain couples of GMV-algebras and to left adjoint mappings of canonical embeddings of GMV-algebras. Furthermore, functional MGMV-algebras are studied and polyadic GMV-algebras are introduced and discussed.
    Download  
     
    Export citation  
     
    Bookmark   2 citations  
  • The Universal Theory of First Order Algebras and Various Reducts.Lawrence Valby - 2015 - Logica Universalis 9 (4):475-500.
    First order formulas in a relational signature can be considered as operations on the relations of an underlying set, giving rise to multisorted algebras we call first order algebras. We present universal axioms so that an algebra satisfies the axioms iff it embeds into a first order algebra. Importantly, our argument is modular and also works for, e.g., the positive existential algebras and the quantifier-free algebras. We also explain the relationship to theories, and indicate how to add in function symbols.
    Download  
     
    Export citation  
     
    Bookmark  
  • Complexity of Equational Theory of Relational Algebras with Standard Projection Elements.Szabolcs Mikulás, Ildikó Sain & András Simon - 2015 - Synthese 192 (7):2159-2182.
    The class $$\mathsf{TPA}$$ TPA of t rue p airing a lgebras is defined to be the class of relation algebras expanded with concrete set theoretical projection functions. The main results of the present paper is that neither the equational theory of $$\mathsf{TPA}$$ TPA nor the first order theory of $$\mathsf{TPA}$$ TPA are decidable. Moreover, we show that the set of all equations valid in $$\mathsf{TPA}$$ TPA is exactly on the $$\Pi ^1_1$$ Π 1 1 level. We consider the class $$\mathsf{TPA}^-$$ (...)
    Download  
     
    Export citation  
     
    Bookmark  
  • Undecidable Theories of Lyndon Algebras.Vera Stebletsova & Yde Venema - 2001 - Journal of Symbolic Logic 66 (1):207-224.
    With each projective geometry we can associate a Lyndon algebra. Such an algebra always satisfies Tarski's axioms for relation algebras and Lyndon algebras thus form an interesting connection between the fields of projective geometry and algebraic logic. In this paper we prove that if G is a class of projective geometries which contains an infinite projective geometry of dimension at least three, then the class L(G) of Lyndon algebras associated with projective geometries in G has an undecidable equational theory. In (...)
    Download  
     
    Export citation  
     
    Bookmark  
  • On the Equational Theory of Representable Polyadic Equality Algebras.István Németi & Gábor Sági - 2000 - Journal of Symbolic Logic 65 (3):1143-1167.
    Among others we will prove that the equational theory of ω dimensional representable polyadic equality algebras (RPEA ω 's) is not schema axiomatizable. This result is in interesting contrast with the Daigneault-Monk representation theorem, which states that the class of representable polyadic algebras is finite schema-axiomatizable (and hence the equational theory of this class is finite schema-axiomatizable, as well). We will also show that the complexity of the equational theory of RPEA ω is also extremely high in the recursion theoretic (...)
    Download  
     
    Export citation  
     
    Bookmark   10 citations  
  • Turning Decision Procedures Into Disprovers.André Rognes - 2009 - Mathematical Logic Quarterly 55 (1):87-104.
    A class of many-sorted polyadic set algebras is introduced. These generalise structure and model in a way that is relevant in regards to the Entscheidungsproblem and to automated reasoning.A downward Löwenheim-Skolem property is shown in that each satisfiable finite conjunction of purely relational first-order prenex sentences has a finite generalised model. This property does, together with a construction related to doubling the size of a finite structure, provide several strict generalisations of the strategy of finite model search for disproving.
    Download  
     
    Export citation  
     
    Bookmark  
  • On Fork Arrow Logic and Its Expressive Power.Paulo A. S. Veloso, Renata P. De Freitas, Petrucio Viana, Mario Benevides & Sheila R. M. Veloso - 2007 - Journal of Philosophical Logic 36 (5):489 - 509.
    We compare fork arrow logic, an extension of arrow logic, and its natural first-order counterpart (the correspondence language) and show that both have the same expressive power. Arrow logic is a modal logic for reasoning about arrow structures, its expressive power is limited to a bounded fragment of first-order logic. Fork arrow logic is obtained by adding to arrow logic the fork modality (related to parallelism and synchronization). As a result, fork arrow logic attains the expressive power of its first-order (...)
    Download  
     
    Export citation  
     
    Bookmark  
  • Sahlqvist's Theorem for Boolean Algebras with Operators with an Application to Cylindric Algebras.Maarten De Rijke & Yde Venema - 1995 - Studia Logica 54 (1):61 - 78.
    For an arbitrary similarity type of Boolean Algebras with Operators we define a class of Sahlqvist identities. Sahlqvist identities have two important properties. First, a Sahlqvist identity is valid in a complex algebra if and only if the underlying relational atom structure satisfies a first-order condition which can be effectively read off from the syntactic form of the identity. Second, and as a consequence of the first property, Sahlqvist identities are canonical, that is, their validity is preserved under taking canonical (...)
    Download  
     
    Export citation  
     
    Bookmark   9 citations