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  1. Algebraic Logic, Where Does It Stand Today?Tarek Sayed Ahmed - 2005 - Bulletin of Symbolic Logic 11 (3):465-516.
    This is a survey article on algebraic logic. It gives a historical background leading up to a modern perspective. Central problems in algebraic logic (like the representation problem) are discussed in connection to other branches of logic, like modal logic, proof theory, model-theoretic forcing, finite combinatorics, and Gödel’s incompleteness results. We focus on cylindric algebras. Relation algebras and polyadic algebras are mostly covered only insofar as they relate to cylindric algebras, and even there we have not told the whole story. (...)
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  • On Generalization of Definitional Equivalence to Non-Disjoint Languages.Koen Lefever & Gergely Székely - 2019 - Journal of Philosophical Logic 48 (4):709-729.
    For simplicity, most of the literature introduces the concept of definitional equivalence only for disjoint languages. In a recent paper, Barrett and Halvorson introduce a straightforward generalization to non-disjoint languages and they show that their generalization is not equivalent to intertranslatability in general. In this paper, we show that their generalization is not transitive and hence it is not an equivalence relation. Then we introduce another formalization of definitional equivalence due to Andréka and Németi which is equivalent to the Barrett–Halvorson (...)
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  • Modal Cognitivism and Modal Expressivism.Hasen Khudairi - manuscript
    This paper aims to provide a mathematically tractable background against which to model both modal cognitivism and modal expressivism. I argue that epistemic modal algebras comprise a materially adequate fragment of the language of thought, and endeavor to show how such algebras provide the resources necessary to resolve Russell's paradox of propositions. I demonstrate, then, how modal expressivism can be regimented by modal coalgebraic automata, to which the above epistemic modal algebras are dually isomorphic. I examine, in particular, the virtues (...)
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  • Forms of Luminosity.Hasen Khudairi - 2017
    This dissertation concerns the foundations of epistemic modality. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The dissertation demonstrates how phenomenal consciousness and gradational possible-worlds models in Bayesian perceptual psychology relate to epistemic modal space. The dissertation demonstrates, then, how epistemic modality relates to the computational theory of mind; metaphysical modality; deontic modality; logical modality; the types of mathematical modality; to the (...)
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  • Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism.Hasen Khudairi - 2019 - In Matteo Vincenzo D'Alfonso & Don Berkich (eds.), On the Cognitive, Ethical, and Scientific Dimensions of Artificial Intelligence. Springer.
    This essay examines the philosophical significance of Ω-logic in Zermelo-Fraenkel set theory with choice (ZFC). The dual isomorphism between algebra and coalgebra permits Boolean-valued algebraic models of ZFC to be interpreted as coalgebras. The modal profile of Ω-logical validity can then be countenanced within a coalgebraic logic, and Ω-logical validity can be defined via deterministic automata. I argue that the philosophical significance of the foregoing is two-fold. First, because the epistemic and modal profiles of Ω-logical validity correspond to those of (...)
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  • 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.
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  • Duality in Logic and Language.Iep Author - 2016
    Duality in Logic and Language [draft--do not cite this article] Duality phenomena occur in nearly all mathematically formalized disciplines, such as algebra, geometry, logic and natural language semantics. However, many of these disciplines use the term ‘duality’ in vastly different senses, and while some of these senses are intimately connected to each other, others seem to be entirely … Continue reading Duality in Logic and Language →.
     
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  • First Order Logic Without Equality on Relativized Semantics.Amitayu Banerjee & Mohamed Khaled - 2018 - Annals of Pure and Applied Logic 169 (11):1227-1242.
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  • Structuralism and Representation Theorems.George Weaver - 1998 - Philosophia Mathematica 6 (3):257-271.
    Much of the inspiration for structuralist approaches to mathematics can be found in the late nineteenth- and early twentieth-century program of characterizing various mathematical systems upto isomorphism. From the perspective of this program, differences between isomorphic systems are irrelevant. It is argued that a different view of the import of the differences between isomorphic systems can be obtained from the perspective of contemporary discussions of representation theorems and that from this perspective both the identification of isomorphic systems and the reduction (...)
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  • Truth Theories, Competence, and Semantic Computation.Peter Pagin - 2012 - In Gerhard Preyer (ed.), Donald Davidson on Truth, Meaning, and the Mental. Oxford University Press. pp. 49.
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  • On the Unusual Effectiveness of Logic in Computer Science.Joseph Y. Halpern, Robert Harper, Neil Immerman, Phokion G. Kolaitis, Moshe Y. Vardi & Victor Vianu - 2001 - Bulletin of Symbolic Logic 7 (2):213-236.
  • Omitting Types for Algebraizable Extensions of First Order Logic.Tarek Sayed Ahmed - 2005 - Journal of Applied Non-Classical Logics 15 (4):465-489.
    We prove an Omitting Types Theorem for certain algebraizable extensions of first order logic without equality studied in [SAI 00] and [SAY 04]. This is done by proving a representation theorem preserving given countable sets of infinite meets for certain reducts of ?- dimensional polyadic algebras, the so-called G polyadic algebras (Theorem 5). Here G is a special subsemigroup of (?, ? o) that specifies the signature of the algebras in question. We state and prove an independence result connecting our (...)
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  • Epistemic Modality, Mind, and Mathematics.Hasen Khudairi - 2017 - Gutenberg.
    This book concerns the foundations of epistemic modality. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how phenomenal consciousness and gradational possible-worlds models in Bayesian perceptual psychology relate to epistemic modal space. The book demonstrates, then, how epistemic modality relates to the computational theory of mind; metaphysical modality; deontic modality; logical modality; the types of mathematical modality; to the (...)
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  • 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 (...)
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  • Copeland Algebras.Daniel B. Demaree - 1972 - Journal of Symbolic Logic 37 (4):646-656.
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  • Undecidable Properties of Finite Sets of Equations.George F. McNulty - 1976 - Journal of Symbolic Logic 41 (3):589-604.
  • Projective Algebra and the Calculus of Relations.A. R. Bednarek & S. M. Ulam - 1978 - Journal of Symbolic Logic 43 (1):56-64.
  • Transcendental Ethics: Hyperintensional Foundations of Value Theory.Hasen Khudairi - 2017 - Gutenberg.
    This book concerns the modal and hyperintensional foundations of ethics. The book provides three metaphysical frameworks in which to examine the nature of ethical value, obligation, cognition, and consciousness. The first framework is an algebraic semantics, and provides support for the claim that truths about modality are metaphysically fundamental. The second framework is a multi-dimensional hyperintensional semantics, and is shown to provide a natural setting in which to address issues both in decision theory and which concern intentional action. The third (...)
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  • Isomorphic but Not Lower Base-Isomorphic Cylindric Set Algebras.B. Biró & S. Shelah - 1988 - Journal of Symbolic Logic 53 (3):846-853.
    This paper belongs to cylindric-algebraic model theory understood in the sense of algebraic logic. We show the existence of isomorphic but not lower base-isomorphic cylindric set algebras. These algebras are regular and locally finite. This solves a problem raised in [N 83] which was implicitly present also in [HMTAN 81]. This result implies that a theorem of Vaught for prime models of countable languages does not continue to hold for languages of any greater power.
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  • Expressibility of Properties of Relations.Hajnal Andréka, Ivo Düntsch & István Németi - 1995 - Journal of Symbolic Logic 60 (3):970-991.
    We investigate in an algebraic setting the question of which logical languages can express the properties integral, permutational, and rigid for algebras of relations.
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  • Complete Representations in Algebraic Logic.Robin Hirsch & Ian Hodkinson - 1997 - Journal of Symbolic Logic 62 (3):816-847.
    A boolean algebra is shown to be completely representable if and only if it is atomic, whereas it is shown that neither the class of completely representable relation algebras nor the class of completely representable cylindric algebras of any fixed dimension (at least 3) are elementary.
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  • The Cylindric Algebras of Three-Valued Logic.Norman Feldman - 1998 - Journal of Symbolic Logic 63 (4):1201-1217.
  • Relation Algebra Reducts of Cylindric Algebras and an Application to Proof Theory.Robin Hirsch, Ian Hodkinson & Roger D. Maddux - 2002 - Journal of Symbolic Logic 67 (1):197-213.
    We confirm a conjecture, about neat embeddings of cylindric algebras, made in 1969 by J. D. Monk, and a later conjecture by Maddux about relation algebras obtained from cylindric algebras. These results in algebraic logic have the following consequence for predicate logic: for every finite cardinal α ≥ 3 there is a logically valid sentence X, in a first-order language L with equality and exactly one nonlogical binary relation symbol E, such that X contains only 3 variables (each of which (...)
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  • 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 (...)
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  • The Equational Theory of CA 3 is Undecidable.Roger Maddux - 1980 - Journal of Symbolic Logic 45 (2):311 - 316.
  • Varieties of Complex Algebras.Robert Goldblatt - 1989 - Annals of Pure and Applied Logic 44 (3):173-242.
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  • On Generalization of Definitional Equivalence to Languages with Non-Disjoint Signatures.Koen Lefever & Gergely Székely - unknown
    For simplicity, most of the literature introduces the concept of definitional equivalence only to languages with disjoint signatures. In a recent paper, Barrett and Halvorson introduce a straightforward generalization to languages with non-disjoint signatures and they show that their generalization is not equivalent to intertranslatability in general. In this paper,we show that their generalization is not transitive and hence it is not an equivalence relation. Then we introduce the Andréka and Németi generalization as one of the many equivalent formulations for (...)
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  • Probabilities Defined on Standard and Non-Standard Cylindric Set Algebras.Miklós Ferenczi - 2015 - Synthese 192 (7):2025-2033.
    Cylindric set algebras are algebraizations of certain logical semantics. The topic surveyed here, i.e. probabilities defined on cylindric set algebras, is closely related, on the one hand, to probability logic (to probabilities defined on logical formulas), on the other hand, to measure theory. The set algebras occuring here are associated, in particular, with the semantics of first order logic and with non-standard analysis. The probabilities introduced are partially continous, they are continous with respect to so-called cylindric sums.
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  • Algebraization of Quantifier Logics, an Introductory Overview.István Németi - 1991 - Studia Logica 50 (3-4):485 - 569.
    This paper is an introduction: in particular, to algebras of relations of various ranks, and in general, to the part of algebraic logic algebraizing quantifier logics. The paper has a survey character, too. The most frequently used algebras like cylindric-, relation-, polyadic-, and quasi-polyadic algebras are carefully introduced and intuitively explained for the nonspecialist. Their variants, connections with logic, abstract model theory, and further algebraic logics are also reviewed. Efforts were made to make the review part relatively comprehensive. In some (...)
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  • On the Predecessor Relation in Abstract Algebras.Karl-Heinz Diener - 1993 - Mathematical Logic Quarterly 39 (1):492-514.
    We show the existence of a high r. e. degree bounding only joins of minimal pairs and of a high2 nonbounding r. e. degree. MSC: 03D25.
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  • Zero-Place Operations and Functional Completeness, and the Definition of New Connectives.I. L. Humberstone - 1993 - History and Philosophy of Logic 14 (1):39-66.
    Tarski 1968 makes a move in the course of providing an account of ?definitionally equivalent? classes of algebras with a businesslike lack of fanfare and commentary, the significance of which may accordingly be lost on the casual reader. In ?1 we present this move as a response to a certain difficulty in the received account of what it is to define a function symbol (or ?operation symbol?). This difficulty, which presents itself as a minor technicality needing to be got around (...)
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  • Universal Classes of Monadic Algebras.Th Lucas - 1976 - Mathematical Logic Quarterly 22 (1):35-44.
  • On a Theorem of Vaught for First Order Logic with Finitely Many Variables.Tarek Sayed Ahmed - 2009 - Journal of Applied Non-Classical Logics 19 (1):97-112.
    We prove that the existence of atomic models for countable atomic theories does not hold for Ln the first order logic restricted to n variables for finite n > 2. Our proof is algebraic, via polyadic algebras. We note that Lnhas been studied in recent times as a multi-modal logic with applications in computer science. 2000 MATHEMATICS SUBJECT CLASSIFICATION. 03C07, 03G15.
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  • 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.) (...)
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  • On the Transitive Hull of a Κ‐Narrow Relation.Karl‐Heinz Diener & K.‐H. Diener - 1992 - Mathematical Logic Quarterly 38 (1):387-398.
    We will prove in Zermelo-Fraenkel set theory without axiom of choice that the transitive hull R* of a relation R is not much “bigger” than R itself. As a measure for the size of a relation we introduce the notion of κ+-narrowness using surjective Hartogs numbers rather than the usul injective Hartogs values. The main theorem of this paper states that the transitive hull of a κ+-narrow relation is κ+-narrow. As an immediate corollary we obtain that, for every infinite cardinal (...)
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  • On the Transitive Hull of a Κ-Narrow Relation.Karl-Heinz Diener & K. -H. Diener - 1992 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 38 (1):387-398.
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