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This paper is a continuation of earlier work by the author on the connection between the logic KR and projective geometry. It contains a new, simplified construction of KR model structures; as a consequence, it extends the previous results to a much more extensive class of projective spaces and the corresponding modular lattices.
Studia Logica, 1977
Annals of Pure and Applied Logic, 2010
We introduce a new and general notion of canonical extension for algebras in the algebraic counterpart AlgS of any finitary and congruential logic S. This definition is logic-based rather than purely order-theoretic and is in general different from the one given e.g. in , but it agrees with it whenever the algebras in AlgS are based on lattices. As a case study on logics purely based on implication, we prove that the varieties of Hilbert and Tarski algebras are canonical in this new sense.
Mathematical Proceedings of the Cambridge Philosophical Society, 2008
The Kripke semantics of classical propositional normal modal logic is made algebraic via an embedding of Kripke structures into the larger class of pointed stably supported quantales. This algebraic semantics subsumes the traditional algebraic semantics based on lattices with unary operators, and it suggests natural interpretations of modal logic, of possible interest in the applications, in structures that arise in geometry and analysis, such as foliated manifolds and operator algebras, via topological groupoids and inverse semigroups. We study completeness properties of the quantale based semantics for the systems K, T, K4, S4 and S5, in particular obtaining an axiomatization for S5 which does not use negation or the modal necessity operator. As additional examples we describe intuitionistic propositional modal logic, the logic of programs PDL and the ramified temporal logic CTL.
St. Petersburg Mathematical Journal, 2008
Let Θ be an arbitrary variety of algebras and H an algebra in Θ. Along with algebraic geometry in Θ over the distinguished algebra H, a logical geometry in Θ over H is considered. This insight leads to a system of notions and stimulates a number of new problems. Some logical invariants of algebras H ∈ Θ are introduced and logical relations between different H 1 and H 2 in Θ are analyzed. The paper contains a brief review of ideas of logical geometry (§1), the necessary material from algebraic logic (§2), and a deeper introduction to the subject (§3). Also, a list of problems is given. 0.1. Introduction. The paper consists of three sections. A reader wishing to get a feeling of the subject and to understand the logic of the main ideas can confine himself to §1. A more advanced look at the topic of the paper is presented in § §2 and 3. In §1 we give a list of the main notions, formulate some results, and specify problems. Not all the notions used in §1 are well formalized and commonly known. In particular, we operate with algebraic logic, referring to §2 for precise definitions. However, §1 is self-contained from the viewpoint of ideas of universal algebraic geometry and logical geometry. Old and new notions from algebraic logic are collected in §2. Here we define the Halmos categories and multisorted Halmos algebras related to a variety Θ of algebras. §3 is a continuation of §1. Here we give necessary proofs and discuss problems. The main problem we are interested in is what are the algebras with the same geometrical logic. The theory described in the paper has deep ties with model theory, and some problems are of a model-theoretic nature. We emphasize once again that §1 gives a complete insight on the subject, while §2 and §3 describe and decode the material of §1. §1. Preliminaries. General view 1.1. Main idea. We fix an arbitrary variety Θ of algebras. Throughout the paper we consider algebras H in Θ. To each algebra H ∈ Θ one can attach an algebraic geometry (AG) in Θ over H and a logical geometry (LG) in Θ over H. In algebraic geometry we consider algebraic sets over H, while in logical geometry we consider logical (elementary) sets over H. These latter sets are related to the elementary logic, i.e., to the first order logic (FOL). Consideration of these sets gives grounds to geometries in an arbitrary variety of algebras. We distinguish algebraic and logical geometries in Θ. However, there is very 2000 Mathematics Subject Classification. Primary 03G25.
Studia Logica, 1977
Post Mgebras were introduced for the s time by l~osenbloom [6] and investigated after that by many autors (for the full bibliography and historical remarks see and ). These Mgcbras play such a role for the m-vMued logics of E. Post as Boolean algebras for the classical logic. The first standaxd systems (in the sense of YCasiowa [4]) of m-valued pro-positionM calculi, complete with respect to the class of Post algebras, were constructed by Rousseau [7], . The logic con'esponding to these calculi was called by Rousseau ch~ssical many-vMued logic. In [7] and [8] l~ousseau introduced also the notions of intuitionistic many-vMued logic and pseudo-Post algebras as a semantic basis of this logic.
A structure here is a non-empty universe together with a collection of functions and predicates. Such a structure is considered as a generalized Kripke frame whose set of possible worlds is endowed with a specific algebraic structure. Thus, a class of similar structures induces a certain multimodal logic. The authors axiomatize the basic modal logic of the class of algebras of arbitrary signature and give universal schemes for axiomatization of modal logic for universal and for Π 2 0 -classes of structures. They also discuss the connections of their approach with modal languages for complex algebras and promise to discuss a number of logical and algebraical consequences in a forthcoming full paper.
2013
The main objective of this paper is to show that the notion of type which was developed within the frames of logic and model theory has deep ties with geometric properties of algebras. These ties go back and forth from universal algebraic geometry to the model theory through the machinery of algebraic logic. We show that types appear naturally as logical kernels in the Galois correspondence between filters in the Halmos algebra of first order formulas with equalities and elementary sets in the corresponding affine space.
2020
1.1 Most of the material below is known. See [2] and [5] for a modeltheoretic approach which we further pursue here. The community of anabelian geometers prefers to speak in terms of Galois categories, see e.g. [1]. One of the aims of the current project is to demonstrate advantages of the model-theoretic point of view. Unlike the above publications we do not apriori restrict the power of the language to first order. The default assumptions is that the expressive power of the language is such that the following holds:
Publications of the Research Institute For Mathematical Sciences, 1971
Relational Methods in Computer Science, 2006
2020
In this paper, we introduce the concept of fuzzy congruence relations on a pseudo BE-algebra and some of properties are investigated. We show that the set of all fuzzy congruence relations is a modular lattice and the quotient structure induced by fuzzy congruence relations is studied. A Title Article Information Corresponding Author: A. Borumand Saeid; Received: April 2020; Accepted: Invited paper.
Studies in Universal Logic, 2015
The aim of this paper is to discuss and extend some of Béziau's (published and unpublished) results on the logical geometry of the modal logic S5 and the subjective quantifiers many and few. After reviewing some of the basic notions of logical geometry, we discuss Béziau's work on visualising the Aristotelian relations in S5 by means of two-and three-dimensional diagrams, such as hexagons and a stellar rhombic dodecahedron. We then argue that Béziau's analysis is incomplete, and show that it can be completed by considering another three-dimensional Aristotelian diagram, viz. a rhombic dodecahedron. Next, we discuss Béziau's proposal to transpose his results on the logical geometry of the modal logic S5 to that of the subjective quantifiers many and few. Finally, we propose an alternative analysis of many and few, and compare it with that of Béziau's. While the two analyses seem to fare equally well from a strictly logical perspective, we argue that the new analysis is more in line with certain linguistic desiderata.
Journal of Logic and Computation, 2012
Let K be a variety of (commutative, integral) residuated lattices. The substructural logic usually associated with K is an algebraizable logic that has K as its equivalent algebraic semantics, and is a logic that preserves truth, i.e., 1 is the only truth value preserved by the inferences of the logic. In this paper we introduce another logic associated with K, namely the logic that preserves degrees of truth, in the sense that it preserves lower bounds of truth values in inferences. We study this second logic mainly from the point of view of abstract algebraic logic. We determine its algebraic models and we classify it in the Leibniz and the Frege hierarchies: we show that it is always fully selfextensional, that for most varieties K it is non-protoalgebraic, and that it is algebraizable if and only K is a variety of generalized Heyting algebras, in which case it coincides with the logic that preserves truth. We also characterize the new logic in three ways: by a Hilbert style axiomatic system, by a Gentzen style sequent calculus, and by a set of conditions on its closure operator. Concerning the relation between the two logics, we prove that the truth preserving logic is the extension of the one that preserves degrees of truth with either the rule of Modus Ponens or the rule of Adjunction for the fusion connective.
2001
The notion of projective formula was introduced by Ghilardi [8] in 1999. Let us denote by Pn a set of fixed p1, . . . , pn propositional variables and by Φn – all equivalence classes of intuitionistic formulas with variables in Pn. Consider a substitution σ : Pn → Φn and extend it to all of Φn by σ(φ(p1, . . . , pn)) = φ(σ(p1), . . . , σ(pn)). Now, a formula φ ∈ Φn is called projective if there exists a substitution σ : Φn → Φn such that ⊢ σ(φ) and φ ⊢ ψ ↔ σ(ψ), for all ψ ∈ Φn. In this paper we study projective formulas from the relational and algebraic semantical point of view. We show a close connection between projective formulas and projective Heyting algebras (for definition see Section 4). Namely, to each finitely generated projective Heyting algebra there corresponds a projective formula; to non-isomorphic finitely generated projective algebras there correspond nonequivalent projective formulas, but there can be non-equvalent projective formulas which correspond to isomorphic...
Transactions of the American Mathematical Society, 1973
This paper initiates a series of papers which will reexamine some problems and results of classical invariant theory, within the framework of modern first-order logic. In this paper the notion that an equation is of invariant significance for the general linear group is extended in two directions. It is extended to define invariance of an arbitrary first-order formula for a category of linear transformations between vector spaces of dimension n. These invariant formulas are characterized by equivalence to formulas of a particular syntactic form: homogeneous formulas in determinants or "brackets". The fuller category of all semilinear transformations is also introduced in order to cover all changes of coordinates in a projective space. Invariance for this category is investigated. The results are extended to cover invariant formulas with both covariant and contravariant vectors. Finally, Klein's Erlanger Program is reexamined in the light of the extended notion of invariance as well as some possible geometric categories. Introduction. Classical invariant theory, after a series of ups and downs, has recently enjoyed new vogue, first with the well-known book of Hermann Weyl [lO] and more recently with the extensive survey article by Dieudonné and Carrell [2], In this series of articles we will employ the tools of the first-order predicate calculus, both model theory and proof theory, to recast and extend some of the basic results of the invariant theory. Invariant theory considered polynomials and polynomial equations in the coordinates of points in a projective space and asked which such expressions are "invariant" under changes of coordinates for the space? It was noticed quite
Theoria a Swedish Journal of Philosophy, 2008
2005
In this paper we introduce a new natural deduction system for the logic of lattices, and a number of extensions of lattice logic with different negation connectives. We provide the class of natural deduction proofs with both
Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1974
Publications of the Research Institute for Mathematical Sciences, 1972
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