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1977, Studia Logica
AI
This work merges abstract model theory and algebraic logic to propose a unified framework for understanding and constructing logics. A universal algebraic approach is employed to investigate different types of logics systematically and to establish representability criteria. Key results include the development of a new algebraic system for logics that enhances existing models while emphasizing the ongoing relevance of algebraic principles in logics.
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.
2007
Since all the algebras connected to logic have, more or less explicitely, an associated order relation, it follows that they have two presentations, dual to each other. We classify these dual presentations in "left" and "right" ones and we consider that, when dealing with several algebras in the same research, it is useful to present them unitarily, either as "left" algebras or as "right" algebras. In some circumstances, this choice is essential, for instance if we want to build the ordinal sum (product) between a BL algebra and an MV algebra. We have chosen the "left" presentation and several algebras of logic have been redefined as particular cases of BCK algebras. We introduce several new properties of algebras of logic, besides those usually existing in the literature, which generate a more refined classification, depending on the pro- perties satisfied. In this work (Parts I-V) we make an exhaustive study of these algebras - wit...
2009
In this chapter we determine the classes of S-algebras and of full models for several logics, especially for some which do not fit into the classical approaches to the algebraization of logic. We classify them according to several of the criteria we have been considering, i.e., the properties of the Leibniz, Tarski and Frege operators, which determine the classes of selfextensional logics, Fregean logics, strongly selfextensional logics, protoalgebraic logics, etc. We also study the counterexamples promised in the preceding chapters of this monograph. It goes without saying that the number of cases we have examined is limited, and that many more are waiting to be studied 32. In our view this is an interesting program, especially for non-algebraizable logics. Among those already proven in Blok and Pigozzi [1989a] not to be algebraizable we find many quasi-normal and other modal logics like Lewis' S1, S2 and S3, entailment system E, several purely implicational logics like BCI, the system R → of relevant implication, the "pure entailment" system E → , the implicative fragment S5 → of the Wajsbergstyle version of S5, etc. Other non-algebraizable logics not treated in the present monograph are Da Costa's paraconsistent logics C n (see Lewin, Mikenberg, and Schwarze [1991]), and the "logic of paradox" of Priest [1979] (see Pynko [1995]). This program is also interesting for some algebraizable logics whose class of Salgebras is already known, but whose full models have not yet been investigated; this includes Łukasiewicz many-valued logics (see Rodríguez, Torrens, and Verdú [1990]), BCK logic and some of its neighbours (see Blok and Pigozzi [1989a] Theorem 5.10), the equivalential fragments of classical and intuitionistic logics 32 The full models of several subintuitionistic logics have been determined in Bou [2001]; those of
2009
In this chapter we include the main definitions, notations, and general properties concerning logical matrices, abstract logics and sentential logics. Most of the results reproduced here are not new; however, those concerning abstract logics are not well-known, so it seems useful to recall them in some detail, and to prove some of the ones that are new. Useful references on these topics are Brown and Suszko [1973], Burris and Sankappanavar [1981] and Wójcicki [1988]. Algebras In this monograph (except in Chapter 5, where we deal with examples) we will always work with algebras A = A,. .. of the same, arbitrary, similarity type; thus, when we say "every/any/some algebra" we mean "of the same type". By Hom(A, B) we denote the set of all homomorphisms from the algebra A into the algebra B. The set of congruences of the algebra A will be denoted by ConA. Many of the sets we will consider have the structure of a (often complete, or even algebraic) lattice, but we will not use a different symbol for the lattice and for the underlying set, since no confusion is likely to arise. Given any class K of algebras, the set Con K A = {θ ∈ ConA : A/θ ∈ K} is called the set of K-congruences of A; while this set is ordered under ⊆, in general it is not a lattice. This set will play an important role in this monograph.
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.
Bulletin of Symbolic Logic, 2005
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. We relate the algebraic notion of neat embeddings (a notion special to cylindric algebras) to the metalogical ones of provability, interpolation and omitting types in variants of first logic. Another novelty that occurs here is relating the algebraic notion of atom-canonicity for a class of boolean algebras with operators to the metalogical one of omitting types for the corresponding logic. A hitherto unpublished application of algebraic logic...
Studia Logica, Volume 56, 263-276, 1996
Journal of Algebra and Its Applications, 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.
Logic Journal of IGPL, 2000
We give an algebraic version of rst order logic without equality in which the class of representable algebras forms a nitely based equational class. Further, the representables are de ned in terms of set algebras, and all operations of the latter are permutation invariant. The algebraic form of this result is Theorem 1 (a concrete version of which is given by Theorems 1.8 and 3.2), while its logical form is Corollary 4.2. For rst order logic with equality we give a result weaker than the one for rst order logic without equality. Namely, in this case | instead of nitely axiomatizing the corresponding class of all representable algebras | we nitely axiomatize only the equational theory of that class. See subsection 5.1, especially Remark 5.5 there. The proof of Theorem 1 is elaborated in sections 2 and 3. These sections contain theorems which are interesting of their own rights, too, e.g. Theorem 3.2 is a purely semigroup theoretic result. Cf. also \Further main results" in the Introduction.
2003
Classical propositional logic is one of the earliest formal systems of logic, with its origins in the work of Boole and De Morgan. The algebraic semantics of this logic is given by Boolean algebras. Both, the logic and the algebraic semantics have been generalized in many directions over the last 150 years. In this talk we primarily take the algebraic point of view, but we will also use the powerful framework of algebraic logic to clarify the close relationship between algebra and logic. In various applications (such as fuzzy logic) the properties of Boolean conjunction are too stringent, hence a new binary connective •, usually called fusion, is introduced. In Boolean algebra the relationship between conjunction and implication is given by the residuation equivalences x ∧ y ≤ z ⇐⇒ x ≤ y → z ⇐⇒ y ≤ x → z. These equivalences imply many of the properties of ∧ and → (such as commutativity of ∧, distributivity of ∧ over ∨, left-distributivity of → over ∨ and right-distributivity of → over ∧) so one wishes to retain some aspects of these equivalences, but with conjunction replaced by fusion. To avoid imposing commutativity on the fusion operation one has to introduce two implications: x • y ≤ z ⇐⇒ x ≤ y → z ⇐⇒ y ≤ x z.
Studia Logica - An International Journal for Symbolic Logic, 2003
2000
Connections between Algebraic Logic and (ordinary) Logic. Algebraic co- unterpart of model theoretic semantics, algebraic counterpart of proof theory, and their connections. The class Alg(L) of algebras associated to any logic L. Equivalence theorems stating that L has a certain logical property iff Alg(L) has a certain algebraic property. (E.g. L admits a strongly complete Hilbert- style inference system iff
2017
Introduction 1 1 Generalities on abstract logics and sentential logics 13 2 Abstract logics as models of sentential logics 29 2.1 Models and full models 29 2.2 5-algebras 34 2.3 The lattice of full models over an algebra 38 2.4 Full models and metalogical properties 42 3 Applications to protoalgebraic and algebraizable logics 55 4 Abstract logics as models of Gentzen systems 69 4.1 Gentzen systems and their models 70 4.2 Selfextensional logics with Conjunction 80 4.3 Selfextensional logics having the Deduction Theorem \ • • • 89 5 Applications to particular sentential logics 97 5.1 Some non-protoalgebraic logics 99 5.1.1 CPC AV , the {A, V}-fragment of Classical Logic 99 5.1.2 The logic of lattices 101 5.1.3 Belnap's four-valued logic, and other related logics 102 5.1.4 The implication-less fragment of IPC and its extensions .... 104 5.2 Some Fregean algebraizable logics 105 5.2.1 Alternative Gentzen systems adequate for IPC_ not having the full Deduction Theorem 107 5.3 Some modal logics 108 5.3.1 A logic without a strongly adequate Gentzen system Ill vi Contents 5.4 Other miscellaneous examples. .. Ill 5.4.1 Two relevance logics 112 5.4.2 Sette's paraconsistent logic 113 5.4.3 Tetravalent modal logic 114 5.4.4 Logics related to cardinality restrictions in the Deduction Theorem 115 Bibliography 119
Mathematical Logic Quarterly, 1999
In this paper we address our efforts to extend the well-known connection in equational logic between equational theories and fully invariant congruences to otherpossibly infinitarylogics. In the special case of algebras, this problem has been formerly treated by H. J. Hoehnke [lo] and R. W. Quackenbush . Here we show that the connection extends at least up to the universal fragment of logic. Namely, we establish that the concept of (infinitary) universal theory matches the abstract notion of fully invariant system. We also prove that, inside this wide group of theories, the ones which are strict universal Horn correspond to fully invariant closure systems, whereas those which are universal atomic can be characterized as principal fully invariant systems.
2018
The major concern in the study of categories of logics is to describe condition for preservation, under the a method of combination of logics, of meta-logical properties. Our complementary approach to this field is study the ”global” aspects of categories of logics in the vein of the categories Ss,Ls,As studied in [AFLM3]. All these categories have good properties however the category of logics L does not allow a good treatment of the ”identity problem” for logics ([Bez]): for instance, the presentations of ”classical logics” (e.g., in the signature {¬,∨} and {¬,→}) are not Ls-isomorphic. In this work, we sketch a possible way to overcome this ”defect” (and anothers) by a mathematical device: a representation theory of logics obtained from category theoretic aspects on (Blok-Pigozzi) algebraizable logics. In this setting we propose the study of (left and right) ”Morita equivalence” of logics and variants. We introduce the concepts of logics (left/right)-(stably) -Morita-equivalent a...
arXiv (Cornell University), 2023
Algebras of Logic deal with some algebraic structures, often bounded lattices, considered as models of certain logics, including logic as a domain of order theory. There are well known their importance and applications in social life to advance useful concepts, as for example computer algebra. In this paper we reffer in specially to BL-algebras and we present properties of finite rings or rings with a finite number of ideals in their connections with BL-rings.
Logic Journal of IGPL, 1997
In this work, we attempt to alleviate three (more or less) equivalent negative results. These are (i) non-axiomatizability (by any finite schema) of the valid formula schemas of first order logic, (ii) non-axiomatizability (by finite schema) of any propositional logic equivalent with classical first order logic (i.e., modal logic of quantification and substitution), and (iii) non-axiomatizability (by finite schema) of the class of representable cylindric algebras (i.e., of the algebraic counterpart of first order logic). Here we present two finite schema axiomatizable classes of algebras that contain, as a reduct, the class of representable quasi-polyadic algebras and the class of representable cylindric algebras, respectively. We establish positive results in the direction of finitary algebraization of first order logic without equality as well as that with equality. Finally, we will indicate how these constructions can be applied to turn negative results (i), (ii) above to positive ones. 1
Outstanding contributions to logic, 2018
We establish some relations between the class of truth-equational logics, the class of assertional logics, other classes in the Leibniz hierarchy, and the classes in the Frege hierarchy. We argue that the class of assertional logics belongs properly in the Leibniz hierarchy. We give two new characterizations of truth-equational logics in terms of their full generalized models, and use them to obtain further results on the internal structure of the Frege hierarchy and on the relations between the two hierarchies. Some of these results and several counterexamples contribute to answer a few open problems in abstract algebraic logic, and open a new one.
Reports on Mathematical Logic
In this short note we show that the full generalized models of any extension of a logic can be determined from the full generalized models of the base logic in a simple way. The result is a consequence of two central theorems of the theory of full generalized models of sentential logics. As applications we investigate when the full generalized models of an extension can also be full generalized models of the base logic, and we prove that each Suszko filter of a logic determines a Suszko filter of each of its extensions, also in a simple way. We use the terminology and notations that are standard in abstract algebraic logic, as given for instance in [7]. We identify a sentential logic S with a structural (i.e., substitution invariant) consequence relation S on the set of formulas of some algebraic
The major concern in the study of categories of logics is to describe condition for preservation, under the a method of combination of logics, of meta-logical properties. Our complementary approach to this field is study the "global" aspects of categories of logics in the vein of the categories Ss, Ls, As studied in [AFLM3]. All these categories have good properties however the category of logics L does not allow a good treatment of the "identity problem" for logics ([Bez]): for instance, the presentations of "classical logics" (e.g., in the signature {¬, ∨} and {¬ ′ , → ′ }) are not Ls-isomorphic. In this work, we sketch a possible way to overcome this "defect" (and anothers) by a mathematical device: a representation theory of logics obtained from category theoretic aspects on (Blok-Pigozzi) algebraizable logics. In this setting we propose the study of (left and right) "Morita equivalence" of logics and variants. We introduce the concepts of logics (left/right)-(stably)-Morita-equivalent and show that the presentations of classical logics are stably Morita equivalent but classical logics and intuitionist logics are not stably-Morita-equivalent: they are only stably-Morita-adjointly related.
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