Behavioral Algebraization of Logics

Journal of Symbolic Logic, 2016

This paper presents a unified framework that explains and extends the already successful applications of the Leibniz operator, the Suszko operator, and the Tarski operator in recent developments in abstract algebraic logic. To this end, we refine Czelakowski's notion of an S-compatibility operator, and introduce the notion of coherent family of S-compatibility operators, for a sentential logic S. The notion of coherence is a restricted property of commutativity with inverse images by surjective homomorphisms, which is satisfied by both the Leibniz and the Suszko operators. We generalize several constructions and results already existing for the mentioned operators; in particular, the well-known classes of algebras associated with a logic through each of them, and the notions of full generalized model of a logic and a special kind of S-filters (which generalizes the less-known notion of Leibniz filter). We obtain a General Correspondence Theorem, extending the well-known one from the theory of protoalgebraic logics to arbitrary logics and to more general operators, and strengthening its formulation. We apply the general results to the Leibniz and the Suszko operators, and obtain several characterizations of the main classes of logics in the Leibniz hierarchy by the form of their full generalized models, by old and new properties of the Leibniz operator, and by the behaviour of the Suszko operator. Some of these characterizations complete or extend known ones, for some classes in the hierarchy, thus offering an integrated approach to the Leibniz hierarchy that uncovers some new, nice symmetries.

Studia Logica - An International Journal for Symbolic Logic, 2003

This book presents an algebraic treatment of propositional and first-order logic. The presentation is unconventional. We start from a Boolean congruence on a pre-Boolean algebra as an algebraic model of propositional logic, define a propositional theory by the hull of a Boolean congruence, and then derive deductive systems from the properties of a propositional theory. For this approach, we provide a new characterization of a Boolean algebra that closely parallels our definition of propositional theories. The algebraic method adopted for propositional logic is then extended to first-order logic. Again, we start from a first-order Boolean congruence, define a first-order theory by the hull of a first-order Boolean congruence, and then derive deductive systems from the properties of a first-order theory. To reconcile the algebra and semantics of first-order logic, an isomorphism is established between the Lindenbaum algebra of a firstorder theory and a field of sets of that theory. The completeness theorem and the compactness theorem are then derived from this representation theorem.

2011

Algebraic Logic and Computer Science Jacinta Poças, Manuel Martins, and Carlos Caleiro 1 Dep. Mathematics, IST TU Lisbon, Portugal 2 Dep. Mathematics, U Aveiro, Portugal 3 SQIG Instituto de Telecomunicações Abstract. Abstract algebraic logic (AAL) is a branch of logic that Abstract algebraic logic (AAL) is a branch of logic that uses universal algebra to study the properties of logical systems by associating them with representative classes of algebras, thus generalizing the Lindenbaum-Tarski process that leads to linking Boolean algebras with classical propositional logic. The theory of AAL then classifies logical systems and their metaproperties, systematically, along bridge theorems that relate them with properties of the associated algebras. Recently, concepts of behavioral algebraic specification have influenced the development of the behavioral approach to AAL. Namely, the notion of behavioral equivalence, imported from computer science, has been used to weaken the traditional...

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

Journal of Applied Logic, 2016

This work advances a research agenda which has as its main aim the application of Abstract Algebraic Logic (AAL) methods and tools to the specification and verification of software systems. It uses a generalization of the notion of an abstract deductive system to handle multi-sorted deductive systems which differentiate visible and hidden sorts. Two main results of the paper are obtained by generalizing properties of the Leibniz congruencethe central notion in AAL. In this paper we discuss a question we posed in [2] about the relationship between the behavioral equivalences of equivalent hidden logics. We also present a necessary and sufficient intrinsic condition for two hidden logics to be equivalent.

Theoretical Computer Science, 1990

Equational type logic is an extension of (conditional) equational logic, that enables one to deal in a single, unified framework with diverse phenomena such as partiality, type polymorphism and dependent types. In this logic, terms may denote types as well as elements, and atomic formulae are either equations or type assignments. Models of this logic are type algebras, viz. universal algebras equipped with a binary relation-to support type assignment. Equational type logic has a sound and complete calculus, and initial models exist. The use of equational type logic is illustrated by means of simple examples, where all of the aforementioned phenomena occur. Formal notions of reduction and extension are introduced, and their relationship to free constructions is investigated. Computational aspects of equational type logic are investigated in the framework of conditional term rewriting systems, genera!izing known results on confluence of these systems. Finally, some closely related work is reviewed and future research directions are outlined in the conclusions. * 41, Ird.wy'~~-rted (conditional) equational logic is the most established basis to the algebraic approach to abstract data type (ADT) specification [ 15,9]. algebras [ 17,4] are the standard models of this lo&c, extending u structures in a straightforward way. In algebraic specification, however, several phenomena indicate that this logic encounters limitations in practice. We mention a few, most interesting of these phenomena (which are discussed in Section 2): partiaiity, exception handling, extension, type polymorphism, dependent types. Several formal frameworks have been designed to solve the problems that are raised by e&r& of these phenomena. In particular, many of these frameworks are based on extensions of equational logic in various forms. Most of these approaches address the phenomena of their interest at a rather high level of generality. Yet, a unifying approach, where all of these phenomena can be dealt with, does not seem to have emerged. The following problem is addressed in this paper: to nd and investigate a parsimonious logic of types where al2 of the aforementioned phenomena can be dealt with in an algebraic setting. In Section 2 we further motivate our irwstigation ng, for each of those phenomena, rt discussion an a simple exam@ational type logic

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.

1998

Brouwer{Zadeh MV-algebras and de Morgan BZMV-algebras are the result of a natural \pasting" between Brouwer{Zadeh algebras and MValgebras. Such structures are characterized by a splitting of the operations that correspond to the basic logical connectives (not, and, or). In this framework, we prove the categorical equivalence between di erent kinds of structures that have been introduced in order to semantically characterize some many-valued logics. In particular, we investigate the cases of Chang's MV-algebras, Cignoli{Monteiro Lukasiewicz algebras, Stonian MV-algebras.

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