On the Algebraization of Many-Sorted Logics

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.

Studia Logica, 2009

We introduce and study a new approach to the theory of abstract algebraic logic (AAL) that explores the use of many-sorted behavioral logic in the role traditionally played by unsorted equational logic. Our aim is to extend the range of applicability of AAL towards providing a meaningful algebraic counterpart also to logics with a manysorted language, and possibly including non-truth-functional connectives. The proposed behavioral approach covers logics which are not algebraizable according to the standard approach, while also bringing a new algebraic perspective to logics which are algebraizable using the standard tools of AAL. Furthermore, we pave the way towards a robust behavioral theory of AAL, namely by providing a behavioral version of the Leibniz operator which allows us to generalize the traditional Leibniz hierarchy, as well as several well-known characterization results. A number of meaningful examples will be used to illustrate the novelties and advantages of the approach.

Studia Logica, 1977

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.

2022

This short text contains a rigorous introduction to mathematical logic: Many-sorted Languages are the natural way of formulating mathematical theories. E.g. Category Theory uses two kinds (= sorts) of objects: Sets and classes. Incidence Geometry uses points and lines, and so on. We present a formalism of deduction in many-sorted languages and introduce the notion of models. The main results are the correctness and completeness theorems, the compactness theorem and the theorem of Lindenbaum, all in the many-sorted case. This introduction follows the lines of U. Felgner of the Univerisity of Tübingen, who found this extremely elegant formalism. The autor's contribution was to expand this formalism from one-sorted logic to many-sorted logic.

2014

Algebraization of first order logic and its deduction are introduced according to Halmos approach. Application to functional polyadic algebra is done. Index Term: Polyadic algebra, Polyadic ideal, Polyadic filter, Functional polyadic algebra.

Notre Dame Journal of Formal Logic, 1984

In this paper, by developing a study of first-order logic through many-sorted algebras, we show that every first-order theory is a particular algebra verifying axioms in equational form (see Section 2); therefore we are able to apply Birkhoff s theorems concerning the varieties (see Section 1 and [7] and [8]) and to obtain the Henkin models algebraically, whence the completeness theorem of first-order logic (see Section 3). The many-sorted (or heterogeneous) algebras, systematized by Birkhoff and Lipson (see [4] and [10]), find a natural application in investigating programming languages: several approaches to the formal definition of semantics for such languages can be developed by means of morphisms between manysorted algebras (see [2] and [6]). This paper shows the analogous possibility of algebraizing linguistic features of logic, thus yielding a unique framework for the formal analysis of both programming and logical languages within universal algebra. The analysis here developed is strongly related to those of [1], [2], [11], and [12]. Moreover, an approach to the topic of present paper is elaborated in the monographs [8] and [9]. Knowledge of these works is not necessary to understand what follows, though it would better enable one to appreciate our results. This work was inspired by some lectures given by Professor E. de Giorgi in 1977-78 at Scuola Normale Superiore of Pisa. Astute comments from Professors C. Traverso and M. Forti of the University of Pisa helped to clarify the paper's focus. Finally, the authors are grateful to Professor I. Nemeti of the Hungarian Academy of Sciences of Budapest, whose corrections and suggestions improved the first draft of the paper immensely, and who indicated how to relate the paper to a more general research field.

Studia Logica - An International Journal for Symbolic Logic, 2003

Archive for Mathematical Logic, 2012

The best known algebraizable logics with a conjunction and an implication have the property that the conjunction defines a meet semi-lattice in the algebras of their algebraic counterpart. This property makes it possible to associate with them a semi-lattice based deductive system as a companion. Moreover, the order of the semi-lattice is also definable using the implication. This makes that the connection between the properties of the logic and the properties of its semi-lattice based companion is strong. We introduce a class of algebraizable deductive systems that includes those systems, and study some of their properties and of their semi-lattice based companions. We also study conditions which, when satisfied by a deductive system in the class, imply that it is strongly algebraizable. This brings some information on the open area of research of Abstract Algebraic Logic which consists in finding interesting characterizations of classes of algebraizable logics that are strongly algebraizable.

Logic Journal of the IGPL, 2017

The present work presents some results about the categorial relation between logics and its categories of structures. A (propositional, finitary) logic is a pair given by a signature and Tarskian consequence relation on its formula algebra. The logics are the objects in our categories of logics; the morphisms are certain signature morphisms that are translations between logics ([AFLM1],[AFLM2],[AFLM3] [FC]). Morphisms between algebraizable logics ([BP]) are translations that preserves algebraizing pairs ([MaMe]): they can be completely encoded by certain functors defined on the quasi-variety canonically associated to the algebraizable logics. This kind of results will be useful in the development of a categorial approach to the representation theory of general logics ([MaPi1], [MaPi2], [AJMP]).