Categorical Syntax and Consequence Relations

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.

2010

Fujiwara defined, through the concept of family of basic mappingformulas between single-sorted signatures, a notion of morphism which generalizes the ordinary notion of homomorphism between algebras. Subsequently he also defined an equivalence relation, the relation of conjugation, on the families of basic mapping-formulas. In this article we extend the theory of Fujiwara to the, not necessarily similar, many-sorted algebras, by defining the concept of polyderivor between many-sorted signatures under which are subsumed the standard signature morphisms, the derivors of Goguen-Thatcher-Wagner, and the basic mapping-formulas of Fujiwara.

Proof theory is the result of a short and tumultuous history, developed on the periphery of mainstream mathematics. Hence, its language is often idiosyn- cratic: sequent calculus, cut-elimination, subformula property, etc. This survey is designed to guide the novice reader and the itinerant mathematician along a smooth and consistent path, investigating the symbolic mechanisms of cut- elimination, and their algebraic transcription as coherence diagrams in cate- gories with structure. This spiritual journey at the meeting point of linguistic and algebra is demanding at times, but also pleasantly rewarding: to date, no language (either formal or informal) has been studied by mathematicians as thoroughly as the language of proofs. We start the survey by a short introduction to proof theory (Chapter 1) followed by an informal explanation of the principles of denotational semantics (Chapter 2) which we express as a representation theory for proofs - generat- ing algebraic invariants modu...

Studia Logica, 1977

Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1985

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.

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.

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

Mathematical Structures in Computer Science, 2002

2007

Appetizer. Algebraic logic (AL) is a well established discipline yet some natural questions remain out of its scope. For instance, it is well known that the Horn and universal fragments of FOL are original logics close to equational logic rather than just sublogics of FOL, and, very similarly, Gentzen's axiomatization of FOL seems has much in common with the universal equational logic: how c a n these phenomena be placed in the AL framework? In general, what is equational-like logic, and how are these logics related to propo-sitional logics? Another block of questions is whether cylindric or polyadic algebraization of FOL make it possible to consider it as a kind of (complex yet) propositional logic, or there are principal diierences? Has it sense to ask about the extent to which a given logic is propositional-like? And if even the questions above c a n b e treated formally, will such an eeort be helpful in logic as such o r will remain a purely metalogical achievement? A more t...

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.

2014

The definition of identity in terms of other logical symbols is a recurrent issue in logic. In particular, in First-Order Logic (FOL) there is no way of defining the global relation of identity, while in standard Second-Order Logic (SOL) this definition is not only possible, but widely used. In this paper, the reverse question is posed and affirmatively answered: Can we define with only equality and abstraction the remaining logical symbols? Our present work is developed in the context of an equational hybrid logic (i.e. a modal logic with equations as propositional atoms enlarged with the hybrid expressions: nominals and the @ operator). Our logical base is propositional type theory. We take the propositional equality, λ abstraction, nominals, ♦ and @ operators as primitive symbols and we demonstrate that all of the remaining logical symbols can be defined,ion the remaining logical symbols? Our present work is developed in the context of an equational hybrid logic (i.e. a modal log...

Information Processing Letters, 1990

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.

Logic and Logical Philosophy, 2003