Abstract Algebraic Logic Research Articles

Categorical Abstract Algebraic Logic: Prealgebraicity and Protoalgebraicity

Two classes of π are studied whose properties are similar to those of the protoalgebraic deductive systems of Blok and Pigozzi. The first is the class of N-protoalgebraic π-institutions and the second is the wider class of N-prealgebraic π-institutions. Several characterizations are provided. For instance, N-prealgebraic π-institutions are exactly those π-institutions that satisfy monotonicity of the N-Leibniz operator on theory systems and N-protoalgebraic π-institutions those that satisfy monotonicity of the N-Leibniz operator on theory families. Analogs of the correspondence property of Blok and Pigozzi for π-institutions are also introduced and their connections with preand protoalgebraicity are explored. Finally, relations of these two classes with the (\({\mathcal{I}}\), N)-algebraic systems, introduced previously by the author as an analog of the \({\mathcal{S}}\) -algebras of Font and Jansana, and with an analog of the Suszko operator of Czelakowski for π-institutions are also investigated.

Categorical Abstract Algebraic Logic: Ordered Equational Logic and Algebraizable PoVarieties

A syntactic apparatus is introduced for the study of the algebraic properties of classes of partially ordered algebraic systems (a.k.a. partially ordered functors (pofunctors)). A Birkhoff-style order HSP theorem and a Mal’cev-style order SLP theorem are proved for partially ordered varieties and partially ordered quasivarieties, respectively, of partially ordered algebraic systems based on this syntactic apparatus. Finally, the notion of a finitely algebraizable partially-ordered quasi-variety, in the spirit of Palasinska and Pigozzi, is introduced and some of the properties of these quasi-povarieties are explored in the categorical framework.

Categorical Abstract Algebraic Logic: Subdirect Representation of Pofunctors

Pałasińska and Pigozzi developed a theory of partially ordered varieties and quasi-varieties of algebras with the goal of addressing issues pertaining to the theory of algebraizability of logics involving an abstract form of the connective of logical implication. Following their lead, the author has abstracted the theory to cover the case of algebraic systems, systems that replace algebras in the theory of categorical abstract algebraic logic. In this note, an order subdirect representation theorem for partially ordered algebraic systems is proven. This is an analog of the Order Subdirect Representation Theorem of Pałasińska and Pigozzi, which, in turn, generalizes the well-known Subdirect Representation Theorem of Universal Algebra.

Categorical Abstract Algebraic Logic: Leibniz Equality and Homomorphism Theorems

The study of structure systems, an abstraction of the concept of first-order structures, is continued. Structure systems have algebraic systems rather than universal algebras as their algebraic reducts. Moreover, their relational component consists of a collection of relation systems on the underlying functors rather than simply a system of relations on a single set. Congruence systems of structure systems are introduced and the Leibniz congruence system of a structure system is defined. Analogs of the Homomorphism, the Second Isomorphism and the Correspondence Theorems of Universal Algebra are provided in this more abstract context. These results generalize corresponding results of Elgueta for equality-free first-order logic. Finally, a version of Godel’s Completeness Theorem is provided with reference to structure systems.

Categorical Abstract Algebraic Logic: More on Protoalgebraicity

Protoalgebraic logics are characterized by the monotonicity of the Leibniz operator on their theory lattices and are at the lower end of the Leibniz hierarchy of abstract algebraic logic. They have been shown to be the most primitive among those logics with a strong enough algebraic character to be amenable to algebraic study techniques. Protoalgebraic π-institutions were introduced recently as an analog of protoalgebraic sentential logics with the goal of extending the Leibniz hierarchy from the sentential framework to the π-institution framework. Many properties of protoalgebraic logics, studied in the sentential logic framework by Blok and Pigozzi, Czelakowski, and Font and Jansana, among others, have already been adapted in previous work by the author to the categorical level. This work aims at further advancing that study by exploring in this new level some more properties of protoalgebraic sentential logics.

Categorical abstract algebraic logic: The largest theory system included in a theory family

AbstractIn this note, it is shown that, given a π ‐institution ℐ = 〈Sign, SEN, C 〉, with N a category of natural transformations on SEN, every theory family T of ℐ includes a unique largest theory system $ \overleftarrow T $ of ℐ. $ \overleftarrow T $ satisfies the important property that its N ‐Leibniz congruence system always includes that of T . As a consequence, it is shown, on the one hand, that the relation ΩN ($ \overleftarrow T $) = ΩN (T ) characterizes N ‐protoalgebraicity inside the class of N ‐prealgebraic π ‐institutions and, on the other, that all N ‐Leibniz theory families associated with theory families of a protoalgebraic π ‐institution ℐ are in fact N ‐Leibniz theory systems. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Categorical Abstract Algebraic Logic: Partially Ordered Algebraic Systems

An extension of parts of the theory of partially ordered varieties and quasivarieties, as presented by Palasinska and Pigozzi in the framework of abstract algebraic logic, is developed in the more abstract framework of categorical abstract algebraic logic. Algebraic systems, as introduced in previous work by the author, play in this more abstract framework the role that universal algebras play in the more traditional treatment. The aim here is to build the generalized framework and to formulate and prove abstract versions of the ordered homomorphism theorems in this framework.

Corrigendum to “Categorical abstract algebraic logic: The criterion for deductive equivalence”

We give a correction to the paper [2] mentioned in the title. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Categorical Abstract Algebraic Logic: Models of π-Institutions

An important part of the theory of algebraizable sentential logics consists of studying the algebraic semantics of these logics. As developed by Czelakowski, Blok, and Pigozzi and Font and Jansana, among others, it includes studying the properties of logical matrices serving as models of deductive systems and the properties of abstract logics serving as models of sentential logics. The present paper contributes to the development of the categorical theory by abstracting some of these model theoretic aspects and results from the level of sentential logics to the level of π-institutions.

Categorical abstract algebraic logic: Gentzenπ -institutions and the deduction-detachment property

Given a π -institution I , a hierarchy of π -institutions I (n ) is constructed, for n ≥ 1. We call I (n ) the n-th order counterpart of I . The second-order counterpart of a deductive π -institution is a Gentzen π -institution, i.e. a π -institution associated with a structural Gentzen system in a canonical way. So, by analogy, the second order counterpart I (2) of I is also called the “Gentzenization” of I . In the main result of the paper, it is shown that I is strongly Gentzen , i.e. it is deductively equivalent to its Gentzenization via a special deductive equivalence, if and only if it has the deduction-detachment property . (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Categorical Abstract Algebraic Logic: (ℐ,N)-Algebraic Systems

Algebraic systems play in the theory of algebraizability of π-institutions the role that algebras play in the theory of algebraizable sentential logics. In this same sense, ℐ-algebraic systems are to a π-institution ℐ what \(\mathcal{S}\) -algebras are to a sentential logic \(\mathcal{S}\) . More precisely, an (ℐ,N)-algebraic system is the sentence functor reduct of an N′-reduced (N,N′)-full model of a π-institution ℐ. Algebraic systems are formally introduced and their relationship with full models and with bilogical morphisms is investigated.

Fregean logics with the multiterm deduction theorem and their algebraization

A deductive system $$\mathcal{S}$$ (in the sense of Tarski) is Fregean if the relation of interderivability, relative to any given theory T, i.e., the binary relation between formulas $$\{ \left\langle {\alpha ,\beta } \right\rangle :T,\alpha \vdash s \beta and T,\beta \vdash s \alpha \} ,$$ is a congruence relation on the formula algebra. The multiterm deduction-detachment theorem is a natural generalization of the deduction theorem of the classical and intuitionistic propositional calculi (IPC) in which a finite system of possibly compound formulas collectively plays the role of the implication connective of IPC. We investigate the deductive structure of Fregean deductive systems with the multiterm deduction-detachment theorem within the framework of abstract algebraic logic. It is shown that each deductive system of this kind has a deductive structure very close to that of the implicational fragment of IPC. Moreover, it is algebraizable and the algebraic structure of its equivalent quasivariety is very close to that of the variety of Hilbert algebras. The equivalent quasivariety is however not in general a variety. This gives an example of a relatively point-regular, congruence-orderable, and congruence-distributive quasivariety that fails to be a variety, and provides what apparently is the first evidence of a significant difference between the multiterm deduction-detachment theorem and the more familiar form of the theorem where there is a single implication connective.

Categorical abstract algebraic logic categorical algebraization of first-order logic without terms

An algebraization of multi-signature first-order logic without terms is presented. Rather than following the traditional method of choosing a type of algebras and constructing an appropriate variety, as is done in the case of cylindric and polyadic algebras, a new categorical algebraization method is used: The substitutions of formulas of one signature for relation symbols in another are treated in the object language. This enables the automatic generation via an adjunction of an algebraic theory. The algebras of this theory are then used to algebraize first-order logic.

Categorical Abstract Algebraic Logic: Categorical Algebraization of Equational Logic

This paper deals with the algebraization of multi-signature equational logic in the context of the modern theory of categorical abstract algebraic logic. Two are the novelties compared to traditional treatments: First, interpretations between different algebraic types are handled in the object language rather than the metalanguage. Second, rather than constructing the type of the algebraizing class of algebras explicitly in an ad-hoc universal algebraic way, the whole clone is naturally constructed using categorical algebraic techniques.

Categorical abstract algebraic logic: The criterion for deductive equivalence

AbstractEquivalent deductive systems were introduced in [4] with the goal of treating 1‐deductive systems and algebraic 2‐deductive systems in a uniform way. Results of [3], appropriately translated and strengthened, show that two deductive systems over the same language type are equivalent if and only if their lattices of theories are isomorphic via an isomorphism that commutes with substitutions. Deductive equivalence of π‐institutions [14, 15] generalizes the notion of equivalence of deductive systems. In [15, Theorem 10.26] this criterion for the equivalence of deductive systems was generalized to a criterion for the deductive equivalence of term π‐institutions, forming a subclass of all π‐institutions that contains those π‐institutions directly corresponding to deductive systems. This criterion is generalized here to cover the case of arbitrary π‐institutions.

Categorical Abstract Algebraic Logic: Equivalent Institutions

A category theoretic generalization of the theory of algebraizable deductive systems of Blok and Pigozzi is developed. The theory of institutions of Goguen and Burstall is used to provide the underlying framework which replaces and generalizes the universal algebraic framework based on the notion of a deductive system. The notion of a term π-institution is introduced first. Then the notions of quasi-equivalence, strong quasi-equivalence and deductive equivalence are defined for π-institutions. Necessary and sufficient conditions are given for the quasi-equivalence and the deductive equivalence of two term π-institutions, based on the relationship between their categories of theories. The results carry over without any complications to institutions, via their associated π-institutions. The π-institution associated with a deductive system and the institution of equational logic are examined in some detail and serve to illustrate the general theory.

A Survey of Abstract Algebraic Logic

A Survey of Abstract Algebraic Logic

An abstract algebraic logic approach to tetravalent modal logics

AbstractThis paper contains a joint study of two sentential logics that combine a many-valued character, namely tetravalence, with a modal character; one of them is normal and the other one quasinormal. The method is to study their algebraic counterparts and their abstract models with the tools of Abstract Algebraic Logic, and particularly with those of Brown and Suszko's theory of abstract logics as recently developed by Font and Jansana in their “A General Algebraic Semantics for Sentential Logics”. The logics studied here arise from the algebraic and lattice-theoretical properties we review of Tetravalent Modal Algebras, a class of algebras studied mainly by Loureiro, and also by Figallo. Landini and Ziliani, at the suggestion of the late Antonio Monteiro.