Abstract Algebraic Logic Research Articles

Algebraic logic for the negation fragment of classical logic

Abstract The general aim of this article is to study the negation fragment of classical logic within the framework of contemporary (Abstract) Algebraic Logic. More precisely, we shall find the three classes of algebras that are canonically associated with a logic in Algebraic Logic, i.e. we find the classes $\textrm{Alg}^*$, $\textrm{Alg}$ and the intrinsic variety of the negation fragment of classical logic. In order to achieve this, firstly, we propose a Hilbert-style axiomatization for this fragment. Then, we characterize the reduced matrix models and the full generalized matrix models of this logic. Also, we classify the negation fragment in the Leibniz and Frege hierarchies.

AN ABSTRACT ALGEBRAIC LOGIC STUDY OF DA COSTA’S LOGIC AND SOME OF ITS PARACONSISTENT EXTENSIONS

AbstractTwo famous negative results about da Costa’s paraconsistent logic ${\mathscr {C}}_1$ (the failure of the Lindenbaum–Tarski process [44] and its non-algebraizability [39]) have placed ${\mathscr {C}}_1$ seemingly as an exception to the scope of Abstract Algebraic Logic (AAL). In this paper we undertake a thorough AAL study of da Costa’s logic ${\mathscr {C}}_1$ . On the one hand, we strengthen the negative results about ${\mathscr {C}}_1$ by proving that it does not admit any algebraic semantics whatsoever in the sense of Blok and Pigozzi (a weaker notion than algebraizability also introduced in the monograph [6]). On the other hand, ${\mathscr {C}}_1$ is a protoalgebraic logic satisfying a Deduction-Detachment Theorem (DDT). We then extend our AAL study to some paraconsistent axiomatic extensions of ${\mathscr {C}}_1$ covered in the literature. We prove that for extensions ${\mathcal {S}}$ such as ${\mathcal {C}ilo}$ [26], every algebra in ${\mathsf {Alg}}^*({\mathcal {S}})$ contains a Boolean subalgebra, and for extensions ${\mathcal {S}}$ such as , , or [16, 53], every subdirectly irreducible algebra in ${\mathsf {Alg}}^*({\mathcal {S}})$ has cardinality at most 3. We also characterize the quasivariety ${\mathsf {Alg}}^*({\mathcal {S}})$ and the intrinsic variety $\mathbb {V}({\mathcal {S}})$ , with , , and .

Completely distributive completions of posets

A $$\Delta_1$$ -completion of a poset is a completion for which, simultaneously, every element is reachable as a join of meets and a meet of joins from the original poset. We focus our attention on $$\Delta_1$$ -completions that can be obtained from polarities $$\langle\mathcal{F},\mathcal{I},\mathcal{R}\rangle$$ where $$\mathcal{F}$$ is a collection of upsets containing the principal upsets, $$\mathcal{I}$$ is a collection of downsets containing the principal downsets of the original poset, and $$R\subseteq\mathcal{F}\times\mathcal{I}$$ is the relation of nonempty intersection. These $$\Delta_1$$ -completions are called $$\langle\mathcal{F},\mathcal{I}\rangle$$ -completions, and they satisfy a compactness property. In this paper, we show that if a pair $$\langle\mathcal{F},\mathcal{I}\rangle$$ satisfies a separating condition (similar to the Prime Filter Theorem for distributive lattices), then the $$\langle\mathcal{F},\mathcal{I}\rangle$$ -completion of the original poset is a completely distributive algebraic lattice. Given a poset P and an algebraic closure system $$\mathcal{F}$$ of upsets of P satisfying a distributivity condition, we show how to choose a collection of downsets $$\mathcal{I}$$ of P such that the $$\langle\mathcal{F},\mathcal{I}\rangle$$ -completion of P is a completely distributive algebraic lattice. Then, we study the extensions of additional operations on posets to their corresponding $$\langle\mathcal{F},\mathcal{I}\rangle$$ -completions. Finally, we use the previous results to obtain adequate $$\langle\mathcal{F},\mathcal{I}\rangle$$ -completions for the classes of Tarski algebras and Hilbert algebras, and for the classes of algebras that are canonically associated (in the sense of abstract algebraic logic) with some propositional logics.

Selfextensional logics with a distributive nearlattice term

We define when a ternary term m of an algebraic language $$\mathcal {L}$$ is called a distributive nearlattice term ( $$\mathrm {DN}$$ -term) of a sentential logic $$\mathcal {S}$$ . Distributive nearlattices are ternary algebras generalising Tarski algebras and distributive lattices. We characterise the selfextensional logics with a $$\mathrm {DN}$$ -term through the interpretation of the DN-term in the algebras of the algebraic counterpart of the logics. We prove that the canonical class of algebras (under the point of view of Abstract Algebraic Logic) associated with a selfextensional logic with a $$\mathrm {DN}$$ -term is a variety, and we obtain that the logic is in fact fully selfextensional.

Extension Properties and Subdirect Representation in Abstract Algebraic Logic

This paper continues the investigation, started in Lavicka and Noguera (Stud Log 105(3): 521–551, 2017), of infinitary propositional logics from the perspective of their algebraic completeness and filter extension properties in abstract algebraic logic. If follows from the Lindenbaum Lemma used in standard proofs of algebraic completeness that, in every finitary logic, (completely) intersection-prime theories form a basis of the closure system of all theories. In this article we consider the open problem of whether these properties can be transferred to lattices of filters over arbitrary algebras of the logic. We show that in general the answer is negative, obtaining a richer hierarchy of pairwise different classes of infinitary logics that we separate with natural examples. As by-products we obtain a characterization of subdirect representation for arbitrary logics, develop a fruitful new notion of natural expansion, and contribute to the understanding of semilinear logics.

Categorical Abstract Algebraic Logic: Wojcicki's Conjecture and Malinowski's Theorem

Categorical Abstract Algebraic Logic: Wojcicki's Conjecture and Malinowski's Theorem

A New Hierarchy of Infinitary Logics in Abstract Algebraic Logic

In this article we investigate infinitary propositional logics from the perspective of their completeness properties in abstract algebraic logic. It is well-known that every finitary logic is complete with respect to its relatively (finitely) subdirectly irreducible models. We identify two syntactical notions formulated in terms of (completely) intersection-prime theories that follow from finitarity and are sufficient conditions for the aforementioned completeness properties. We construct all the necessary counterexamples to show that all these properties define pairwise different classes of logics. Consequently, we obtain a new hierarchy of logics going beyond the scope of finitarity.

The semantic isomorphism theorem in abstract algebraic logic

One of the most interesting aspects of Blok and Pigozzi's algebraizability theory is that the notion of algebraizable logic L can be characterised by means of Syntactic and Semantic Isomorphism Theorems. While the Syntactic Isomorphism Theorem concerns the relation between the theories of the algebraizable logic L and those of the equational consequence relative to its equivalent algebraic semantics K, the Semantic Isomorphism Theorem describes the interplay between the filters of L on an arbitrary algebra A and the congruences of A relative to K. The pioneering insight of Blok and Jónsson, and the further generalizations by Galatos, Tsinakis, Gil-Férez and Russo, showed that the concept of algebraizability was not intrinsic to the connection between a logic and an equational consequence, thus inaugurating the abstract theory of equivalence between structural closure operators. However all these works focus only on the Syntactic Isomorphism Theorem, disregarding the semantic aspects present in the original theory. In this paper we fill this gap by introducing the notion of compositional lattice, which acts on a category of evaluational frames. In this new framework the non-linguistic flavour of the Semantic Isomorphism Theorem can be naturally recovered. In particular, we solve the problem of finding sufficient and necessary conditions for transferring a purely syntactic equivalence to the semantic level as in the Semantic Isomorphism Theorem.

Categorical abstract algebraic logic: skywatching in semilattice systems

Font and Moraschini established a bijective correspondence between congruences of semilattices with sectionally finite height and certain special subsets of their universes, called clouds. They provided a characterization of clouds and showed that the correspondence is given by the Leibniz operator of abstract algebraic logic. We extend the bijection to one between congruence systems on the semilattice systems of categorical abstract algebraic logic and what we call cloud families. In this context, the categorical analog of the Leibniz operator plays a similar role. In addition, we show that, even though the exact analogue of the Font–Moraschini condition fails in general, a more complex variant provides an analogous characterization of cloud families.

Categorical Abstract Algebraic Logic: Truth-Equational π-Institutions

Categorical Abstract Algebraic Logic: Truth-Equational π-Institutions

Categorical Abstract Algebraic Logic: Referential π-Institutions

Wojcicki introduced in the late 1970s the concept of a referential semantics for propositional logics. Referential semantics incorporate features of the Kripke possible world semantics for modal logics into the realm of algebraic and matrix semantics of arbitrary sentential logics. A well-known theorem of Wojcicki asserts that a logic has a referential semantics if and only if it is selfextensional. Referential semantics was subsequently studied in detail by Malinowski and the concept of selfextensionality has played, more recently, an important role in the field of abstract algebraic logic in connection with the operator approach to algebraizability. We introduce and review some of the basic definitions and results pertaining to the referential semantics of π-institutions, abstracting corresponding results from the realm of propositional logics.

A Note on Natural Extensions in Abstract Algebraic Logic

Transfer theorems are central results in abstract algebraic logic that allow to generalize properties of the lattice of theories of a logic to any algebraic model and its lattice of filters. Their proofs sometimes require the existence of a natural extension of the logic to a bigger set of variables. Constructions of such extensions have been proposed in particular settings in the literature. In this paper we show that these constructions need not always work and propose a wider setting (including all finitary logics and those with countable language) in which they can still be used.

Categorical Abstract Algebraic Logic: Referential Algebraic Semantics

Wojcicki has provided a characterization of selfextensional logics as those that can be endowed with a complete local referential semantics. His result was extended by Jansana and Palmigiano, who developed a duality between the category of reduced congruential atlases and that of reduced referential algebras over a fixed similarity type. This duality restricts to one between reduced atlas models and reduced referential algebra models of selfextensional logics. In this paper referential algebraic systems and congruential atlas systems are introduced, which abstract referential algebras and congruential atlases, respectively. This enables the formulation of an analog of Wojcicki's Theorem for logics formalized as ?-institutions. Moreover, the results of Jansana and Palmigiano are generalized to obtain a duality between congruential atlas systems and referential algebraic systems over a fixed categorical algebraic signature. In future work, the duality obtained in this paper will be used to obtain one between atlas system models and referential algebraic system models of an arbitrary selfextensional ?-institution. Using this latter duality, the characterization of fully selfextensional deductive systems among the selfextensional ones, that was obtained by Jansana and Palmigiano, can be extended to a similar characterization of fully selfextensional ?-institutions among appropriately chosen classes of selfextensional ones.

Categorical Abstract Algebraic Logic: Meet-Combination of Logical Systems

The widespread and rapid proliferation of logical systems in several areas of computer science has led to a resurgence of interest in various methods for combining logical systems and in investigations into the properties inherited by the resulting combinations. One of the oldest such methods isfibring. In fibring the shared connectives of the combined logics inherit properties frombothcomponent logical systems, and this leads often to inconsistencies. To deal with such undesired effects, Sernadas et al. (2011, 2012) have recently introduced a novel way of combining logics, calledmeet-combination, in which the combined connectives share only thecommonlogical properties they enjoy in the component systems. In their investigations they provide a sound and concretely complete calculus for the meet-combination based on available sound and complete calculi for the component systems. In this work, an effort is made to abstract those results to a categorical level amenable tocategorical abstract algebraic logictechniques.

Update to “A Survey of Abstract Algebraic Logic”

A definition and some inaccurate cross-references in the paper A Survey of Abstract Algebraic Logic, which might confuse some readers, are clarified and corrected; a short discussion of the main one is included. We also update a dozen of bibliographic references.

Categorical Abstract Algebraic Logic: Structurality, protoalgebraicity, and correspondence

AbstractThe notion of an ℐ ‐matrix as a model of a given π ‐institution ℐ is introduced. The main difference from the approach followed so far in Categorical Abstract Algebraic Logic (CAAL) and the one adopted here is that an ℐ ‐matrix is considered modulo the entire class of morphisms from the underlying N ‐algebraic system of ℐ into its own underlying algebraic system, rather than modulo a single fixed (N,N ′)‐logical morphism. The motivation for introducing ℐ ‐matrices comes from a desire to formulate a correspondence property for N ‐protoalgebraic π ‐institutions closer in spirit to the one for sentential logics than that considered in CAAL before. As a result, in the previously established hierarchy of syntactically protoalgebraic π ‐institutions, i. e., those with an implication system, and of protoalgebraic π ‐institutions, i. e., those with a monotone Leibniz operator, the present paper interjects the class of those π ‐institutions with the correspondence property, as applied to ℐ ‐matrices. Moreover, this work on ℐ ‐matrices enables us to prove many results pertaining to the local deduction‐detachment theorems, paralleling classical results in Abstract Algebraic Logic formulated, first, by Czelakowski and Blok and Pigozzi. Those results will appear in a sequel to this paper. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Categorical Abstract Algebraic Logic: Local Characterization Theorems for Classes of Systems

Let ℒ = ⟨F, R, ρ⟩ be a system language. Given a class of ℒ-systems K and an ℒ-algebraic system A = ⟨SEN,⟨N,F⟩⟩, i.e., a functor SEN: Sign → Set, with N a category of natural transformations on SEN, and F:F → N a surjective functor preserving all projections, define the collection K A of A-systems in K as the collection of all members of K of the form 𝔄 = ⟨ SEN,⟨N,F⟩,R 𝔄 ⟩, for some set of relation systems R 𝔄 on SEN. Taking after work of Czelakowski and Elgueta in the context of the model theory of equality-free first-order logic, several relationships between closure properties of the class K, on the one hand, and local properties of K A and global properties connecting K A and K A′, whenever there exists an ℒ-morphism ⟨ F,α⟩ : A → A′, on the other, are investigated. In the main result of the article, it is shown, roughly speaking, that K A is an algebraic closure system, for every ℒ-algebraic system A, provided that K is closed under subsystems and reduced products.

Categorical Abstract Algebraic Logic: Equivalential π-Institutions

The theory of equivalential deductive systems, as introduced by Prucnal and Wroński and further developed by Czelakowski, is abstracted to cover the case of logical systems formalized as π-Institutions. More precisely, the notion of an N-equivalence system for a given π-Institutions is introduced. A characterization theorem for N-equivalence systems, previously proven for N-parameterized equivalence systems, is revisited and a “transfer theorem” for N-equivalence systems is proven. For a π-Institutions I having an N-equivalence system, the maximum such system is singled out and, then, an analog of Herrmann’s Test, characterizing those N-protoalgebraic π-Institutions having an N-equivalence system, is formulated. Finally, some of the rudiments of matrix theory are revisited in the context of π-Institutions, as they relate to the existence of N-equivalence systems.

Categorical Abstract Algebraic Logic: Bloom's Theorem for Rule-Based -Institutions

Categorical Abstract Algebraic Logic: Bloom's Theorem for Rule-Based -Institutions

Categorical abstract algebraic logic: The Diagram and the Reduction Operator Lemmas

AbstractThe study of structure systems, an abstraction of the concept of first‐order structures, is continued. Structure systems have algebraic systems as their algebraic reducts and their relational component consists of a collection of relation systems on the underlying functors. An analog of the expansion of a first‐order structure by constants is presented. Furthermore, analogs of the Diagram Lemma and the Reduction Operator Lemma from the theory of equality‐free first‐order structures are provided in the framework of structure systems. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)