Carles Noguera

This paper presents a unified algebraic study of a family of logics related to Abelian logic (Ab), the logic of Abelian lattice-ordered groups. We treat Ab as the base system and refer to its expansions as superabelian logics. The paper focuses on two main families of expansions. First, we investigate the rich landscape of infinitary extensions of Ab, providing an axiomatization for the infinitary logic of real numbers and showing that there exist 2 Superscript 2 Super Superscript omega $2^{2^\omega }$ 22ω distinct logics in this family. Second, we introduce pointed Abelian logic (pAb ${\text {pAb}}$ pAb), the logic of pointed Abelian lattice-ordered groups, by adding a new constant to the language. This framework includes Łukasiewicz unbound logic. We provide axiomatizations for its finitary and infinitary versions as extensions of pAb ${\text {pAb}}$ pAb and establish their precise relationship with standard Łukasiewicz logic via a formal translation. Finally, the methods developed for this analysis are generalized to axiomatize the logics of other prominent pointed groups.

In this paper, we study two operators, inquisitive disjunction and inquisitive existential quantifier, in the context of first-order intuitionistic logic with constant domains. We explain that these operators allow us to express types of intuitionistic propositions. We first provide this language with a relational semantics and formulate a sound axiomatic system. Completeness of the system for the full language is presented as an open problem but we prove completeness for a rich fragment of the language adapting the methods developed by Gianluca Grilleti in the context of classical inquisitive logic. We also develop a general algebraic framework in which we characterize the class of “Kripkean algebras” generated by the relational semantics. In this way, we obtain our first algebraic characterization of the logic of intuitionistic types. We further characterize the class of all homomorphic images of “Kripkean algebras” that we call “inquisitive algebras”, thus obtaining a second, more general algebraic semantics for this logic.

Clearly, not only implication, but also other logical connectives are crucial for the theory and the applications of particular logics. Hence, this chapter is devoted to the study of two groups of important connectives: lattice and residuated connectives.We start by exploring the logical and algebraic properties of these connectives. In the style of this book, we introduce them in terms of Hilbert-style rules which enforce the expected semantical behavior and study consequences of their presence in a logic.After that, we introduce and study two minimal logics containing certain collections of these connectives: Lambek logic LL (the least weakly implicative logic where all the residuated connectives have the minimal intended behavior) and the logic SL (the least expansion of LL where also the lattice connectives have the minimal intended behavior and furthermore the truth constant 1 behaves as the unit). We axiomatize them by means of strongly separable Hilbert-style calculi and describe their classes of reduced models and show that they admit regular completions. These two logics also serve as bases for our study of substructural logics. Indeed, we define them as the class of weakly implicative logics expanding the implicative fragment of LL and then we build a hierarchy of substructural logics expanding SL, which contains the most prominent members of this family.The rest of the chapter is devoted to the study of deduction theorems in substructural logics, capitalizing on the presence of implication and residuated conjunction. We introduce a general notion of implicational deduction theorem and provide a characterization in terms of the existence of a presentation that only has one binary rule (modus ponens) and unary rules of a certain simple form. As interesting byproducts, for any logic enjoying such deduction theorem, we obtain a description of filter generation (in its reduced models) and a construction technique for a new form of generalized disjunction connective (given by a set of formulas) with the proof by cases property.

The last chapter gives a short introduction to the study of first-order predicate logics built over weakly implicative logics. We follow a semantics-first approach in which we start from semantically defined predicate logics and then propose suitable Hilbert-style axiomatizations and prove corresponding completeness theorems by following non-trivial generalizations of Henkin’s proof of completeness of classical first-order logic. More precisely, to introduce a general semantics of predicate models we utilize the fact that any reduced model of a given weakly implicative logic is ordered and define the truth-value of a universal (resp. existentially) quantified formula as the infimum (resp. supremum) of the truth-values of its instances.Then, for any given class of reduced models of the logic in question, we define a corresponding consequence relation on predicate formulas. We focus on three particular meaningful logics: the predicate logic of all reduced models, the subdirectly irreducible ones, and the restriction of the latter to witnessed predicate models in which quantifiers are actually computed as minima and maxima. We propose axiomatic systems for these three logics and, under certain conditions, prove the corresponding completeness theorems.While the first completeness result is relatively straightforward and works in absolute generality, the other two require non-trivial modifications of Henkin’s proof by making use of a suitable disjunction connective which needs to be added as a requirement for the propositional logic. As a by-product, we also obtain, for certain logics, a form of Skolemization. The relatively modest assumptions on the propositional side allow for a wide generalization of previous approaches and help to illuminate the ‘essentially first-order’ steps in the classical Henkin’s proof.

The general kind of disjunctions enjoying the proof by cases property obtained for substructural logics in the previous chapter (given by a set of formulas instead of a single disjunction connective) motivates the abstract study of generalized disjunctions that we present in the fifth chapter.First, we introduce several forms of proof by cases property and corresponding generalized disjunctions, yielding a hierarchy of logics that we illustrate and separate with suitable examples. We continue by providing several (groups of) characterizations of generalized disjunctions.The first characterization is based on the notion of special forms of a rule, which is obtained from the original rule by disjuncting its premises and conclusion with an arbitrary formula. This notion allows us to characterize logics with a strong p-disjunction as those closed under the formation of these forms of its rules. This characterization is then used to study the preservation of the proof by cases property in expansions and to prove its transfer to the general matrix semantics in terms of generated filters. The second group of characterizations is based on various generalized distributivity properties of the lattice of filters. The third characterization is based on prime filters, a generalization of the well-known notion for classical and intuitionistic logic and its relation to the intersection-prime filters. Finally, we use generalized disjunctions in order to: (1) obtain some axiomatizations of interesting extensions of a given logic, (2) introduce a symmetric notion of consequence relation with a disjunctive reading on the right-hand side, (3) improve some of the characterizations of completeness properties seen in Chapter 3, and, finally, (4) study completeness with respect to finite matrices and matrices with prime filters.

This chapter focuses on the other family that motivated the general study of logics with implication: semilinear logics, defined as logics complete with respect to linearly ordered reduced matrices.We start by formulating and proving useful characterizations of semilinear logics in terms of linear filters, a syntactical metarule akin to the proof by cases property, and the coincidence of finitely subdirectly irreducible and linearly ordered reduced matrices. We use these characterizations to show which of the examples of weakly implicative logics considered in the previous chapters are actually semilinear logics and prove which of them are not semilinear with respect to any possible implication. Then we study the problem of, given an arbitrary weakly implicative logic (in particular, given a substructural logic), finding its least semilinear extension. We also use the presence of disjunction and the results obtained in the previous chapter to prove better characterizations of semilinearity leading to axiomatizations of the least semilinear extension of a given logic. Finally, we focus on completeness with respect to the subclass of linear models in which the order is dense and discuss its relation with completeness with respect to reduced matrices defined over the rational and the real unit intervals.

This chapter is the real beginning of our story and hence it is devoted to presenting its main object of study: the class of weakly implicative logics.We start by introducing basic syntactical notions (variables, connectives, formulas, Hilbert-style proof systems, etc.) and giving a purely syntactical definition of logics as mathematical objects (namely, as structural consequence relations). After testing the definition with three extreme, mostly uninteresting examples, we immediately present some of the most important logics that one can find in the literature and that will accompany the reader throughout the book: classical logic, intuitionisticlogic, Łukasiewicz logic (both its finitary and infinitary version), Gödel-Dummett logic, and the implicational logics known as BCI or BCK. We conclude the syntactical part of the chapter with a brief study of important metalogical properties of these logics such as (variants of) the deduction theorem and the proof by cases property; these properties are later, in Chapters 4 and 5, studied in a much more abstract setting.Next, we start introducing basic semantical notions. The fundamental one is that of a logical matrix, which is an arbitrary algebra equipped with a set of designated truth-values, called the filter of the matrix. Any logic can be assigned a class of logical matrices, whose members we call models of the logic, which can be shown to provide a complete semantics, albeit a very uninformative one with a very loose connection to the logic in question. In order to obtain a more meaningful semantics,we introduce the notion of Leibniz congruence of a given matrix, which allows us to define, for any given logic, a class of reduced models which provide a more refined complete semantics and are more tightly related to the logic in question.Finally, we give a purely syntactical definition of the class of weakly implicative logics and show that these are exactly the logics where the Leibniz congruence admits a simple description using the implication connective. This connective can also be used to define another fundamental notion of the book: the matrix order in the reduced models of weakly implicative logics. The chapter is concluded by introducing and exploring the class algebraically implicative logics, in which one can disregard logical matrices and work just with algebras instead.

This chapter presents the foundations of the theory of logical matrices with a special focus on the question of which classes of matrices provide a complete semantics for a given logic. We identify three kinds of completeness based on how we restrict the cardinality of the sets of premises: we distinguish strong completeness, where there is no restriction, finite strong completeness, where we restrict ourselves to finite sets of premises, and weak completeness, where we disregard premises altogether and speak about theorems and tautologies only. Throughout the chapter we introduce increasingly complex model-theoretic constructions on (classes of) matrices and use them to characterize not only the completeness properties but also other important notions. We start by introducing submatrices, homomorphisms and direct products of matrices. We use these tools to improve our understanding of the Leibniz congruence and obtain an important semantical characterization of the notion of conservative expansions and of the class of algebraically implicative logics. Our next tools are the subdirect products and subdirectly irreducible matrices. We show that each finitary logic is strongly complete with respect to such matrices, in particular recovering the completeness of classical logic with respect to the two-element Boolean algebra. Finally, the most complex construction we use are the filtered products, in particular countably-filtered products and ultraproducts, whichwe use to give a purely semantical characterization of finitary logics and completeness properties.

Many-valued logics, in general, and real-valued logics, in particular, usually focus on a notion of consequence based on preservation of full truth, typically represented by the value $1$ in the semantics given in the real unit interval $[0,1]$. In a recent paper [Foundations of Reasoning with Uncertainty via Real-valued Logics, Proceedings of the National Academy of Sciences 121(21): e2309905121, 2024], Ronald Fagin, Ryan Riegel, and Alexander Gray have introduced a new paradigm that allows to deal with inferences in propositional real-valued logics based on a rich class of sentences, multi-dimensional sentences, that talk about combinations of any possible truth values of real-valued formulas. They have proved a strong completeness result that allows one to derive exactly what information can be inferred about the combinations of truth values of a collection of formulas given information about the combinations of truth values of a finite number of other collections of formulas. In this paper, we extend that work to the first-order (as well as modal) logic of multi-dimensional sentences. We give a parameterized axiomatic system that covers any reasonable logic and prove a corresponding completeness theorem, first assuming that the structures are defined over a fixed domain, and later for the logics of varying domains. As a by-product, we also obtain a zero-one law for finitely-valued versions of these logics. Since several first-order real-valued logics are known not to have recursive axiomatizations but only infinitary ones, our system is by force akin to infinitary systems.

This paper studies which truth-values are most likely to be taken on finite models by arbitrary sentences of a many-valued predicate logic. The classical zero-one law (independently proved by Fagin and Glebskiĭ et al.) states that every sentence in a purely relational language is almost surely false or almost surely true, meaning that the probability that the formula is true in a randomly chosen finite structures of cardinal n is asymptotically $0$ or $1$ as n grows to infinity. We obtain generalizations of this result for any logic with values in a finite lattice-ordered algebra, and for some infinitely valued logics, including Łukasiewicz logic. The finitely valued case is reduced to the classical one through a uniform translation and Oberschelp’s generalization of the zero-one law. Moreover, it is shown that the complexity of determining the almost sure value of a given sentence is PSPACE-complete (generalizing Grandjean’s result for the classical case), and for some logics we describe completely the set of truth-values that can be taken by sentences almost surely.

Lindström’s theorem obviously fails as a characterization of first-order logic without identity ( $\mathcal {L}_{\omega \omega }^{-} $ ). In this note, we provide a fix: we show that $\mathcal {L}_{\omega \omega }^{-} $ is a maximal abstract logic satisfying a weak form of the isomorphism property (suitable for identity-free languages and studied in [11]), the Löwenheim–Skolem property, and compactness. Furthermore, we show that compactness can be replaced by being recursively enumerable for validity under certain conditions. In the proofs, we use a form of strong upwards Löwenheim–Skolem theorem not available in the framework with identity.

In the literature on vagueness one finds two very different kinds of degree theory. The dominant kind of account of gradable adjectives in formal semantics and linguistics is built on an underlying framework involving bivalence and classical logic: its degrees are not degrees of truth. On the other hand, fuzzy logic based theories of vagueness—largely absent from the formal semantics literature but playing a significant role in both the philosophical literature on vagueness and in the contemporary logic literature—are logically nonclassical and give a central role to the idea of degrees of truth. Each kind of degree theory has a strength: the classical kind allows for rich and subtle analyses of the comparative form of gradable adjectives and of various types of gradable precise adjectives, while the fuzzy kind yields a compelling solution to the sorites paradox. This paper argues that the fuzzy kind of theory can match the benefits of the classical kind and hence that the burden is on the latter to match the advantages of the former. In particular, we develop a new version of the fuzzy logic approach that—unlike existing fuzzy theories—yields a compelling analysis of the comparative as well as an adequate account of gradable precise predicates, while still retaining the advantage of genuinely solving the sorites paradox.

This monograph presents a general theory of weakly implicative logics, a family covering a vast number of non-classical logics studied in the literature, concentrating mainly on the abstract study of the relationship between logics and their algebraic semantics. It can also serve as an introduction to algebraic logic, both propositional and first-order, with special attention paid to the role of implication, lattice and residuated connectives, and generalized disjunctions. Based on their recent work, the authors develop a powerful uniform framework for the study of non-classical logics. In a self-contained and didactic style, starting from very elementary notions, they build a general theory with a substantial number of abstract results. The theory is then applied to obtain numerous results for prominent families of logics and their algebraic counterparts, in particular for superintuitionistic, modal, substructural, fuzzy, and relevant logics. The book may be of interest to a wide audience, especially students and scholars in the fields of mathematics, philosophy, computer science, or related areas, looking for an introduction to a general theory of non-classical logics and their algebraic semantics.

Stemming from the works of Petr Hájek on mathematical fuzzy logic, graded model theory has been developed by several authors in the last two decades as an extension of classical model theory that studies the semantics of many-valued predicate logics. In this paper we take the first steps towards an abstract formulation of this model theory. We give a general notion of abstract logic based on many-valued models and prove six Lindström-style characterizations of maximality of first-order logics in terms of metalogical properties such as compactness, abstract completeness, the Löwenheim–Skolem property, the Tarski union property, and the Robinson property, among others. As necessary technical restrictions, we assume that the models are valued on finite MTL-chains and the language has a constant for each truth-value.

This paper is devoted to the problem of existence of saturated models for first-order many-valued logics. We consider a general notion of type as pairs of sets of formulas in one free variable that express properties that an element of a model should, respectively, satisfy and falsify. By means of an elementary chains construction, we prove that each model can be elementarily extended to a $\kappa $-saturated model, i.e. a model where as many types as possible are realized. In order to prove this theorem we obtain, as by-products, some results on tableaux (understood as pairs of sets of formulas) and their consistency and satisfiability and a generalization of the Tarski–Vaught theorem on unions of elementary chains. Finally, we provide a structural characterization of $\kappa $-saturation in terms of the completion of a diagram representing a certain configuration of models and mappings.

This paper is a contribution to graded model theory, in the context of mathematical fuzzy logic. We study characterizations of classes of graded structures in terms of the syntactic form of their first-order axiomatization. We focus on classes given by universal and universal-existential sentences. In particular, we prove two amalgamation results using the technique of diagrams in the setting of structures valued on a finite MTL-algebra, from which analogues of the Łoś–Tarski and the Chang–Łoś–Suszko preservation theorems follow.

It is well known that MTL satisfies the finite embeddability property. Thus MTL is complete w. r. t. the class of all finite MTL-chains. In order to reach a deeper understanding of the structure of this class, we consider the extensions of MTL by adding the generalized contraction since each finite MTL-chain satisfies a form of this generalized contraction. Simultaneously, we also consider extensions of MTL by the generalized excluded middle laws introduced in [9] and the axiom of weak cancellation defined in [31]. The algebraic counterpart of these logics is studied characterizing the subdirectly irreducible, the semisimple, and the simple algebras. Finally, some important algebraic and logical properties of the considered logics are discussed: local finiteness, finite embeddability property, finite model property, decidability, and standard completeness

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 subdirectly irreducible models. We identify two syntactical notions formulated in terms of 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.

This paper presents an abstract study of completeness properties of non-classical logics with respect to matricial semantics. Given a class of reduced matrix models we define three completeness properties of increasing strength and characterize them in several useful ways. Some of these characterizations hold in absolute generality and others are for logics with generalized implication or disjunction connectives, as considered in the previous papers. Finally, we consider completeness with respect to matrices with a linear dense order and characterize it in terms of an extension property and a syntactical metarule. This is the final part of the investigation started and developed in the papers :417–446, 2010; Arch Math Logic 53:353–372, 2016).

This paper continues the investigation, started in Lávička and Noguera : 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, 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.

In this position paper we present a logical framework for modelling reasoning with graded predicates. We distinguish several types of graded predicates and discuss their ubiquity in rational interaction and the logical challenges they pose. We present mathematical fuzzy logic as a set of logical tools that can be used to model reasoning with graded predicates, and discuss a philosophical account of vagueness that makes use of these tools. This approach is then generalized to other kinds of graded predicates. Finally, we propose a general research program towards a logic-based account of reasoning with graded predicates.

This paper aims at being a systematic investigation of different completeness properties of first-order predicate logics with truth-constants based on a large class of left-continuous t-norms . We consider standard semantics over the real unit interval but also we explore alternative semantics based on the rational unit interval and on finite chains. We prove that expansions with truth-constants are conservative and we study their real, rational and finite chain completeness properties. Particularly interesting is the case of considering canonical real and rational semantics provided by the algebras where the truth-constants are interpreted as the numbers they actually name. Finally, we study completeness properties restricted to evaluated formulae of the kind , where φ has no additional truth-constants

This paper is a contribution to Mathematical fuzzy logic, in particular to the algebraic study of t-norm based fuzzy logics. In the general framework of propositional core and Δ-core fuzzy logics we consider three properties of completeness with respect to any semantics of linearly ordered algebras. Useful algebraic characterizations of these completeness properties are obtained and their relations are studied. Moreover, we concentrate on five kinds of distinguished semantics for these logics–namely the class of algebras defined over the real unit interval, the rational unit interval, the hyperreals, the strict hyperreals and finite chains, respectively–and we survey the known completeness methods and results for prominent logics. We also obtain new interesting relations between the real, rational and hyperreal semantics, and good characterizations for the completeness with respect to the semantics of finite chains. Finally, all completeness properties and distinguished semantics are also considered for the first-order versions of the logics where a number of new results are proved.

Substructural logics extending the full Lambek calculus FL have largely benefited from a systematical algebraic approach based on the study of their algebraic counterparts: residuated lattices. Recently, a nonassociative generalization of FL has been studied by Galatos and Ono as the logic of lattice-ordered residuated unital groupoids. This paper is based on an alternative Hilbert-style presentation for SL which is almost MP -based. This presentation is then used to obtain, in a uniform way applicable to most substructural logics, a form of local deduction theorem, description of filter generation, and proper forms of generalized disjunctions. A special stress is put on semilinear substructural logics. Axiomatizations of the weakest semilinear logic over SL and other prominent substructural logics are provided and their completeness with respect to chains defined over the real unit interval is proved.