Nowadays is critical to complement Artificial Intelligence (AI) systems, such as those based on Deep Learning and Generative AI, by robust and trustworthy methodologies like different useful mathematical tools, such as (fuzzy) logic, formal concept analysis, rough set theory, etc. Multi-adjoint algebras are flexible structures considered in many of these mathematical tools, which are fundamental for obtaining traceable and reliable information from data sets. Along this line, this paper aims at introducing an algebraizable logic having the quasi-variety of these algebras as its associated equivalent algebraic semantics. To do so, we first introduce a logic associated with bounded lattices with an implication and show it is algebraizable in the sense of Blok-Pigozzi. We then expand this base logic to another algebraizable logic able to properly capture the multi-adjoint framework. Furthermore, we also consider different extensions of the logics and their properties analysed.

In this paper we study the definition and axiomatisation of different (finitary) threshold preserving companions of several extensions of the Monoidal t-norm based fuzzy logic MTL. More in detail, we first focus on the non-falsity preserving logics, where a conclusion follows from a set of premises if, whenever the premises are non-false (i.e., have a truth degree greater than 0), the conclusion is also non-false. We then introduce a new type of companions, namely logics that preserve the notion of acceptability , where a formula is called acceptable whenever it is deemed to be more true than false, or in other words it is more true than its negation. Finally, we also consider a more general stance and consider logics that preserve some intermediate truth-value 0 < a < 1, so that a can be understood as a (strict or non-strict) threshold above which a formula is considered as valid. All these types of threshold-preserving companions of a given MTL logic can be seen as particular cases of matrix logics defined by lattice filters.

Abstract The aim of the paper is to analyze the expressive power of the square operator of Łukasiewicz logic: $\ast x=x\odot x$, where $\odot $ is the strong Łukasiewicz conjunction. In particular, we aim at understanding and characterizing those cases in which the square operator is enough to construct a finite MV-chain from a finite totally ordered set endowed with an involutive negation. The first of our main results shows that, indeed, the whole structure of MV-chain can be reconstructed from the involution and the Łukasiewicz square operator if and only if the obtained structure has only trivial subalgebras and, equivalently, if and only if the cardinality of the starting chain is of the form $n+1$ where $n$ belongs to a class of prime numbers that we fully characterize. Secondly, we axiomatize the algebraizable matrix logic whose semantics is given by the variety generated by a finite totally ordered set endowed with an involutive negation and Łukasiewicz square operator. Finally, we propose an alternative way to account for Łukasiewicz square operator on involutive Gödel chains. In this setting, we show that such an operator can be captured by a rather intuitive set of equations.

Abstract The aim of this paper is to develop an algebraic and logical study of certain paraconsistent systems, from the family of the logics of formal inconsistency (LFIs), which are definable from the degree-preserving companions of logics of distributive involutive residuated lattices ($\textrm {dIRL}$s) with a consistency operator, the latter including as particular cases, Nelson logic ($\textsf {NL}$), involutive monoidal t-norm based logic ($\textsf {IMTL}$) or nilpotent minimum ($\textsf {NM}$) logic. To this end, we first algebraically study enriched dIRLs with suitable consistency operators. In fact, we consider three classes of consistency operators, leading respectively to three subquasivarieties of such expanded residuated lattices. We characterize the simple and subdirectly irreducible members of these quasivarieties, and we extend Sendlewski’s representation results for the case of Nelson lattices with consistency operators. Finally, we define and axiomatize the logics of three quasivarieties of $ \textrm {dIRL}$s and their corresponding degree-preserving companions that belong to the family of LFIs.

In this paper we consider the logics |$\mathsf{L}_n^i$| obtained from the |$(n+1)$|-valued Łukasiewicz logics Ł|$_{n+1}$| by taking the order filter generated by |$i/n$| as the set of designated elements. In particular, the conditions of maximality and strong maximality among them are analyzed. We present a very general theorem that provides sufficient conditions for maximality between logics. As a consequence of this theorem, it is shown that |$\mathsf{L}_n^i$| is maximal w.r.t. CPL whenever |$n$| is prime. Concerning strong maximality (i.e. maximality w.r.t. rules instead of only axioms), we provide algebraic arguments in order to show that the logics |$\mathsf{L}_n^i$| are not strongly maximal w.r.t. CPL, even for |$n$| prime. Indeed, in such case, we show that there is just one extension between |$\mathsf{L}_n^i$| and CPL obtained by adding to |$\mathsf{L}_n^i$| a kind of graded explosion rule. Finally, using these results, we show that the logics |$\mathsf{L}_n^i$| with |$n$| prime and |$i/n < 1/2$| are ideal paraconsistent logics.

In a previous paper, it was shown that the (minimal) modal logic MŁncwith fuzzy accessibility relations over the finite-valued Łukasiewicz logic Łnand a corresponding multi-modal logic mMŁnc(with a modality □afor each value a in the n-valued Łn-chain) had the same expressive power when the language is extended with truth-constants. In this paper we partially extend these results when replacing the underlying logic Łnby the infinite-valued Łukasiewicz logic (with rational truth constants in the language). We prove that the (standard) tautologies of the modal logic MŁnc(resp. mMŁc) are in fact the common tautologies of all the logics MŁnc(resp. all the logics mMŁn) when letting n vary over N. This fact opens the door to show an alternative proof of the finite model property for these logics and hence their decidability.

In this article, we study modal extensions of Product fuzzy logic with both algebraic semantics and relational semantics based on Kripke structures with crisp accessibility relations, when the underlying product fuzzy logic is expanded with truth-constants, the Δ operator and with two infinitary inference rules. We provide completeness results for both kinds of semantics. Finally, we also consider a generalization of possibilistic logic evaluated over product algebras.

A wrong argument in the proof of one of the main results in the paper is corrected.The result itself remains true.The right proof incorporates the basic ideas in the originally alleged proof, but in a more restricted construction.

It is well-known that satisfiability (and hence validity) in the minimal classical modal logic is a PSPACE-complete problem. In this paper we consider the satisfiability and validity problems (here they are not dual, although mutually reducible) for the minimal modal logic over a finite Lukasiewicz chain, and show that they also are PSPACE-complete. This result is also true when adding either the Delta operator or truth constants in the language, i.e. in all these cases it is PSPACE-complete.

This article deals with many-valued modal logics, based only on the necessity operator, over a residuated lattice. We focus on three basic classes, according to the accessibility relation, of Kripke frames: the full class of frames evaluated in the residuated lattice (and so defining the minimum modal logic), the ones only evaluated in the idempotent elements and the ones evaluated in 0 and 1. We show how to expand an axiomatization, with canonical truth-constants in the language, of a finite residuated lattice into one of the modal logic, for each one of the three basic classes of Kripke frames. We also provide axiomatizations for the case of a finite MV chain but this time without canonical truth-constants in the language.

Description Logics (DLs) are knowledge representation languages built on the basis of classical logic. DLs allow the creation of knowledge bases and provide ways to reason on the contents of these bases. Fuzzy Description Logics (FDLs) are natural extensions of DLs for dealing with vague concepts, commonly present in real applications. Following the ideas of Hájek in [17] and García-Cerdaña et al. in [15] we develop a family of FDLs whose underlying logic is the fuzzy logic of a finite linearly ordered residuated lattice, that is, an n-graded fuzzy logic defined by a divisible finite t-norm over a finite chain. Moreover, the role of the constructor of implication in the languages for FDLs is discussed, and a hierarchy of AL-languages adapted to the behavior of the connectives in the fuzzy logics underlying these description languages is proposed. Finally, we deal with reasoning tasks within the framework of finitely valued DLs.

Journal Article Logics Preserving Degrees of Truth from Varieties of Residuated Lattices Get access Félix Bou, Félix Bou Artificial Intelligence Research Institute (IIIA - CSIC), Bellaterra, Spain.E-mail: [email protected]; [email protected] Search for other works by this author on: Oxford Academic Google Scholar Francesc Esteva, Francesc Esteva Artificial Intelligence Research Institute (IIIA - CSIC), Bellaterra, Spain.E-mail: [email protected]; [email protected] Search for other works by this author on: Oxford Academic Google Scholar Josep Maria Font, Josep Maria Font Department of Probability, Logic and Statistics, Faculty of Mathematics, University of Barcelona, Spain.E-mail: [email protected] Search for other works by this author on: Oxford Academic Google Scholar Àngel J. Gil, Àngel J. Gil Departament d'Economia i Empresa, Universitat Pompeu Fabra, Barcelona, Spain.E-mail: [email protected] Search for other works by this author on: Oxford Academic Google Scholar Lluís Godo, Lluís Godo Artificial Intelligence Research Institute (IIIA - CSIC), Bellaterra, Spain.E-mail: [email protected] Search for other works by this author on: Oxford Academic Google Scholar Antoni Torrens, Antoni Torrens Department of Probability, Logic and Statistics, Faculty of Mathematics, University of Barcelona, Spain.E-mail: [email protected]; [email protected] Search for other works by this author on: Oxford Academic Google Scholar Ventura Verdú Ventura Verdú Department of Probability, Logic and Statistics, Faculty of Mathematics, University of Barcelona, Spain.E-mail: [email protected]; [email protected] Search for other works by this author on: Oxford Academic Google Scholar Journal of Logic and Computation, Volume 19, Issue 6, December 2009, Pages 1031–1069, https://doi.org/10.1093/logcom/exp030 Published: 26 June 2009 Article history Received: 03 March 2008 Published: 26 June 2009

Product Logic Π is an axiomatic extension of Hájek's Basic Fuzzy Logic BL coping with the 1-tautologies when the strong conjunction & and implication → are interpreted by the product of reals in [0, 1] and its residuum respectively. In this paper we investigate expansions of Product Logic by adding into the language a countable set of truth-constants (one truth-constant r\#304; for each r in a countable Π-subalgebra 𝒞 of [0, 1]) and by adding the corresponding book-keeping axioms for the truthconstants. We first show that the corresponding logics Π(𝒞) are algebraizable, and hence complete with respect to the variety of Π(𝒞)-algebras. The main result of the paper is the canonical standard completeness of these logics, that is, theorems of Π(𝒞) are exactly the 1-tautologies of the algebra defined over the real unit interval where the truth-constants are interpreted as their own values. It is also shown that they do not enjoy the canonical strong standard completeness, but they enjoy it for finite theories when restricted to evaluated Π-formulas of the kind r\#304; → φ, where r\#304; is a truth-constant and φ a formula not containing truth-constants. Finally we consider the logics ΠΔ(𝒞), the expansion of Π(𝒞) with the well-known Baaz's projection connective Δ, and we show canonical finite strong standard completeness for them.

In this paper we investigate the falsehood-free fragments of main residuated fuzzy logics related to continuous t-norms (Hájek's Basic fuzzy logic BL and some well-known axiomatic extensions), and we relate them to the varieties of 0-free subreducts of the corresponding algebras. These turn out to be classes of algebraic structures known as hoops. We provide axiomatizations of all these fragments and we call them hoop logics; we prove they are strongly complete with respect to their corresponding classes of hoops, and that each fuzzy logic is a conservative extension of the corresponding hoop logic. Analogously, we also study the falsehood-free fragment of a weaker logic than BL, called MTL, which is the logic of left-continuous t-norms and their residua, and we introduce the related algebraic structures which are called semihoops. Moreover, we also consider the falsehood-free fragments of the fuzzy predicate calculi of the above logics and show completeness and conservativeness results. The role of axiom (∀3) in these predicate logics is studied. Finally, computational complexity issues of the prepositional logics are also addressed.

The issue of understanding and modelling vagueness has been addressed by many authors, especially in the second half of the 20th Century. They were mainly philosophers, logicians, psychologists and computer scientists. Often, these authors have preferred one type or one view of vagueness, and have tried to provide some representation scheme for this view using some formalism at hand. In this paper, we provide an organized discussion of different categories of vagueness, pointing out in what circumstances they appear, with a unified view of the formalisms which can be used. Basic representation frameworks are proposed for each case. However, what they have in common leads to a trichotomy of the universe of discourse, which seems to be the common feature of the different forms of vagueness. In each situation, we examine how a fuzzy set-based representation can take place, which gives birth to a different type of fuzzy set-based construction in each case.

Introduces a class of fuzzy closure operators called implicative closure operators, which generalize some notions of fuzzy closure operators already introduced by different authors. We show that implicative closure operators capture some usual consequence relations used in approximate reasoning. We study the relation of the implicative closure operators to other existing fuzzy closure operators as the natural inference operators defined by Boixader and Jacas (1998) and the canonical extension of a classical closure operator defined by Gerla (1996).

This paper is an attempt at providing a fuzzy set formalization of case-based reasoning and decision. Learning aspects are not considered here. The proposed approach assumes a principle stating that “the more similar are the problem description attributes, the more similar are the outcome attributes.” A weaker form of this principle concluding only on the graded possibility of the similarity of the outcome attributes, is also considered. These two forms of the case-based reasoning principle are modelled in terms of fuzzy rules. Then an approximate reasoning machinery taking advantage of this principle enables us to apply the information stored in the memory of previous cases to the current problem. A particular instance of case-based reasoning, named case-based decision, is especially investigated. A logical formalization of the basic case-based reasoning inference is also proposed. Extensions of the proposed approach in order to handle imprecise or fuzzy descriptions or to manage more general forms of the principle underlying case-based reasoning are briefly discussed in the conclusion. © 1998 John Wiley & Sons, Inc.

Within the many-valued approach for approximate reasoning, the aim of this paper is two-fold. First, to extend truth-values lattices to cope with the imprecision due to possible incompleteness of the available information. This is done by considering two bilattices of truth-value intervals corresponding to the so-called weak and strong truth orderings. Based on the use of interval bilattices, the second aim is to introduce what we call partial many-valued logics. The (partial) models of such logics may assign intervals of truth-values to formulas, and so they stand for representations of incomplete states of knowledge. Finally, the relation between partial and complete semantical entailment is studied, and it is provedtheir equivalence for a family of formulas, including the so-called free well formed formulas.

. In this paper we describe an approach to the problem of dealing with uncertainty by means of finite multi-valued logics in modular expert systems, and the results obtained. The modularity of the systems allows us to address two main characteristics of human problem-solving: the adaptation of general knowledge to particular problems and the dependency of the management of uncertainty on the different subtasks being implemented in the modules of the system, i.e. different modules can have different local multiple-valued logics as part of their local deductive mechanisms. Although the results obtained are general, we use, throughout the paper, examples of a medical expert system that has been designed using a modular language called MILORD-II, that implements them showing the practical interest of the theoretical concepts involved.