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 This paper investigates recovery operators in quasi-Nelson (QN) logic, the algebraizable logical counterpart of QN algebras. These form a variety of three-potent, distributive, but not necessarily involutive residuated lattices that may be regarded as a common generalization of Nelson and Heyting algebras. We consider both consistency and determinedness operators, with a particular focus on logics and algebras that satisfy the prelinearity condition, which is well-known in the area of mathematical fuzzy logics. We show that, essentially, all algebraic and logical results already proved for (prelinear, distributive) involutive residuated lattice-based logics of formal inconsistency/logics of formal undeterminedness can be recovered in the QN setting, where one dispenses with the involutivity assumption. In this setting, consistency and undeterminedness operators are no longer duals of one another, and hence call for a more fine-grained algebraic and logical formalization.

Lewis-Gärdenfors imaging is an updating procedure for probability functions that generalizes Bayesian conditionalization, allowing to approach the probability of conditionals and counterfactual formulas without incurring in Lewis well-known triviality result. Precisely, while the probability of a so called Stalnaker conditional (as formalizable in Lewis logic C2) was proved to be an imaged probability by Lewis in his celebrated paper from 1976, a variant of Gärdenfors generalized imaging (proposed by Dubois and Prade in 1994) has been recently proved to characterize the probability of Lewis counterfactuals, the latter refer to those conditionals of Lewis’s logic C1. The present contribution extends the analysis on Lewis’s triviality result, imaging and generalized imaging to cope with possibility and necessity measures. In particular, after showing that the triviality result also holds in the possibilistic framework, we introduce a way to define the possibility measure of conditional and counterfactual formula. Then we prove that the possibilistic version of Lewis-Gärdenfors imaging (that is inspired by a definition given again by Dubois and Prade in 1994) actually characterizes, as in the aforementioned cases, the possibility of Stalnaker conditionals and Lewis counterfactuals. Furthermore, we show that possibilistic imaging can also be described within the setting of Boolean algebras of conditionals and Lewis algebras. These are algebraic models for conditional and counterfactual formulas recently introduced by two of the present authors. On these structures one can (canonically) define a notion of possibility measure that turns out to be the conditional possibility and imaged possibility mentioned above, respectively, and hence it represents the possibility of conditional and counterfactual formulas.

In this paper we investigate canonical extensions of conditional probabilities to Boolean algebras of conditionals. Before entering into the probabilistic setting, we first prove that the lattice order relation of every Boolean algebra of conditionals can be characterized in terms of the well-known order relation given by Goodman and Nguyen. Then, as an interesting methodological tool, we show that canonical extensions behave well with respect to conditional subalgebras. As a consequence, we prove that a canonical extension and its original conditional probability agree on basic conditionals. Moreover, we verify that the probability of conjunctions and disjunctions of conditionals in a recently introduced framework of Boolean algebras of conditionals are in full agreement with the corresponding operations of conditionals as defined in the approach developed by two of the authors to conditionals as three-valued objects, with betting-based semantics, and specified as suitable random quantities. Finally we discuss relations of our approach with nonmonotonic reasoning based on an entailment relation among conditionals.

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.

This paper presents an investigation on the structure of conditional events and on the probability measures which arise naturally in that context. In particular we introduce a construction which defines a (finite) Boolean algebra of conditionals from any (finite) Boolean algebra of events. By doing so we distinguish the properties of conditional events which depend on probability and those which are intrinsic to the logico-algebraic structure of conditionals. Our main result provides a way to regard standard two-place conditional probabilities as one-place probability functions on conditional events. We also consider a logical counterpart of our Boolean algebras of conditionals with links to preferential consequence relations for non-monotonic reasoning. The overall framework of this paper provides a novel perspective on the rich interplay between logic and probability in the representation of conditional knowledge.

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 this article, we study a dialogue-based approach to multi-agent collaborative plan search in the framework of t-DeLP, an extension of DeLP for defeasible temporal reasoning. In t-DeLP programs, temporal facts and rules combine into arguments, which compare against each other to decide which of their conclusions are to prevail. By adding temporal actions for multiple agents to this argumentative logic programming framework, one obtains a centralized planning framework. In this planning system, it can be shown that breadth-first search is sound and complete for both forward and backward planning. The main contribution is to extend these results in centralized planning to cooperative planning tasks, where the executing agents themselves are assumed to have reasoning and planning abilities. In particular, we propose a planning algorithm where agents exchange information on plans using suitable dialogues. We show that the soundness and completeness properties of centralized t-DeLP plan search are preserved, so the dialoguing agents will reach an agreement upon a joint plan if and only if some solution exists.

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.

The Belief-Desire-Intention (BDI) architecture is a practical approach for modelling large-scale intelligent systems. In the BDI setting, a complex system is represented as a network of interacting agents - or components - each one modelled based on its beliefs, desires and intentions. However, current BDI implementations are not well-suited for modelling more realistic intelligent systems which operate in environments pervaded by different types of uncertainty. Furthermore, existing approaches for dealing with uncertainty typically do not offer syntactical or tractable ways of reasoning about uncertainty. This complicates their integration with BDI implementations, which heavily rely on fast and reactive decisions. In this paper, we advance the state-of-the-art w.r.t. handling different types of uncertainty in BDI agents. The contributions of this paper are, first, a new way of modelling the beliefs of an agent as a set of epistemic states. Each epistemic state can use a distinct underlying uncertainty theory and revision strategy, and commensurability between epistemic states is achieved through a stratification approach. Second, we present a novel syntactic approach to revising beliefs given unreliable input. We prove that this syntactic approach agrees with the semantic definition, and we identify expressive fragments that are particularly useful for resource-bounded agents. Third, we introduce full operational semantics that extend CAN, a popular semantics for BDI, to establish how reasoning about uncertainty can be tightly integrated into the BDI framework. Fourth, we provide comprehensive experimental results to highlight the usefulness and feasibility of our approach, and explain how the generic epistemic state can be instantiated into various representations.

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.

Possibilistic defeasible logic programming (P-DeLP) is a logic programming framework which combines features from argumentation theory and logic programming, in which defeasible rules are attached with weights expressing their relative belief or preference strength. In P-DeLP,a conclusion succeeds if there exists an argument that entails the conclusion and this argument is found to be undefeated by a warrant procedure that systematically explores the universe of arguments in order to present an exhaustive synthesis of the relevant chains of pros and cons for the given conclusion. Recently, we have proposed a new warrant recursive semantics for P-DeLP, called Recursive P-DeLP (RP-DeLP for short), based on the claim that the acceptance of an argument should imply also the acceptance of all its sub-arguments which reflect the different premises on which the argument is based. This paper explores the relationship between the exhaustive dialectical analysis-based semantics of P-DeLP and the recursive-based semantics of RP-DeLP, and analyses a non-monotonic inference operator for RP-DeLP which models the expansion of a given program by adding new weighted facts associated with warranted conclusions. Given the recursive-based semantics of RP-DeLP, we have also implemented an argumentation framework for RP-DeLP that is able to compute not only the output of warranted and blocked conclusions, but also explain the reasons behind the status of each conclusion. We have developed this framework as a stand-alone application with a simple text-based input/output interface to be able to use it as part of other artificial intelligence systems.

In this paper we first define a recursive semantics for warranted formulas in a general defeasible argumentation framework by formalizing a notion of collective (non-binary) conflict among arguments. The recursive semantics for warranted formulas is based on the fact that if the argument is rejected, then all arguments built on it should also be rejected. The main characteristic of our recursive semantics is that an output (extension) of a knowledge base is a pair of sets of warranted and blocked formulas. Arguments for both warranted and blocked formulas are recursively based on warranted formulas but, while warranted formulas do not generate any collective conflict, blocked conclusions do. Formulas that are neither warranted nor blocked correspond to rejected formulas. Second we extend the general defeasible argumentation framework by attaching levels of preference to defeasible knowledge items and by providing a level-wise definition of warranted and blocked formulas. Third we formalize the warrant recursive semantics for the particular framework of Possibilistic Defeasible Logic Programming, we call this particular framework Recursive Possibilistic Defeasible Logic Programming (\\mbox{RP-DeLP} for short), and we show its relevance in the scope of Political debates. An RP-DeLP program may have multiple outputs in case of circular definitions of conflicts among arguments. So, we tackle the problem of which output one should consider for an RP-DeLP program with multiple outputs. To this end we define the maximal ideal output of an RP-DeLP program as the set of conclusions which are ultimately warranted and we present an algorithm for computing them in polynomial space and with an upper bound on complexity equal to P^{NP}. Finally, we propose an efficient and scalable implementation of this algorithm that is based on implementing the two main queries of the system, looking for valid arguments and collective conflicts between arguments, using SAT encodings. We perform an experimental evaluation of our SAT based approach when solving test sets of instances with single and multiple preference levels for defeasible knowledge.

Revising its beliefs when receiving new information is an important ability of any intelligent system. However, in realistic settings the new input is not always certain. A compelling way of dealing with uncertain input in an agent-based setting is to treat it as unreliable input, which may strengthen or weaken the beliefs of the agent. Recent work focused on the postulates associated with this form of belief change and on finding semantical operators that satisfy these postulates. In this paper we propose a new syntactic approach for this form of belief change and show that it agrees with the semantical definition. This makes it feasible to develop complex agent systems capable of efficiently dealing with unreliable input in a semantically meaningful way. Additionally, we show that imposing restrictions on the input and the beliefs that are entailed allows us to devise a tractable approach suitable for resource-bounded agents or agents where reactive ness is of paramount importance.

When an agent wants to fulfill its desires about the world, the agent usually has multiple plans to choose from and these plans have different pre-conditions and additional effects in addition to achieving its goals. Therefore, for further reasoning and interaction with the world, a plan selection strategy (usually based on plan cost estimation) is mandatory for an autonomous agent. This demand becomes even more critical when uncertainty on the observation of the world is taken into account, since in this case, we consider not only the costs of different plans, but also their chances of success estimated according to the agent's beliefs. In addition, when multiple goals are considered together, different plans achieving the goals can be conflicting on their preconditions (contexts) or the required resources. Hence a plan selection strategy should be able to choose a subset of plans that fulfills the maximum number of goals while maintaining context consistency and resource-tolerance among the chosen plans. To address the above two issues, in this paper we first propose several principles that a plan selection strategy should satisfy, and then we present selection strategies that stem from the principles, depending on whether a plan cost is taken into account. In addition, we also show that our selection strategy can partially recover intention revision.

In previous works, a recursive warrant semantics for Defeasible Logic Programming extended with levels of possibilistic uncertainty for defeasible rules was introduced. The resulting argumentation framework, called RP-DeLP, is based on a general notion of collective (non-binary) conflict among arguments allowing to ensure direct and indirect consistency properties with respect to the strict knowledge. In this paper we propose an efficient and scalable implementation of an interpreter for RP-DeLP using Answer Set Programming (ASP) encodings for the two main queries of the system: looking for valid arguments and finding collective conflicts among arguments. We perform an experimental evaluation of our ASP approach and we compare the results with a previously proposed SAT based approach. The results show that with ASP we are able to scale up to bigger problem instances.

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.

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.

Possibility and necessity measures are commonly defined over Boolean algebras. This work considers a generalization of these kinds of measures over MV-algebras as a possibilistic counterpart of the (probabilistic) notion of state on MV-algebras. Two classes of possibilistic states over MV-algebras of functions are characterized in terms of (generalized) Sugeno integrals. For reasoning about these representable classes of possibilistic states, we introduce many-valued modal logics based on the Rational Łukasiewicz Logic, that are shown to be complete with respect to corresponding classes of Kripke models equipped with those states.

In this paper we define a recursive semantics for warrant in a general defeasible argumentation framework by formalizing a notion of collective (non-binary) conflict among arguments. This allows us to ensure direct and indirect consistency (in the sense of Caminada and Amgoud) without distinguishing between direct and indirect conflicts. Then, the general defeasible argumentation framework is extended by allowing to attach levels of preference to defeasible knowledge items and by providing a level-wise definition of warranted and blocked conclusions. Finally, we formalize the warrant recursive semantics for the particular framework of Possibilistic Defeasible Logic Programming, characterize the unique output program property and design an efficient algorithm for computing warranted conclusions in polynomial space.

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.

De Morgan duality between a t-norm T and a t-conorm S is defined with respect to an involutive (or strong) negation N, so that S(x, y) = N(T(N(x), N(y))), and vice versa T(x,y) = N(S(N(x),N(y))). A weaker form of duality can be met when the negation operator N is only required to be a decreasing bijection such that S = N/sup -1/ T(N x N). In this paper we address some issues about the (general) duality between continuous t-norms and t-conorms. Given such a t-norm T and such a t-conorm S, we show how to construct a negation N (possibly non-involutive) that makes T and S become N-dual.

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.

We propose a solution for overcoming possible lacks of covering of the input space in fuzzy rule bases. Our approach is based on similarity-based reasoning and considers a kind of extrapolative inference rule which enlarges the range of applicability of a fuzzy rule by replacing conditions in the premise of the form "X is A" by "X is approximately-A", where approximately-A is the image of A by a suitable fuzzy similarity relation.

In this article, we present a first-order logic programming language for fuzzy reasoning under possibilistic uncertainty and poorly known information. Formulas are represented by a pair (φ, α), in which φ is a first-order Horn clause or a query with fuzzy constants and regular predicates, and α ∈ [0, 1] is a lower bound on the belief on φ in terms of necessity measures. Since fuzzy constants can occur in the logic component of formulas, the truth value of formulas is many-valued instead of Boolean. Moreover, since we have to reason about the possibilistic uncertainty of formulas with fuzzy constants, belief states are modeled by normalized possibility distributions on a set of many-valued interpretations. In this framework, (1) we define a syntax and a semantics of the underlying logic; (2) we give a sound modus ponens-style calculus by derivation based on a semantic unification pattern of fuzzy constants; (3) we develop a directional fuzzy unification algorithm based on the distinction between general and specific object constants; and (4) we describe a backward first-order proof procedure oriented to queries that is based on the calculus of the language and the computation of the unification degree between fuzzy constants in terms of a necessity measure for fuzzy events. © 2002 Wiley Periodicals, Inc.

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).

Representational issues of preferences in the framework of a possibilistic ordinal decision model under uncertainty were introduced by Dubois and Prade a few years ago. In this framework, finite, linear, commensurate uncertainty and preference scales are assumed and decisions are ranked according to their expected utility in terms of Sugeno integrals. In this paper we generalise the model by allowing (i) to measure uncertainty and preferences on finite, distributive lattices with involution, and (ii) to formulate the utility expectations in terms of generalized Sugeno integrals where t-norms and t-conorms play a role. For these generalized utility functions we provide axiomatic characterisations. Finally, we also briefly comment the case of working with belief states that may be partially inconsistent, which is usual in case-based decision making problems or when several inconsistent sources of information are available.

In many fuzzy systems applications, values to be aggregated are of a qualitative nature. In that case, if one wants to compute some type of average, the most common procedure is to perform a numerical interpretation of the values, and then apply one of the well-known (the most suitable) numerical aggregation operators. However, if one wants to stick to a purely qualitative setting, choices are reduced to either weighted versions of max-min combinations or to a few existing proposals of qualitative versions of ordered weighted average (OWA) operators. In this paper, we explore the feasibility of defining a qualitative counterpart of the weighted mean operator without having to use necessarily any numerical interpretation of the values. We propose a method to average qualitative values, belonging to a (finite) ordinal scale, weighted with natural numbers, and based on the use of finite t-norms and t-conorms defined on the scale of values. Extensions of the method for other OWA-like and Choquet integral-type aggregations are also considered.

This paper improves a previously proposed axiomatic setting for qualitative decision under uncertainty in the von Neumann and Morgenstern' style, where only ordinal linear scales are required for assessing uncertainty and utility. Two qualitative criteria are axiomatized in a finite setting: a pessimistic one and an optimistic one, respectively obeying an uncertainty aversion axiom and an uncertainty-attraction axiom. These criteria generalize the well-known maximin and maximax criteria, making them more realistic. They are suited to one-shot decisions and they are not based on the notion of mean value, but take the form of medians. Elements for a qualitative case-based decision methodology are also proposed, with pessimistic and optimistic evaluations formally similar to the expressions which cope with uncertainty, up to modifying factors which cope with the lack of normalization of similarity evaluations. Finally two extensions of the model are analysed: (i) the case of generalized possibilistic mixtures, using a t-norm instead of min, and (ii) the case of evaluating either preferences or uncertainty on Cartesian products of ordinal scales.

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.

In dealing with representing knowledge under uncertainty there is a sustained tendency to increase flexibility in order to avoid problems of inconsistency in the knowledge. Early uncertainty management systems dealt with single real values within a predefined range, soon interval valued approaches were proposed and more recently we have witnessed the introduction of fuzzy-interval valued approaches, i.e., possibility distributions and fuzzy truth-values. In this paper we describe these fuzzy set based approaches with an emphasis on the concept of fuzzy truth-value.

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.

The objective of this article is to describe the MILORD Shell and particularly its architecture and its management of uncertainty. MILORD is an expert systems building tool consisting of two inference engines and an explanation module. the system allows one to perform different calculi of uncertainty on an expert defined set of linguistic terms expressing uncertainty. Each calculus corresponds to specific conjunction, disjunction, and implication operators. the internal representation of each linguistic uncertainty value is a fuzzy subset of the interval [0,1]. the different calculi of uncertainty applied to the set of linguistic terms give, as a result, a fuzzy subset that is approximated, by means of a linguistic approximation process, to a linguistic certainty value belonging to the set of linguistic terms. This linguistic approximation keeps the calculus of uncertainty closed. This has the advantage that, once the linguistic certainty values have been defined, the system computes, off-line, the conjunction, disjunction, and implication operations for all the pairs of linguistic uncertainty values in the term set and stores the results in matrices. Therefore, when MILORD is run, the propagation and combination of uncertainty is performed by simply accessing these precomputed matrices. MILORD also deals with nonmonotonic reasoning in the same framework of uncertainty management. Finally, an application to the diagnosis and treatment of pneumoniae is presented.