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The present paper is devoted to study the effect of connected and disconnected rotations of G\"odel algebras with operators grounded on directly indecomposable structures. The structures resulting from this construction we will present are nilpotent minimum (with or without negation fixpoint, depending on whether the rotation is connected or disconnected) with special modal operators defined on a directly indecomposable algebra. In this paper we will present a (quasi-)equational definition of these latter structures. Our main results show that directly indecomposable nilpotent minimum algebras (with or without negation fixpoint) with modal operators are fully characterized as connected and disconnected rotations of directly indecomposable G\"odel algebras endowed with modal operators.

Non-additive uncertainty theories, typically possibility theory, belief functions and imprecise probabilities share a common feature with modal logic: the duality properties between possibility and necessity measures, belief and plausibility functions as well as between upper and lower probabilities extend the duality between possibility and necessity modalities to the graded environment. It has been shown that the all-or-nothing version of possibility theory can be exactly captured by a minimal epistemic logic (MEL) that uses a very small fragment of the KD modal logic, without resorting to relational semantics. Besides, the case of belief functions has been studied independently, and a belief function logic has been obtained by extending the modal logic S5 to graded modalities using {\L}ukasiewicz logic, albeit using relational semantics. This paper shows that a simpler belief function logic can be devised by adding {\L}ukasiewicz logic on top of MEL. It allows for a more natural semantics in terms of Shafer basic probability assignments.

Conditionals play a key role in different areas of logic and probabilistic reasoning, and they have been studied and formalized from different angles. In this paper we focus on the de Finetti's notion of conditional as a three-valued object, with betting-based semantics, and its related approach as random quantity as mainly developed by two of the authors. Compound conditionals have been studied in the literature, but not in full generality. In this paper we provide a natural procedure to explicitly attach conditional random quantities to arbitrary compound conditionals that also allows us to compute their previsions. By studying the properties of these random quantities, we show that, in fact, the set of compound conditionals can be endowed with a Boolean algebraic structure. In doing so, we pave the way to build a bridge between the long standing tradition of three-valued conditionals and a more recent proposal of looking at conditionals as elements from suitable Boolean algebras.

G\"odel modal logics can be seen as extenions of intutionistic modal logics with the prelinearity axiom. In this paper we focus on the algebraic and relational semantics for G\"odel modal logics that leverages on the duality between finite G\"odel algebras and finite forests, i.e. finite posets whose principal downsets are totally ordered. We consider different subvarieties of the basic variety of G\"odel algebras with two modal operators (GAOs for short) and their corresponding classes of forest frames, either with one or two accessibility relations. These relational structures can be considered as prelinear versions of the usual relational semantics of intuitionistic modal logic. More precisely we consider two main extensions of finite G\"odel algebras with operators: the one obtained by adding Dunn axioms, typically studied in the fragment of positive classical (and intuitionistic) logic, and the one determined by adding Fischer Servi axioms. We present J\'onsson-Tarski like representation theorems for the different types of finite GAOs considered in the paper., Comment: 26 pages,10 figures

In this paper, we provide simplified semantics for the logic K45(G), i.e. the many-valued Godel counterpart of the classical modal logic K45. More precisely, we characterize K45(G) as the set of valid formulae of the class of possibilistic Godel Kripke Frames where W is a non-empty set of worlds and \pi: W \to [0, 1] is a possibility distribution on W., Comment: arXiv admin note: text overlap with arXiv:1611.04444

The aim of the paper is to analyze the expressive power of the square operator of Lukasiewicz logic: $\ast x=x\odot x$, where $\odot$ is the strong Lukasiewicz 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 Lukasiewicz square 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 Lukasiewicz's square operator. Finally, we propose an alternative way to account for Lukasiewicz square operator on involutive G\"odel chains. In this setting, we show that such an operator can be captured by a rather intuitive set of equations.

This paper presents an investigation on the structure of conditional events and on the probability measures which arise naturally in this context. In particular we introduce a construction which defines a (finite) {\em 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.

The aim of this paper is to propose a many-valued modal framework to formalize reasoning with both graded preferences and propositions, in the style of van Benthem et al.'s classical modal logics for preferences. To do so, we start from Bou et al.'s minimal modal logic over a finite and linearly ordered residuated lattice. We then define appropriate extensions on a multi-modal language with graded modalities, both for weak and strict preferences, and with truth-constants. Actually, the presence of truth-constants in the language allows us to show that the modal operators Box and Diamond of the minimal modal logic are inter-definable. Finally, we propose an axiomatic system for this logic in an extended language (where the preference modal operators are definable), and prove completeness with respect to the intended graded preference semantics.

Motivated by Gentzen disjunction elimination rule in his Natural Deduction calculus and reading inequalities with meet in a natural way, we conceive a notion of distributivity for join-semilattices. We prove that it is equivalent to a notion present in the literature. In the way, we prove that those notions are linearly ordered. We finally consider the notion of distributivity in join-semilattices with arrow, that is, the algebraic structure corresponding to the disjunction-conditional fragment of intuitionistic logic.

The Logic of Approximate Entailment (LAE) is a graded counterpart of classical propositional calculus, where conclusions that are only approximately correct can be drawn. This is achieved by equipping the underlying set of possible worlds with a similarity relation. When using this logic in applications, however, a disadvantage must be accepted; namely, in LAE it is not possible to combine conclusions in a conjunctive way. In order to overcome this drawback, we propose in this paper a modification of LAE where, at the semantic level, the underlying set of worlds is moreover endowed with an order structure. The chosen framework is designed in view of possible applications.

In this paper we consider the logics $L_n^i$ obtained from the (n+1)-valued Lukasiewicz logics $L_{n+1}$ 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 analysed. We present a very general theorem which provides sufficient conditions for maximality between logics. As a consequence of this theorem it is shown that $L_n^i$ is maximal w.r.t. CPL whenever n is prime. Concerning strong maximality between the logics $L_n^i$ (that is, maximality w.r.t. rules instead of axioms), we provide algebraic arguments in order to show that the logics $L_n^i$ are not strongly maximal w.r.t. CPL, even for n prime. Indeed, in such case, we show there is just one extension between $L_n^i$ and CPL obtained by adding to $L_n^i$ a kind of graded explosion rule. Finally, using these results, we show that the logics $L_n^i$ with n prime and i/n < 1/2 are ideal paraconsistent logics.

The aim of this paper is to extend probability theory from the classical to the product t-norm fuzzy logic setting. More precisely, we axiomatize a generalized notion of finitely additive probability for product logic formulas, called state, and show that every state is the Lebesgue integral with respect to a unique regular Borel probability measure. Furthermore, the relation between states and measures is shown to be one-one. In addition, we study geometrical properties of the convex set of states and show that extremal states, i.e., the extremal points of the state space, are the same as the truth-value assignments of the logic. Finally, we axiomatize a two-tiered modal logic for probabilistic reasoning on product logic events and prove soundness and completeness with respect to probabilistic spaces, where the algebra is a free product algebra and the measure is a state in the above sense., Comment: 27 pages, 1 figure

We expand FLew with a unary connective whose algebraic counterpart is the operation that gives the greatest complemented element below a given argument. We prove that the expanded logic is conservative and has the Finite Model Property. We also prove that the corresponding expansion of the class of residuated lattices is an equational class., Comment: 15 pages, 4 figures in Soft Computing, published online 23 July 2016

In this paper we provide a simplified semantics for the logic KD45(G), i.e. the many-valued G\"odel counterpart of the classical modal logic KD45. More precisely, we characterize KD45(G) as the set of valid formulae of the class of possibilistic G\"odel Kripke Frames $\langle W,\pi \rangle$, where $W$ is a non-empty set of worlds and $\pi: W \longrightarrow [0, 1]$ is a normalized possibility distribution on $W$., Comment: 12 pages

We introduce a new class of games where each player's aim is to randomise her strategic choices in order to affect the other players' expectations aside from her own. The way each player intends to exert this influence is expressed through a Boolean combination of polynomial equalities and inequalities with rational coefficients. We offer a logical representation of these games as well as a computational study of the existence of equilibria., Comment: In Proceedings SR 2014, arXiv:1404.0414

This paper proposes the meeting of fuzzy logic with paraconsistency in a very precise and foundational way. Specifically, in this paper we introduce expansions of the fuzzy logic MTL by means of primitive operators for consistency and inconsistency in the style of the so-called Logics of Formal Inconsistency (LFIs). The main novelty of the present approach is the definition of postulates for this type of operators over MTL-algebras, leading to the definition and axiomatization of a family of logics, expansions of MTL, whose degree-preserving counterpart are paraconsistent and moreover LFIs., Comment: Revised and improved final version. 33 pages, 3 figures

The way experts manage uncertainty usually changes depending on the task they are performing. This fact has lead us to consider the problem of communicating modules (task implementations) in a large and structured knowledge based system when modules have different uncertainty calculi. In this paper, the analysis of the communication problem is made assuming that (i) each uncertainty calculus is an inference mechanism defining an entailment relation, and therefore the communication is considered to be inference-preserving, and (ii) we restrict ourselves to the case which the different uncertainty calculi are given by a class of truth functional Multiple-valued Logics., Comment: Appears in Proceedings of the Seventh Conference on Uncertainty in Artificial Intelligence (UAI1991)

This paper investigates the possibility of performing automated reasoning in probabilistic logic when probabilities are expressed by means of linguistic quantifiers. Each linguistic term is expressed as a prescribed interval of proportions. Then instead of propagating numbers, qualitative terms are propagated in accordance with the numerical interpretation of these terms. The quantified syllogism, modelling the chaining of probabilistic rules, is studied in this context. It is shown that a qualitative counterpart of this syllogism makes sense, and is relatively independent of the threshold defining the linguistically meaningful intervals, provided that these threshold values remain in accordance with the intuition. The inference power is less than that of a full-fledged probabilistic con-quaint propagation device but better corresponds to what could be thought of as commonsense probabilistic reasoning., Comment: Appears in Proceedings of the Eighth Conference on Uncertainty in Artificial Intelligence (UAI1992)

Within the possibilistic approach to uncertainty modeling, the paper presents a modal logical system to reason about qualitative (comparative) statements of the possibility (and necessity) of fuzzy propositions. We relate this qualitative modal logic to the many--valued analogues MVS5 and MVKD45 of the well known modal logics of knowledge and belief S5 and KD45 respectively. Completeness results are obtained for such logics and therefore, they extend previous existing results for qualitative possibilistic logics in the classical non-fuzzy setting., Comment: Appears in Proceedings of the Tenth Conference on Uncertainty in Artificial Intelligence (UAI1994)

In this paper we deal with a new approach to probabilistic reasoning in a logical framework. Nearly almost all logics of probability that have been proposed in the literature are based on classical two-valued logic. After making clear the differences between fuzzy logic and probability theory, here we propose a {em fuzzy} logic of probability for which completeness results (in a probabilistic sense) are provided. The main idea behind this approach is that probability values of crisp propositions can be understood as truth-values of some suitable fuzzy propositions associated to the crisp ones. Moreover, suggestions and examples of how to extend the formalism to cope with conditional probabilities and with other uncertainty formalisms are also provided., Comment: Appears in Proceedings of the Eleventh Conference on Uncertainty in Artificial Intelligence (UAI1995)

Possibilistic logic is a well-known graded logic of uncertainty suitable to reason under incomplete information and partially inconsistent knowledge, which is built upon classical first order logic. There exists for Possibilistic logic a proof procedure based on a refutation complete resolution-style calculus. Recently, a syntactical extension of first order Possibilistic logic (called PLFC) dealing with fuzzy constants and fuzzily restricted quantifiers has been proposed. Our aim is to present steps towards both the formalization of PLFC itself and an automated deduction system for it by (i) providing a formal semantics; (ii) defining a sound resolution-style calculus by refutation; and (iii) describing a first-order proof procedure for PLFC clauses based on (ii) and on a novel notion of most general substitution of two literals in a resolution step. In contrast to standard Possibilistic logic semantics, truth-evaluation of formulas with fuzzy constants are many-valued instead of boolean, and consequently an extended notion of possibilistic uncertainty is also needed., Comment: Appears in Proceedings of the Fifteenth Conference on Uncertainty in Artificial Intelligence (UAI1999)

In this paper we present a propositional logic programming language for reasoning under possibilistic uncertainty and representing vague knowledge. Formulas are represented by pairs (A, c), where A is a many-valued proposition and c is value in the unit interval [0,1] which denotes a lower bound on the belief on A in terms of necessity measures. Belief states are modeled by possibility distributions on the set of all many-valued interpretations. In this framework, (i) we define a syntax and a semantics of the general underlying uncertainty logic; (ii) we provide a modus ponens-style calculus for a sublanguage of Horn-rules and we prove that it is complete for determining the maximum degree of possibilistic belief with which a fuzzy propositional variable can be entailed from a set of formulas; and finally, (iii) we show how the computation of a partial matching between fuzzy propositional variables, in terms of necessity measures for fuzzy sets, can be included in our logic programming system., Comment: Appears in Proceedings of the Sixteenth Conference on Uncertainty in Artificial Intelligence (UAI2000)

Defeasible argumentation frameworks have evolved to become a sound setting to formalize commonsense, qualitative reasoning from incomplete and potentially inconsistent knowledge. Defeasible Logic Programming (DeLP) is a defeasible argumentation formalism based on an extension of logic programming. Although DeLP has been successfully integrated in a number of different real-world applications, DeLP cannot deal with explicit uncertainty, nor with vague knowledge, as defeasibility is directly encoded in the object language. This paper introduces P-DeLP, a new logic programming language that extends original DeLP capabilities for qualitative reasoning by incorporating the treatment of possibilistic uncertainty and fuzzy knowledge. Such features will be formalized on the basis of PGL, a possibilistic logic based on Godel fuzzy logic., Comment: Appears in Proceedings of the Twentieth Conference on Uncertainty in Artificial Intelligence (UAI2004)