We study the problem of explaining observations about the probabilities of events such as ‘it rains 20% of the time’, ‘rain and snow are equally likely’, etc. We explain these statements with a probability distribution or a statement about probabilities of (other) events that are consistent with our knowledge and entail the observation. We formalise this problem in a fuzzy probabilistic logic FP. We define and motivate the notions of abduction problems and their solutions. We analyse the complexity of solution recognition and existence for a given abduction problem in FP for the case of full language and its disjunctive-clause fragments. We also obtain a translation of classical probabilistic abduction (finding the most likely explanation of a given event) to FP.

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.

A proof for the NP-containment for the probabilistic coherence problem over events represented by formulas of the infinite-valued Łukasiewicz logic was proposed in [1] . The geometric and combinatorial argument to prove that complexity bound contains a mistake that is fixed in the present paper. Actually we present two ways to restore that imprecise claim and, by doing so, we show that the main result of that paper is indeed valid.

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.

We investigate the notion of coherence for (non-)additive uncertainty measures from a logico-geometric point of view. Our main result is to the effect that distinct criteria for coherence are not always matched by axiomatically distinct measures of uncertainty. In addition we introduce a metalogic within which this kind of result can be captured formally.

The present paper investigates proof-theoretical and algebraic properties for the probability logic FP(Ł,Ł), meant for reasoning on the uncertainty of Łukasiewicz events. Methodologically speaking, we will consider a translation function between formulas of FP(Ł,Ł) to the propositional language of Łukasiewicz logic that allows us to apply the latter and the well-developed theory of MV-algebras directly to probabilistic reasoning. More precisely, leveraging on such translation map, we will show proof-theoretical properties for FP(Ł,Ł) and introduce a class of algebras with respect to which FP(Ł,Ł) will be proved to be locally sound and complete. Finally, we will apply these previous results to investigate what we called “probabilistic unification problem”. In this respect, we will prove that Ghilardi's algebraic view on unification can be extended to our case and, on par with the Łukasiewicz propositional case, we show that probabilistic unification is of nullary type.

For functional representation in an algebraizable logic we mean a representation of the algebras of formulas of the logic by means of (possibly real-valued) functions. Functional representations have shown to be a key tool for the study of non-classical logics, since they allow to regard formulas as functions and, by means of them, to approach the study of typical proof theoretical properties of the logics by means of their functional semantics. In the realm of (algebraizable) fuzzy logics, possibly the most well-known result in this respect is McNaughton theorem that shows formulas of the infinite-valued Lukasiewicz calculus to correspond, up to logical equivalence, to real valued continuous and piecewise linear functions. The functional representation for many-valued logics has been very recently shown in a paper by the second author to have an impact outside the purely logical realm and, in particular, they can be applied to study properties of artificial neural networks. In this contribution, we will provide a functional representation for the probability modal logic FP(L) that builds on Lukasiewicz calculus by adding to it a unary operator P that reads “it is probable that”. While the logic FP(L) is not algebraizable, at least not in the usual sense due to Blok and Pigozzi, we still can provide a functional representation result for its modal formulas. In order to do so, we adapt the usual universal algebraic methods to this peculiar setting, and moreover we make use of some techniques developed in a recent paper by two of the authors where a class of purely algebraic models for FP(L) based on de Finetti's coherence criterion have been introduced and studied. Our contribution will present two ways of providing a functional representation of the algebras of formulas of the modal logic FP(L): a local one, that relies on de Finetti's coherence argument; and a global one that, instead, relies on probability distributions on a finite domain.

In this paper we propose a semantic analysis of Lewis' counterfactuals. By exploiting the structural properties of the recently introduced boolean algebras of conditionals, we show that counterfactuals can be expressed as formal combinations of a conditional object and a normal necessity modal operator. Specifically, we introduce a class of algebras that serve as modal expansions of boolean algebras of conditionals, together with their dual relational structures. Moreover, we show that Lewis' semantics based on sphere models can be reconstructed in this framework. As a consequence, we establish the soundness and completeness of a slightly stronger variant of Lewis' logic for counterfactuals with respect to our algebraic models. In the second part of the paper, we present a novel approach to the probability of counterfactuals showing that it aligns with the uncertainty degree assigned by a belief function, as per Dempster-Shafer theory, to its associated conditional formula. Furthermore, we characterize the probability of a counterfactual in terms of Gärdenfors' imaging rule for the probabilistic update.

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.

The probabilistic logic FP(Ł,Ł) was axiomatized with the aim of presenting a formal setting for reasoning about the probability of infinite-valued Łukasiewicz events. Besides several attempts, proving that axiomatic system to be complete with respect to a class of standard models, remained an open problem since the first paper on FP(Ł,Ł) was published in 2007. In this article we give a solution to it. In particular we introduce two semantics for that probabilistic system: a first one based on Łukasiewicz states and a second one based on regular Borel measures and we prove that FP(Ł,Ł) is complete with respect to both these classes of models. Further, we will show that the finite model property holds for FP(Ł,Ł).

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.

We take a glimpse at the relation between WNM-algebras (algebraic models of the well-known Weak Nilpotent Minimum logic) and quasi-Nelson algebras, a non-involutive generalisation of Nelson algebras (models of Nelson's constructive logic with strong negation) that was introduced in a recent paper. We show that the two varieties can be related via the twist-structure construction, obtaining a new representation for a subvariety of WNM-algebras that includes the involutive ones (i.e. NM-algebras). Our results imply, in particular, that every pre-linear quasi-Nelson algebra is a WNM-algebra; we thus generalize the known result that the class of pre-linear Nelson algebras coincides with that of NM-algebras (models of Nilpotent Minimum logic).

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.

Abstract We investigate the property of strict coherence in the setting of many-valued logics. Our main results read as follows: (i) a map from an MV-algebra to [0,1] is strictly coherent if and only if it satisfies Carnap’s regularity condition, and (ii) a [0,1]-valued book on a finite set of many-valued events is strictly coherent if and only if it extends to a faithful state of an MV-algebra that contains them. Remarkably this latter result allows us to relax the rather demanding conditions for the Shimony-Kemeny characterisation of strict coherence put forward in the mid 1950s in this Journal.

This article studies the class of strongly perfect MTL-algebras, i.e. MTL-algebras having an involutive co-radical, and the variety they generate, namely |$\mathbb{SBP}_0$|⁠. Once these structures will be introduced, we will first establish categorical equivalences for several of their relevant proper subvarieties by employing a generalized notion of triplets whose main components are a Boolean algebra and a prelinear semihoop. When triplets are further expanded by a suitable operation between their semihoop reducts, we define a category of quadruples that are equivalent to the whole category of SBP|$_0$|-algebras. Finally, we will provide an explicit representation of SBP|$_0$|-algebras in terms of (weak) Boolean products.

As is well-known standard MTL-algebras in general do not define the t-conorm +*associated with their t-norm *. As +*is defined by x+*y = 1-((1-x) * (1-y)), we address the generalised problem of characterising those MTL-algebras with monoidal operation * that define the dual monoidal operation x+*y = ~(~x*~y) for some order-reversing involution ~. The barest requirement on such structures is clearly that they are subdirect products of order-anti-automorphic chains (o.a.a., for short). We deal with the case of BL-algebras, stating two properties of involutions and fully characterising those BL-algebras defining the dual monoidal operation when the involution satisfies both properties. We also exhibit a BL-chain defining the dual monoidal operation determined by an involution failing both properties. We further prove that all o.a.a. algebras in the variety generated by EMTL-algebras and IMTL-algebras define the dual monoidal operation uniformly with the same term. By contrast, we present a variety whose o.a.a. chains define the dual monoidal operation, but with distinct terms for distinct algebras, generally. If we require definability of the dual residual operation, too, we are left with IMTL-algebras as the only known examples.

In this paper we present, by way of case studies, a proof of concept, based on a prototype working on a automotive data set, aimed at showing the potential usefulness of using formulas of Łukasiewicz propositional logic to query databases in a fuzzy way. Our approach distinguishes itself for its stress on the purely linguistic, contraposed with numeric, formulations of queries. Our queries are expressed in the pure language of logic, and when we use (integer) numbers, these stand for shortenings of formulas on the syntactic level, and serve as linguistic hedges on the semantic one. Our case-study queries aim first at showing that each numeric-threshold fuzzy query is simulated by a Łukasiewicz formula. Then they focus on the expressing power of Łukasiewicz logic which easily allows for updating queries by clauses and for modifying them through a potentially infinite variety of linguistic hedges implemented with a uniform syntactic mechanism. Finally we shall hint how, already at propositional level, Łukasiewicz natural semantics enjoys a degree of reflection, allowing to write syntactically simple queries that semantically work as meta-queries weighing the contribution of simpler ones.

We introduce a class of algebraic structures, finite GBL-pairs, as pairs made of a finite Heyting algebra and a subgroup of its automorphism group. The group determines an equivalence relation on the Heyting algebra: we prove that the quotient, when endowed with suitable operations, is a GBL-algebra, and the operations can be interpreted as infima or suprema of equivalence classes. Conversely, we prove that every finite GBL-algebra can be represented as a GBL-pair. The motivation is to provide models for a fuzzy extension of intuitionistic propositional logic.

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 article, we compare models for many-valued probabilistic reasoning from the point of view of the sets of satisfiable formulas, positive satisfiable formulas, and tautologies. The results arising from this comparison will be used in the final part of the present article to provide results about the computational complexity for the problem of deciding if a formula belongs to one of the previously discussed sets.