A simple logic for reasoning about incomplete knowledge

International audienceThe semantics of modal logics for reasoning about belief or knowledge is often described in terms of accessibility relations, which is too expressive to account for mere epistemic states of an agent. This paper proposes a simple logic whose atoms express epistemic attitudes about formulae expressed in another basic propositional language, and that allows for conjunctions, disjunctions and negations of belief or knowledge statements. It allows an agent to reason about what is known about the beliefs held by another agent. This simple epistemic logic borrows its syntax and axioms from the modal logic KD. It uses only a fragment of the S5 language, which makes it a two-tiered propositional logic rather than as an extension thereof. Its semantics is given in terms of epistemic states understood as subsets of mutually exclusive propositional interpretations. Our approach offers a logical grounding to uncertainty theories like possibility theory and belief functions. In fact, we define the most basic logic for possibility theory as shown by a completeness proof that does not rely on accessibility relations

Nonmonotonic Desires: A Possibility Theory Viewpoint

International audienceIf an agent desires that ϕ and desires that ψ, this agent often also desires that ϕ and ψ hold at the same time (ϕ ∧ ψ). However, there are cases where ϕ ∧ ψ may be found less satisfactory for the agent than each of ϕ or ψ alone.This paper is a first attempt at modeling such nonmonotonic desires. The approach is developed in the setting of possibility theory, since it has been recently pointed out that guaranteed (or strong) possibility measures are a good candidate for modeling graded desires. Although nonmonotonic reasoning has been studied extensively for knowledge, and that preferential nonmonotonic consequence relations can be faithfully represented in the possibilistic setting, nonmonotonic desires appear to require a different approach

Weighted logics for artificial intelligence : an introductory discussion

International audienceBefore presenting the contents of the special issue, we propose a structured introductory overview of a landscape of the weighted logics (in a general sense) that can be found in the Artificial Intelligence literature, highlighting their fundamental differences and their application areas

Three-valued logics, uncertainty management and rough sets

This paper is a survey of the connections between three-valued logics and rough sets from the point of view of incomplete information management. Based on the fact that many three-valued logics can be put under a unique algebraic umbrella, we show how to translate three-valued conjunctions and implications into operations on ill-known sets such as rough sets. We then show that while such translations may provide mathematically elegant algebraic settings for rough sets, the interpretability of these connectives in terms of an original set approximated via an equivalence relation is very limited, thus casting doubts on the practical relevance of truth-functional logical renderings of rough sets

Borderline vs. unknown: comparing three-valued representations of imperfect information

International audienceIn this paper we compare the expressive power of elementary representation formats for vague, incomplete or conflicting information. These include Boolean valuation pairs introduced by Lawry and González-Rodríguez, orthopairs of sets of variables, Boolean possibility and necessity measures, three-valued valuations, supervaluations. We make explicit their connections with strong Kleene logic and with Belnap logic of conflicting information. The formal similarities between 3-valued approaches to vagueness and formalisms that handle incomplete information often lead to a confusion between degrees of truth and degrees of uncertainty. Yet there are important differences that appear at the interpretive level: while truth-functional logics of vagueness are accepted by a part of the scientific community (even if questioned by supervaluationists), the truth-functionality assumption of three-valued calculi for handling incomplete information looks questionable, compared to the non-truth-functional approaches based on Boolean possibility–necessity pairs. This paper aims to clarify the similarities and differences between the two situations. We also study to what extent operations for comparing and merging information items in the form of orthopairs can be expressed by means of operations on valuation pairs, three-valued valuations and underlying possibility distributions

Possibilistic reasoning with partially ordered beliefs

International audienceThis paper presents the extension of results on reasoning with totally ordered belief bases to the partially ordered case. The idea is to reason from logical bases equipped with a partial order expressing relative certainty and to construct a partially ordered deductive closure. The difficult point lies in the fact that equivalent definitions in the totally ordered case are no longer equivalent in the partially ordered one. At the syntactic level we can either use a language expressing pairs of related formulas and axioms describing the properties of the ordering, or use formulas with partially ordered symbolic weights attached to them in the spirit of possibilistic logic. A possible semantics consists in assuming the partial order on formulas stems from a partial order on interpretations. It requires the capability of inducing a partial order on subsets of a set from a partial order on its elements so as to extend possibility theory functions. Among different possible definitions of induced partial order relations, we select the one generalizing necessity orderings (closely related to epistemic entrenchments). We study such a semantic approach inspired from possibilistic logic, and show its limitations when relying on a unique partial order on interpretations. We propose a more general sound and complete approach to relative certainty, inspired by conditional modal logics, in order to get a partial order on the whole propositional language. Some links between several inference systems, namely conditional logic, modal epistemic logic and non-monotonic preferential inference are established. Possibilistic logic with partially ordered symbolic weights is also revisited and a comparison with the relative certainty approach is made via mutual translations

Counterfactuals as modal conditionals, and their probability

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

A Probabilistic Modelling Approach for Rational Belief in Meta-Epistemic Contexts

This work is part of the larger project INTEGRITY. Integrity develops a conceptual frame integrating beliefs with individual (and consensual group) decision making and action based on belief awareness. Comments and criticisms are most welcome via email. The text introduces the conceptual (internalism, externalism), quantitative (probabilism) and logical perspectives (logics for reasoning about probabilities by Fagin, Halpern, Megiddo and MEL by Banerjee, Dubois) for the framework

A Probabilistic Modelling Approach for Rational Belief in Meta-Epistemic Contexts

This work is part of the larger project INTEGRITY. Integrity develops a conceptual frame integrating beliefs with individual (and consensual group) decision making and action based on belief awareness. Comments and criticisms are most welcome via email. Starting with a thorough discussion of the conceptual embedding in existing schools of thought and liter- ature we develop a framework that aims to be empirically adequate yet scalable to epistemic states where an agent might testify to uncertainly believe a propositional formula based on the acceptance that a propositional formula is possible, called accepted truth. The familiarity of human agents with probability assignments make probabilism particularly appealing as quantitative modelling framework for defeasible reasoning that aspires empirical adequacy for gradual belief expressed as credence functions. We employ the inner measure induced by the probability measure, going back to Halmos, interpreted as estimate for uncertainty. Doing so omits generally requiring direct probability assignments testi�ed as strength of belief and uncertainty by a human agent. We provide a logical setting of the two concepts uncertain belief and accepted truth, completely relying on the the formal frameworks of 'Reasoning about Probabilities' developed by Fagin, Halpern and Megiddo and the 'Metaepistemic logic MEL' developed by Banerjee and Dubois. The purport of Probabilistic Uncertainty is a framework allowing with a single quantitative concept (an inner measure induced by a probability measure) expressing two epistemological concepts: possibilities as belief simpliciter called accepted truth, and the agents' credence called uncertain belief for a criterion of evaluation, called rationality. The propositions accepted to be possible form the meta-epistemic context(s) in which the agent can reason and testify uncertain belief or suspend judgement