In this paper, we introduce a syntactic framework for analyzing and handling inconsistencies in propositional bases. Our approach focuses on examining the relationships between variable occurrences within conflicts. We propose two dual concepts: Minimal Inconsistency Relation (MIR) and Maximal Consistency Relation (MCR). Each MIR is a minimal equivalence relation on variable occurrences that results in inconsistency, while each MCR is a maximal equivalence relation designed to prevent inconsistency. Notably, MIRs capture conflicts overlooked by minimal inconsistent subsets. Using MCRs, we develop a series of non-explosive inference relations. The main strategy involves restoring consistency by modifying the propositional base according to each MCR, followed by employing the classical inference relation to derive conclusions. Additionally, we propose an unusual semantics that assigns truth values to variable occurrences instead of the variables themselves. The associated inference relations are established through Boolean interpretations compatible with the occurrence-based models.

We analyse a specific instance of the general approach of reasoning based on forgetting by Lang and Marquis. More precisely, we discuss an approach for reasoning with inconsistent information using maximal consistent subsignatures, where a maximal consistent subsignature is a maximal set of propositions such that forgetting the remaining propositions restores consistency. We analyse maximal consistent subsignatures and the corresponding minimal inconsistent subsignatures in-depth and show, among others, that the hitting set duality applies for them as well. We further analyse inference relations based on maximal consistent subsignatures wrt. rationality postulates from non-monotonic reasoning and computational complexity. We also consider the relationship of our approach with inconsistency measurement and paraconsistent reasoning.

Inconsistency handling is an important issue in knowledge management. Especially in ontology engineering, logical inconsistencies may occur during ontology construction. A natural way to reason with an inconsistent ontology is to utilize the maximal consistent subsets of the ontology. However, previous studies on selecting maximum consistent subsets have rarely considered the semantics of the axioms, which may result in irrational inference. In this paper, we propose a novel approach to reasoning with inconsistent ontologies in description logics based on the embeddings of axioms. We first give a method for turning axioms into distributed semantic vectors to compute the semantic connections between the axioms. We then define an embedding-based method for selecting the maximum consistent subsets and use it to define an inconsistency-tolerant inference relation. We show the rationality of our inference relation by considering some logical properties. Finally, we conduct experiments on several ontologies to evaluate the reasoning power of our inference relation. The experimental results show that our embedding-based method can outperform existing inconsistency-tolerant reasoning methods based on maximal consistent subsets.

Reasoning in the context of a conditional knowledge base containing rules of the form'If A then usually B' can be defined in terms of preference relations on possible worlds. These preference relations can be modeled by ranking functions that assign a degree of disbelief to each possible world. In general, there are multiple ranking functions that accept a given knowledge base. Several nonmonotonic inference relations have been proposed using c-representations, a subset of all ranking functions. These inference relations take subsets of all c-representations based on various notions of minimality into account, and they operate in different inference modes, i.e., skeptical, weakly skeptical, or credulous. For nonmonotonic inference relations, weaker versions of monotonicity like rational monotony (RM) and weak rational monotony (WRM) have been developed. In this paper, we investigate which of the inference relations induced by sets of minimal c-representations satisfy rational monotony or weak rational monotony.

A conditional knowledge base R is a set of conditionals of the form "If A, the usually B". Using structural information derived from the conditionals in R, we introduce the preferred structure relation on worlds. The preferred structure relation is the core ingredient of a new inference relation called system W inference that inductively completes the knowledge given explicitly in R. We show that system W exhibits desirable inference properties like satisfying system P and avoiding, in contrast to e.g. system Z, the drowning problem. It fully captures and strictly extends both system Z and skeptical c-inference. In contrast to skeptical c-inference, it does not require to solve a complex constraint satisfaction problem, but is as tractable as system Z.

Reasoning in the context of a conditional knowledge base containing rules of the form ’If A then usually B’ can be defined in terms of preference relations on possible worlds. These preference relations can be modeled by ranking functions that assign a degree of disbelief to each possible world. In general, there are multiple ranking functions that accept a given knowledge base. Several nonmonotonic inference relations have been proposed using c-representations, a subset of all ranking functions. These inference relations take subsets of all c-representations based on various notions of minimality into account, and they operate in different inference modes, i.e., skeptical, weakly skeptical, or credulous. For nonmonotonic inference relations, weaker versions of monotonicity like rational monotony (RM) and weak rational monotony (WRM) have been developed. In this paper, we investigate which of the inference relations induced by sets of minimal c-representations satisfy rational monotony or weak rational monotony.

Forgetting as a knowledge management operation has received much less attention than operations like inference, or revision. It was mainly in the area of logic programming that techniques and axiomatic properties have been studied systematically. However, at least from a cognitive view, forgetting plays an important role in restructuring and reorganizing a human's mind, and it is closely related to notions like relevance and independence which are crucial to knowledge representation and reasoning. In this paper, we propose axiomatic properties of (intentional) forgetting for general epistemic frameworks which are inspired by those for logic programming, and we evaluate various forgetting operations which have been proposed recently by Beierle et al. according to them. The general aim of this paper is to advance formal studies of (intentional) forgetting operators while capturing the many facets of forgetting in a unifying framework in which different forgetting operators can be contrasted and distinguished by means of formal properties.

We propose a general framework for inconsistency-tolerant query answering within existential rule setting. This framework unifies the main semantics proposed by the state of art and introduces new ones based on cardinality and majority principles. It relies on two key notions: modifiers and inference strategies. An inconsistency-tolerant semantics is seen as a composite modifier plus an inference strategy. We compare the obtained semantics from a productivity point of view.

In a standard possibilistic logic, prioritized information are encoded by means of weighted knowledge base. This paper proposes an extension of possibilistic logic for dealing with partially ordered information. We Show that all basic notions of standard possibilitic logic (sumbsumption, syntactic and semantic inference, etc.) have natural couterparts when dealing with partially ordered information. We also propose an algorithm which computes possibilistic conclusions of a partial knowledge base of a partially ordered knowlege base.

Recently, (Dunne et al. 2009; 2011) have suggested to weight attacks within Dung’s abstract argumentation frameworks, and introduced the concept of WAF (Weighted Argumentation Framework). However, they use WAFs in a very specific way for relaxing attacks. The aim of this paper is to explore ways to take advantage of attacks weights within an argumentation process. Two different approaches are considered: The first one extends the proposal by (Dunne et al. 2011) and accounts for other aggregation functions than sum in the objective of relaxing attacks. The second one shows how weights can be exploited to strengthen the usual notion of defence, leading to new concepts of extensions.