In this paper, we introduce novel methods for drawing conclusions from inconsistent information in a highly cautious manner. While standard paraconsistent approaches typically rely on the formulas in the intersection of maximally consistent subsets, known as the free formulas, we argue that not all these formulas share the same degree of reliability. Our refined reasoning frameworks distinguish between free formulas, based on their actual involvement in the inconsistency, an so enabling inference only when conclusions are robustly supported. These methods are particularly valuable in high-stakes contexts where decisions carry irreversible or far-reaching consequences. We present several implementation techniques grounded in multi-valued semantics and syntactic independence, analyze their fundamental logical properties, and establish a hierarchy of their inferential strength.

Logic-based argumentation is a formal method for constructing, evaluating and comparing arguments. In this paper we address two (related) key issues concerning the representation of logical argumentation frameworks: how to describe them in a compact way, and how to move from one framework to another while preserving their basic logical characteristics. The results are applied to various forms of attack rules and different kinds of argumentative semantics, and are demonstrated for transitions between several 3-valued logics and classical logic. As a byproduct, our results are also used for converting logic-based argumentation frameworks to assumption-based argumentation frameworks.

Abstract This paper continues an established line of research about the relations between argumentation theory, particularly assumption-based argumentation, and different kinds of logic programs. In particular, we extend known result of Bondarenko, Dung, Kowalski and Toni, and of Caminada and Schulz, by showing that assumption-based argumentation can represent not only normal logic programs, but also disjunctive logic programs under the stable model semantics. For this, we consider some inference rules for disjunction that the core logic of the argumentation frameworks should respect, and show the correspondence to the handling of disjunctions in the heads of the logic programs’ rules.

We present a novel computational approach to resolving conflicts among norms by nonmonotonic normative reasoning (in constrained I/O logics). Our approach extends standard sequent-based proof systems and makes them more adequate to nonmonotonic reasoning by adding to the sequents annotations that keep track of what is known about the defeasible status of the derived sequents. This makes transparent the reasons according to which norms should be applicable or inapplicable, and accordingly the sequents that make use of such norms are accepted or retracted. We also show that this proof theoretic method has tight links to the semantics of formal argumentation frameworks. The outcome of this paper is thus a threefold characterization result that relates, in the context of nonmonotonic normative reasoning, three traditional ingredients of AI-based reasoning methods: maximally consistent sets of premises (in constrained I/O logics), derived sequents (which are accepted in corresponding annotated sequent calculi), and logical arguments (that belong to the grounded extensions of the induced logical argumentation frameworks).

Conflicts are inherent to normative systems. In this paper, we explore a novel approach to normative reasoning by quantifying the amount of conflicts within normative systems. We refine the idea from classical logic, according to which a formula is a consequence of a knowledge base in case its negation renders the knowledge base inconsistent. In our approach, whether a formula is a logical consequence depends, for instance, on its negation's marginal contribution to the inconsistency of the given knowledge base. Accordingly, various inconsistency measures and corresponding (nonmonotonic and paraconsistent) normative entailment relations are analyzed relative to a number of logical properties. To illustrate our approach, we adopt Input/Output logic, a renowned formalism in deontic logic, specifically designed for defeasible normative reasoning. As an application, the resulting entailment relations provide recommendations to agents for minimizing norm conflicts, and may be incorporated in a number of implementations (like the Tweety libraries and the LogiKey framework) by involving inconsistency measurements in normative reasoning.

Approximation fixpoint theory (AFT) is an abstract and general algebraic framework for studying the semantics of nonmonotonic logics. It provides a unifying study of the semantics of different formalisms for nonmonotonic reasoning, such as logic programming, default logic and autoepistemic logic. In this paper, we extend AFT to dealing with non-deterministic constructs that allow to handle indefinite information, represented e.g. by disjunctive formulas. This is done by generalizing the main constructions and corresponding results of AFT to non-deterministic operators, whose ranges are sets of elements rather than single elements. The applicability and usefulness of this generalization is illustrated in the context of disjunctive logic programming.

We show that assumption-based argumentation frameworks, based on contrapositive logics and partially-ordered preference functions, provide a solid platform for argumentation-based reasoning. Two useful properties of the preference functions are identified (selectivity and max-lower-boundedness), and extended forms of attacks relations are supported (exists-attacks and forall-attacks), which assure several desirable properties and a variety of reasoning modes.

Logical argumentation is a well-known approach to modeling non-monotonic reasoning with conflicting information. In this paper we provide a comprehensive postulate-based study of properties of logical argumentation frameworks and a full characterization of their semantics and inference relations. In this way we identify well-behaved formal argumentative models of drawing logically justified inferences from a given set of possibly conflicting defeasible, as well as strict assumptions. Given some desiderata in terms of rationality postulates, we consider the conditions that an argumentation framework should fulfill for the desiderata to hold. One purpose of this approach is to assist designers to “plug-in” pre-defined formalisms according to actual needs. To this end, we present a classification of argumentation frameworks relative to the types of attacks they implement. In turn, for each class we determine which desiderata are satisfied. Our study is highly abstract, supposing only a minimal set of requirements on the considered underlying deductive systems, and in this way covering a broad range of formalisms, including classical, intuitionistic and modal logics.

Explainable artificial intelligence (XAI) has gained increasing interest in recent years in the argumentation community. In this paper we consider this topic in the context of logic-based argumentation, showing that the latter is a particularly promising paradigm for facilitating explainable AI. In particular, we provide two representations of abductive reasoning by sequent-based argumentation frameworks and show that such frameworks successfully cope with related challenges, such as the handling of synonyms, justifications, and logical equivalences.

We introduce annotated sequent calculi, which are extensions of standard sequent calculi, where sequents are combined with annotations that represent their derivation statuses. Unlike in ordinary calculi, sequents that are derived in annotated calculi may still be retracted in the presence of conflicting sequents, thus inferences are made under stricter conditions. Conflicts in the resulting systems are handled like in adaptive logics and argumentation theory. The outcome is a robust family of proof systems for non-monotonic reasoning with inconsistent information, where revision considerations are fully integrated into the object level of the proofs. These systems are shown to be strongly connected to logical argumentation.

In this paper we provide a detailed analysis of the inference process induced by logical argumentation frameworks. The frameworks may be defined with respect to any propositional language and logic, different arguments that represent deductions in the logic, various support-based attack relations between arguments, and all the complete Dung-style semantics for the frameworks. We show that, ultimately, for characterizing the inference process with respect to a given framework, extension-based semantics may be divided into two types: single-extension and multiple-extension, which induce respective kinds of entailment relations. These entailments are further classified by the way they tolerate new information (nonmonotonicity-related properties) and maintain conflicts among arguments (inconsistency-related properties).

Approximation fixpoint theory (AFT) constitutes an abstract and general algebraic framework for studying the semantics of nonmonotonic logics. It provides a unifying study of the semantics of different formalisms for nonmonotonic reasoning, such as logic programming, default logic and autoepistemic logic. In this paper, we extend AFT to non-deterministic constructs such as disjunctive information. This is done by generalizing the main constructions and corresponding results to non-deterministic operators, whose ranges are sets of elements rather than single elements. The applicability and usefulness of this generalization is illustrated in the context of disjunctive logic programming.

In this article, we present an argumentative approach to normative reasoning. Special attention is paid to deontic conflicts, contrary-to-duty and specificity cases, which are modelled by means of argumentative attacks. For this, we adopt a recently proposed framework for logical argumentation in which arguments are generated by a sequent calculus of a given base logic (see the papers of the second author in CLIMA'2013 and of the two authors in Volume 6 (No. 1) of Argument & Computation ), and use standard deontic logic as our base logic. Argumentative attacks are realized by elimination rules that allow to discharge specific sequents. We demonstrate the usefulness of our approach by means of various well-known benchmark examples, and show that this approach is rich enough to capture a variety of paradigms for handling conflicting norms such as reasoning with maximally consistent sets, prioritized norms and deontic formalisms based on I/O logic.

We investigate Dung's semantics for assumption-based frameworks (ABFs) that are induced by contrapositive logics. We show that unless the falsity propositional constant is part of the defeasible assumptions, the grounded semantics lacks most of the desirable properties it has in abstract argumentation frameworks (AAFs), and that for simple definitions of the contrariness operator and the attacks relations, preferred and stable semantics are reduced to naive semantics. We also show the tight relations of this framework to reasoning with maximally consistent sets, and consider some properties of the induced entailments, such as being cumulative or preferential relations that satisfy non-interference.

Reasoning with the maximally consistent subsets (MCS) of the premises is a well-known approach for handling contradictory information. In this paper we consider several variations of this kind of reasoning, for each one we introduce two complementary computational methods that are based on logical argumentation theory. The difference between the two approaches is in their ways of making consequences: one approach is of a declarative nature and is related to Dung-style semantics for abstract argumentation, while the other approach has a more proof-theoretical flavor, extending Gentzen-style sequent calculi. The outcome of this work is a new perspective on reasoning with MCS, which shows a strong link between the latter and argumentation systems, and which can be generalized to some related formalisms. As a by-product of this we obtain soundness and completeness results for the dynamic proof systems with respect to several of Dung’s semantics. In a broader context, we believe that this work helps to better understand and evaluate the role of logic-based instantiations of argumentation frameworks.

Current approaches for giving semantics to abstract argumentation frameworks dismiss altogether any possibility of having conflicts among accepted arguments by requiring that the latter should be ‘conflict free’. In reality, however, contradictory phenomena coexist, or it may happen that one cannot make a choice between conflicting indications but still would like to keep track to all of them. For this purpose we introduce in this article a new kind of argumentation semantics, called ‘conflict-tolerant’, in which all the accepted arguments must be justified (in the sense that each one of them can be defended), but some of them may still attack each other. In terms of graphical representation of argumentation systems, where attacks are represented by directed edges, this means that the possibility of accepting ‘loops’ of arguments is not automatically ruled out without any further considerations. To provide conflict-tolerant semantics, we enhance the two standard approaches for defining coherent (conflict-free) semantics for argumentation frameworks. The extension-based approach is generalized by relaxing the ‘conflict-freeness’ requirement of the chosen sets of arguments, and the three-valued labelling approach is replaced by a four-valued labelling system that allows to capture mutual attacks among accepted arguments. We show that our setting is not a substitute of standard (conflict-free) semantics, but rather a generalized framework that accommodates both conflict-free and conflict-tolerant semantics. Moreover, the one-to-one relationship between extensions and labellings of conflict-free semantics is carried on to a similar correspondence between the extended approaches for providing conflict-tolerant semantics. Thus, in our setting as well, these are essentially two points of views for the same thing.

We introduce a preferential-based setting for reasoning with different types of argumentation-based semantics, including those that are not necessarily conflict-free or admissible. The induced entailments are defined by n-valued labeling and may be computed by answer-set programs.

We introduce a general approach for representing and reasoning with argumentation-based systems. In our framework arguments are represented by Gentzen-style sequents, attacks (conflicts) between arguments are represented by sequent elimination rules, and deductions are made by dynamic proof systems. This framework accommodates different languages and logics in which arguments may be represented, supports a variety of attack relations, and tolerates dynamic changes in the argumentation setting by revising derivations of assertions in light of new information.

We introduce a unified logical approach, based on signed theories and Quantified Boolean Formulas (QBFs), that can serve as a basis for representing and reasoning with various argumentation-based decision problems. By this, we are able to represent, in a uniform and simple way, a wide range of extension-based semantics for argumentation theory, including complete, grounded, preferred, semi-stable, stage, ideal and eager semantics. Furthermore, our approach involves only propositional languages and quantifications over propositional variables, making decision problems like skeptical and credulous acceptance of arguments simply a matter of logical entailment and satisfiability, which can be verified by existing QBF-solvers.

Maximality is a desirable property of paraconsistent logics, motivated by the aspiration to tolerate inconsistencies, but at the same time retain as much as possible from classical logic. In this paper we introduce a new, strong notion of maximal paraconsistency, which is based on possible extensions of the consequence relation of a logic. We investigate this notion in the framework of finite-valued paraconsistent logics, and show that for every n > 2 there exists an extensive family of n-valued logics, each of which is maximally paraconsistent in our sense, is partial to classical logic, and is not equivalent to any k-valued logic with k <; n. On the other hand, we specify a natural condition that guarantees that a paraconsistent logic is contained in a logic in the class of three-valued paraconsistent logics, and show that all reasonably expressive logics in this class are maximal.

The Closed World Assumption (CWA) on databases expresses the assumption that an atom not in the database is false. This assumption is applicable only in cases where the database has complete knowledge about the domain of discourse. In this article, we investigate locally closed databases, that is: databases that are sound but partially incomplete about their domain. Such databases consist of a standard database instance, augmented with a collection of Local Closed World Assumptions (LCWAs). A LCWA is a “local” form of the CWA, expressing that a database relation is complete in a certain area, called a window of expertise . In this work, we study locally closed databases both from a knowledge representation and from a computational perspective. At the representation level, the approach taken in this article distinguishes between the data that is conveyed by a database and the metaknowledge about the area in which the data is complete. We study the semantics of the LCWA's and relate it to several knowledge representation formalisms. At the reasoning level, we study the complexity of, and algorithms for two basic reasoning tasks: computing certain and possible answers to queries and determining whether a database has complete knowledge on a query. As the complexity of these tasks is unacceptably high, we develop efficient approximate methods for query answering. We also prove that for useful classes of queries and locally closed databases, these methods are optimal , and thus they solve the original query in a tractable way. As a result, we obtain classes of queries and locally closed databases for which query answering is tractable.

We introduce a uniform distance semantics for different paradigms that require nonmonotonic and paraconsistent reasoning, among which are mediators of independent data-sources, integrators of prioritized knowledge-bases, operators for iterated belief revision, and analytic tools of decision support systems. We show that the consequence relations that are induced by our framework share some desirable properties and demonstrate this by relevant applications.

We present a family of algebraic structures, called rectangular bilattices, which serve as a natural accommodation and powerful generalization to both intuitionistic fuzzy sets (IFSs) and interval-valued fuzzy sets (IVFSs). These structures are useful on one hand to clarify the exact nature of the relationship between the above two common extensions of fuzzy sets, and on the other hand provide an intuitively attractive framework for the representation of uncertain and potentially conflicting information. We also provide these structures with adequately defined graded versions of the basic logical connectives, and study their properties and relationships. Application potential and intuitive appeal of the proposed framework are illustrated in the context of preference modeling.

In this paper, Ginsberg's/Fitting's theory of bilattices, and in particular the associated constructs of bilattice-based squares and triangles, is introduced as an attractive framework for the representation of uncertain and potentially conflicting information, paralleling Goguen's L-fuzzy set theory. We recall some of the advantages of bilattice-based frameworks for handling fuzzy sets and systems, provide the related structures with adequately defined graded versions of the basic logical connectives, and study their properties and relationships

We introduce a uniform approach of representing a variety of paraconsistent nonmonotonic formalisms by quantified Boolean formulae (QBFs) in the context of multiple-valued logics. We show that this framework provides a useful platform for capturing, in a simple and natural way, a wide range of methods for preferential reasoning. The outcome is a subtle approach to represent the underlying formalisms, which induces a straightforward way to compute the corresponding entailments: By incorporating off-the-shelf QBF solvers it is possible to simulate within our framework various kinds of preferential formalisms, among which are Priest's logic LPm of reasoning with minimal inconsistency, Batens' adaptive logic ACLuNs2, Besnard and Schaub's inference relation &vbar;= n , a variety of formula-preferential systems, some bilattice-based preferential relations (e.g., &vbar;= I 1 and &vbar;= I 2 ), and consequence relations for reasoning with graded uncertainty, such as the four-valued logic &vbar;= 4 c .

Abstract: We introduce an abductive method for a coherent integration of independent data-sources. The idea is to compute a list of data-facts that should be inserted to the amalgamated database or retracted from it in order to restore its consistency. This method is implemented by an abductive solver, called Asystem, that applies SLDNFA-resolution on a meta-theory that relates different, possibly contradicting, input databases. We also give a pure model-theoretic analysis of the possible ways to `recover' consistent data from an inconsistent database in terms of those models of the database that exhibit as minimal inconsistent information as reasonably possible. This allows us to characterize the `recovered databases' in terms of the `preferred' (i.e., most consistent) models of the theory. The outcome is an abductive-based application that is sound and complete with respect to a corresponding model-based, preferential semantics, and -- to the best of our knowledge -- is more expressive (thus more general) than any other implementation of coherent integration of databases.

We introduce a general method for paraconsistent reasoning in the context of classical logic. A standard technique for paraconsistent reasoning on inconsistent classical theories is by shifting to multiple-valued logics. We show how these multiple-valued theories can be ‘shifted back’ to two-valued classical theories through a polynomial transformation, and how preferential reasoning based on multiple-valued logic can be represented by classical circumscription-like axioms. By applying this process we provide new ways of implementing multiple-valued paraconsistent reasoning. Standard multiple-valued reasoning can thus be performed through theorem provers for classical logic, and multiple-valued preferential reasoning can be implemented using algorithms for processing circumscriptive theories (such as DLS and SCAN).

In his well-known paper "How computer should think" (1977) Belnap argues that four-valued semantics is a very suitable setting for computerized reasoning. In this paper we vindicate this thesis by showing that the logical role that the four-valued structure has among Ginsberg's well-known bilattices is similar to the role that the two-valued algebra has among Boolean algebras.

The authors present a mechanism for recovering consistent data from an inconsistent set of assertions. For a common family of knowledge bases they also provide an efficient algorithm for doing so automatically. This method is nonmonotonic and paraconsistent. It is particularly useful for making diagnoses on faulty devices.

The notion of a bilattice was first proposed by Ginsberg (1988) as a general framework for many applications. This notion was further investigated and applied for various goals by Fitting (1989, 1990, 1991, 1993). In this paper, we develop proof systems which correspond to bilattices in an essential way. We then show how to use those bilattices for efficient inferences from possibly inconsistent data. For this, we incorporate certain ideas of Kifer and Lozinskii (1992) concerning inconsistencies, which happen to well suit the framework of bilattices. The outcome is a paraconsistent logic with many desirable properties.>