Search

Conditionals are defeasible rules of the form If A then usually B, and they play a central role in many approaches to nonmonotonic reasoning. Normal forms of conditional knowledge bases consisting of a set of such conditionals are useful to create, process, and compare the knowledge represented by them. In this article, we propose several new normal forms for conditional knowledge bases. Compared to the previously introduced antecedent normal form, the reduced antecedent normal form (RANF) represents conditional knowledge with significantly fewer conditionals by taking nonmonotonic entailments licenced by system P into account. The renaming normal form(ρNF) addresses equivalences among conditional knowledge bases induced by renamings of the underlying signature. Combining the concept of renaming normal form with other normal forms yields the renaming antecedent normal form (ρ ANF) and the renaming reduced antecedent normal form (ρ RANF). For all newly introduced normal forms, we show their key properties regarding, existence, uniqueness, model equivalence, and inferential equivalence, and we develop algorithms transforming every conditional knowledge base into an equivalent knowledge base being in the respective normal form. For the most succinct normal form, the ρ RANF, we present an algorithm KBρra systematically generating knowledge bases over a given signature in ρ RANF. We show that the generated knowledge bases are consistent, pairwise not antecedentwise equivalent, and pairwise not equivalent under signature renaming. Furthermore, the algorithm is complete in the sense that, when taking signature renamings and model equivalence into account, every consistent knowledge base is generated. Observing that normalizing the set of all knowledge bases over a signature Σ to ρ RANF yields exactly the same result as KBρra (Σ), highlights the interrelationship between normal form transformations on the one hand and systematically generating knowledge bases in normal form on the other hand.

We discuss nonmonotonic reasoning in terms of consequence relations and corresponding operators. Based on the matrix consequence that gives the monotonic case, we define a restricted matrix consequence that illustrates the nonmonotonic case. The latter is a generalization of the relation of logical friendliness introduced by D. Makinson. We prove that any restricted single matrix consequence, although it may be nonmonotonic, is always weakly monotonic and, in the case of a finite matrix, the restricted matrix consequence is very strongly finitary. Further, by modifying the definition of logical friendliness relation formulated specifically in a proof-theoretic manner, we show a possibility of obtaining other reflexive nonmonotonic consequence relations, for which a limited result towards finitariness is proved. This leads to numerous questions about nonmonotonic consequence relations in the segment between the monotonic consequence relation based on intuitionistic propositional logic and logical friendliness.

The principle of tolerance characteristic of vague predicates is sometimes presented as a soft rule, namely as a default which we can use in ordinary reasoning, but which requires care in order to avoid paradoxes. We focus on two ways in which the tolerance principle can be modeled in that spirit, using special consequence relations. The first approach relates tolerant reasoning to nontransitive reasoning; the second relates tolerant reasoning to nonmonotonic reasoning. We compare the two approaches and examine three specific consequence relations in relation to those, which we call: strict-to-tolerant entailment, pragmatic-to-tolerant entailment, and pragmatic-to-pragmatic entailment. The first two are nontransitive, whereas the latter two are nonmonotonic.

Description Logics (DLs) under Rational Closure (RC) is a well-known framework for non-monotonic reasoning in DLs. In this paper, we address the concept subsumption decision problem under RC for nominal safe E L O ⊥ , a notable and practically important DL representative of the OWL 2 profile OWL 2 EL. Our contribution here is to define a polynomial time subsumption procedure for nominal safe E L O ⊥ under RC that relies entirely on a series of classical, monotonic E L ⊥ subsumption tests. Therefore, any existing classical monotonic E L ⊥ reasoner can be used as a black box to implement our method. We then also adapt the method to one of the known extensions of RC for DLs, namely Defeasible Inheritance-based DLs without losing the computational tractability.

In this paper we present a duality between nonmonotonic consequence relations and well-founded convex geometries. On one side of the duality we consider nonmonotonic consequence relations satisfying the axioms of an infinitary variant of System P, which is one of the most studied axiomatic systems for nonmonotonic reasoning, conditional logic and belief revision. On the other side of the duality we consider well-founded convex geometries, which are infinite convex geometries that generalize well-founded posets. Since there is a close correspondence between nonmonotonic consequence relations and path independent choice functions one can view our duality as an extension of an existing duality between path independent choice functions and convex geometries that has been developed independently by Koshevoy and by Johnson and Dean.

We introduce a nonmonotonic procedure for preferential Description Logics in order to reason about typicality by taking probabilities of exceptions into account. We consider an extension, called ALC + T R P , of the logic of typicality ALC + T R by inclusions of the form T ( C ) ⊑ p D with probability p, whose intuitive meaning is that “all the typical Cs are Ds, and the probability that a C is not a D is 1 − p ”. We consider a notion of extension of an ABox containing only some typicality assertions, then we equip each extension with a probability. We then restrict entailment of a query F to those extensions whose probabilities belong to a given and fixed range. We propose a decision procedure for reasoning in ALC + T R P and we exploit it to show that entailment is ExpTime -complete as for the underlying ALC .

In order to characterize causal relation, different scholars have studied it from multiple perspectives. In the field of logic, the causal theory proposed by Alexander elaborated the causal relationship from the perspective of nonmonotonic reasoning. The Horn causal theory and the concept of equivalence relationship are defined in his article, but the author did not give the determination method of equivalence relationship. In order to make up for this deficiency, this paper introduces the concept of bisimulation to describe the equivalence relation between Horn causal theories and proves the consistency between bisimulation relation and the equivalence of models. Then the corresponding algorithm is proposed to determine the equivalence. An example is also given to illustrate the application of this method in determining equivalence relationship.

An important concept for nonmonotonic reasoning in the context of a conditional belief base \(\mathcal R\) is syntax splitting, essentially stating that taking only the syntactically relevant part of \(\mathcal R\) into account should be sufficient. In this paper, for the semantics of ordinal conditional functions (OCF) of \(\mathcal R\), we introduce the notion of semantic splitting of \(\mathcal R\) where the combination of models of sub-belief bases of \(\mathcal R\) corresponds to the models of \(\mathcal R\). While this is not the case for all OCF models, we show that for c-representations which are a subclass of all OCFs governed by the principle of conditional preservation, every syntactic splitting is also a semantic splitting. Furthermore, for the semantics of c-representations, we introduce constraint splittings of a conditional belief base and show that they fully capture and go beyond syntax splittings, thus allowing for additional belief base splittings that enable the computation of models locally from sub-belief bases.

Most (if not all) nonmonotonic logics are assumed to be cumulative, which is often regarded as the minimum requirement for a logic. We argue that cumulativity, in particular, cumulative transitivity can be abandoned, in order to better characterize reasoning in uncertainty. But giving up cumulative transitivity makes it hard to obtain representation results for these logics. Borrowing the idea from strict-tolerant logics, we give some representation results for nonmonotonic logics that are not cumulatively transitive.

This chapter addresses the main approaches to formalization of default reasoning in first-order framework. The first part of the chapter overviews the historical development of nonmonotonic reasoning systems. After discussing three wellknown early formalisms (Default Logic, Nonmonotonic modal logic and Circumscription), we focus on nonmonotonic consequence relations and their close connections with conditional logics and probabilistic reasoning systems. The logic \( \mathit L ^{P,\mathbb I}_{\mathit \omega\omega}\) ww occupies the central position in this chapter. This logic extends classical first-order logic with a list of Keisler-style conditional probability quantifiers with the intention to express statistical knowledge as well as approximate probabilities. We provide a sound and complete deductive system with respect to semantics based on first-order structures endowed with suitable probability spaces. In addition, we describe two decidable fragments of \( \mathit L ^{P,\mathbb I}_{\mathit \omega\omega}\) ww that seem to be rich enough for many practical purposes. Finally, we discuss how the logic \( \mathit L ^{P,\mathbb I}_{\mathit \omega\omega}\) ww and its suitable extensions can be used to model default reasoning and analyze some properties of nonmonotonic consequence relations in the light of probabilistic inference.

The negative zombie argument states that p&~q is ideally negatively conceivable and, therefore, possible, what would entail that physicalism is false (Chalmers, 2002, 2010}. In the argument, p is the conjunction of the fundamental physical truths and laws and $q$ is a phenomenal truth. A sentence phi is ideally negatively conceivable iff phi cannot be ruled out a priori on ideal rational reflection. In this paper, I argue that if its premises are true, the negative zombie argument is neither conclusive (valid) nor a priori. First, I argue that the argument is sound iff there exists a finite ideal reasoner R for a logic x with the relevant properties which believes <>(p&~q) on the basis of not believing p->q on a priori basis. A finite reasoner is a reasoner with finite memory and finite computational power. I argue that if x has the relevant properties and R is finite, then x must be nonmonotonic and R may only approach ideallity at the limit of a reasoning sequence. This would render the argument nonconclusive. Finally, I argue that, for some q, R does not believe <>(p&~q) on the basis of not believing p->q on a priori basis. For example, for q=`someone is conscious'. I conclude that the negative zombie argument (and, maybe, all zombie arguments) is neither conclusive nor a priori (the choice of q relies on empirical information). The negative zombie argument states that p&~q is ideally negatively conceivable and, therefore, possible, what would entail that physicalism is false (Chalmers, 2002, 2010}. In the argument, p is the conjunction of the fundamental physical truths and laws and $q$ is a phenomenal truth. A sentence phi is ideally negatively conceivable iff phi cannot be ruled out a priori on ideal rational reflection. In this paper, I argue that if its premises are true, the negative zombie argument is neither conclusive (valid) nor a priori. First, I argue that the argument is sound iff there exists a finite ideal reasoner R for a logic x with the relevant properties which believes <>(p&~q) on the basis of not believing p->q on a priori basis. A finite reasoner is a reasoner with finite memory and finite computational power. I argue that if x has the relevant properties and R is finite, then x must be nonmonotonic and R may only approach ideallity at the limit of a reasoning sequence. This would render the argument nonconclusive. Finally, I argue that, for some q, R does not believe <>(p&~q) on the basis of not believing p->q on a priori basis. For example, for q=`someone is conscious'. I conclude that the negative zombie argument (and, maybe, all zombie arguments) is neither conclusive nor a priori (the choice of q relies on empirical information).

A propositional conditional of the form ( B | A ) , representing the default rule “If A , then usually B ”, goes beyond the limits of classical logic, and the semantics of a knowledge base consisting of such conditionals must take into account the three-valued nature of conditionals. Ordinal conditional functions (OCF), also called ranking functions, assign a degree of implausibility to possible worlds and provide a convenient semantic framework for conditionals. To each consistent knowledge base R , system Z associates a unique minimal OCF accepting R , enabling inductive reasoning from this model. In this article, we present an approach of transforming the ideas of the popular system Z to the first-order case. We develop system Z FO that considers first-order conditionals of the form ( B ( x ) | A ( x ) ) . The notion of tolerance used in system Z is extended to the notion of tolerance pairs, taking also a partition of the domain elements into account. We show that each tolerance pair induces a system Z-like ranking function, and we provide an algorithm systematically generating all tolerance pairs. We introduce the notion of minimal tolerance pairs and a corresponding refinement of the algorithm computing all minimal tolerance pairs. An inference relation based on minimal tolerance pairs is presented, and it is shown that system Z FO properly generalizes the basic ideas underlying system Z.

I extend theories of nonmonotonic reasoning to account for reasons allowing free choice. My approach works with a wide variety of approaches to nonmonotonic reasoning and explains the connection between reasons for kinds of action and reasons for actions or subkinds falling under them. I use an Anderson–Kanger reduction of reason statements, identifying key principles in the logic of reasons.

Most of reasoning for decision making in daily life is based on preferences. As other kinds of reasoning processes, there are many formalisms trying to capture preferences, however none of them is able to capture all the subtleties of the human reasoning. In this paper we analyze how to formalize the preferences expressed by humans and how to reason with them to produce rankings. Particularly, we show that qualitative preferences are best represented with a combination of reward logics and conditional logics. We propose a new algorithm based on ideas of similarity between objects commonly used in case-based reasoning. We see that the new approach produces rankings close to the ones expressed by users. © 2016 Elsevier B.V., This research is partially funded by the CSIC Intramural projects RPREF (201650E044) and AMIDA (201450E009), the Spanish MICINN projects EdeTRI (TIN2012-39348-C02-01) and MILESS (TIN2013-45039-P), and the grant 2014-SGR-118 from the Generalitat de Catalunya

This paper extends the theory of situated inference from Mares (Relevant logic: a philosophical interpretation. Cambridge University Press, Cambrdge, 2004) to treat two weak relevant logics, B and DJ. These logics are interesting because they can be used as bases for consistent naive theories, such as naive set theory. The concepts of a situation and of information that are employed by the theory of situated inference are used to justify various aspects of these logics and to give an interpretation of the notion of set that is represented in the naive set theories that are based on them.

Formalising deontic concepts, such as obligation, prohibition and permission, is normally carried out in a modal logic with a possible world semantics, in which some worlds are better than others. The main focus in these logics is on inferring logical consequences, for example inferring that the obligation O q is a logical consequence of the obligations O p and O (p → q). In this paper we propose a non-modal approach in which obligations are preferred ways of satisfying goals expressed in first-order logic. To say that p is obligatory, but may be violated, resulting in a less than ideal situation s, means that the task is to satisfy the goal p ∨ s, and that models in which p is true are preferred to models in which s is true. Whereas, in modal logic, the preference relation between possible worlds is part of the semantics of the logic, in this non-modal approach, the preference relation between first-order models is external to the logic. Although our main focus is on satisfying goals, we also formulate a notion of logical consequence, which is comparable to the notion of logical consequence in modal deontic logic. In this formalisation, an obligation O p is a logical consequence of goals G, when p is true in all best models of G. We show how this non-modal approach to the treatment of deontic concepts deals with problems of contrary-to-duty obligations and normative conflicts, and argue that the approach is useful for many other applications, including abductive explanations, defeasible reasoning, combinatorial optimisation, and reactive systems of the production system variety.

Problems of automatic action planning and optimal control of unmanned vehicles in an adversarial environment are considered. An approach based on nonmonotonic logic in the language of positively formed formulas for automatically planning a route to a chosen target area is developed. The choice of this area, as well as classification and determination of priority targets, is implemented by means of logical inference with classical semantics. The optimal pursuit problem for the example of a group of three targets is solved.

Justification logic originated from the study of the logic of proofs. However, in a more general setting, it may be regarded as a kind of explicit epistemic logic. In such logic, the reasons a fact is believed are explicitly represented as justification terms. Traditionally, the modeling of uncertain beliefs is crucially important for epistemic reasoning. Graded modal logics interpreted with possibility theory semantics have been successfully applied to the representation and reasoning of uncertain beliefs; however, they cannot keep track of the reasons an agent believes a fact. This article is aimed at extending the graded modal logics with explicit justifications. We introduce a possibilistic justification logic, present its syntax and semantics, and investigate its metaproperties, such as soundness, completeness, and realizability.

The paper suggests a modal predicate logic that deals with classical quantification and modalities as well as intermediate operators, like “most” and “mostly”. Following up the theory of generalized quantifiers, we will understand them as two-placed operators and call them determiners. Quantifiers as well as modal operators will be constructed from them. Besides the classical deduction, we discuss a weaker probabilistic inference “therefore, probably” defined by symmetrical probability measures in Carnap’s style. The given probabilistic inference relates intermediate quantification to singular statements: “Most S are P” does not logically entail that a particular individual S is also P, but it follows that this is probably the case, where the probability is not ascribed to the propositions but to the inference. We show how this system deals with single case expectations while predictions of statistical statements remain generally problematic.

Defeasible conditionals are statements of the form `ifAthen normallyB'. One plausible interpretation introduced in nonmonotonic reasoning dictates that ($$A\Rightarrow B$$AźB) is true iff B is true in `most' A-worlds. In this paper, we investigate defeasible conditionals constructed upon a notion of `overwhelming majority', defined as `truth in a cofinite subset of$$\omega $$ź', the first infinite ordinal. One approach employs the modal logic of the frame $$(\omega

Judgment aggregation studies how to aggregate individual judgments on logically correlated propositions into collective judgments. Different logics can be used in judgment aggregation, for which Dietrich and Mongin have proposed a generalized model based on general logics. Despite its generality, however, all nonmonotonic logics are excluded from this model. This paper argues for using nonmonotonic logic in judgment aggregation. Then it generalizes Dietrich and Mongin’s model to incorporate a large class of nonmonotonic logics. This generalization broadens the theoretical boundaries of judgment aggregation by proving that, even if these nonmonotonic logics are employed, certain typical impossibility results still hold.

A consequence relation $$\vdash $$ is monotonic iff for premise sets $$\Gamma , \Delta $$ and conclusion $$\varphi $$ , if $$\Gamma \vdash \varphi $$ , $$\Gamma \subseteq \Delta $$ , then $$\Delta \vdash \varphi $$ ; and non-monotonic if this fails in some instance. More plainly, a consequence relation is monotonic when whatever is entailed by a premise set remains entailed by any of its supersets. From the High Middle Ages through the Early Modern period, consequence in theology is assumed to be monotonic. Concomitantly, to the degree the argument formulated by Anselm at Proslogion 2–4 is taken up by later commentators, it is accepted or rejected in accordance with a monotonic notion of consequence. Examining Anselm’s use of parallelism in the Proslogion, I show Anselm embeds his famous argument in Proslogion 2–4 in a non-monotonic context. The results here presented challenge some deeply ingrained ideas governing the historiography of the long twelfth century, particularly concerning how the theology of the later eleventh through the twelfth century relates to the scholasticism of the thirteenth.

The knowledge representation and reasoning of both humans and artificial systems often involves conditionals. A conditional connects a consequence which holds given a precondition. It can be easily recognized in natural languages with certain key words, like "if" in English. A vast amount of literature in both fields, both artificial intelligence and psychology, deals with the questions of how such conditionals can be best represented and how these conditionals can model human reasoning. On the other hand, findings in the psychology of reasoning, such as those in the Suppression Task, have led to a paradigm shift from the monotonicity assumptions in human inferences towards nonmonotonic reasoning. Nonmonotonic reasoning is sensitive for information change, that is, inferences are drawn cautiously such that retraction of previous information is not required with the addition of new information. While many formalisms of nonmonotonic reasoning have been proposed in the field of Artificial Intelligence, their capability to model properties of human reasoning has not yet been extensively investigated. In this paper, we analyzed systematically from both a formal and an empirical perspective the power of formal nonmonotonic systems to model (i) possible explicit defeaters, as in the Suppression Task, and (ii) more implicit conditional rules that trigger nonmonotonic reasoning by the keywords in such rules. The results indicated that the classical evaluation for the correctness of inferences has to be extended in the three major aspects (i) regarding the inference system, (ii) the knowledge base, and (iii) possible assumed exceptions for the rule.

A range of formal models of human reasoning have been proposed in a number of fields such as philosophy, logic, artificial intelligence, computer science, psychology, cognitive science, etc.: various logics (epistemic logics; non-monotonic logics), probabilistic systems (most notably, but not exclusively, Bayesian probability theory), belief revision systems, neural networks, among others. Now, it seems reasonable to require that formal models of human reasoning be (minimally) empirically adequate if they are to be viewed as models of the phenomena in question. How are formal models of human reasoning typically put to empirical test? One way to do so is to isolate a number of key principles of the system, and design experiments to gauge the extent to which participants do or do not follow them in reasoning tasks. Another way is to take relevant existing results and check whether a particular formal model predicts these results. The present investigation provides an illustration of the second kind of empirical testing by comparing two formal models for reasoning – namely the non-monotonic logic known as preferential logic; and a particular version of belief revision theories, screened belief revision – against the reasoning phenomenon known as belief bias in the psychology of reasoning literature: human reasoners typically seek to maintain the beliefs they already hold, and conversely to reject contradicting incoming information. The conclusion of our analysis will be that screened belief revision is more empirically adequate with respect to belief bias than preferential logic and non-monotonic logics in general, as what participants seem to be doing is above all a form of belief management on the basis of background knowledge. The upshot is thus that, while it may offer valuable insights into the nature of human reasoning, preferential logic (and non-monotonic logics in general) is ultimately inadequate as a formal model of the phenomena in question.

Combining Description Logic (DL) ontologies and nonmonotonic rules has gained increasing attention in the past decade, due to the growing range of applications of DLs. A well-known proposal for such a combination are non-monotonic DL-programs, which support rule-based reasoning on top of DL ontologies in a loose coupling, using a well-defined query interface. However, inconsistency may easily arise as a result of the interaction of the rules and the ontology, such that no answer set (i.e., model) of a DL-program exists; this makes the program useless. To overcome this problem, we present a framework for repairing inconsistencies in DL-programs by exchanging formulas of an ontology formulated in DL- Lite A , which is a prominent DL that allows for tractable reasoning. Viewing the data part of the ontology as a source of inconsistency, we define program repairs and repair answer sets based on them. We analyze the complexity of the notion, and we extend an algorithm for evaluating DL-programs to compute repair answer sets, under optional selection of preferred repairs that satisfy additional constraints. The algorithm induces a generalized ontology repair problem, in which the entailment respectively non-entailment of queries to the ontology, subject to possible updates, must be achieved by a data change. While this problem is intractable in general, we identify several tractable classes of preferred repairs that are useful in practice. For the class of deletion repairs among them, we optimize the algorithm by reducing query evaluation to constraint matching, based on the novel concept of support set, which roughly speaking is a portion of the data from which entailment of an ontology query follows. Our repair approach is implemented within an answer set program system, using a declarative method for repair computation. An experimental evaluation on a suite of benchmark problems shows the effectiveness of our approach and promising results, both regarding performance and quality of the obtained repairs. While we concentrate on DL- Lite A ontologies, our notions extend to other DLs, for which more general computation approaches may be used.

© 2016 ACM. In the past, compelling arguments in favour of the well-founded semantics for autoepistemic logic have been presented. In this article, we show that for certain classes of theories, this semantics fails to identify the unique intended model. We solve this problem by refining the well-founded semantics. We develop our work in approximation fixpoint theory, an abstract algebraical study of semantics of nonmonotonic logics. As such, our results also apply to logic programming, default logic, Dung's argumentation frameworks, and abstract dialectical frameworks. ispartof: ACM Transactions on Computational Logic vol:17 issue:4 pages:27- status: published

This paper is about the logical properties of sentences of the form S's are normally P, and starts from the idea that any logical theory for such sentences should meet the following simple requirement: If the only available information about some object x is that x has property S, it must be valid to infer by default that x has all the properties P that objects with property S normally have.We investigate how this requirement can be met by theories developed within the framework of circumscription, and specify a constraint – the exemption principle – that must be satisfied to do so. This principle determines in cases of conflicting default rules which objects are exempted from which rules, and, as such, is the main source for the capricious logical behavior of the sentences we are interested in.To facilitate comparison (and implementation) we supply an algorithm for inheritance networks and prove that arguments that can be expressed in both frameworks are valid on the circumscriptive account if and only if the inheritance algorithm has a positive outcome.

Paraconsistent logics are the formal systems in which absurdities do not trivialise the logic. In this paper, we give Hintikka-style game theoretical semantics for a variety of paraconsistent and non-classical logics. For this purpose, we consider Priest’s Logic of Paradox, Dunn’s First-Degree Entailment, Routleys’ Relevant Logics, McCall’s Connexive Logic and Belnap’s four-valued logic. We also present a game theoretical characterisation of a translation between Logic of Paradox/Kleene’s K3 and S5. We underline how non-classical logics require different verification games and prove the correctness theorems of their respective game theoretical semantics. This allows us to observe that paraconsistent logics break the classical bidirectional connection between winning strategies and truth values.

The critical behaviors of NP-complete problems have been studied extensively, and numerous results have been obtained for Boolean formula satisfiability (SAT) and constraint satisfaction (CSP), among others. However, few results are known for the critical behaviors of NP-hard nonmonotonic reasoning problems so far; in particular, a mathematical model for phase transition in nonmonotonic reasoning is still missing. In this article, we investigate the phase transition of negative two-literal logic programs under the answer-set semantics. We choose this class of logic programs since it is the simplest class for which the consistency problem of deciding if a program has an answer set is still NP-complete. We first introduce a new model, called quadratic model for generating random logic programs in this class. We then mathematically prove that the consistency problem for this class of logic programs exhibits a phase transition. Furthermore, the phase-transition follows an easy-hard-easy pattern. Given the correspondence between answer sets for negative two-literal programs and kernels for graphs, as a corollary, our result significantly generalizes de la Vega's well-known theorem for phase transition on the existence of kernels in random graphs. We also report some experimental results. Given our mathematical results, these experimental results are not really necessary. We include them here as they suggest that our phase-transition result is more general and likely holds for more general classes of logic programs.

We translate unconstrained and constrained input/output logics as introduced by Makinson and van der Torre to modal logics, using adaptive logics for the constrained case. The resulting reformulation has some additional benefits. First, we obtain a proof-theoretic (dynamic) characterization of input/output logics. Second, we demonstrate that our framework naturally gives rise to useful variants and allows to express important notions that go beyond the expressive means of input/output logics, such as violations and sanctions.

The aim of this paper is to define the logical system Sm4 characterised by the degree of truth-preserving consequence relation defined on the ordered set of values of Smiley’s four-element matrix MSm4. The matrix MSm4 has been of considerable importance in the development of relevant logics and it is at the origin of bilattice logics. It will be shown that Sm4 is a most interesting paraconsistent logic which encloses a sound theory of logical necessity similar to that of Anderson and Belnap’s logic of entailment E. Intuitively, Sm4 can be described as a four-valued expansion of the positive fragment of Lewis’ S5 or, alternatively, as a four-valued version of S5.

Gentzen?s and Prawitz?s approach to deductive systems, and Carnap?s and Popper?s treatment of probability in logic were two fruitful ideas of logic in the mid-twentieth century. By combining these two concepts, the notion of sentence probability, and the deduction relation formalized by means of inference rules, we introduce a system of inference rules based on the traditional proof-theoretic principles enabling to work with each form of probabilized propositional formulae. Namely, for each propositional connective, we define at least one introduction and one elimination rule, over the formulae of the form A[a,b] with the intended meaning that ?the probability c of truthfulness of a sentence A belongs to the interval [a,b] ?[0,1]?. It is shown that our system is sound and complete with respect to the Carnap-Poper-type probability models.

This reply to Oaksford and Chater’s (O&C)’s critical discussion of our use of logic programming (LP) to model and predict patterns of conditional reasoning will frame the dispute in terms of the semantics of the conditional. We begin by outlining some common features of LP and probabilistic conditionals in knowledge-rich reasoning over long-term memory knowledge bases. For both, context determines causal strength; there are inferences from the absence of certain evidence; and both have analogues of the Ramsey test. Some current work shows how a combination of counting defeaters and statistics from network monitoring can provide the information for graded responses from LP reasoning. With this much introduction, we then respond to O&C’s specific criticisms and misunderstandings.

Graham Priest defends the use of a nonmonotonic logic, LPm, in his analysis of reasoning in the face of true contradictions, such as those arising from the paradoxes of self-reference. In the course of defending this choice of logic in the face of the criticism that this logic is not truth preserving, Priest argued (2012) that requirement is too much to ask: since LPm is a nonmonotonic logic, it necessarily fails to preserve truth. In this article, I show that this assumption is incorrect, and I explain why nonmonotonic logics can nonetheless be truth preserving. Finally, I diagnose Priest’s error, to explain when nonmonotonic logics do indeed fail to preserve truth.

Due to an oversight by the Publisher during the typesetting stage, an uncorrected version of the paper was published.

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. © 2017 Elsevier Inc., The authors acknowledge partial support by the SYSMICS project ( EU H2020-MSCA-RISE-2015 Project 689176). Also, Flaminio and Godo acknowledge partial support by the FEDER/MINECO project TIN2015-71799-C2-1-P

In this paper we investigate the links between instantiated argumentation systems and the axioms for non-monotonic reasoning described in [15] with the aim of characterising the nature of argument based reasoning. In doing so, we consider two possible interpretations of the consequence relation, and describe which axioms are met by \({{\textsc {aspic}}^\mathbf {+}}\) under each of these interpretations. We then consider the links between these axioms and the rationality postulates. Our results indicate that argument based reasoning as characterised by \({{\textsc {aspic}}^\mathbf {+}}\) is—according to the axioms of [15]—non-cumulative and non-monotonic, and therefore weaker than the weakest non-monotonic reasoning systems considered in [15]. This weakness underpins \({{\textsc {aspic}}^\mathbf {+}}\)’s success in modelling other reasoning systems. We conclude by considering the relationship between \({{\textsc {aspic}}^\mathbf {+}}\) and other weak logical systems.

Priest and others have presented their “most telling” argument for paraconsistent logic: that only paraconsistent logics allow non-trivial inconsistent theories. This is a very prevalent argument; occurring as it does in the work of many relevant and more generally paraconsistent logicians. However this argument can be shown to be unsuccessful. There is a crucial ambiguity in the notion of non-triviality. Disambiguated the most telling reason for paraconsistent logics is either question-begging or mistaken. This highlights an important confusion about the role of logic in our development of our theories of the world. Does logic chart good reasoning or our commitments? We also consider another abductive argument for paraconsistent logics which also is shown to fail.

We investigate the application of Courcelle's theorem and the logspace version of Elberfeld et al. in the context of non-monotonic reasoning. Here we formalize the implication problem for propositional sets of formulas, the extension existence problem for default logic, the expansion existence problem for autoepistemic logic, the circumscriptive inference problem, as well as the abduction problem in monadic second order logic and thereby obtain fixed-parameter time and space efficient algorithms for these problems. On the other hand, we exhibit, for each of the above problems, families of instances of a very simple structure that, for a wide range of different parameterizations, do not have efficient fixed-parameter algorithms (even in the sense of the large class XPnu, resp., XLnu) under standard complexity assumptions.

Many modern applications of description logics (DLs, for short), such as biomedical ontologies and semantic web policies, provide fresh motivations for extending DLs with nonmonotonic inferences—a topic that has attracted a significant amount of attention along the years. Despite this, nonmonotonic inferences are not yet supported by DL technology due to a number of issues related to expressiveness, computational complexity, and optimizations. This paper contributes to the practical support of nonmonotonic inferences in description logics by introducing a new semantics expressly designed to address knowledge engineering needs. This formalism has appealing expressiveness, enjoys nice computational properties, and constitutes an interesting solution to an ample class of application needs. The formalism is validated through extensive comparison with the other nonmonotonic DLs, and systematic scalability tests. The test case generator and its novel validation methodology constitute a further contribution of this paper.

The traditional connection between logic and reasoning has been under pressure ever since Gilbert Harman attacked the received view that logic yields norms for what we should believe. In this article I first place Harman's challenge in the broader context of the dialectic between logical revisionists like Bob Meyer and sceptics about the role of logic in reasoning like Harman. I then develop a formal model based on contemporary epistemic and doxastic logic in which the relation between logic and norms for belief can be captured.

This paper provides an analysis of contrary-to-duty reasoning from the proof-theoretical perspective of category theory. While Chisholm’s paradox hints at the need of dyadic deontic logic by showing that monadic deontic logics are not able to adequately model conditional obligations and contrary-to-duties, other arguments can be objected to dyadic approaches in favor of non-monotonic foundations. We show that all these objections can be answered at one fell swoop by modeling conditional obligations within a deductive system defined as an instance of a symmetric monoidal closed category. Using category theory as a foundational framework for logic, we show that it is possible to model conditional normative reasoning and conflicting obligations within a monadic approach without adding further operators or considering deontic conditionals as primitive.

Recent experiments have shown that naive speakers find borderline contradictions involving vague predicates acceptable. In Cobreros et al. (Journal of Philosophical Logic, 41, 347–385, 2012a) we proposed a pragmatic explanation of the acceptability of borderline contradictions, building on a three-valued semantics. In a reply, Alxatib et al. (Journal of Philosophical Logic, 42, 619–634, 2013) show, however, that the pragmatic account predicts the wrong interpretations for some examples involving disjunction, and propose as a remedy a semantic analysis instead, based on fuzzy logic. In this paper we provide an explicit global pragmatic interpretation rule, based on a somewhat richer semantics, and show that with its help the problem can be overcome in pragmatics after all. Furthermore, we use this pragmatic interpretation rule to define a new (nonmonotonic) consequence-relation and discuss some of its properties.

In agent-based systems, an agent generally forms her belief based on evidence from multiple sources, such as messages from other agents or perception of the external environment. In this paper, we present a logic for reasoning about evidence and belief. Our framework not only takes advantage of the source-tracking capability of justification logic, but also allows the distinction between the actual observation and simply potential admissibility of evidence. We present the axiomatization for the basic logic and its dynamic extension, investigate its properties, and use a running example to show its applicability to information fusion for autonomous agents.

The theory of mental models accounts for the meanings of causal relations in daily life. They refer to seven temporally-ordered deterministic relations between possibilities, which include causes, prevents, and enables. Various factors—forces, mechanisms, interventions—can enter into the interpretation of causal assertions, but they are not part of their core meanings. Mental models represent only salient possibilities, and so they are identical for causes and enables, which may explain failures to distinguish between their meanings. Yet, reasoners deduce different conclusions from them, and distinguish between them in scenarios, such as those in which one event enables a cause to have its effect. Neither causation itself nor the distinction between causes and enables can be captured in the pure probability calculus. Statistical regularities, however, often underlie the induction of causal relations. The chapter shows how models help to resolve inconsistent causal scenarios and to reverse engineer electrical circuits.

The development of recursion theory motivated Kleene to create regular three-valued logics. Remove it taking his inspiration from the computer science, Fitting later continued to investigate regular three-valued logics and defined them as monotonic ones. Afterwards, Komendantskaya proved that there are four regular three-valued logics and in the three-valued case the set of regular logics coincides with the set of monotonic logics. Next, Tomova showed that in the four-valued case regularity and monotonicity do not coincide. She counted that there are 6400 four-valued regular logics, but only six of them are monotonic. The purpose of this paper is to create natural deduction systems for them. We also describe some functional properties of these logics.

As agents faced with fallible information, we frequently find ourselves in situations where we are forced to base our beliefs on evidence which is in some way or another contradictory. We nevertheless want these beliefs to be rational. This paper presents a simple probabilistic model of what it means for a belief based on a contradictory body of evidence to be rational. In this approach, we model contradictions in the evidence available to us as resulting from random noise, and we model our task as rational agents as reconstructing the most likely states of affairs given the evidence available to us. Our main result consists in providing several equivalent descriptions of the non-reflexive and non-monotonic consequence relation which formalizes the notion that it is reasonable to accept that a proposition is true given good evidence supporting some set of propositions.

First-order typed model counting extends first-order model counting by the ability to distinguish between different types of models. In this paper, we exploit this benefit in order to calculate weighted conditional impacts (WCIs) which play a central role in nonmonotonic reasoning based on conditionals. More precisely, WCIs store information about the verification and the falsification of conditionals with respect to a possible worlds semantics, and therefore serve as sufficient statistics for maximum entropy (ME) distributions as models of probabilistic conditional knowledge bases. Formally, we annotate formulas with algebraic types that encode concisely all structural information needed to compute WCIs, while allowing for a systematic and efficient counting of models. In this way, our approach to typed model counting for ME-reasoning integrates both structural and counting aspects in the same framework.