This paper studies the properties of eight semantic consequence relations defined from a Tarski-logic L and a preference relation ≺. They are equivalent to Shoham’s so-called preferential entailment for smooth model structures, but avoid certain problems of the latter in non-smooth configurations. Each of the logics can be characterized in terms of what we call multi-selection semantics. After discussing this type of semantics, we focus on some concrete proposals from the literature, checking a number of meta-theoretic properties (...) and elaborating on their intuitive motivation. As it turns out, many of their meta-properties only hold in case ≺ is transitive. To tackle this problem, we propose slight modifications of each of the systems, showing the resulting logics to behave better at the intuitive level and in metatheoretic terms, for arbitrary ≺. (shrink)

This paper is concerned with removing the identity schema from the axiomatic basis of deontic conditionals. This is in order to allow a stipulated ideal to be contrary or opposite in nature to the fact it is predicated upon. It is desirable, or so it is argued, to retain the order-theoretic orientation of preferential semantics towards the analysis of deontic conditionals, more specifically of maximality semantics in the tradition from Bengt Hansson. So understood, the problem involves abstracting (...) away the settledness assumption that is built in to maximality semantics. This is the assumption that what is optimal given ϕ is that which all the best ϕ-states have in common, notably ϕ itself. We propose a solution based on a strict and finite preference relation over which deontic conditionals are evaluated by letting ϕ-states evolve freely, as fate or fortune would have it, into different possibly ensuing optima that may but need not be ϕ-states themselves. The result is a deontic conditional that does not have identity. This new conditional is shown to be a proper generalization of the Hansson conditional. Hansson’s conditional can be recovered in the new idiom as a special case. Indeed, the new semantics is general enough to cover several apparently very different conceptions of deontic conditionality. For instance, the input/output logic known as basic output is a sublogic of the new system. This is somewhat surprising and suggests that there may yet be unity to be had in the field of deontic logic. (shrink)

Handling exceptions in a knowledge-based system is an important issue in many application domains, such as medical domain. Recently, there is an increasing interest in nonmonotonic extension of description logics to handle exceptions in ontologies. In this paper, we propose three preferential semantics for plausible subsumption to deal with exceptions in description logic-based knowledge bases. Our preferential semantics are defined in the framework of possibility theory, which is an uncertainty theory devoted to handling incomplete information. We (...) consider the properties of these semantics and their relationships. We also discuss the relationship between two of our preferential semantics and two existing preferential semantics. We extend a description logic-based knowledge base by adding preferential subsumptions. Entailment of plausible subsumptions relative to an extended knowledge base is defined. Properties of the preferential subsumption relations relative to an extended description logic-based knowledge base are discussed. Finally, we show that our semantics for plausible subsumption can be reduced to standard semantics of an expressive description logic. Thus, the problem of plausible subsumption checking under our semantics can be reduced to the problem of subsumption checking under the classical semantics. (shrink)

We introduce and explore the notion of duality for entailment relations induced by preference orderings on states. We discuss the relationship between these preferential entailment relations from the perspectives of Boolean algebra, inference rules, and modal axiomatisation. Interpreting the preference relations as accessibility relations establishes modular Gödel-Löb logic as a suitable modal framework for rational preferential reasoning.

This paper presents a uniform semantic treatment of nonmonotonic inference operations that allow for inferences from infinite sets of premises. The semantics is formulated in terms of selection functions and is a generalization of the preferential semantics of Shoham (1987), (1988), Kraus, Lehman, and Magidor (1990) and Makinson (1989), (1993). A selection function picks out from a given set of possible states (worlds, situations, models) a subset consisting of those states that are, in some sense, the most (...) preferred ones. A proposition α is a nonmonotonic consequence of a set of propositions Γ iff α holds in all the most preferred Γ-states. In the literature on revealed preference theory, there are a number of well-known theorems concerning the representability of selection functions, satisfying certain properties, in terms of underlying preference relations. Such theorems are utilized here to give corresponding representation theorems for nonmonotonic inference operations. At the end of the paper, the connection between nonmonotonic inference and belief revision, in the sense of Alchourrón, Gärdenfors, and Makinson, is explored. In this connection, infinitary belief revision operations that allow for the revision of a theory with a possibly infinite set of propositions are introduced and characterized axiomatically. (shrink)

We introduce Information Bearing Relation Systems (IBRS) as an abstraction of many logical systems. These are networks with arrows recursively leading to other arrows etc. We then define a general semantics for IBRS, and show that a special case of IBRS generalizes in a very natural way preferential semantics and solves open representation problems for weak logical systems. This is possible, as we can the strong coherence properties of preferential structures by higher arrows, that is, arrows, (...) which do not go to points, but to arrows themselves. (shrink)

In this article, we redefine classical notions of theory reduction in such a way that model-theoretic preferential semantics becomes part of a realist depiction of this aspect of science. We offer a model-theoretic reconstruction of science in which theory succession or reduction is often better - or at a finer level of analysis - interpreted as the result of model succession or reduction. This analysis leads to 'defeasible reduction', defined as follows: The conjunction of the assumptions of a (...) reducing theory T with the definitions translating the vocabulary of a reduced theory T' to the vocabulary of T, defeasibly entails the assumptions of reduced T'. This relation of defeasible reduction offers, in the context of additional knowledge becoming available, articulation of a more flexible kind of reduction in theory development than in the classical case. Also, defeasible reduction is shown to solve the problems of entailment that classical homogeneous reduction encounters. Reduction in the defeasible sense is a practical device for studying the processes of science, since it is about highlighting different aspects of the same theory at different times of application, rather than about naive dreams concerning a metaphysical unity of science. (shrink)

Modal accounts of normality in non-monotonic reasoning traditionally have an underlying semantics based on a notion of preference amongst worlds. In this paper, we motivate and investigate an alternative semantics, based on ordered accessibility relations in Kripke frames. The underlying intuition is that some world tuples may be seen as more normal, while others may be seen as more exceptional. We show that this delivers an elegant and intuitive semantic construction, which gives a new perspective on defeasible necessity. (...) Technically, the revisited logic does not change the expressive power of our previously defined preferential modalities. This conclusion follows from an analysis of both semantic constructions via a generalisation of bisimulations to the preferential case. Reasoners based on the previous semantics therefore also suffice for reasoning over the new semantics. We complete the picture by investigating different notions of defeasible conditionals in modal logic that can also be captured within our framework. (shrink)

Preferential model is one of the important semantical structures in nonmonotonic logic. This paper aims to establish an isomorphism theorem for preferential models, which gives us a purely algebraic characterization of the equivalence of preferential models. To this end, we present the notions of local similarity and local simulation. Based on these notions, two operators Δ(·) and μ(·) over preferential models are introduced and explored respectively. Together with other two existent operators ρ(·) and ΠD(·), we introduce (...) an operator ∂D(·). Then the isomorphism theorem is obtained in terms of ∂D(·), which asserts that for any two preferential models M₁ and M₂, they generate the same preferential inference if and only if ∂D(M₁) and ∂D(M₂) are isomorphic. Based on ∂D(·), we also get an alternative model-theoretical characterization of the well-known postulate "Weaken Disjunctive Rationality". Moreover, in the finite language framework, we show that Δ(μ(·)) is competent for the task of eliminating redundancy, and provide a representation result for k-relations. (shrink)

We give a systematic overview of semantical and logical rules in non monotonic and related logics. We show connections and sometimes subtle differences, and also compare such rules to uses of the notion of size.

ABSTRACT The major approaches to belief revision and non monotonic reasoning proposed in the literature differ along a number of dimensions, including whether they are ?syntax- based? or ?semantic-based?, ?foundational? or ?coherentist?, ?consistence-restoring? or ?inconsistency-tolerant?. Our contribution towards clarifying the connections between these various approaches is threefold: ?We show that the two main approaches to belief revision, the foundations and coherence theories, are mathematically equivalent, thus answering a question left open in [Gar90, Doy92], The distinction between syntax-based approaches to revision (...) and approaches based on (semantic) preferential structures falls along similar lines, and their expressive equivalence is a consequence of this result. ?We formally clarify the connection between belief revision and non monotonic reasoning, in a particularly simple way which also throws light on the connection between consistence- restoring and reasoning-from-inconsistency approaches [BDP95]. ?As a direct application of the above, we show that Poole's (syntax-based) system of default reasoning and Shoham's preferential semantic for non monotonic reasoning are also expressively equivalent, in that they can represent the same set of non monotonic consequence relations. (shrink)

This multiplex semantics incorporates multiple relations of deontic accessibility or multiple preference rankings on alternative worlds to represent distinct normative standards. This provides a convenient framework for deontic logic that allows conflicts of obligation, due either to conflicts between normative standards or to incoherence within a single standard. With the multiplex structures, two general senses of "ought" may be distinguished, an indefinite sense under which something is obligatory when it is enjoined by some normative standard and a core sense (...) for when something is enjoined by all normative standards. Multiple normative standards may themselves be given a preferential order; this leads to a concept of ranked obligation. This paper presents the foundations of this multiplex semantics and the propositional deontic logics they define. (shrink)

Preferential structures are probably the best examined semantics for nonmonotonic and deontic logics; in a wider sense, they also provide semantical approaches to theory revision and update, and other fields where a preference relation between models is a natural approach. They have been widely used to differentiate the various systems of such logics, and their construction is one of the main subjects in the formal investigation of these logics. We introduce new techniques to construct preferential structures for (...) completeness proofs. Since our main interest is to provide general techniques, which can be applied in various situations and for various base logics (propositional and other), we take a purely algebraic approach, which can be translated into logics by easy lemmata. In particular, we give a clean construction via indexing by trees for transitive structures, this allows us to simplify the proofs of earlier work by the author, and to extend the results given there. (shrink)

The problem of interpolation is a classical problem in logic. Given a consequence relation |~ and two formulas φ and ψ with φ |~ ψ we try to find a “simple" formula α such that φ |~ α |~ ψ. “Simple" is defined here as “expressed in the common language of φ and ψ". Non-monotonic logics like preferential logics are often a mixture of a non-monotonic part with classical logic. In such cases, it is natural examine also variants of (...) the interpolation problem, like: is there “simple" α such that φ ⊢ α |~ ψ where ⊢ is classical consequence? We translate the interpolation problem from the syntactic level to the semantic level. For example, the classical interpolation problem is now the question whether there is some “simple" model set X such that M(φ) ⫅ X ⫅ M(ψ). We can show that such X always exist for monotonic and antitonic logics. The case of non-monotonic logics is more complicated, there are several variants to consider, and we mostly have only partial results. (shrink)

Research using the intermodal preferential looking paradigm (IPLP) has consistently shown that English‐learning children aged 2 can associate transitive argument structure with causal events. However, studies using the same methodology investigating 2‐year‐old children’s knowledge of the conjoined agent intransitive and semantic role assignment have reported inconsistent findings. The aim of the present study was to establish at what age English‐learning children have verb‐general knowledge of both transitive and intransitive argument structure using a new method: the forced‐choice pointing paradigm. The (...) results suggest that young 2‐year‐olds can associate transitive structures with causal (or externally caused) events and can use transitive structure to assign agent and patient roles correctly. However, the children were unable to associate the conjoined agent intransitive with noncausal events until aged 3;4. The results confirm the pattern from previous IPLP studies and indicate that children may develop the ability to comprehend different aspects of argument structure at different ages. The implications for theories of language acquisition and the nature of the language acquisition mechanism are discussed. (shrink)

The amount of information conveyed by linguistic conventions depends on their precision, yet the codes that humans and other animals use to communicate are quite ambiguous: they may map several vague meanings to the same symbol. How does semantic precision evolve, and what are the constraints that limit it? We address this question using a multiplayer gaming app, where individuals communicate with one another in a scaled-up referential game. Here, the goal is for a sender to use black and white (...) symbols to communicate colors. We expected that the players’ mappings between symbols and colors would grow more specific over time, through a selection process whereby precise mappings are preferentially copied. We found that players become increasingly more precise in their use of symbols over the course of their interactions. This trend did not, however, result from selective copying of precise mappings. We explore the implications of this result for the study of lexical ambiguity, Zipf's Law of Meaning, and disagreements over semantic conventions. (shrink)

The concept of inconsistency has become recently the subject of many studies focused on the principle ex contradictione sequitur quodlibet which is a hallmark of the classical inconsistency. Stanisław Jaśkowski was the first who took a non-classical standpoint toward this principle building a system of propositional logic which rejects this classical principle. Rejecting it implies important consequences for the concept of classical negation, and poses the question in which properties the op-eration of negation should be endowed. The intention of this (...) article is to define the concept of inconsistency as well as the concept of negation in a way satisfying the main intuitions natural language users connect with the two notions occurring in propositions containing vague concepts. The vagueness of natural language dis-course leads to the phenomenon of the seeming contradiction characteristic of natural language. A non-standard consequence relation for such a language has been defined in terms of preferential semantics making use of the concept of most classical model. This non-standard consequence relation is applied to the ethical discourse. The concepts introduced in this article have been used to an interpretation of the contradictories of Plato's Parmenides, as well as to the rejection of dialetheism. (shrink)

A consequence relation in the framework of preferential semantics based on the four-valued Belnap-Dunn logic is constructed which proves that the sorites paradoxes are unsound or invalid inferences .

We investigate an aspect of defeasibility that has somewhat been overlooked by the non-monotonic reasoning community, namely that of defeasible modes of reasoning. These aim to formalise defeasibility of the traditional notion of necessity in modal logic, in particular of its different readings as action, knowledge and others in specific contexts, rather than defeasibility of conditional forms. Building on an extension of the preferential approach to modal logics, we introduce new modal osperators with which to formalise the notion of (...) defeasible necessity and distinct possibility, and that can be used to represent expected effects, refutable knowledge, and so on. We show how KLM-style conditionals can smoothly be integrated with our richer language. We also propose a tableau calculus which is sound and complete with respect to our modal preferential semantics, and of which the computational complexity remains in the same class as that of the underlying classical modal logic. (shrink)

We introduce a family of preferential logics that are useful for handling information with different levels of uncertainty. The corresponding consequence relations are nonmonotonic, paraconsistent, adaptive, and rational. It is also shown that the formalisms in this family can be embedded in corresponding four-valued logics with at most three uncertainty levels, and that reasoning with these logics can be simulated by algorithms for processing circumscriptive theories, such as DLS and SCAN.

In this work, we propose a meaningful extension of description logics for non-monotonic reasoning. We introduce \, a logic allowing for the representation of and reasoning about both typical class-membership and typical instances of a relation. We propose a preferential semantics for \ in terms of partially-ordered DL interpretations which intuitively captures the notions of typicality we are interested in. We define a tableau-based algorithm for checking \ knowledge-base consistency that always terminates and we show that it is (...) sound and complete w.r.t. our preferential semantics. The general framework we here propose can serve as the foundation for further exploration of non-monotonic reasoning in description logics and similarly structured logics. (shrink)

We present two defeasible logics of norm-propositions (statements about norms) that (i) consistently allow for the possibility of normative gaps and normative conflicts, and (ii) map each premise set to a sufficiently rich consequence set. In order to meet (i), we define the logic LNP, a conflict- and gap-tolerant logic of norm-propositions capable of formalizing both normative conflicts and normative gaps within the object language. Next, we strengthen LNP within the adaptive logic framework for non-monotonic reasoning in order to meet (...) (ii). This results in the adaptive logics LNPr and LNPm, which interpret a given set of premises in such a way that normative conflicts and normative gaps are avoided ‘whenever possible’. LNPr and LNPm are equipped with a preferential semantics and a dynamic proof theory. (shrink)

The paradoxes of vagueness are unsound or invalid reasoning on the ground of preferential semantics based on the four-valued Belnap-Dunn logic.

In this inaugural lecture I offer, against the background of a discussion of knowledge representation and its tools, an overview of my research in the philosophy of science. I defend a relational model-theoretic realism as being the appropriate meta-stance most congruent with the model-theoretic view of science as a form of human engagement with the world. Making use of logics with preferential semantics within a model-theoretic paradigm, I give an account of science as process and product. I demonstrate (...) the power of the full-blown employment of this paradigm in the philosophy of science by discussing the main applications of model-theoretic realism to traditional problems in the philosophy of science. I discuss my views of the nature of logic and of its role in the philosophy of science today. I also specifically offer a brief discussion on the future of cognitive philosophy in South Africa. My conclusion is a general look at the nature of philosophical inquiry and its significance for philosophers today. South African Journal of Philosophy Vol. 25 (4) 2006: pp. 275-289. (shrink)

The year 2012 has witnessed worldwide celebrations of Alan Turing’s 100th birthday. A great number of conferences and workshops were organized by logicians, computer scientists and researchers in AI, showing the continued flourishing of computer science, and the fruitful interfaces between logic and computer science. Logic is no longer just the concept that Frege had about one hundred years ago, let alone that of Aristotle twenty centuries before. One of the prominent features of contemporary logic is its interdisciplinary character, connecting (...) across mathematics, philosophy, modern computer science, and even the cognitive and social sciences. -/- China is no exception. This special issue explores interfaces of logic with AI and other disciplines, and it attempts to give a bird’s eye view of the developing landscape of logic and AI in China. Our contributors come from computer science, AI, philosophy and linguistics, united by their interest in using formal logical tools and notions to deal with substantial problems in their fields. We are aware that AI is a vast area, and that the limitations of space in a volume like this may provide a somewhat incomplete or biased view. Still we hope that the six papers selected here illustrate the concerns and curiosity driving Chinese researchers, on topics shared with our colleagues around the world. -/- One of the active research areas in AI is planning under uncertainty, with incomplete knowledge, where a well-established logical framework is the action language of Son and Baral (Son and Baral 2001). This language can express the notions of sensing actions and making conditional plans. However, its current semantics is not yet satisfactory, since according to the semantics an agent would not have knowledge of what happens after a plan is executed, which can be a serious problem in a safety-critical environment. In their paper Proof Systems for Planning under Cautious Semantics (Shen and Zhao 2013), Yuping Shen and Xishun Zhao address this problem and propose a cautious and weakly cautious semantics for the action language. This new semantics enables the agent to know exactly what happens after executing an action. Hoare style proof systems are then proposed and proved to be sound and complete. What is more, the proposed new system reduces the reasoning complexity of the action language. -/- In Preferential Semantics for Plausible Subsumption in Possibility Theory (Qi and Zhang 2013), Guilin Qi and Zhizheng Zhang investigate another important issue in AI: how to handle exceptions in a knowledge-based system. This has become a hot topic in recent years, especially in ontological design, where description logics are widely used. The question then arises whether we can get a nonmonotonic extension of description logic. To answer this, the authors first propose three preferential semantics for plausible subsumption in the framework of possibility theory (Dubois and Prade 1985) and study their properties and their relationships. Then they introduce preferential subsumptions to a description logic-based knowledge system, and define an entailment relation of plausible subsumption. Moreover, they study reductions of the new semantics to standard semantics for an expressive description logic. -/- Counterfactual reasoning has been explored extensively by philosophers, cf. Lewis (1973), Stalnaker (1968). But in the last two decades, causal models have been proposed for modeling counterfactual reasoning by researchers in AI. In A Lewisian Logic of Causal Counterfactuals (Zhang 2013), Jiji Zhang studies classes of causal models and provides formal characterizations. In particular, a new logic system is proposed and proved complete for the class of ‘Lewisian models’ satisfying all the principles of David Lewis’s logic of counterfactuals. Moreover, Zhang compares his new framework with those of Lewis and Stalnaker, and also establishes formal connections with more recent approaches by Galles and Pearl, and by Halpern. -/- Consistency is a fundamental notion in logic, but it also plays a key role in many techniques for solving constraint satisfaction problems (Mackworth 1977). In Variable-Centered Consistency in Model RB (Li et al. 2013), Liang Li, Tian Liu and Ke Xu start with a discussion of i-consistency in ‘Model RB’ and ‘Model RD’, two models of random constraint satisfaction problems with growing domain size, then compare it with the notion of t-consistency and strong t-consistency in well-known earlier work by Freuder. After that, the authors define a new kind of variable-centered consistency, and show an upper bound on a suitably chosen parameter in Model RB and Model RD, such that up to this bound, all typical instances of Model RB are variable-centered consistent. -/- Language understanding is a cognitive process that human beings carry out every day. Metaphor is a pervasive phenomenon in natural languages, and its cognitive role calls for interpretation. There is a growing body of literature on cognitive mechanisms underlying metaphor. The paper An Ontology-Based Approach to Metaphor Cognitive Computation (Huang et al. 2013) by Xiaoxi Huang, Huaxin Huang, Beishui Liao, and Cihua Xu proposes a theoretical framework for metaphor understanding based on recent results from cognitive science (Shutova 2010). They then introduce ontology as a knowledge representation method, and present a new ontological model to formalize metaphor-based knowledge. An algorithmic description is provided using a quantitative measure of integrated degree, and its effectiveness is demonstrated. -/- Puzzles and their solution mechanisms are a recurrent theme in many fields. Dynamic epistemic logic (van Ditmarsch et al. 2007); (van Benthem 2011) is a recent framework that can handle information dynamics in knowledge puzzles. In Reasoning About Agent Types and the Hardest Logic Puzzle Ever (Liu and Wang 2013), Fenrong Liu and Yanjing Wang propose a new dynamic-epistemic logic that can specify different types of agents and analyse agents’ knowledge, communication and reasoning. To demonstrate its power, Smullyan’s well-known Knights and Knaves puzzles are formalized, as well as the Hardest Logic Puzzle Ever (HLPE) of Boolos (Boolos 1996). Moreover, a spectrum of new puzzles is found involving knowledge-based agent types, and their solution complexity is shown to be harder than that for existing logic puzzles. It is also shown that a version of HLPE in which the agents do not know the others’ types does not have a solution at all. -/- Finally, we would like to thank the editor of Minds and Machines, Gregory Wheeler, for his efficient assistance and kind support. We also thank Ties Nijssen at Springer for his encouragement to start this project at the first place. And of course, our appreciation goes to our authors, as well as all the colleagues who have helped us review the papers in this collection. (shrink)

We investigate a possible universal constraint on spatial meaning. It has been proposed that people attend preferentially to the endpoints of spatial motion events, and that languages may therefore make finer semantic distinctions at event endpoints than at event beginnings. We test this proposal. In Experiment 1, we show that people discriminate the endpoints of spatial motion events more readily than they do event beginnings—suggesting a non-linguistic attentional bias toward endpoints. In Experiment 2, speakers of Arabic, Chinese, and English each (...) described a set of spatial events displayed in video clips. Although the spatial systems of these languages differ, speakers of all three languages made finer semantic distinctions at event endpoints, compared to event beginnings. These findings are consistent with the proposal that event endpoints are privileged over event beginnings, in both language and perception. (shrink)

According to a simple theory of the relationship between 'want' ascriptions and the desires they ascribe, when I learn that ⌜A wants p⌝ is true, I learn that the truth of p is necessary and sufficient for satisfying one of A’s desires. I argue that this simple theory is false: ⌜A wants p⌝ can be true and underspecific: p may be necessary but not sufficient for the satisfaction of one of A’s desires. I show that existing semantics for 'want' (...) cannot account for this kind of underspecificity, and I propose a desire-based semantics for ‘want’ that can. I go on to argue that my semantics has empirical and methodological advantages over existing theories of ‘want’ that give truth conditions in terms of an agent’s preferential and doxastic state rather than their desires. -/- . (shrink)

The language comprehension system preferentially assumes that agents come first during incremental processing. While this might reflect a biologically fixed bias, shared with other domains and other species, the evidence is limited to languages that place agents first, and so the bias could also be learned from usage frequency. Here, we probe the bias with electroencephalography (EEG) in Äiwoo, a language that by default places patients first, but where sentence-initial nouns are still locally ambiguous between patient or agent roles. Comprehenders (...) transiently interpreted nonhuman nouns as patients, eliciting a negativity when disambiguation was toward the less common agent-initial order. By contrast and against frequencies, human nouns were transiently interpreted as agents, eliciting an N400-like negativity when the disambiguation was toward patient-initial order. Consistent with the notion of a fixed property, the agent bias is robust against usage frequency for human referents. However, this bias can be reversed by frequency experience for nonhuman referents. (shrink)

We prove completeness of preferential conditional logic with respect to convexity over finite sets of points in the Euclidean plane. A conditional is defined to be true in a finite set of points if all extreme points of the set interpreting the antecedent satisfy the consequent. Equivalently, a conditional is true if the antecedent is contained in the convex hull of the points that satisfy both the antecedent and consequent. Our result is then that every consistent formula without nested (...) conditionals is satisfiable in a model based on a finite set of points in the plane. The proof relies on a result by Richter and Rogers showing that every finite abstract convex geometry can be represented by convex polygons in the plane. (shrink)

The two volumes in this advanced textbook present results, proof methods, and translations of motivational and philosophical considerations to formal constructions. In this Vol. I the author explains preferential structures and abstract size. In the associated Vol. II he presents chapters on theory revision and sums, defeasible inheritance theory, interpolation, neighbourhood semantics and deontic logic, abstract independence, and various aspects of nonmonotonic and other logics. In both volumes the text contains many exercises and some solutions, and the author (...) limits the discussion of motivation and general context throughout, offering this only when it aids understanding of the formal material, in particular to illustrate the path from intuition to formalisation. Together these books are a suitable compendium for graduate students and researchers in the area of computer science and mathematical logic. (shrink)

Epistemic logic has grown from its philosophical beginnings to find diverse applications in computer science, and as a means of reasoning about the knowledge and belief of agents. This book provides a broad introduction to the subject, along with many exercises and their solutions. The authors begin by presenting the necessary apparatus from mathematics and logic, including Kripke semantics and the well-known modal logics K, T, S4 and S5. Then they turn to applications in the context of distributed systems (...) and artificial intelligence. These include the notions of common knowledge, distributed knowledge, explicit and implicit belief, the interplays between knowledge and time, and knowledge and action, as well as a graded (or numerical) variant of the epistemic operators. The authors also discuss extensively the problem of logical omniscience. They cover Halpern & Moses' theory of honest formulas, and they make a digression into the realm of nonmonotonic reasoning and preferential entailment. They discuss Moore's autoepistemic logic, together with Levesque's related logic of "all I know". Furthermore, they show how one can base default and counterfactual reasoning on epistemic logic. Graduate students in philosophy or in computer science, especially those with an interest in AI, will find this book useful. (shrink)

Argumentation is the most representative of intelligent activities of humans. Therefore, it is natural to think that it could have many implications for artificial intelligence and computer science as well. Specifically, argumentation may be considered a most primitive capability for interaction among computational agents. In this paper we present an argumentation framework based on the four-valued paraconsistent logic. Tolerance and acceptance of inconsistency that this logic has as its logical feature allow for arguments on inconsistent knowledge bases with which we (...) are often confronted. We introduce various concepts for argumentation, such as arguments, attack relations, argument justification, preferential criteria of arguments based on social norms, and so on, in a way proper to the four-valued paraconsistent logic. Then, we provide the fixpoint semantics and dialectical proof theory for our argumentation framework. We also give the proofs of the soundness and completeness. (shrink)

Belief fusion, instead of AGM belief revision, was first proposed to solve the problem of inconsistency, that arised from repetitive application of the operation when agents' knowledge were amalgamated. In the preceding work of Maynard-Reid II and Shoham, the fusion operator is applied to belief states, which is total preorders over possible worlds which is based on the semantics of belief revision. Moreover, they introduced the pedigreed belief state, which represented multiple sources of belief states, ordered by a credibility (...) ranking. However in the theory, all the sources must be totally ordered and thus applicable area is quite restrictive. In this paper, we realize the fusion operator of multiple agents for partially ordered sources. When we consider such a partial ranking over sources, there is no need to restrict that each agent has total preorders over possible worlds. The preferential model, based on the semantics on nonmonotonic reasoning, allows each agent to have strict partial orders over possible worlds. Especially, such an order is called a preferential relation, that prescribes a world is more plausible than the other. Therefore, we introduce an operation which combines multiple preferential relations of agents. In addition, we show that our operation can properly include the ordinary belief fusion. (shrink)

Logics are like shadows on a wall; to understand why they dance as they do, and how they can be made to move differently, one needs to look at the mathematical structures from which they can be projected. That is a methodology that has long proven its value for classical and other forms of deductive inference; this book manifests its pertinence to logics of uncertain qualitative reasoning. It draws together and refines work from the literature on preferential and other (...) quite different semantics for such reasoning and then, in its central chapter, integrates the leading ideas from them into a single multi-dimensional semantics. It enters new territory with investigations of interpolation and qualitative independence in non-monotonic contexts as well as connections between non-monotonic logic and neuroscience, specifically with Edelman's theory of organization of the brain. Finally, it proposes novel applications such as equational modellings for contrary-to-duty normative reasoning and mathematical analyses of some kinds of casuistic reasoning in the Talmudic tradition. Profusely signposted at every turn, the volume is accessible to readers equipped with a modicum of set theory and abstract algebra. Visitors to the area can come in by the front gate and work their way, in various orders, through as much of the garden as strikes their fancy; specialists will enter at Chap. 11 or at whatever special topic interests them, well assured that they can easily backtrack to whatever bits and pieces are needed from what they skipped. A volume with a clear vision, a book to return to many times." (David Makinson). (shrink)

This text centers around three main subjects. The first is the concept of modularity and independence in classical logic and nonmonotonic and other nonclassical logic, and the consequences on syntactic and semantical interpolation and language change. In particular, we will show the connection between interpolation for nonmonotonic logic and manipulation of an abstract notion of size. Modularity is essentially the ability to put partial results achieved independently together for a global result. The second aspect of the book is the authors' (...) uniform picture of conditionals, including many-valued logics and structures on the language elements themselves and on the truth value set. The third topic explained by the authors is neighbourhood semantics, their connection to independence, and their common points and differences for various logics, e.g., for defaults and deontic logic, for the limit version of preferential logics, and for general approximation"--Publisher. (shrink)

The two volumes in this advanced textbook present results, proof methods, and translations of motivational and philosophical considerations to formal constructions. In the associated Vol. I the author explains preferential structures and abstract size. In this Vol. II he presents chapters on theory revision and sums, defeasible inheritance theory, interpolation, neighbourhood semantics and deontic logic, abstract independence, and various aspects of nonmonotonic and other logics. In both volumes the text contains many exercises and some solutions, and the author (...) limits the discussion of motivation and general context throughout, offering this only when it aids understanding of the formal material, in particular to illustrate the path from intuition to formalisation. Together these books are a suitable compendium for graduate students and researchers in the area of computer science and mathematical logic. (shrink)

In modal subordination, a modal sentence is interpreted relative to a hypothetical scenario introduced in an earlier sentence. In this paper, I argue that this phenomenon reflects the fact that the interpretation of modals is an ANAPHORIC process. Modal morphemes introduce sets of possible worlds, representing alternative hypothetical scenarios, as entities into the discourse model. Their interpretation depends on evoking sets of worlds recording described and reference scenarios, and relating such sets to one another using familiar notions of restricted, (...) class='Hi'>preferential quantification. This proposal relies on an extended model of environments in dynamic semantics to keep track of associations between possible worlds and ordinary individuals; it assumes that modal meanings and other lexical meanings encapsulate quantification over possible worlds. These two innovations are required in order for modals to refer to sets of possible worlds directly as static objects in place of the inherently dynamic objects—quite different from the referents of pronouns and tenses—used in previous accounts. The simpler proposal that results offers better empirical coverage and suggests a new parallel between modal and temporal interpretation. (shrink)

Clause-embedding predicates come in three major varieties: responsive predicates are compatible with both declarative and interrogative complements; rogative predicates are only compatible with interrogative complements; and anti-rogative predicates are only compatible with declarative complements. It has been suggested that these selectional properties are at least partly semantic in nature. In particular, it has been proposed that the anti-rogativity of neg-raising predicates like believe comes from the triviality in meaning that would arise with interrogative complements. This paper puts forward a similar (...) semantic explanation for non-veridical preferential predicates such as hope, which are anti-rogative, unlike their veridical counterparts such as be happy, which are responsive. (shrink)

Agents act on the basis of their beliefs and these beliefs change as they interact with other agents. In this book the authors propose and explain general logical tools for handling change. These tools include preferential reasoning, theory revision, and reasoning in inheritance systems, and the authors use these tools to examine nonmonotonic logic, deontic logic, counterfactuals, modal logic, intuitionistic logic, and temporal logic. This book will be of benefit to researchers engaged with artificial intelligence, and in particular agents, (...) multiagent systems and nonmonotonic logic. (shrink)

This paper has two goals. First, we develop frameworks for logical systems which are able to reflect not only non-monotonic patterns of reasoning, but also paraconsistent reasoning. Our second goal is to have a better understanding of the conditions that a useful relation for nonmonotonic reasoning should satisfy. For this we consider a sequence of generalizations of the pioneering works of Gabbay, Kraus, Lehmann, Magidor and Makinson. These generalizations allow the use of monotonic nonclassical logics as the underlying logic upon (...) which nonmonotonic reasoning may be based. Our sequence of frameworks culminates in what we call plausible, nonmonotonic, multiple-conclusion consequences relations . Our study yields intuitive justifications for conditions that have been proposed in previous frameworks and also clarifies the connections among some of these systems. In addition, we present a general method for constructing plausible nonmonotonic relations. This method is based on a multiple-valued semantics, and on Shoham's idea of preferential models. (shrink)

This article is about conditionalized modal statements whose antecedents concern a preferential attitude of an agent. The focus is on anankastic conditionals or, as they are known in the philosophical literature, hypothetical imperatives. We present a linguistically-motivated analysis of anankastic and related conditionals and use it to address challenges for semantic theories of natural language conditionals motivated by certain philosophical concerns about practical reasoning and the requirements of rationality.

After the seminal work of Kraus, Lehmann and Magidor (formally known as the KLM approach) on conditionals and preferential models, many aspects of defeasibility in more complex formalisms have been studied in recent years. Examples of these aspects are the notion of typicality in description logic and defeasible necessity in modal logic. We discuss a new aspect of defeasibility that can be expressed in the case of temporal logic, which is the normality in an execution. In this contribution, we (...) take Linear Temporal Logic (LTL) as case study for this defeasible aspect. LTL has found extensive applications in Computer Science and Artificial Intelligence, notably as a formal framework for representing and verifying computer systems that vary over time. However, some systems may presents exceptions at some innocuous time points where they can be tolerated, or conversely, exceptions at other crucial time points where they need to be addressed. In order to ensure the reliability of such systems, we study a preferential extension of LTL, called defeasible linear temporal logic (LTL ~ ). In the first part of this paper, we show how semantics of KLM's preferential models can be integrated with LTL. We also discuss the addition of non-monotonic temporal operators as a way to formalise defeasible properties of these systems. The second part of this paper is a study of the satisfiability problem of LTL ~ sentences. Based on Sistla and Clarke's work on the complexity of the classical LTL language, we show the bounded-model property of two fragments of LTL ~ language. Moreover, we provide a procedure to check the satisfiability of sentences in both of these fragments. (shrink)

This article addresses Thomas More's use of an especially complex Latin predicate, fio, as a means of examining the degree of classicism in this aspect of his writing. To this end, the main lexical-semantic and syntactic features of the verb in Classical Latin are presented, and a comparative review is made of More's use of the predicate—and also its use in texts contemporaneous to More, as well as in Late and Medieval Latin—in both prose and poetry. The analysis shows that (...) he works within a general framework of classicism, although he introduces some of his own idiosyncrasies, these essentially relating to the meaning of the verb that he employs in a preferential way and to the variety of verbal forms that occur in his poetic text. (shrink)

The paper presents a non-monotonic inference relation on a language containing a conditional that satisfies the Ramsey Test. The logic is a weakening of classical logic and preserves many of the ‘paradoxes of implication’ associated with the material implication. It is argued, however, that once one makes the proper distinction between supposing that something is the case and accepting that it is the case, these ‘paradoxes’ cease to be counterintuitive. A representation theorem is provided where conditionals are given a non-bivalent (...) semantics and epistemic states are represented via preferential models. (shrink)

We prove some embedding theorems for classical conditional logic, covering 'finitely cumulative' logics, 'preferential' logics and what we call 'semi-monotonic' logics. Technical tools called 'partial frames' and 'frame morphisms' in the context of neighborhood semantics are used in the proof.

This essay offers a defence of the non-cognitivist approach to the interpretation of moral judgments as disguised imperatives corresponding to social rules. It addresses the body of criticism that faced R. M. Hare, and that currently faces moral anti-realists, on two levels, by providing a full semantic analysis of evaluative judgments and by arguing that anti-realism is compatible with moral aspiration despite the non-existence of obligations as the externalist imagines them. A moral judgment consists of separate descriptive and prescriptive components (...) and is to be understood as a declarative statement prefaced by an 'ideality operator'. Moral beliefs are genuinely representational, but their truth conditions can only be stated with reference to imaginary ideal worlds. Moral judgments are neither confirmed nor verified, but alternative moral positions are preferentially endorsed and adopted by individual agents on the basis of their perceived all-things-considered optimality. High aspiration moralities are normally very costly to agents in terms of their prudential and aesthetic interests, but they are theoretically as eligible as the adoption of other, less demanding sets of rules. (shrink)