Nonmonotonic reasoning in its broadest sense is reasoning to conclusions on the basis of incomplete information. Given more information, previously drawn inferences may be retracted. Commonsense reasoning has a nonmonotonic component; it has been argued that almost all commonsense inferences are of this sort. From the end of the 1980s to the present there has been an explosion in research in nonmonotonic reasoning. It is now possible to understand more clearly the properties of the major formalisms from (...) a metatheoretical point of view, the relationships among the formalisms and their connection to independently developed proof methods. The goal of this monograph is to make this understanding more accessible. (shrink)

Reasoning almost always occurs in the face of incomplete information. Such reasoning is nonmonotonic in the sense that conclusions drawn may later be withdrawn when additional information is obtained. There is an active literature on the problem of modeling such nonmonotonic reasoning, yet no category of method-let alone a single method-has been broadly accepted as the right approach. This paper introduces a new method, called sweeping presumptions, for modeling nonmonotonic reasoning. The main goal of the paper is (...) to provide an example-driven, nontechnical introduction to the method of sweeping presumptions, and thereby to make it plausible that sweeping presumptions can usefully be applied to the problems of nonmonotonic reasoning. The paper discusses a representative sample of examples that have appeared in the literature on nonmonotonic reasoning, and discusses them from the point of view of sweeping presumptions. (shrink)

Nonmonotonic logics allow—contrary to classical (monotone) logics— for withdrawing conclusions in the light of new evidence. Nonmonotonic reasoning is often claimed to mimic human common sense reasoning. Only a few studies, though, have investigated this claim empirically. system p is a central, broadly accepted nonmonotonic reasoning system that proposes basic rationality postulates. We previously investigated empirically a probabilistic interpretation of three selected rules of system p. We found a relatively good agreement of human reasoning and principles of (...) nonmonotonic reasoning according to the coherence interpretation of system p. This study reports an experiment on the cautious monotonicity Rule and its “incautious” counterpart that is not contained in system p, namely the monotonicity Rule. In accordance with our previous results, the data suggest that people reason nonmonotonically: the subjects in the cautious monotonicity condition infer significantly tighter intervals close to the coherence interpretation of system p compared with the subjects in the incautious monotonicity condition where rather wide (and hence non-informative) intervals are inferred. (shrink)

I am presenting a sequent calculus that extends a nonmonotonic consequence relation over an atomic language to a logically complex language. The system is in line with two guiding philosophical ideas: (i) logical inferentialism and (ii) logical expressivism. The extension defined by the sequent rules is conservative. The conditional tracks the consequence relation and negation tracks incoherence. Besides the ordinary propositional connectives, the sequent calculus introduces a new kind of modal operator that marks implications that hold monotonically. Transitivity fails, (...) but for good reasons. Intuitionism and classical logic can easily be recovered from the system. (shrink)

Nonmonotonic logics were created as an abstraction of some types of common sense reasoning, analogous to the way classical logic serves to formalize ideal reasoning about mathematical objects. These logics are nonmonotonic in the sense that enlarging the set of axioms does not necessarily imply an enlargement of the set of formulas deducible from these axioms. Such situations arise naturally, for example, in the use of information of different degrees of reliability. This book emphasizes basic concepts by outlining (...) connections between different formalisms of nonmonotonic logic, and gives a coherent presentation of recent research results and reasoning techniques. It provides a self-contained state-of-the-art survey of the area addressing researchers in AI lo. (shrink)

Using Gärdenfors's notion of epistemic entrenchment, we develop the semantics of a logic which accounts for the following points. It explains why we may generally infer `If ~A then B´ if all we know is AvB while must not generally infer `If ~A then B´ if all we know is {AvB, A}. More generally, it explains the nonmonotonic nature of the consequence relation governing languages which contain conditionals, and it explains how we can deduce conditionals from premise sets without (...) conditionals. Depending on the language at hand, our logic provides different ways of keeping the Ramsey test and getting round the Gärdenfors triviality theorem. I argue that consistent additions of new items of belief are not to be performed by transitions to logical expansions. (shrink)

An graduate level introduction to nonmonotonic reasoning, emphasizing structures and spirit common to different formulations, with exercises.

The properties of monotonic inference systems and the properties of their theories are strongly linked. These links, however, are much weaker in nonmonotonic inference systems. In this paper we introduce the notion of anaxiomatic variety for a theory and show how this notion, instead of the notion of a theory, can be used for the syntactic and semantic analysis of nonmonotonic inferences.

In this book the author gives a broad overview of different areas of research in nonmonotonic reasoning, and presents some new results and ideas based on his research. The guiding principles are: clarification of the different research activities in the area, which have sometimes been undertaken independently of each other; and appreciation of the fact that these research activities often represent different means to the same ends, namely sound theoretical foundations and efficient computation. The book begins with a discussion (...) of the various types of nonmonotonic reasoning, their applications and their logics. Theorem proving techniques for these logics are also described. The following chapters deal with formulations of nonmonotonic inheritance, and nonmonotonic reasoning based on nonmonotonic rules. The final chapter discusses the achievements in the field in the light of the Yale shooting example. The book will be welcomed by researchers in theoretical computer science and artificial intelligence. (shrink)

Machine learning and nonmonotonic reasoning are closely related, both concerned with making plausible as well as certain inferences based on available data. In this document a brief overview of different approaches to nonmonotonic reasoning is presented, and it is shown how the concept of argumentation systems arises. The relationship with machine learning work is also discussed. The document aims to highlight the links between nonmonotonic reasoning, argumentation and machine learning and as a result propose some potentially useful (...) directions for new research. (shrink)

Three studies of human nonmonotonic reasoning are described. The results show that people find such reasoning quite difficult, although being given problems with known subclass-superclass relationships is helpful. The results also show that recognizing differences in the logical strengths of arguments is important for the nonmonotonic problems studied. For some of these problems, specificity – which is traditionally considered paramount in drawing appropriate conclusions – was irrelevant and so should have lead to a “can’t tell” response; however, people (...) could give rational conclusions based on differences in the logical consequences of arguments. The same strategy also works for problems where specificity is relevant, suggesting that in fact specificity is not paramount. Finally, results showed that subjects’ success at responding appropriately to nonmonotonic problems involving conflict relies heavily on the ability to appreciate differences in the logical strength of simple, non-conflicting, statements. (shrink)

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. (shrink)

Conclusions reached using common sense reasoning from a set of premises are often subsequently revised when additional premises are added. Because we do not always accept previous conclusions in light of subsequent information, common sense reasoning is said to be nonmonotonic. But in the standard formal systems usually studied by logicians, if a conclusion follows from a set of premises, that same conclusion still follows no matter how the premise set is augmented; that is, the consequence relations of standard (...) logics are monotonic. Much recent research in AI has been devoted to the attempt to develop nonmonotonic logics. After some motivational material, we give four formal proofs that there can be no nonmonotonic consequence relation that is characterized by universal constraints on rational belief structures. In other words, a nonmonotonic consequence relation that corresponds to universal principles of rational belief is impossible. We show that the nonmonotonicity of common sense reasoning is a function of the way we use logic, not a function of the logic we use. We give several examples of how nonmonotonic reasoning systems may be based on monotonic logics. (shrink)

cuted actions. It has been applied to several challenge problems in the theory of commonsense knowledge. We study the relationship between this formalism and other work on nonmonotonic reasoning and knowl-.

Reviews the connections between different kinds of nonmonotonic logic and the general idea of varying degrees of belief.

I’ll describe a range of systems for nonmonotonic conditionals that behave like conditional probabilities above a threshold. The rules that govern each system are probabilistically sound in that each rule holds when the conditionals are interpreted as conditional probabilities above a threshold level specific to that system. The well-known preferential and rational consequence relations turn out to be special cases in which the threshold level is 1. I’ll describe systems that employ weaker rules appropriate to thresholds lower than 1, (...) and compare them to these two standard systems. (shrink)

It is suggested that taking into account considerations that traditionally fall within the scope of computer science in general, and artificial intelligence in particular, sheds new light on the subject of causation. It is argued that adopting causal notions con be viewed as filling a computational need: They allow reasoning with incomplete information, facilitate economical representations, and afford relatively efficient methods for reasoning about those representations. Specifically, it is proposed that causal reasoning is intimately bound to nonmonotonic reasoning. An (...) account of causation is offered that relies upon this connection, and compares this proposal to previous accounts within philosophy and artificial intelligence. (shrink)

Parametric logic is a framework that generalises classical first-order logic. A generalised notion of logical consequence—a form of preferential entailment based on a closed world assumption—is defined as a function of some parameters. A concept of possible knowledge base—the counterpart to the consistent theories of first-order logic—is introduced. The notion of compactness is weakened. The degree of weakening is quantified by a nonnull ordinal—the larger the ordinal, the more significant the weakening. For every possible knowledge base T, a hierarchy of (...) sentences that are generalised logical consequences of T is built. The first layer of the hierarchies corresponds to sentences that can be obtained by a deductive inference, characterised by the compactness property. The second layer of the hierarchies corresponds to sentences that can be obtained by an inductive inference, characterised by the property of weak compactness quantified by 1. Weaker forms of compactness—quantified by nonnull ordinals—determine higher layers in the hierarchies, corresponding to more complex inferences. The naturalness of the hierarchies built over the possible knowledge bases is attested by fundamental connections with notions from Learning theory and from topology. The naturalness of the hierarchies built over the possible knowledge bases is attested by fundamental connections with notions from Learning theory—classification in the limit, with or without a bounded number of mind changes—and from topology—in reference to the Borel and the difference hierarchies. In this paper, we introduce the key model-theoretic aspects of Parametric logic, justify the concept of the knowledge base, define the hierarchies of generalised logical consequences and illustrate their relevance to Nonmonotonic reasoning. More specifically, we show that the degree of nonmonotonicity that is required to infer a sentence can be characterised by the least nonnull ordinal that quantifies the weakening of compactness used to locate the inferred sentence in the hierarchies. (shrink)

Oaksford & Chater (O&C) advocate Bayesian probability as a way to deal formally with the pervasive nonmonotonicity of common sense reasoning. We show that some forms of nonmonotonicity cannot be treated by Bayesian methods.

In Gärdenfors and Makinson :197–245, 1994) and Gärdenfors it was shown that it is possible to model nonmonotonic inference using a classical consequence relation plus an expectation-based ordering of formulas. In this article, we argue that this framework can be significantly enriched by adopting a conceptual spaces-based analysis of the role of expectations in reasoning. In particular, we show that this can solve various epistemological issues that surround nonmonotonic and default logics. We propose some formal criteria for constructing (...) and updating expectation orderings based on conceptual spaces, and we explain how to apply them to nonmonotonic reasoning about objects and properties. (shrink)

Nonmonotonic reasoning is a theory of the rational use of assumptions. We describe the relations between NMR and Logic, and two main paradigms of NMR, preferential and explanatory one.

A. Tarski [21] proposed the study of infinitary consequence operations as the central topic of mathematical logic. He considered monotonicity to be a property of all such operations. In this paper, we weaken the monotonicity requirement and consider more general operations, inference operations. These operations describe the nonmonotonic logics both humans and machines seem to be using when infering dofeasible information from incomplete knowledge. We single out a number of interesting families of inference operations. This study of infinitary inference (...) operations is inspired by the results of [11] on finitary nonmonotonic operations, but this paper is self-contained. (shrink)

In this paper first order theories for nonmonotone inductive definitions are introduced, and a proof-theoretic analysis for such theories based on combined operator forms a la Richter with recursively inaccessible and Mahlo closure ordinals is given.

An extended review of what is known about the formal behaviour of nonmonotonic inference operations, including those generated by the principal systems in the artificial intelligence literature. Directed towards computer scientists and others with some background in logic.

Nonmonotonic conditionals (A |∼ B) are formalizations of common sense expressions of the form “if A, normally B”. The nonmonotonic conditional is interpreted by a “high” coherent conditional probability, P(B|A) > .5. Two important properties are closely related to the nonmonotonic conditional: First, A |∼ B allows for exceptions. Second, the rules of the nonmonotonic system p guiding A |∼ B allow for withdrawing conclusions in the light of new premises. This study reports a series of (...) three experiments on reasoning with inference rules about nonmonotonic conditionals in the framework of coherence. We investigated the cut, and the right weakening rule of system p. As a critical condition, we investigated basic monotonic properties of classical (monotone) logic, namely monotonicity, transitivity, and contraposition. The results suggest that people reason nonmonotonically rather than monotonically. We propose nonmonotonic reasoning as a competence model of human reasoning. (shrink)

Motivated by semantic inferentialism and logical expressivism proposed by Robert Brandom, in this paper, I submit a nonmonotonic modal relevant sequent calculus equipped with special operators, □ and R. The base level of this calculus consists of two different types of atomic axioms: material and relevant. The material base contains, along with all the flat atomic sequents (e.g., Γ0, p |~0 p), some non-flat, defeasible atomic sequents (e.g., Γ0, p |~0 q); whereas the relevant base consists of the local (...) region of such a material base that is sensitive to relevance. The rules of the calculus uniquely and conservatively extend these two types of nonmonotonic bases into logically complex material/relevant consequence relations and incoherence properties, while preserving Containment in the material base and Reflexivity in the relevant base. The material extension is supra-intuitionistic, whereas the relevant extension is stronger than a logic slightly weaker than R. The relevant extension also avoids the fallacies of relevance. Although the extended material consequence relation is defeasible and insensitive to relevance, it has local regions of indefeasibility and relevance (the latter of which is marked by the relevant extension). The newly introduced operators, □ and R, codify these local regions within the same extended material consequence relation. (shrink)

The problems we deal with concern reasoning about incomplete knowledge. Knowledge is understood as ability of an ideal rational agent to make decisions about pieces of information. The formalisms we are particularly interested in are Moore's autoepistemic logic (AEL) and its variant, the logic of acceptance and rejection (AEL2). It is well-known that AEL may be seen as the nonmonotonic KD45 modal logic. The aim is to give an appropriate modal formalization for AEL2.

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 possible explicit defeaters, as in the Suppression Task, and 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 regarding the inference system, the knowledge base, and possible assumed exceptions for the rule. (shrink)

A. Tarski [22] proposed the study of infinitary consequence operations as the central topic of mathematical logic. He considered monotonicity to be a property of all such operations. In this paper, we weaken the monotonicity requirement and consider more general operations, inference operations. These operations describe the nonmonotonic logics both humans and machines seem to be using when infering defeasible information from incomplete knowledge. We single out a number of interesting families of inference operations. This study of infinitary inference (...) operations is inspired by the results of [12] on nonmonotonic inference relations, and relies on some of the definitions found there. (shrink)

Nonmonotonic reasoning is often claimed to mimic human common sense reasoning. Only a few studies, though, investigated this claim empirically. In the present paper four psychological experiments are reported, that investigate three rules of system p, namely the and, the left logical equivalence, and the or rule. The actual inferences of the subjects are compared with the coherent normative upper and lower probability bounds derived from a non-infinitesimal probability semantics of system p. We found a relatively good agreement of (...) human reasoning and principles of nonmonotonic reasoning according to the coherence interpretation of system p. Contrary to the results reported in the “heuristics and biases” tradition, the subjects committed relatively few upper bound violations (conjunction fallacies). More lower than upper bound violations were observed. When the premises were presented in terms of intervals higher mean lower bounds were observed as when the premises were presented in terms of point percentages. (shrink)

Nonmonotonic reasoning is often claimed to mimic human common sense reasoning. Only a few studies, though, have investigated this claim empirically. We report four experiments which investigate three rules of SYSTEMP, namely the AND, the LEFT LOGICAL EQUIVALENCE, and the OR rule. The actual inferences of the subjects are compared with the coherent normative upper and lower probability bounds derived from a non-infinitesimal probability semantics of SYSTEM P. We found a relatively good agreement of human reasoning and principles of (...) nonmonotonic reasoning. Contrary to the results reported in the ‘heuristics and biases’ tradition, the subjects committed relatively few upper bound violations (conjunction fallacies). (shrink)

The paper presents an exhaustive menu of nonmonotonic logics. The options are individuated in terms of the principles they reject. I locate, e.g., cumulative logics and relevance logics on this menu. I highlight some frequently neglected options, and I argue that these neglected options are particularly attractive for inferentialists.

This proceedings volume contains a selection of revised and extended papers presented at the Second International Workshop on Nonmonotonic and InductiveLogic, NIL '91, which took place at Reinhardsbrunn Castle, December 2-6, 1991. The volume opens with an extended version of a tutorial on nonmonotonic logic by G. Brewka, J. Dix, and K. Konolige. Fifteen selected papers follow, on a variety of topics. The majority of papers belong either to the area of nonmonotonic reasoning or to the field (...) of inductive inference, but some papers integrate research from both areas. The first workshop in this series was held at the University of Karlsruhe in December 1990 and its proceedings were published as Lecture Notes in Artificial Intelligence Volume 543. The series of workshops was made possible by financial support from Volkswagen Stiftung, Hannover. This workshop was also supported by IBM Deutschland GmbH and Siemens AG. (shrink)

Interpretation is the process whereby a hearer reasons to an interpretation of a speaker's discourse. The hearer normally adopts a credulous attitude to the discourse, at least for the purposes of interpreting it. That is to say the hearer tries to accommodate the truth of all the speaker's utterances in deriving an intended model. We present a nonmonotonic logical model of this process which defines unique minimal preferred models and efficiently simulates a kind of closed‐world reasoning of particular interest (...) for human cognition. Byrne's “suppression” data (Byrne, 1989) are used to illustrate how variants on this logic can capture and motivate subtly different interpretative stances which different subjects adopt, thus indicating where more fine‐grained empirical data are required to understand what subjects are doing in this task. We then show that this logical competence model can be implemented in spreading activation network models. A one pass process interprets the textual input by constructing a network which then computes minimal preferred models for (3‐valued) valuations of the set of propositions of the text. The neural implementation distinguishes easy forward reasoning from more complex backward reasoning in a way that may be useful in explaining directionality in human reasoning. (shrink)

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. (shrink)

The goal of a logic is to define a consequence relation between a set of formulas Γ and, in most cases, an individual formula A. This definition generally takes one of two forms. From a proof theoretic standpoint, A is said to be a consequence of Γ whenever there is a deduction of A from the set Γ, viewed as a set of premises; from a model theoretic standpoint, A is said to be a consequence of Γ whenever A holds (...) in every model that satisfies each formula in Γ. (shrink)

Examines the link between nonmonotonic inference relations and theory revision operations, focusing on the correspondence between abstract properties which each may satisfy.

Recently, John Bell has proposed that a specific conditional logic, C, be considered as a serious candidate for formally representing and faithfully capturing various (possibly all) formalized notions of nonmonotonic inference. The purpose of the present paper is to develop evaluative criteria for critically assessing such claims. Inference patterns are described in terms of the presence or absence of residual classical monotonicity and intrinsic nonmonotonicity. The concept of a faithful representation is then developed for a formalism purported to encode (...) a pattern of nonmonotonic inference already captured by another. In the main body of the paper these evaluative criteria are applied to assess (negatively) whether C or any conditional logic provides a faithful representation for nonmonotonic patterns of inference captured by inference operators and relations modeling the dynamics of belief change. (shrink)

Nonmonotonic consequence is the subject of a vast literature, but the idea of a nonmonotonic counterpart of logical inconsistency—the idea of a defeasible property representing internal conflict of an inductive or evidential nature—has been entirely neglected. After considering and dismissing two possible analyses relating nonmonotonic consequence and a nonmonotonic counterpart of logical inconsistency, this paper offers a set of postulates for nonmonotonic inconsistency, an analysis of nonmonotonic inconsistency in terms of nonmonotonic consequence, and (...) a series of results showing that nonmonotonic inconsistency conforms to these postulates given the analysis of nonmonotonic inconsistency presented here and certain postulates for nonmonotonic consequence. The results presented here establish the interest of certain previously undiscussed postulates of nonmonotonic consequence. These results also show that nonmonotonicity, which has never seemed useful in the formulation of general principles governing nonmonotonic reasoning, is relevant to the positive characterization of nonmonotonic inference after all. (shrink)

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 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. (shrink)

From a philosophical standpoint, the work presented here is based on van Fraassen [26]. The bulk of that paper is organized around a series of arguments against the assumption, built into standard deontic logic, that moral dilemmas are impossible; and van Fraassen only briefly sketches his alternative approach. His paper ends with the conclusion that “the problem of possibly irresolvable moral conflict reveals serious flaws in the philosophical and semantic foundations of ‘orthodox’ deontic logic, but also suggests a rich set (...) of new problems and methods for such logic.” My goal has been to suggest that some of these methods might be found in current research on nonmonotonic reasoning, and that some of the problems may have been confronted there as well.I have shown that nonmonotonic logics provide a natural framework for reasoning about moral dilemmas, perhaps even more useful than the ordinary modal framework, and that the issues surrounding the treatment of exceptional information within these logics run parallel to some of the problems posed by conditional oughts. However, there is also another way in which deontic logic might benefit from a connection to nonmonotonic reasoning. A familiar criticism among ethicists of work in deontic logic is that it is too abstract, and too far removed from the kind of problems confronted by real agents in moral deliberation. It must be said that similar criticisms of abstraction and irrelevance are often lodged against work in nonmonotonic reasoning by more practically minded researchers in artificial intelligence; but here, at least, the criticisms are taken seriously. Nonmonotonic logic aims at a qualitative account of commonsense reasoning, which can be used to relate planning and action to defeasible goals and beliefs; and at least some of the theories developed in this area have been tested in realistic situations. By linking the subject of deontic logic to this research, it may be possible also to relate the idealized study of moral reasoning typical of the field to a more robust treatment of practical deliberation. (shrink)