We study the problem of embedding Halpern and Moses's modal logic of minimal knowledge states into two families of modal formalism for nonmonotonic reasoning, McDermott and Doyle's nonmonotonic modal logics and ground nonmonotonic modal logics. First, we prove that Halpern and Moses's logic can be embedded into all ground logics; moreover, the translation employed allows for establishing a lower bound (3p) for the problem of skeptical reasoning in all ground logics. Then, we show a translation of Halpern and Moses's logic (...) into a significant subset of McDermott and Doyle's formalisms. Such a translation both indicates the ability of Halpern and Moses's logic of expressing minimal knowledge states in a more compact way than McDermott and Doyle's logics, and allows for a comparison of the epistemological properties of such nonmonotonic modal formalisms. (shrink)

Argumentation represents a way of reasoning over a knowledge base containing possibly incomplete and/or inconsistent information, to obtain useful conclusions. As a reasoning mechanism, the way an argumentation reasoning engine reaches these conclusions resembles the cognitive process that humans follow to analyze their beliefs; thus, unlike other computationally reasoning systems, argumentation offers an intellectually friendly alternative to other defeasible reasoning systems. LogicProgrammingisacomputationalparadigmthathasproducedcompu- tationallyattractivesystemswithremarkablesuccessinmanyapplications. Merging ideas from both areas, Defeasible Logic Programming offers a computational reasoning system that uses an argumentation engine (...) to obtain answers from a knowledge base represented using a logic programming language extended with defeasible rules. This combination of ideas brings about a computationally effective system together with a human-like reasoning model facilitating its use in applications. (shrink)

We present a Hilbert-style axiomatization of a recently introduced logic, called G'3 G'3 is based on a 3-valued semantics. We prove a soundness and completeness theorem. The replacement theorem holds in G'3. As it has already been shown in previous work, G'3 can express some non-monotonic semantics. We prove that G'3can define the same class of functions as Lukasiewicz 3 valued logic. Moreover, we identify some normal forms for this logic.

Lifschitz introduced the notion of defining extensions of predicate default theories not as absolute, but relative to a specified domain. We look specifically at default theories over a countable domain and show the set of default theories which possess an ω -extension is Σ 2 1 -complete. That the set is in Σ 2 1 is shown by writing a nearly circumscriptive formula whose ω -models correspond to the ω -extensions of a given default theory; similarly, Σ 2 1 -hardness (...) is established by a method for translating formulas into default theories in such a way that ω -models of the circumscriptive formula correspond to ω -extensions of the default theory. (shrink)

A number of nonmonotonic reasoning formalisms have been introduced to model the set of beliefs of an agent. These include the extensions of a default logic, the stable models of a general logic program, and the extensions of a truth maintenance system among others. In [13] and [16], the authors introduced nonmonotomic rule systems as a nonlogical generalization of all essential features of such formulisms so that theorems applying to all could be proven once and for all. In this paper, (...) we extend Rieter's normal default theories, which have a number of the nice properties which make them a desirable context for belief revision, to the setting of nonmonotonic rule systems. Reiter defined a default theory to be normal if all the rules of the default theory satisfied a simple syntatic condition. However, this simple syntatic condition has no obvious analogue in the setting of nonmonotonic rule systems. Nevertheless, an analysis of the proofs of the main results on normal default theories reveals that the proofs do not rely on the particular syntactic form of the rules but rather on the fact that all rules have a certain consistency property. This led us to extend the notion of normal default theories with respect to a general consistency property. This extended notion of normal default theories, which we call Forward Chaining normal , is easily lifted to nonmonotomic rule systems and hence applies to general logic programs and truth maintenance. (shrink)

We investigate expressiveness and definability issues with respect to minimal models, particularly in the scope of Circumscription. First, we give a proof of the failure of the Löwenheim-Skolem Theorem for Circumscription. Then we show that, if the class of P; Z-minimal models of a first-order sentence is Δ-elementary, then it is elementary. That is, whenever the circumscription of a first-order sentence is equivalent to a first-order theory, then it is equivalent to a finitely axiomatizable one. This means that classes of (...) models of circumscribed theories are either elementary or not Δ-elementary. Finally, using the previous result, we prove that, whenever a relation Pi is defined in the class of P; Z-minimal models of a first-order sentence Φ and whenever such class of P; Z-minimal models is Δ-elementary, then there is an explicit definition ψ for Pi such that the class of P; Z-minimal models of Φ is the class of models of Φ ∧ ψ. In order words, the circumscription of P in Φ with Z varied can be replaced by Φ plus this explicit definition ψ for Pi. (shrink)

The paper provides a preliminary study of algebraic semantics for modal and superintuitionistic non-monotonic logics. The main question answered is: how can non-monotonic inference be understood algebraically?

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)

Much previous work in artificial intelligence has neglected representing time in all its complexity. In particular, it has neglected continuous change and the indeterminacy of the future. To rectify this, I have developed a first‐order temporal logic, in which it is possible to name and prove things about facts, events, plans, and world histories. In particular, the logic provides analyses of causality, continuous change in quantities, the persistence of facts (the frame problem), and the relationship between tasks and actions. It (...) may be possible to implement a temporal‐inference machine based on this logic, which keeps track of several “maps” of a time line, one per possible history. (shrink)

ABSTRACT What characterizes human reasoning is the ability of dealing with incomplete information. Incomplete logic is developed for modeling incomplete knowledge. The most distinctive feature of incomplete logic is its semantics. This is an alternative presentation of partial semantics. In this paper, we will introduce the general notion of incomplete logic (ICL), compare it with partial logic, and give the resolution method for it. We will also show how ICL can be applied to nonmonotonic reasoning. We define nonmonotonic derivation as (...) monotonic derivation in ICL from the database and some consistent assumptions. The mechanism of ICL makes it easy to assert the consistency of an assumption without asserting the assumption itself. (shrink)

The applicability of Bayesian conditionalization in setting one’s posterior probability for a proposition, α, is limited to cases where the value of a corresponding prior probability, PPRI(α|∧E), is available, where ∧E represents one’s complete body of evidence. In order to extend probability updating to cases where the prior probabilities needed for Bayesian conditionalization are unavailable, I introduce an inference schema, defeasible conditionalization, which allows one to update one’s personal probability in a proposition by conditioning on a proposition that represents a (...) proper subset of one’s complete body of evidence. While defeasible conditionalization has wider applicability than standard Bayesian conditionalization (since it may be used when the value of a relevant prior probability, PPRI(α|∧E), is unavailable), there are circumstances under which some instances of defeasible conditionalization are unreasonable. To address this difficulty, I outline the conditions under which instances of defeasible conditionalization are defeated. To conclude the article, I suggest that the prescriptions of direct inference and statistical induction can be encoded within the proposed system of probability updating, by the selection of intuitively reasonable prior probabilities. (shrink)

In this paper we discuss Gabbay's idea of basing nonmonotonic deduction on semantic consequence in intuitionistic logic extended by a consistency operator and Turner's suggestion of replacing the intuitionistic base system by Kleene's three-valued logic. It is shown that a certain counterintuitive feature of these approaches can be avoided by using Nelson's constructive logic N instead of intuitionistic logic or Kleene's system. Moreover, in N a more general notion of consistency can be defined and nonmonotonic deduction can thus be based (...) on a logical system satisfying the Deduction Theorem. (shrink)

inheritance reasoning in semantic networks allowing for multiple inheritance with exceptions. The approach leads to a definition of iaheritance that is..

We discuss O&C's probabilistic approach from a probability logical point of view. Specifically, we comment on subjective probability, the indispensability of logic, the Ramsey test, the consequence relation, human nonmonotonic reasoning, intervals, generalized quantifiers, and rational analysis.

Default reasoning occurs when the available information does not deductively guarantee the truth of the conclusion; and the conclusion is nonetheless correctly arrived at. The formalisms that have been developed in Artificial Intelligence to capture this mode of reasoning have suffered from a lack of agreement as to which non-monotonic inferences should be considered correct; and so Lifschitz 1989 produced a set of “Nonmonotonic Benchmark Problems” which all future formalisms are supposed to honor. The present work investigates the extent to (...) which humans follow the prescriptions set out in these Benchmark Problems. (shrink)

There has long been a history of studies investigating how people (“ordinary people”) perform on tasks that involve deductive reasoning. The upshot of these studies is that people characteristically perform some deductive tasks well but others badly. For instance, studies show that people will typically perform MP (“modus ponens”: from ‘If A then B’ and ‘A’, infer ‘B’) and bi-conditional MP (from: ‘A if and only if B’ and ‘A’, infer ‘B’) correctly when invited to make the inference and additionally (...) can discover of their own accord when such inferences are appropriate. On the other hand, the same studies show that people typically perform MT (“modus tollens”: from ‘If A then B’ and ‘not-B’, infer ‘not-A’) and biconditional MT badly. They not only do not recognize when it is appropriate to draw such inferences, but also they will balk at doing them even when they are told that they can make it. Related to these shortcomings seems to be the inability of people to understand that contrapositives are equivalent (that ‘If A then B’ is equivalent to ‘If not-B then not-A’). [Studies of people’s deductive inference-drawing abilities have a long history, involving many studies in the early 20th century concerning Aristotelian syllogisms. But the current spate of studies draws much of its impetus from Wason (Wason, 1968; see also Wason & Johnson-Laird, 1972). ] The general conclusion seems to be that there are specific areas where “ordinary people” do not perform very logically. This conclusion will not come as a surprise to teachers of elementary logic, who have long thought that the majority of “ordinary people” are inherently illogical and need deep and forceful schooling in order to overcome this flaw. (shrink)

This is a position paper concerning the role of empirical studies of human default reasoning in the formalization of AI theories of default reasoning. We note that AI motivates its theoretical enterprise by reference to human skill at default reasoning, but that the actual research does not make any use of this sort of information and instead relies on intuitions of individual investigators. We discuss two reasons theorists might not consider human performance relevant to formalizing default reasoning: (a) that intuitions (...) are sufficient to describe a model, and (b) that human performance in this arena is irrelevant to a competence model of the phenomenon. We provide arguments against both these reasons. We then bring forward three further considerations against the use of intuitions in this arena: (a) it leads to an unawareness of predicate ambiguity, (b) it presumes an understanding of ordinary language statements of typicality, and (c) it is similar to discredited views in other fields. We advocate empirical investigation of the range of human phenomena that intuitively embody default reasoning. Gathering such information would provide data with which to generate formal default theories and against which to test the claims of proposed theories. Our position is that such data are the very phenomena that default theories are supposed to explain. (shrink)

This essay seeks to give a contractarian foundation to the concept of Corporate Social Responsibility (CSR), meant as an extended model of corporate governance of the firm. Whereas, justificatory issues have been discussed in a related paper (Sacconi, L.: 2006b, this journal), in this essay I focus on the implementation of and compliance with this normative model. The theory of reputation games, with reference to the basic game of trust, is introduced in order to make sense of self-regulation as a (...) way to implement the social contract on the multi-fiduciary model of corporate governance. This affords understanding of why self-regulation, meant as mere recourse to a long-run strategy in a repeated trust game, fails. Two basic problems for the functioning of the reputation mechanism are examined: the cognitive fragility problem, and the motivational problem. As regards the cognitive fragilities of reputation (which result from the impact of unforeseen contingencies and from bounded rationality), the paper develops the logic and the structure that self-regulatory norms must satisfy if they are to serve as gap-filling tools with which to remedy cognitive limitations in the reputation mechanism. The motivation problem then arises from the possibility of sophisticated abuse by the firm. Developed in this case is an entirely new application of the theory of conformism-and-reciprocity-based preferences, the result of which is that the stakeholders refuse to acquiesce to sophisticated abuse on the part of the firm. (shrink)

The basic theory of scientific understanding presented in Sections 1–2 exploits three main ideas.First, that to understand a phenomenonP (for a given agent) is to be able to fitP into the cognitive background corpusC (of the agent).Second, that to fitP intoC is to connectP with parts ofC (via arguments in a very broad sense) such that the unification ofC increases.Third, that the cognitive changes involved in unification can be treated as sequences of shifts of phenomena inC. How the theory fits (...) typical examples of understanding and how it excludes spurious unifications is explained in detail. Section 3 gives a formal description of the structure of cognitive corpuses which contain descriptive as well as inferential components. The theory of unification is then refined in the light of so called puzzling phenomena, to enable important distinctions, such as that between consonant and dissonant understanding. In Section 4, the refined theory is applied to several examples, among them a case study of the development of the atomic model. The final part contains a classification of kinds of understanding and a discussion of the relation between understanding and explanation. (shrink)

This article sketches a theory of objective probability focusing on nomic probability, which is supposed to be the kind of probability figuring in statistical laws of nature. The theory is based upon a strengthened probability calculus and some epistemological principles that formulate a precise version of the statistical syllogism. It is shown that from this rather minimal basis it is possible to derive theorems comprising (1) a theory of direct inference, and (2) a theory of induction. The theory of induction (...) is not of the familiar Bayesian variety, but consists of a precise version of the traditional Nicod Principle and its statistical analogues. (shrink)

Probabilities are important in belief updating, but probabilistic reasoning does not subsume everything else (as the Bayesian would have it). On the contrary, Bayesian reasoning presupposes knowledge that cannot itself be obtained by Bayesian reasoning, making generic Bayesianism an incoherent theory of belief updating. Instead, it is indefinite probabilities that are of principal importance in belief updating. Knowledge of such indefinite probabilities is obtained by some form of statistical induction, and inferences to non-probabilistic conclusions are carried out in accordance with (...) the statistical syllogism. Such inferences have been the focus of much attention in the nonmonotonic reasoning literature, but the logical complexity of such inference has not been adequately appreciated. (shrink)

This paper is an attempt to bring together two separated areas of research: classical mathematics and metamathematics on the one side, non-monotonic reasoning on the other. This is done by simulating nonmonotonic logic through antitonic theory extensions. In the first half, the specific extension procedure proposed here is motivated informally, partly in comparison with some well-known non-monotonic formalisms. Operators V and, more generally, U are obtained which have some plausibility when viewed as giving nonmonotonic theory extensions. In the second half, (...) these operators are treated from a mathematical and metamathematical point of view. Here an important role is played by U -closed theories and U -fixed points. The last section contains results on V-closed theories which are specific for V. (shrink)

It is shown that the constructive four-valued logic N4 can be faithfully embedded into the modal logic S4. This embedding is used to obtain complete, cut-free display sequent calculi for N4 and C4, the modal logic of consistency over N4. C4 is a natural monotonic base system for semantics-based non-monotonic reasoning.

There was a long tradition in philosophy according to which good reasoning had to be deductively valid. However, that tradition began to be questioned in the 1960’s, and is now thoroughly discredited. What caused its downfall was the recognition that many familiar kinds of reasoning are not deductively valid, but clearly confer justification on their conclusions. Here are some simple examples.

We identify two pragmatic problems in temporal reasoning, the qualification problem and the extended prediction problem, the latter subsuming the infamous frame problem. Solutions to those seem to call for nonmonotonic inferences, and yet naive use of standard nonmonotonic logics turns out to be inappropriate. Looking for an alternative, we first propose a uniform approach to constructing and understanding nonmonotonic logics. This framework subsumes many existing nonmonotonic formalisms, and yet is remarkably simple, adding almost no extra baggage to traditional logic. (...) We then use this framework to propose a solution to the qualification problem. We also outline a solution to the extended prediction problem, and speculate on a possible connection between the above discussion and the concept of causation. (shrink)

We present a straightforward embedding of modal nonmonotonic logics into default logic.

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)

There has been considerable discussion in the past about the assumptions and basis of different ethical rules. For instance, it is commonplace to say that ethical rules are defaults rules, which means that they tolerate exceptions. Some authors argue that morality can only be grounded in particular cases while others defend the existence of general principles related to ethical rules. Our purpose here is not to justify either position, but to try to model general ethical rules with artificial intelligence formalisms (...) and to compute logical consequences of different ethical theories. More precisely, this is an attempt to show that progress in non-monotonic logics, which simulates default reasoning, could provide a way to formalize different ethical conceptions. From a technical point of view, the model developed in this paper makes use of the Answer Set Programming (ASP) formalism. It is applied comparatively to different ethical systems with respect to their attitude towards lying. The advantages of such formalization are two-fold: firstly, to clarify ideas and assumptions, and, secondly, to use solvers to derive consequences of different ethical conceptions automatically, which can help in a rigorous comparison of ethical theories. (shrink)

When reasoning about complex domains, where information available is usually only partial, nonmonotonic reasoning can be an important tool. One of the formalisms introduced in this area is Reiter's Default Logic (1980). A characteristic of this formalism is that the applicability of default (inference) rules can only be verified in the future of the reasoning process. We describe an interpretation of default logic in temporal epistemic logic which makes this characteristic explicit. It is shown that this interpretation yields a semantics (...) for default logic based on temporal epistemic models. A comparison between the various semantics for default logic will show the differences and similarities of these approaches and ours. (shrink)

Heuristics can be regarded as justifying the actions and beliefs of problem-solving agents. I use an analysis of heuristics to argue that a symbiotic relationship exists between traditional epistemology and contemporary artificial intelligence. On one hand, the study of models of problem-solving agents usingquantitative heuristics, for example computer programs, can reveal insight into the understanding of human patterns of epistemic justification by evaluating these models'' performance against human problem-solving. On the other hand,qualitative heuristics embody the justifying ability of defeasible rules, (...) the understanding of which is provided by traditional epistemology. (shrink)

The general conditions of epistemic defeat are naturally represented through the interplay of two distinct kinds of entailment, deductive and defeasible. Many of the current approaches to modeling defeasible reasoning seek to define defeasible entailment via model-theoretic notions like truth and satisfiability, which, I argue, fails to capture this fundamental distinction between truthpreserving and justification-preserving entailments. I present an alternative account of defeasible entailment and show how logic programming offers a paradigm in which the distinction can be captured, allowing for (...) the modeling of a larger range of types of defeat. This is possible through a natural extension of the declarative and procedural semantics of Horn clauses. (shrink)

The notion of the rational closure of a positive knowledge base K of conditional assertions θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$i$$ \end{document} |∼ φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$i$$ \end{document} (standing for if θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$i$$ \end{document} then normally φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$i$$ \end{document}) was first introduced by Lehmann (1989) and developed by Lehmann and Magidor (...) (1992). Following those authors we would also argue that the rational closure is, in a strong sense, the minimal information, or simplest, rational consequence relation satisfying K. In practice, however, one might expect a knowledge base to consist not just of positive conditional assertions, θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$i$$ \end{document} |∼ φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$i$$ \end{document}, but also negative conditional assertions, θi φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$i$$ \end{document} (standing for not if θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$i$$ \end{document} then normally φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$i$$ \end{document}. Restricting ourselves to a finite language we show that the rational closure still exists for satisfiable knowledge bases containing both positive and negative conditional assertions and has similar properties to those exhibited in Lehmann and Magidor (1992). In particular an algorithm in Lehmann and Magidor (1992) which constructs the rational closure can be adapted to this case and yields, in turn, completeness theorems for the conditional assertions entailed by such a mixed knowledge base. (shrink)

Pickering and Chater (P&C) maintain that folk psychology and cognitive science should neither compete nor cooperate. Each is an independent enterprise, with a distinct subject matter and characteristic modes of explanation. P&C''s case depends upon their characterizations of cognitive science and folk psychology. We question the basis for their characterizations, challenge both the coherence and the individual adequacy of their contrasts between the two, and show that they waver in their views about the scope of each. We conclude that P&C (...) do not so muchdiscover ascreate the gap they find between folk psychology and cognitive science. It is an artifact of their implausible and unmotivated attempt to demarcate the two areas, and of the excessively narrow accounts they give of each. (shrink)