The paper argues that cognitive states of biological systems are inherently temporal. Three adequacy conditions for neuronal models of representation are vindicated: the compositionality of meaning, the compositionality of content, and the co-variation with content. Classicist and connectionist approaches are discussed and rejected. Based on recent neurobiological data, oscillatory networks are introduced as a third alternative. A mathematical description in a Hilbert space framework is developed. The states of this structure can be regarded as conceptual representations satisfying the three conditions.

We model three examples of beliefs that agents may have about other agents’ beliefs, and provide motivation for this conceptualization from the theory of mind literature. We assume a modal logical framework for modelling degrees of belief by partially ordered preference relations. In this setting, we describe that agents believe that other agents do not distinguish among their beliefs (‘no preferences’), that agents believe that the beliefs of other agents are in part as their own (‘my preferences’), and the special (...) case that agents believe that the beliefs of other agents are exactly as their own (‘preference refinement’). This multi-agent belief interaction is frame characterizable. We provide examples for introspective agents. We investigate which of these forms of belief interaction are preserved under three common forms of belief revision. (shrink)

Epistemology and epistemic logic At first sight, the modern agenda of epistemology has little to do with logic. Topics include different definitions of knowledge, its basic formal properties, debates between externalist and internalist positions, and above all: perennial encounters with sceptics lurking behind every street corner, especially in the US. The entry 'Epistemology' in the Routledge Encyclopedia of Philosophy (Klein 1993) and the anthology (Kim and Sosa 2000) give an up-to-date impression of the field. Now, epistemic logic started as a (...) contribution to epistemology, or at least a tool in its modus operandi, with the seminal book Knowledge and Belief (Hintikka's 1962, 2005). Formulas like Ki for "the agent i knows that " Bi for "the agent i believes that " provided logical forms for stating and analyzing philosophical propositions and arguments. And more than that, their model-theoretic semantics in terms of ranges of alternatives provided an appealing extensional way of thinking about what agents know or believe in a given situation. In particular, on Hintikka's view, an agent knows those propositions which are true in all situations compatible with what she knows about the actual world; i.e., her current range of uncertainty. (shrink)

We show how belief revision can be treated systematically in the format of dynamicepistemic logic, when operators of conditional belief are added. The core engine consists of definable update rules for changing plausibility relations between worlds, which have been proposed independently in the dynamic-epistemic literature on preference change. Our analysis yields two new types of modal result. First, we obtain complete logics for concrete mechanisms of belief revision, based on compositional reduction axioms. Next, we show how various abstract postulates for (...) belief revision can be analyzed by standard modal frame correspondences for model-changing operations. (shrink)

Statements not only update our current knowledge, but also have other dynamic effects. In particular, suggestions or commands ?upgrade' our preferences by changing the current order among worlds. We present a complete logic of knowledge update plus preference upgrade that works with dynamic-epistemic-style reduction axioms. This system can model changing obligations, conflicting commands, or ?regret'. We then show how to derive reduction axioms from arbitrary definable relation changes. This style of analysis also has a product update version with preferences between (...) actions, as well as worlds. Some illustrations are presented involving defaults and obligations. We conclude that our dynamic framework is viable, while admitting a further extension to more numerical ?utility update'. (shrink)

This paper presents an adaptive logic enhancement of conditional logics of normality that allows for defeasible applications of Modus Ponens to conditionals. In addition to the possibilities these logics already offer in terms of reasoning about conditionals, this way they are enriched by the ability to perform default inferencing. The idea is to apply Modus Ponens defeasibly to a conditional and a fact on the condition that it is ‘safe' to do so concerning the factual and conditional knowledge at hand. (...) It is for instance not safe if the given information describes exceptional circumstances: although birds usually fly, penguins are exceptional to this rule. The two adaptive standard strategies are shown to correspond to different intuitions, a skeptical and a credulous reasoning type, which manifest themselves in the handling of so-called floating conclusions. (shrink)

This is a discussion of the problem of extending the basic AGM belief revision theory to iterated belief revision: the problem of formulating rules, not only for revising a basic belief state in response to potential new information, but also for revising one’s revision rules in response to potential new information. The emphasis in the paper is on foundational questions about the nature of and motivation for various constraints, and about the methodology of the evaluation of putative counterexamples to proposed (...) constraints. Some specific constraints that have been proposed are criticized. The paper emphasizes the importance of meta-information—information about one’s sources of information—and argues that little of substance can be said about constraints on iterated belief revision at a level of abstraction that lacks the resources for explicit representation of meta-information. (shrink)

Conditionals somehow express conditional beliefs. However, conditional belief is a bi-propositional attitude that is generally not truth-evaluable, in contrast to unconditional belief. Therefore, this article opts for an expressivistic semantics for conditionals, grounds this semantics in the arguably most adequate account of conditional belief, that is, ranking theory, and dismisses probability theory for that purpose, because probabilities cannot represent belief. Various expressive options are then explained in terms of ranking theory, with the intention to set out a general interpretive scheme (...) that is able to account for the most variegated usage of conditionals. The Ramsey test is only the first option. Relevance is another, familiar, but little understood item, which comes in several versions. This article adds a further family of expressive options, which is able to subsume also counterfactuals and causal conditionals, and indicates at the end how this family allows for partial recovery of truth conditions for conditionals. (shrink)

Systems of logico-probabilistic reasoning characterize inference from conditional assertions that express high conditional probabilities. In this paper we investigate four prominent LP systems, the systems _O, P_, _Z_, and _QC_. These systems differ in the number of inferences they licence _. LP systems that license more inferences enjoy the possible reward of deriving more true and informative conclusions, but with this possible reward comes the risk of drawing more false or uninformative conclusions. In the first part of the paper, we (...) present the four systems and extend each of them by theorems that allow one to compute almost-tight lower-probability-bounds for the conclusion of an inference, given lower-probability-bounds for its premises. In the second part of the paper, we investigate by means of computer simulations which of the four systems provides the best balance of reward versus risk. Our results suggest that system _Z_ offers the best balance. (shrink)

In this paper I discuss the relation between various properties that have been regarded as important for determining whether or not a belief constitutes a piece of knowledge: its stability, strength and sensitivity to truth, as well as the strength of the epistemic position in which the subject is with respect to this belief. Attempts to explicate the relevant concepts more formally with the help of systems of spheres of possible worlds (à la Lewis and Grove) must take care to (...) keep apart the very different roles that systems of spheres can play. Nozicks sensitivity account turns out to be closer to the stability analysis of knowledge (versions of which I identify in Plato, Descartes, Klein and Lehrer) than one might have suspected. (shrink)

This paper presents the model of ‘bounded revision’ that is based on two-dimensional revision functions taking as arguments pairs consisting of an input sentence and a reference sentence. The key idea is that the input sentence is accepted as far as (and just a little further than) the reference sentence is ‘cotenable’ with it. Bounded revision satisfies the AGM axioms as well as the Same Beliefs Condition (SBC) saying that the set of beliefs accepted after the revision does not depend (...) on the reference sentence (although the posterior belief state does depend on it). Bounded revision satisfies the Darwiche–Pearl (DP) axioms for iterated belief change. If the reference sentence is fixed to be a tautology or a contradiction, two well-known one-dimensional revision operations result. Bounded revision thus naturally fills the space between conservative revision (also known as natural revision) and moderate revision (also known as lexicographic revision). I compare this approach to the two-dimensional model of ‘revision by comparison’ investigated by Fermé and Rott (Artif Intell 157:5–47, 2004 ) that satisfies neither the SBC nor the DP axioms. I conclude that two-dimensional revision operations add substantially to the expressive power of qualitative approaches that do not make use of numbers as measures of degrees of belief. (shrink)

This paper discusses the relation between deontic logic and the study of conversational interactions. Special attention is given to the notion of remedial interchange as analysed by sociologists and linguistic pragmaticians. This notion is close to the one of contrary-to-duty (reparational) obligation, which deontic logicians have been studying in its own right. The present article also investigates the question of whether some of the aspects of conversational interactions can fruitfully be described by using formal tools originally developed in the study (...) of iterated belief change. I here adapt the latter tools to deontic logic, and attempt an account of remedial interchange (and, more generally, contrary-to-duty reasoning) in terms of commutation. This account brings the dynamics of obligations to the fore. (shrink)

We offer a probabilistic model of rational consequence relations (Lehmann and Magidor, 1990) by appealing to the extension of the classical Ramsey-Adams test proposed by Vann McGee in (McGee, 1994). Previous and influential models of nonmonotonic consequence relations have been produced in terms of the dynamics of expectations (Gärdenfors and Makinson, 1994; Gärdenfors, 1993).'Expectation' is a term of art in these models, which should not be confused with the notion of expected utility. The expectations of an agent are some form (...) of belief weaker than absolute certainty. Our model offers a modified and extended version of an account of qualitative belief in terms of conditional probability, first presented in (van Fraassen, 1995). We use this model to relate probabilistic and qualitative models of non-monotonic relations in terms of expectations. In doing so we propose a probabilistic model of the notion of expectation. We provide characterization results both for logically finite languages and for logically infinite, but countable, languages. The latter case shows the relevance of the axiom of countable additivity for our probability functions. We show that a rational logic defined over a logically infinite language can only be fully characterized in terms of finitely additive conditional probability. (shrink)

In this paper it is argued that, in order to solve the problem of iterated belief change, both the belief state and its input should be represented as epistemic entrenchment (EE) relations. A belief revision operation is constructed that updates a given EE relation to a new one in light of an evidential EE relation. It is shown that the operation in question satisfies generalized versions of the Gärdenfors revision postulates. The account offered is motivated by Spohn's ordinal conditionalization functions, (...) and can be seen as the Jeffrization of a proposal considered by Rott. (shrink)

Representation theorems are often taken to provide the foundations for decision theory. First, they are taken to characterize degrees of belief and utilities. Second, they are taken to justify two fundamental rules of rationality: that we should have probabilistic degrees of belief and that we should act as expected utility maximizers. We argue that representation theorems cannot serve either of these foundational purposes, and that recent attempts to defend the foundational importance of representation theorems are unsuccessful. As a result, we (...) should reject these claims, and lay the foundations of decision theory on firmer ground. (shrink)

We propose a minimal logic for interactive epistemology based on a qualitative representation of epistemic individual and group attitudes including knowledge, belief, strong belief, common knowledge and common belief. We show that our logic is sufficiently expressive to provide an epistemic foundation for various game-theoretic solution concepts including “1-round of deletion of weakly dominated strategies, followed by iterated deletion of strongly dominated strategies” ) and “2-rounds of deletion of weakly dominated strategies, followed by iterated deletion of strongly dominated strategies” ). (...) Axiomatization and complexity results for the logic are given in the paper. (shrink)

In this paper, we formulate some approaches to belief fusion and revision using epistemic logic semantics. Fusion operators considered in this paper are majority merging, arbitration, and general merging. Some modalities corresponding to belief fusion and revision operators are incorporated into epistemic logics. The Kripke semantics of these extended logics are presented. While most existing approaches treat belief fusion and revision operators as meta-level constructs, we directly incorporate these operators into our object logic language. By doing so, we both extend (...) the expressive power of epistemic logic and enhance the techniques of information fusion. (shrink)

This is part A of a paper in which we defend a semantics for counterfactuals which is probabilistic in the sense that the truth condition for counterfactuals refers to a probability measure. Because of its probabilistic nature, it allows a counterfactual ‘ifAthenB’ to be true even in the presence of relevant ‘Aand notB’-worlds, as long such exceptions are not too widely spread. The semantics is made precise and studied in different versions which are related to each other by representation theorems. (...) Despite its probabilistic nature, we show that the semantics and the resulting system of logic may be regarded as a naturalistically vindicated variant of David Lewis’ truth-conditional semantics and logic of counterfactuals. At the same time, the semantics overlaps in various ways with the non-truth-conditional suppositional theory for conditionals that derives from Ernest Adams’ work. We argue that counterfactuals have two kinds of pragmatic meanings and come attached with two types of degrees of acceptability or belief, one being suppositional, the other one being truth based as determined by our probabilistic semantics; these degrees could not always coincide due to a new triviality result for counterfactuals, and they should not be identified in the light of their different interpretation and pragmatic purpose. However, for plain assertability the difference between them does not matter. Hence, if the suppositional theory of counterfactuals is formulated with sufficient care, our truth-conditional theory of counterfactuals is consistent with it. The results of our investigation are used to assess a claim considered by Hawthorne and Hájek, that is, the thesis that most ordinary counterfactuals are false. (shrink)

We argue that a semantics for counterfactual conditionals in terms of comparative overall similarity faces a formal limitation due to Arrow’s impossibility theorem from social choice theory. According to Lewis’s account, the truth-conditions for counterfactual conditionals are given in terms of the comparative overall similarity between possible worlds, which is in turn determined by various aspects of similarity between possible worlds. We argue that a function from aspects of similarity to overall similarity should satisfy certain plausible constraints while Arrow’s impossibility (...) theorem rules out that such a function satisfies all the constraints simultaneously. We argue that a way out of this impasse is to represent aspectual similarity in terms of ranking functions instead of representing it in a purely ordinal fashion. Further, we argue against the claim that the determination of overall similarity by aspects of similarity faces a difficulty in addition to the Arrovian limitation, namely the incommensurability of different aspects of similarity. The phenomena that have been cited as evidence for such incommensurability are best explained by ordinary vagueness. (shrink)

We propose in this paper a new family of belief merging operators, that is based on a game between sources : until a coherent set of sources is reached, at each round a contest is organized to find out the weakest sources, then those sources has to concede. This idea leads to numerous new interesting operators and opens new perspectives for belief merging. Some existing operators are also recovered as particular cases. Those operators can be seen as a special case (...) of Booth's Belief Negotiation Models [BOO 02], but the achieved restriction forms a consistent family of merging operators that worths to be studied on its own. (shrink)

The idea of preserving conditional beliefs emerged recently as a new paradigm apt to guide the revision of epistemic states. Conditionals are substantially different from propositional beliefs and need specific treatment. In this paper, we present a new approach to conditionals, capturing particularly well their dynamic part as revision policies. We thoroughly axiomatize a principle of conditional preservation as an indifference property with respect to conditional structures of worlds. This principle is developed in a semi-quantitative setting, so as to reveal (...) its fundamental meaning for belief revision in quantitative as well as in qualitative frameworks. In fact, it is shown to cover other proposed approaches to conditional preservation. (shrink)

This note is a sequel to Huber. It is shown that obeying a normative principle relating counterfactual conditionals and conditional beliefs, viz. the royal rule, is a necessary and sufficient means to attaining a cognitive end that relates true beliefs in purely factual, non-modal propositions and true beliefs in purely modal propositions. Along the way I will sketch my idealism about alethic or metaphysical modality.

This paper starts by indicating the analysis of Hempel's conditions of adequacy for any relation of confirmation (Hempel, 1945) as presented in Huber (submitted). There I argue contra Carnap (1962, Section 87) that Hempel felt the need for two concepts of confirmation: one aiming at plausible theories and another aiming at informative theories. However, he also realized that these two concepts are conflicting, and he gave up the concept of confirmation aiming at informative theories. The main part of the paper (...) consists in working out the claim that one can have Hempel's cake and eat it too - in the sense that there is a logic of theory assessment that takes into account both of the two conflicting aspects of plausibility and informativeness. According to the semantics of this logic, a is an acceptable theory for evidence β if and only if a is both sufficiently plausible given β and sufficiently informative about β. This is spelt out in terms of ranking functions (Spohn, 1988) and shown to represent the syntactically specified notion of an assessment relation. The paper then compares these acceptability relations to explanatory and confirmatory consequence relations (Flach, 2000) as well as to nonmonotonic consequence relations (Kraus et al., 1990). It concludes by relating the plausibility-informativeness approach to Carnap's positive relevance account, thereby shedding new light on Carnap's analysis as well as solving another problem of confirmation theory. (shrink)

Philosophers typically rely on intuitions when providing a semantics for counterfactual conditionals. However, intuitions regarding counterfactual conditionals are notoriously shaky. The aim of this paper is to provide a principled account of the semantics of counterfactual conditionals. This principled account is provided by what I dub the Royal Rule, a deterministic analogue of the Principal Principle relating chance and credence. The Royal Rule says that an ideal doxastic agent’s initial grade of disbelief in a proposition \(A\) , given that the (...) counterfactual distance in a given context to the closest \(A\) -worlds equals \(n\) , and no further information that is not admissible in this context, should equal \(n\) . Under the two assumptions that the presuppositions of a given context are admissible in this context, and that the theory of deterministic alethic or metaphysical modality is admissible in any context, it follows that the counterfactual distance distribution in a given context has the structure of a ranking function. The basic conditional logic V is shown to be sound and complete with respect to the resulting rank-theoretic semantics of counterfactuals. (shrink)

This paper presents a new analysis of C.G. Hempel’s conditions of adequacy for any relation of confirmation [Hempel C. G. (1945). Aspects of scientific explanation and other essays in the philosophy of science. New York: The Free Press, pp. 3–51.], differing from the one Carnap gave in §87 of his [1962. Logical foundations of probability (2nd ed.). Chicago: University of Chicago Press.]. Hempel, it is argued, felt the need for two concepts of confirmation: one aiming at true hypotheses and another (...) aiming at informative hypotheses. However, he also realized that these two concepts are conflicting, and he gave up the concept of confirmation aiming at informative hypotheses. I then show that one can have Hempel’s cake and eat it too. There is a logic that takes into account both of these two conflicting aspects. According to this logic, a sentence H is an acceptable hypothesis for evidence E if and only if H is both sufficiently plausible given E and sufficiently informative about E. Finally, the logic sheds new light on Carnap’s analysis. (shrink)

The problem addressed in this paper is “the main epistemic problem concerning science”, viz. “the explication of how we compare and evaluate theories [...] in the light of the available evidence” (van Fraassen, BC, 1983, Theory comparison and relevant Evidence. In J. Earman (Ed.), Testing scientific theories (pp. 27–42). Minneapolis: University of Minnesota Press). Sections 1– 3 contain the general plausibility-informativeness theory of theory assessment. In a nutshell, the message is (1) that there are two values a theory should exhibit: (...) truth and informativeness—measured respectively by a truth indicator and a strength indicator; (2) that these two values are conflicting in the sense that the former is a decreasing and the latter an increasing function of the logical strength of the theory to be assessed; and (3) that in assessing a given theory by the available data one should weigh between these two conflicting aspects in such a way that any surplus in informativeness succeeds, if the shortfall in plausibility is small enough. Particular accounts of this general theory arise by inserting particular strength indicators and truth indicators. In Section 4 the theory is spelt out for the Bayesian paradigm of subjective probabilities. It is then compared to incremental Bayesian confirmation theory. Section 4 closes by asking whether it is likely to be lovely. Section 5 discusses a few problems of confirmation theory in the light of the present approach. In particular, it is briefly indicated how the present account gives rise to a new analysis of Hempel’s conditions of adequacy for any relation of confirmation (Hempel, CG, 1945, Studies in the logic of comfirmation. Mind, 54, 1–26, 97–121.), differing from the one Carnap gave in § 87 of his Logical foundations of probability (1962, Chicago: University of Chicago Press). Section 6 adresses the question of justification any theory of theory assessment has to face: why should one stick to theories given high assessment values rather than to any other theories? The answer given by the Bayesian version of the account presented in section 4 is that one should accept theories given high assessment values, because, in the medium run, theory assessment almost surely takes one to the most informative among all true theories when presented separating data. The concluding section 7 continues the comparison between the present account and incremental Bayesian confirmation theory. (shrink)

Belief revision theory studies how an ideal doxastic agent should revise her beliefs when she receives new information. In part I, I have first presented the AGM theory of belief revision. Then I have focused on the problem of iterated belief revisions. In part II, I will first present ranking theory (Spohn 1988). Then I will show how it solves the problem of iterated belief revisions. I will conclude by sketching two areas of future research.

Judea Pearl (2000) was the first to propose a definition of actual causation using causal models. A number of authors have suggested that an adequate account of actual causation must appeal not only to causal structure but also to considerations of normality. In Halpern and Hitchcock (2011), we offer a definition of actual causation using extended causal models, which include information about both causal structure and normality. Extended causal models are potentially very complex. In this study, we show how it (...) is possible to achieve a compact representation of extended causal models. (shrink)

The 1985 paper by Carlos Alchourrón (1931–1996), Peter Gärdenfors, and David Makinson (AGM), "On the Logic of Theory Change: Partial Meet Contraction and Revision Functions" was the starting-point of a large and rapidly growing literature that employs formal models in the investigation of changes in belief states and databases. In this review, the first twentyfive years of this development are summarized. The topics covered include equivalent characterizations of AGM operations, extended representations of the belief states, change operators not included in (...) the original framework, iterated change, applications of the model, its connections with other formal frameworks, computatibility of AGM operations, and criticism of the model. (shrink)

The 1985 paper by Carlos Alchourrón, Peter Gärdenfors, and David Makinson, “On the Logic of Theory Change: Partial Meet Contraction and Revision Functions” was the starting-point of a large and rapidly growing literature that employs formal models in the investigation of changes in belief states and databases. In this review, the first twenty-five years of this development are summarized. The topics covered include equivalent characterizations of AGM operations, extended representations of the belief states, change operators not included in the original (...) framework, iterated change, applications of the model, its connections with other formal frameworks, computatibility of AGM operations, and criticism of the model. (shrink)

In the paper "On the role of the research agenda in epistemic change", Olsson and Westlund have suggested that the notion of epistemic state employed in the standard framework of belief revision (Alchourrón et al. 1985; Gärdenfors 1988) should be extended to include a representation of the agent's research agenda (Olsson and Westlund 2006). The resulting framework will here be referred to as interrogative belief revision. In this paper, I attempt to deal with the problem of how research agendas should (...) change in contraction, a problem largely left open by Olsson and Westlund. Two desiderata of an appropriate solution are suggested: one is a principle of continuity, stating that changes in the research agenda should somehow reflect that certain long term research interests are kept fixed. The other desideratum, which is based on part of Olsson and Westlund's motivation for adding research agendas to the epistemic states, is that we should be able to account for how contraction may serve to open up new, fruitful hypotheses for investigation. In order to achieve these desiderata, I base my solution on a revised version of Olsson and Westlund's notion of epistemic state. (shrink)

Glymour’s theory of bootstrap confirmation is a purely qualitative account of confirmation; it allows us to say that the evidence confirms a given theory, but not that it confirms the theory to a certain degree. The present paper extends Glymour’s theory to a quantitative account and investigates the resulting theory in some detail. It also considers the question how bootstrap confirmation relates to justification.

We model three examples of beliefs that agents may have about other agents' beliefs, and provide motivation for this conceptualization from the theory of mind literature. We assume a modal logical framework for modelling degrees of belief by partially ordered preference relations. In this setting, we describe that agents believe that other agents do not distinguish among their beliefs ('no preferences'), that agents believe that the beliefs of other agents are in part as their own ('my preferences'), and the special (...) case that agents believe that the beliefs of other agents are exactly as their own ('preference refinement'). This multi-agent belief interaction is frame characterizable. We provide examples for introspective agents. We investigate which of these forms of belief interaction are preserved under three common forms of belief revision. (shrink)

The connections between nonmonotonic reasoning and belief revision are well-known. A central problem in the area of nonmonotonic reasoning is the problem of default entailment, i.e., when should an item of default information representing “if θ is true then, normally, φ is true” be said to follow from a given set of items of such information. Many answers to this question have been proposed but, surprisingly, virtually none have attempted any explicit connection to belief revision. The aim of this paper (...) is to give an example of how such a connection can be made by showing how the lexicographic closure of a set of defaults may be conceptualised as a process of iterated revision by sets of sentences. Specifically we use the revision method of Nayak. (shrink)

The contributions to the Special Issue on Multiple Belief Change, Iterated Belief Change and Preference Aggregation are divided into three parts. Four contributions are grouped under the heading "multiple belief change" (Part I, with authors M. Falappa, E. Fermé, G. Kern-Isberner, P. Peppas, M. Reis, and G. Simari), five contributions under the heading "iterated belief change" (Part II, with authors G. Bonanno, S.O. Hansson, A. Nayak, M. Orgun, R. Ramachandran, H. Rott, and E. Weydert). These papers do not only pick (...) up the particular questions raised, but also extend and modify the framework of Alchourrón, Gärdenfors and Makinson. Part III deals with preference aggregation and consists of one contribution (by F. Herzberg and D. Eckert). (shrink)

We study probabilistic logic under the viewpoint of the coherence principle of de Finetti. In detail, we explore how probabilistic reasoning under coherence is related to model- theoretic probabilistic reasoning and to default reasoning in System . In particular, we show that the notions of g-coherence and of g-coherent entailment can be expressed by combining notions in model-theoretic probabilistic logic with concepts from default reasoning. Moreover, we show that probabilistic reasoning under coherence is a generalization of default reasoning in System (...) . That is, we provide a new probabilistic semantics for System , which neither uses infinitesimal probabilities nor atomic bound (or big-stepped) probabilities. These results also provide new algorithms for probabilistic reasoning under coherence and for default reasoning in System , and they give new insight into default reasoning with conditional objects. (shrink)

An assertion of high conditional probability or, more briefly, an HCP assertion is a statement of the type: The conditional probability of B given A is close to one. The goal of this paper is to construct logics of HCP assertions whose conclusions are highly likely to be correct rather than certain to be correct. Such logics would allow useful conclusions to be drawn when the premises are not strong enough to allow conclusions to be reached with certainty. This goal (...) is achieved by taking Adams' (1966) logic, changing its intended application from conditionals to HCP assertions, and then weakening its criterion for entailment. According to the weakened entailment criterion, called the Criterion of Near Surety and which may be loosely interpreted as a Bayesian criterion, a conclusion is entailed if and only if nearly every model of the premises is a model of the conclusion. The resulting logic, called NSL, is nonmonotonic. Entailment in this logic, although not as strict as entailment in Adams' logic, is more strict than entailment in the propositional logic of material conditionals. Next, NSL was modified by requiring that each HCP assertion be scaled; this means that to each HCP assertion was associated a bound on the deviation from 1 of the conditional probability that is the subject of the assertion. Scaling of HCP assertions is useful for breaking entailment deadlocks. For example, it it is known that the conditional probabilities of C given A and of - C given B are both close to one but the bound on the former's deviation from 1 is much smaller than the latter's, then it may be concluded that in all likelihood the conditional probability of C given A ∧ B is close to one. The resulting logic, called NSL-S, is also nonmonotonic. Despite great differences in their definitions of entailment, entailment in NSL is equivalent to Lehmann and Magidor's rational closure and, disregarding minor differences concerning which premise sets are considered consistent, entailment in NSL-S is equivalent to entailment in Goldszmidt and Pearl's System-Z⁺. Bacchus, Grove, Halpern, and Koller proposed two methods of developing a predicate calculus based on the Criterion of Near Surety. In their randomstructures method, which assumed a prior distribution similar to that of NSL, it appears possible to define an entailment relation equivalent to that of NSL. In their random-worlds method, which assumed a prior distribution dramatically different from that of NSL, it is known that the entailment relation is different from that of NSL. (shrink)

Following the pioneer work of Bruno De Finetti [12], conditional probability spaces (allowing for conditioning with events of measure zero) have been studied since (at least) the 1950's. Perhaps the most salient axiomatizations are Karl Popper's in [31], and Alfred Renyi's in [33]. Nonstandard probability spaces [34] are a well know alternative to this approach. Vann McGee proposed in [30] a result relating both approaches by showing that the standard values of infinitesimal probability functions are representable as Popper functions, and (...) that every Popper function is representable in terms of the standard real values of some infinitesimal measure. Our main goal in this article is to study the constraints on (qualitative and probabilistic) change imposed by an extended version of McGee's result. We focus on an extension capable of allowing for iterated changes of view. Such extension, we argue, seems to be needed in almost all considered applications. Since most of the available axiomatizations stipulate (definitionally) important constraints on iterated change, we propose a non-questionbegging framework, Iterative Probability Systems (IPS) and we show that every Popper function can be regarded as a Bayesian IPS. A generalized version of McGee's result is then proved and several of its consequences considered. In particular we note that our proof requires the imposition of Cumulativity, i.e. the principle that a proposition that is accepted at any stage of an iterative process of acceptance will continue to be accepted at any later stage. The plausibility and range of applicability of Cumulativity is then studied. In particular we appeal to a method for defining belief from conditional probability (first proposed in [42] and then slightly modified in [6] and [3]) in order to characterize the notion of qualitative change induced by Cumulative models of probability kinematics. The resulting cumulative notion is then compared with existing axiomatizations of belief change and probabilistic supposition. We also consider applications in the probabilistic accounts of conditionals [1] and [30]. (shrink)

Plausibility models are Kripke models that agents use to reason about knowledge and belief, both of themselves and of each other. Such models are used to interpret the notions of conditional belief, degrees of belief, and safe belief. The logic of conditional belief contains that modality and also the knowledge modality, and similarly for the logic of degrees of belief and the logic of safe belief. With respect to these logics, plausibility models may contain too much information. A proper notion (...) of bisimulation is required that characterises them. We define that notion of bisimulation and prove the required characterisations: on the class of image-finite and preimage-finite models, two pointed Kripke models are modally equivalent in either of the three logics, if and only if they are bisimilar. As a result, the information content of such a model can be similarly expressed in the logic of conditional belief, or the logic of degrees of belief, or that of safe belief. This, we found a surprising result. Still, that does not mean that the logics are equally expressive: the logics of conditional and degrees of belief are incomparable, the logics of degrees of belief and safe belief are incomparable, while the logic of safe belief is more expressive than the logic of conditional belief. In view of the result on bisimulation characterisation, this is an equally surprising result. We hope our insights may contribute to the growing community of formal epistemology and on the relation between qualitative and quantitative modelling. (shrink)