Minimal Belief Change and the Pareto Principle

This paper analyzes the notion of a minimal belief change that incorporates new information. I apply the fundamental decision-theoretic principle of Pareto-optimality to derive a notion of minimal belief change, for two different representations of belief: First, for beliefs represented by a theory – a deductively closed set of sentences or propositions – and second for beliefs represented by an axiomatic base for a theory. Three postulates exactly characterize Pareto-minimal revisions of theories, yielding a weaker set of constraints than the standard AGM postulates. The Levi identity characterizes Pareto-minimal revisions of belief bases: a change of belief base is Pareto-minimal if and only if the change satisfies the Levi identity (for “maxichoice” contraction operators). Thus for belief bases, Pareto-minimality imposes constraints that the AGM postulates do not. The Ramsey test is a well-known way of establishing connections between belief revision postulates and axioms for conditionals (“if p, then q”). Pareto-minimal theory change corresponds exactly to three characteristic axioms of counterfactual systems: a theory revision operator that satisfies the Ramsey test validates these axioms if and only if the revision operator is Pareto-minimal. This revised version was published online in June 2006 with corrections to the Cover Date.     

Cornerstones: You’d better believe them

Crispin Wright??s ??Unified Strategy?? for addressing some familiar sceptical paradoxes exploits a subtle distinction between two different ways in which we can be related to a proposition: (full-blown) belief and (mere) acceptance. The importance of the distinction for his strategy stems from his conviction that we cannot acquire any kind of evidence, either empirical or a priori, for the ??cornerstones?? of our cognitive projects, i.e., for those basic presuppositions of our inquiries that we must be warranted to endorse if we are to claim warrant for any of the beliefs formed as a result of such inquiries: regarding the idea of a non-evidential warrant to believe a proposition as a kind of ??conceptual solecism??, he doesn??t set himself the task of showing that we are evidentially warranted to believe such presuppositions, but only that of showing that we are non-evidentially warranted to accept them. In the present paper, I argue that such choice involves a fatal departure from a basic principle governing doxastic commitment??a principle that requires that we regard cornerstones propositions as propositions we are rationally committed to believe, not just entitled to accept. I press the point by presenting the Acceptance Argument, a sceptical paradox whose consideration leads to the conclusion that the Unified Strategy is caught between the Scylla of incoherently invoking a rather dubious form of epistemic alchemy and the Charybdis of placing an unexpected and apparently ad hoc restriction on the doxastic commitments we undertake by believing the things we believe. My final suggestion is that the Unified Strategy might be spared this dilemma only by undergoing a rather radical revision??a revision that would require setting aside the distinction between belief and acceptance to re-conceptualise its goal unabashedly in terms of (non-evidentially) warranted belief.     

Preference-based belief revision for rule-based agents

Agents which perform inferences on the basis of unreliable information need an ability to revise their beliefs if they discover an inconsistency. Such a belief revision algorithm ideally should be rational, should respect any preference ordering over the agent’s beliefs (removing less preferred beliefs where possible) and should be fast. However, while standard approaches to rational belief revision for classical reasoners allow preferences to be taken into account, they typically have quite high complexity. In this paper, we consider belief revision for agents which reason in a simpler logic than full first-order logic, namely rule-based reasoners. We show that it is possible to define a contraction operation for rule-based reasoners, which we call McAllester contraction, which satisfies all the basic Alchourrón, Gärdenfors and Makinson (AGM) postulates for contraction (apart from the recovery postulate) and at the same time can be computed in polynomial time. We prove a representation theorem for McAllester contraction with respect to the basic AGM postulates (minus recovery), and two additional postulates. We then show that our contraction operation removes a set of beliefs which is least preferred, with respect to a natural interpretation of preference. Finally, we show how McAllester contraction can be used to define a revision operation which is also polynomial time, and prove a representation theorem for the revision operation.     

My beliefs about your beliefs: a case study in theory of mind and epistemic logic

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.     

WHEN DEDUCTION LEADS TO BELIEF

The paper questions the common assumption that rational individuals believe all propositions which they know to be logical consequences of their other beliefs: although we must acknowledge the truth of a proposition which is a deductive consequence of our beliefs, we may not genuinely believe it. This conclusion is defended by arguing that some familiar counterexamples to the claim that knowledge is justified true belief fail because they involve propositions which are not really believed. Beliefs guide conduct or issue in assertion by answering questions which arise in the course of deliberation and conversation, but the troublesome cases present propositions which do not present the agent's answer to any question. The paper concludes by sketching the conditions under which the deductive consequences of our beliefs can be believed.¹     

Stratified Belief Bases Revision with Argumentative Inference

We propose a revision operator on a stratified belief base, i.e., a belief base that stores beliefs in different strata corresponding to the value an agent assigns to these beliefs. Furthermore, the operator will be defined as to perform the revision in such a way that information is never lost upon revision but stored in a stratum or layer containing information perceived as having a lower value. In this manner, if the revision of one layer leads to the rejection of some information to maintain consistency, instead of being withdrawn it will be kept and introduced in a different layer with lower value. Throughout this development we will follow the principle of minimal change, being one of the important principles proposed in belief change theory, particularly emphasized in the AGM model. Regarding the reasoning part from the stratified belief base, the agent will obtain the inferences using an argumentative formalism. Thus, the argumentation framework will decide which information prevails when sentences of different layers are used for entailing conflicting beliefs. We will also illustrate how inferences are changed and how the status of arguments can be modified after a revision process.     

Belief Revision And Epistemology

Postulational approaches attempt to understand the dynamics of belief revision by appealing to no more than the set of beliefs held by an agent and the logical relations between them. It is argued there that such an approach cannot work. A proper account of belief revision must also appeal to the arguments supporting beliefs, and recognize that those arguments can be defeasible. If we begin with a mature epistemological theory that accommodates this, it can be seen that the belief revision operators on which the postulational theories are based are ill-defined. It is further argued that there is no way to repair the definitions so as to retain the spirit of those theory. Belief revision is better studied from within an independently motivated epistemological theory.     

A Single Counterexample Leads to Moral Belief Revision

What kind of evidence will lead people to revise their moral beliefs? Moral beliefs are often strongly held convictions, and existing research has shown that morality is rooted in emotion and socialization rather than deliberative reasoning. In addition, more general issues—such as confirmation bias—further impede coherent belief revision. Here, we explored a unique means for inducing belief revision. In two experiments, participants considered a moral dilemma in which an overwhelming majority of people judged that it was inappropriate to take action to maximize utility. Their judgments contradicted a utilitarian principle they otherwise strongly endorsed. Exposure to this scenario led participants to revise their belief in the utilitarian principle, and this revision persisted over several hours. This method provides a new avenue for inducing belief revision.     

Against global method safety

The global method safety account of knowledge states that an agent’s true belief that p is safe and qualifies as knowledge if and only if it is formed by method M, such that her beliefs in p and her beliefs in relevantly similar propositions formed by M in all nearby worlds are true. This paper argues that global method safety is too restrictive. First, the agent may not know relevantly similar propositions via M because the belief that p is the only possible outcome of M. Second, there are cases where there is a fine-grained belief that is unsafe and a relevantly similar coarse-grained belief (with looser truth conditions) that is safe and where both beliefs are based on the same method M. Third, the reliability of conditional reasoning, a basic belief-forming method, seems to be sensitive to fine-grained contents, as suggested by the wide variation in success rates for thematic versions of the Wason selection task.

     

Reversing the Levi identity

The AGM (Alchourrón-GÄrdenfors-Makinson) model of belief change is extended to cover changes on sets of beliefs that arenot closed under logical consequence (belief bases). Three major types of change operations, namely contraction, internal revision, and external revision are axiomatically characterized, and their interrelations are studied. In external revision, the Levi identity is reversed in the sense that onefirst adds the new belief to the belief base, and afterwards contracts its negation. It is argued that external revision represents an intuitively plausible way of revising one's beliefs. Since it typically involves the temporary acceptance of an inconsistent set of beliefs, it can only be used in belief representations that distinguish between different inconsistent sets of belief.     

Can We Infer Naturalism from Scepticism?

Hume, like a number of more recent writers, claims that epistemological scepticism gives us reason to think that our beliefs are non-epistemically determined. Because some body of propositions that we believe are all unjustifiable, the argument goes, our beliefs in those propositions must be determined by non-truth-conducive considerations. I argue initially that scepticism does not by itself entail Humean naturalism. I then then develop an argument from scepticism to naturalism which has considerable promise. This more complex argument is built around two considerations: (i) if a subject accepts a local sceptical argument against one of his beliefs and still does not give it up, then we have very good, if not conclusive, reason to think that his belief is non-epistemically determined; (ii) it seems initially plausible that global scepticism can have no affect on the beliefs it targets, even if we were to accept it. Unfortunately, however, even this argument ultimately fails to establish any connection between scepticism and Humean naturalism.     

The Modal Logic of Bayesian Belief Revision

In Bayesian belief revision a Bayesian agent revises his prior belief by conditionalizing the prior on some evidence using Bayes’ rule. We define a hierarchy of modal logics that capture the logical features of Bayesian belief revision. Elements in the hierarchy are distinguished by the cardinality of the set of elementary propositions on which the agent’s prior is defined. Inclusions among the modal logics in the hierarchy are determined. By linking the modal logics in the hierarchy to the strongest modal companion of Medvedev’s logic of finite problems it is shown that the modal logic of belief revision determined by probabilities on a finite set of elementary propositions is not finitely axiomatizable.

     

Indexical Beliefs and Communication: Against Stalnaker on Self‐Location

Beliefs are commonly analyzed as binary relations between subjects and propositions. Perry and Lewis have shown that the standard account has difficulties in handling self‐locating beliefs. Robert Stalnaker has recently put forward a version of the standard account that is supposed to overcome this problem. Stalnaker's motivation for defending the propositional account of belief is that it comes with a simple and powerful propositional model of communication. In this paper I argue that Stalnaker's proposal fails. The only way of upholding the propositional account of belief is by abandoning the simple account of communication.     

Functionalism and propositions

Conclusion Some have argued, following Stalnaker, that a plausible functionalist account of belief requires coarse-grained propositions. I have explored a class of functionalist accounts, and my argument has been that, in this class, there is no account which meetsall of the following conditions: it is plausible, noncircular, and allows for the validity of the argument to coarse-grained propositions. In producing this argument, I believe that I have shown that it might be open to a functionalist to adopt fine-grained propositions; thus, one might be a functionalist without holding that all mathematical beliefs are about strings of symbols (and that the belief that all bachelors are unmarried men is a belief about words).My project in this paper has been minimal in the following sense. I havenot argued thatno functionalist account of belief which meets the three conditions can be produced; rather, I have simply explored the inadequacies of certain sorts of accounts. I think that this is useful insofar as it makes clear the challenges to be met by an account of belief which can play the required role in the argument to coarse-grained propositions. It is compatible with my position that such an account is forthcoming, insofar as I have not produced a functionalist theory of belief which is clearly non-circular, plausible, and which yields fine-grained propositions. Of course, it is also compatible with my position that no plausible, non-circular functionalist account of belief of any sort can be produced. My argument has been that,if one construes such mental states as belief as functional states, no convincing argument has yet been produced that they require coarse-grained objects.     

A Peircean examination of Gettier’s two cases

If we accept certain Peircean commitments, Gettier’s two cases are not cases of justified true belief because the beliefs are not true. On the Peircean view, propositions are sign substitutes, or “representamens.” In typical cases of thought about the world, propositions represent facts. In each of Gettier’s examples, we have a case in which a person S believes some proposition p, there is some fact F* such that were p to represent F* to S then p would be true, and yet p does not represent F* to S but some other fact F of which p is false. Since truth is a property of propositions with respect to their representational function, it follows that the belief is not true. Although an examination of Gettier’s two cases, this essay is not a defense of the justified true belief (JTB) analysis of knowledge, for there are objections to the JTB analysis other than Gettier’s two cases. Rather, Gettier’s two cases are of particular interest for the light they shed on the nature of truth and representation.

     

Resource Bounded Belief Revision

The AGM paradigm for belief revision provides a very elegant and powerful framework for reasoning about idealized agents. The paradigm assumes that the modeled agent is a perfect reasoner with infinite memory. In this paper we propose a framework to reason about non-ideal agents that generalizes the AGM paradigm. We first introduce a structure to represent an agent's belief states that distinguishes different status of beliefs according to whether or not they are explicitly represented, whether they are currently active and whether they are fully accepted or provisional. Then we define a set of basic operations that change the status of beliefs and show how these operations can be used to model agents with different capacities. We also show how different operations of belief change described in the literature can be seen as special cases of our theory.     

Gricean Belief Change

One of the standard principles of rationality guiding traditional accounts of belief change is the principle of minimal change: a reasoner's belief corpus should be modified in a minimal fashion when assimilating new information. This rationality principle has stood belief change in good stead. However, it does not deal properly with all belief change scenarios. We introduce a novel account of belief change motivated by one of Grice's maxims of conversational implicature: the reasoner's belief corpus is modified in a minimal fashion to assimilate exactly the new information. In this form of belief change, when the reasoner revises by new information p ∨ q their belief corpus is modified so that p∨q is believed but stronger propositions like p∧q are not, no matter what beliefs are in the reasoner's initial corpus. We term this conservative belief change since the revised belief corpus is a conservative extension of the original belief corpus given the new information.     

De se belief and rational choice

The Sleeping Beauty puzzle has dramatized the divisive question of how de se beliefs should be integrated into formal theories of rational belief change. In this paper, I look ahead to a related question: how should de se beliefs be integrated into formal theories of rational choice? I argue that standard decision theoretic frameworks fail in special cases of de se uncertainty, like Sleeping Beauty. The nature of the failure reveals that sometimes rational choices are determined independently of one’s credences in the kinds of ‘narrow’ de se propositions that Sleepy Beauty has set in relief. Consequently, in addition to pinpointing a failure of standard decision theoretic frameworks, this result casts doubt on a large class of strategies for determining principles for the rationally updating de se beliefs in cases like Sleeping Beauty, and also calls into question the importance of making such a determination at all.     

A NEW REVISABILITY PARADOX

Abstract:  In a recent article, Mark Colyvan has criticized Jerrold Katz's attempt to show that Quinean holism is self-refuting. Katz argued that a Quinean epistemology incorporating a principle of the universal revisability of beliefs would have to hold that that and other principles of the system were both revisable and unrevisable. Colyvan rejects Katz's argument for failing to take into account the logic of belief revision. But granting the terms of debate laid down by Colyvan, the universal revisability principle still commits Quineans to holding that one belief is both revisable and unrevisable: the belief that some beliefs are revisable.