Iterative and fixed point common belief

We define infinitary extensions to classical epistemic logic systems, and add also a common belief modality, axiomatized in a finitary, fixed-point manner. In the infinitary K system, common belief turns to be provably equivalent to the conjunction of all the finite levels of mutual belief. In contrast, in the infinitary monotonic system, common belief implies every transfinite level of mutual belief but is never implied by it. We conclude that the fixed- point notion of common belief is more powerful than the iterative notion of common belief.     

A Practical Approach to Revising Prioritized Knowledge Bases

This paper investigates simple syntactic methods for revising prioritized belief bases, that are semantically meaningful in the frameworks of possibility theory and of Spohn's ordinal conditional functions. Here, revising prioritized belief bases amounts to conditioning a distribution function on interpretations. The input information leading to the revision of a knowledge base can be sure or uncertain. Different types of scales for priorities are allowed: finite vs. infinite, numerical vs. ordinal. Syntactic revision is envisaged here as a process which transforms a prioritized belief bases into a new prioritized belief base, and thus allows a subsequent iteration.     

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.

     

Belief Change in Branching Time: AGM-consistency and Iterated Revision

We study belief change in the branching-time structures introduced in Bonanno (Artif Intell 171:144–160, 2007). First, we identify a property of branching-time frames that is equivalent (when the set of states is finite) to AGM-consistency, which is defined as follows. A frame is AGM-consistent if the partial belief revision function associated with an arbitrary state-instant pair and an arbitrary model based on that frame can be extended to a full belief revision function that satisfies the AGM postulates. Second, we provide a set of modal axioms that characterize the class of AGM-consistent frames within the modal logic introduced in Bonanno (Artif Intell 171:144–160, 2007). Third, we introduce a generalization of AGM belief revision functions that allows a clear statement of principles of iterated belief revision and discuss iterated revision both semantically and syntactically.     

Stakes and beliefs

The idea that beliefs may be stake-sensitive is explored. This is the idea that the strength with which a single, persistent belief is held may vary and depend upon what the believer takes to be at stake. The stakes in question are tied to the truth of the belief—not, as in Pascal’s wager and other cases, to the belief’s presence. Categorical beliefs and degrees of belief are considered; both kinds of account typically exclude the idea and treat belief as stake-invariant, though an exception is briefly described. The role of the assumption of stake-invariance in familiar accounts of degrees of belief is also discussed, and morals are drawn concerning finite and countable Dutch book arguments.     

A General Family of Preferential Belief Removal Operators

Most belief change operators in the AGM tradition assume an underlying plausibility ordering over the possible worlds which is transitive and complete. A unifying structure for these operators, based on supplementing the plausibility ordering with a second, guiding, relation over the worlds was presented in Booth et al. (Artif Intell 174:1339–1368, 2010). However it is not always reasonable to assume completeness of the underlying ordering. In this paper we generalise the structure of Booth et al. (Artif Intell 174:1339–1368, 2010) to allow incomparabilities between worlds. We axiomatise the resulting class of belief removal functions, and show that it includes an important family of removal functions based on finite prioritised belief bases.     

Hume on Miracles: Would It Take a Miracle to Believe in a Miracle?

Given Hume's theory of belief and belief production it is no small task to explain how it is possible for a belief in a miracle to be produced. I argue that belief in a miracle cannot be produced through Hume's standard causal mechanisms and that although education, passion, and testimony initially seem to be promising mechanisms for producing belief in a miracle, none of these is able to produce the belief in a miracle. I conclude by explaining how this poses a problem for Hume's theory of belief and I briefly investigate the alternatives available for solving this problem.     

Changes in cooperative behaviour and belief in finitely repeated social dilemmas

Economic and psychological studies have determined the tendency to improve cooperation in initial interactions in which people are required to contribute in public good games. The current study examined the behavioural changes of individuals during finite interaction times. Participants were invited to make decisions twice in a step‐level public good game, and their belief in achieving the collective interest was measured. Participants were divided into three groups based on their first decisions. Results showed that high contributors demonstrated high levels of contribution in their subsequent interactions, whereas low and equal contributors increased their levels of contribution after the initial interaction. Additionally, the cooperative level of low contributors increased along with their cooperative belief. The theoretical and practical implications of changes of behaviour and belief in repeated social dilemmas were identified.     

Taking aim at the truth

One prominent feature of belief is that a belief cannot be formed at will. This paper argues that the best explanation of this fact is that belief formation is a process that takes aim at the truth. Taking aim at the truth is to be understood as causal responsiveness of the processes constituting belief formation to what facilitates achieving true beliefs. The requirement for this responsiveness precludes the possibility of belief formation responding to intentions in a way that would count as forming a belief at will.     

Conditional Ranking Revision

In the context of a general framework for belief dynamics which interprets revision as doxastic constraint satisfaction, we discuss a proposal for revising quasi-probabilistic belief measures with finite sets of graded conditionals. The belief states are ranking measures with divisible values (generalizing Spohn’s epistemology), and the conditionals are interpreted as ranking constraints. The approach is inspired by the minimal information paradigm and based on the principle-guided canonical construction of a ranking model of the input conditionals. This is achieved by extending techniques known from conditional default reasoning. We give an overview of how it handles different principles for conditional and parallel revision and compare it with similar accounts.     

Revocable Belief Revision

Krister Segerberg proposed irrevocable belief revision, to be contrasted with standard belief revision, in a setting wherein belief of propositional formulas is modelled explicitly. This suggests that in standard belief revision is revocable: one should be able to unmake (‘revoke’) the fresh belief in the revision formula, given yet further information that contradicts it. In a dynamic epistemic logical setting for belief revision, for multiple agents, we investigate what the requirements are for revocable belief revision. By this we not merely mean recovering belief in non-modal propositions, as in the recovery principle for belief contraction, but recovering belief in modal propositions: beliefs about beliefs. These requirements are almost never met, a surprising result.     

Accidental Truth and Accidental Justification

The presence of accidentality in a subject's holding a belief is generally thought to be incompatible with that belief7apos;s being an instance of know-ledge. Epistemologists have tended to focus on the accidentality of the truth of a belief. I argue (a) that it is also possible that there be accidentality in the justification of a belief, and (b) that the consequences of this type of accidentality are potentially much more harmful than accidentality only in the truth of a belief.     

Probabilistic dynamic belief revision

We investigate the discrete (finite) case of the Popper–Renyi theory of conditional probability, introducing discrete conditional probabilistic models for knowledge and conditional belief, and comparing them with the more standard plausibility models. We also consider a related notion, that of safe belief, which is a weak (non-negatively introspective) type of “knowledge”. We develop a probabilistic version of this concept (“degree of safety”) and we analyze its role in games. We completely axiomatize the logic of conditional belief, knowledge and safe belief over conditional probabilistic models. We develop a theory of probabilistic dynamic belief revision, introducing probabilistic “action models” and proposing a notion of probabilistic update product, that comes together with appropriate reduction laws.     

Reincarnation and infinite punishment in hell

In the first part of the paper, I argue that Christians should incorporate the theory of reincarnation into their belief system. The problem of the apparent disproportion between finite human sin and infinite punishment in Hell becomes far more tractable against the background of reincarnation. In the second part of the paper, I address and answer three objections that may be raised against a Christian theory of reincarnation. The first objection is based on the role of memory in identity, the second points to the essential unity of body and soul, and the third revolves around the suggestion that living multiple lives may more easily lead to damnation than to salvation.     

Objectivist conditions for defeat and evolutionary debunking arguments

I make a case for distinguishing clearly between subjective and objective accounts of undercutting defeat and for rejecting a hybrid view that takes both subjective and objective elements to be relevant for whether or not a belief is defeated. Moderate subjectivists claim that taking a belief to be defeated is sufficient for the belief to be defeated; subjectivist idealists add that if an idealised agent takes a belief to be defeated then the belief is defeated. Subjectivist idealism evades some of the objections levelled against moderate subjectivism but can be shown to yield inconsistent results in some cases. Both subjectivisms should be rejected. We should be objectivists regarding undercutting defeat. This requirement, however, is likely to be problematic for a popular interpretation of evolutionary debunking arguments in metaethics as it can be shown that existing objectivist accounts of defeat do not support such arguments. I end by discussing the constraints of developing such an account.     

Memory belief is weak

Recently there has been extensive debate over whether “belief is weak”, viz, whether the epistemic standard for belief is lower than for assertion or knowledge. While most current studies focus on notions such as “ordinary belief” and “outright belief”, this paper purports to advance this debate by investigating a specific type of belief; memory belief. It is argued that (outright) beliefs formed on the basis of episodic memories are “weak” due to two forms of “entitlement inequality”. My key argument is thus twofold. First, by rejecting the epistemic theory of memory, I argue that one can be entitled to belief but not to knowledge. Second, by scrutinising a recent defence of the belief norm of assertion, it will be demonstrated that belief is weaker than assertion, as long as we expect one to match words with deeds.     

The Scientific Impotence Excuse:Discounting Belief‐Threatening Scientific Abstracts

The scientific impotence discounting hypothesis predicts that people resist belief‐disconfirming scientific evidence by concluding that the topic of study is not amenable to scientific investigation. In 2 studies, participants read a series of brief abstracts that either confirmed or disconfirmed their existing beliefs about a stereotype associated with homosexuality. Relative to those reading belief‐confirming evidence, participants reading belief‐disconfirming evidence indicated more belief that the topic could not be studied scientifically and more belief that a series of other unrelated topics could not be studied scientifically. Thus, being presented with belief‐disconfirming scientific evidence may lead to an erosion of belief in the efficacy of scientific methods.     

Basic Infobase Change

Generalisations of theory change involving arbitrary sets of wffs instead of belief sets have become known as base change. In one view, a base should be thought of as providing more structure to its generated belief set, and can be used to determine the theory change operation associated with a base change operation. In this paper we extend a proposal along these lines by Meyer et al. We take an infobase as a finite sequence of wffs, with each element in the sequence being seen as an independently obtained bit of information, and define appropriate infobase change operations. The associated theory change operations satisfy the AGM postulates for theory change. Since an infobase change operation produces a new infobase, it allows for iterated infobase change. We measure iterated infobase change against the postulates proposed by Darwiche et al. and Lehmann.     

The interference problem for the betting interpretation of degrees of belief

The paper’s target is the historically influential betting interpretation of subjective probabilities due to Ramsey and de Finetti. While there are several classical and well-known objections to this interpretation, the paper focuses on just one fundamental problem: There is a sense in which degrees of belief cannot be interpreted as betting rates. The reasons differ in different cases, but there’s one crucial feature that all these cases have in common: The agent’s degree of belief in a proposition A does not coincide with her degree of belief in a conditional that A would be the case if she were to bet on A, where the belief in this conditional itself is conditioned on the supposition that the agent will have an opportunity to make such a bet. Even though the two degrees of belief sometimes can coincide (they will coincide in those cases when the bet has no expected causal bearings on the proposition A and the opportunity to bet have no evidential bearings on that proposition), it is the latter belief rather than the former that guides the agent’s rational betting behaviour. The reason is that this latter belief takes into consideration potential interferences that bet opportunities and betting itself might create with regard to the proposition to be betted on. It is because of this interference problem that the agent’s degree of belief in A cannot be interpreted as her betting rate for A.     

Dealing with the Intellectual-Insight Problem in Cognitive and Rational Emotive Behavior Therapy

A client is said to have intellectual, but not emotional, insight when he or she acknowledges that holding a particular belief is irrational, but says that he or she still does not believe, or cannot accept, that fact. The intellectual-insight problem may arise because the client's negative core or irrational belief is embedded in a broader belief system that must be explored before the client will be able to surrender the negative core or irrational belief. In this case one appropriate intervention is to ask what the consequence of giving up the belief would be. Kelly's (1955) theory of personal constructs and DiGiuseppe's (1991) concept of personal paradigms offer ways to conceptualize the problem and its solution.