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This chapter discusses an account of belief-revision that is compatible with the rational belief of contradictions. In the process, a formal account of the model of rationality of the preceding chapter is provided. The account of belief-revision is contrasted with the familiar AGM account.

There is a large spectrum of representations of belief states besides probabilities and ranks. The chapter cannot give a complete survey, but it presents comparisons with a number of important accounts. It first discusses the predecessors of ranking theory: Shackle’s functions of potential surprise, Rescher’s hypothetical reasoning, and Cohen’s account of inductive support. It saves AGM belief revision theory from various critical examples. It carefully discusses Levi’s epistemology and finds discrepancies everywhere. It is critical of Lehrer’s account of knowledge and undefeated justification. Pollock’s theory of defeasible reasoning looks similar, but is quite different. This leads to a discussion of the nature of normative epistemology. The chapter closes with brief comparative remarks on formal learning theory, possibility theory, Dempster-Shafer belief functions, and non-monotonic reasoning.

This chapter introduces and discusses the standard static laws of belief, consistency and deductive closure, as well as the first steps towards a dynamics of belief. The latter amounts to an introduction into AGM belief revision theory and some variant formulations. The chapter thus explains the three kinds of belief change commonly taken as basic: expansions, revisions, and contractions.

The chapter surveys the relevant recent literature in epistemology with two aims in minds. First, the chapter wishes to show that the account of belief revision is orthogonal to, or invariant across, the different ways in which ongoing epistemological debates might be resolved, concerning such matters as the role of experience in founding justifications, and the permissible global patterns of justificatory support (foundationalist, coherentist, or foundherentist?). Secondly, the chapter wishes to show how compatible the new account of belief revision is with various important epistemological accounts that touch on the topic of belief revision, albeit not in any great logical or computational detail. To this end, the chapter examines the writings of Sosa, BonJour, Cornman, Haack, Levi, Quine and Ullian, Harman, Klein and Gupta. This discussion should serve to situate the work for the epistemological reader

This chapter provides further argument justifying the claim that our use of finite dependency networks entails no loss at all of theoretical generality, as far as belief revision on the part of rational creatures is concerned. Some basic concepts in mathematical logic are defined, to lay the groundwork for the metatheorem, due to Harvey Friedman, that is proved in the next chapter.

Section. 8.1 introduces the problem of judgement aggregation and some of the difficulties encountered in trying to solve this problem. Section. 8.2 introduces the theory of belief revision and merging and Pigozzi's application of this theory to judgement aggregation. Section. 8.3 argues that judgements should not be merged directly; rather, one should merge the evidence on which the judgements are based. Given this merged evidence, objective Bayesianism and decision theory can be used to derive an appropriate set of judgements—the resulting judgements should be viewed as the aggregate of the judgements of the original individuals (§8.4).

This chapter turns to the notion of conditional belief: rationally believing a proposition given another proposition. Among other things, conditional belief plays a role for belief revision by entailing a disposition for how to revise one’s belief given new evidence. The numerical version of conditional belief is subjective conditional probability, and first the chapter calls to mind how the concept of conditional probability figures in the standard diachronic norm on degree-of-belief change. Afterwards the same is done for conditional all-or-nothing belief and the standard diachronic norm on all-or-nothing belief change, which rely on the so-called AGM postulates for belief revision. Finally, a stability account of rational conditional belief is developed according to which subjective conditional probability and rational conditional all-or-nothing belief cohere with each other just as their unconditional versions did in Chapters 2 and 3. From this, together with the standard diachronic norms on belief change, it follows that also rational degree-of-belief change and categorical belief change must cohere with each other once the stability theory is in place. Other than extending the stability account to conditional belief, the chapter also serves the purpose of supplying some of the mathematical machinery on which the theory in this book is based.

What we experience of the world—for example, that pig on the lawn—does not bear logical relations to thoughts, or propositions. That pig on the lawn is not of the form of a proposition. Logical relations relate thoughts, or propositions, to one another. Quine thus correctly notes that one could, on no pain of contradiction, respond to any experience with any constellation of beliefs. He incorrectly takes this to mean that any constellation of beliefs could be consistent with one's having experienced what he did. Revision can come anywhere; if it does not, this merely reflects the ways of our people. This is a mistake, and in fact destroys the very possibility of judgement. Quine has given us no reason (even if, in fact, there is one) to think that any proposition may (not incorrectly) be held to have been falsified by experience. So goes the present argument.

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 evidence, but also for revising one’s revision rules in response to potential new evidence. The emphasis in the chapter 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 chapter 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.

Whether a simple reporting practice produces positive changes in knowledge, or V‐value, depends on what reports a message sender sends, who actually receives them, what the receivers’ antecedent degrees of belief are, and how they revise their degrees of belief as a function of the reports. Does any belief‐revision practice have a general propensity to improve a hearer's degrees of knowledge? Assuming objective conditional probabilities, revising degrees of belief by Bayesian conditionalization yields (objectively) expected increases in degrees of knowledge (V‐value), as long as subjective likelihoods match objective likelihoods. This general theorem applies to the special case of reasoning from social communications. As a separate subtopic, theories of justification for testimonial belief are briefly reviewed.

This chapter provides replies to criticisms on the concept of dialetheism or the existence of true contradictions. It clarifies that dialetheism is not incompatible with predication because the former is concerned with the simultaneous truth of statements while the latter is an independent issue. In reply to a criticism on account of negation, it explains that dialetheism only rejects paraconsistent logic, and that there is no logical bar to dialetheism. On the issue of belief-revision, the chapter clarifies that dialetheism does not require abandonment of theories, only the relocation of beliefs to either low or high probabilities.

This chapter considers how the proposed account of the epistemology of mental action, and the proposed account of the self-knowledge involved in conscious thinking, bear on an account of one's knowledge of what one believes. The chapter also explores some of the ways in which the ethics of belief, and accounts of the suspension of judgement and belief revision, may be affected by the fact that we are, as self-conscious subjects, capable of engaging in agential epistemic conscious mental activity, including self-critical reflection.

We review findings from the psychology of science that are relevant to understanding or explaining peoples’ tendencies to believe both scientific and pseudoscientific claims. We discuss relevant theoretical frameworks and empirical findings to support the proposal that pseudoscientific beliefs arise in much the same way as other scientific and non-scientific beliefs do. In particular, we focus on (a) cognitive and metacognitive factors at the individual level; (b) trust in testimony and judgments of expertise at the social level; and (c) personal identity and the public’s relationship with the scientific community at a cultural level.

The chapter presents an account of the meaning of “must” and “can” that relies on general principles for reasoning from premise sets. Premise sets can be inconsistent, so the account has to allow for conflict resolution. There are many areas in human cognition where inconsistencies arise. Inconsistent beliefs are the topic of theories of belief revision; conflicting grammatical constraints are investigated in Optimality Theory; and clashes between presuppositions, conversational implicatures, and at issue meanings have been explored in formal pragmatics since the late 1970s. With potentially inconsistent premise sets, the presentation of premises matters. Deductively equivalent premise sets like {p&q} and {p, q} can show different behavior when a conflicting premise is added. The chapter takes this fact as an invitation for a promising research program that seeks evidence for the way the premises from which we reason are split up and lumped together. The semantics of counterfactuals is a rich source for such evidence, as shown in subsequent chapters.

We situate Core Logic and Classical Core Logic within a wider logical landscape. Core Logic lies at the intersection of two orthogonal lines of reform of Classical Logic—constructivization and relevantization. We explain the genesis of Core Logic and describe its carefully formulated rules of inference. We reveal how Core Logic arises as a smooth generalization of the proto-logic involved in working out the truth values of sentences under particular interpretations; and the case for the complete methodological adequacy of Core Logic for constructive deductive reasoning, and of Classical Core Logic for non-constructive deductive reasoning. Core Logic deserves the label ‘Core’, because it is both fully employed, and sufficient, as the metalogic involved in any process of rational belief revision. No rule of Core Logic can be surrendered. We end by speculating on two possible explanations—semantic and methodological—of how Core Logic might have been bloated to Classical Logic.