Justifying Objective Bayesianism on Predicate Languages

Objective Bayesianism says that the strengths of one’s beliefs ought to be probabilities, calibrated to physical probabilities insofar as one has evidence of them, and otherwise sufficiently equivocal. These norms of belief are often explicated using the maximum entropy principle. In this paper we investigate the extent to which one can provide a unified justification of the objective Bayesian norms in the case in which the background language is a first-order predicate language, with a view to applying the resulting formalism to inductive logic. We show that the maximum entropy principle can be motivated largely in terms of minimising worst-case expected loss

Where do we stand on maximal entropy?

Edwin Jaynes’ principle of maximum entropy holds that one should use the probability distribution with maximum entropy, from all those that fit the evidence, to draw inferences, because that is the distribution that is maximally non-committal with respect to propositions that are underdetermined by the evidence. The principle was widely applied in the years following its introduction in 1957, and in 1978 Jaynes took stock, writing the paper ‘Where do we stand on maximum entropy?’ to present his view of the state of the art. Jaynes’ principle needs to be generalised to a principle of maximal entropy if it is to be applied to first-order inductive logic, where there may be no unique maximum entropy function. The development of this objective Bayesian inductive logic has also been very fertile and it is the task of this chapter to take stock. The chapter provides an introduction to the logic and its motivation, explaining how it overcomes some problems with Carnap’s approach to inductive logic and with the subjective Bayesian approach. It also describes a range of recent results that shed light on features of the logic, its robustness and its decidability, as well as methods for performing inference in the logic

There Is No Pure Empirical Reasoning

The justificatory force of empirical reasoning always depends upon the existence of some synthetic, a priori justification. The reasoner must begin with justified, substantive constraints on both the prior probability of the conclusion and certain conditional probabilities; otherwise, all possible degrees of belief in the conclusion are left open given the premises. Such constraints cannot in general be empirically justified, on pain of infinite regress. Nor does subjective Bayesianism offer a way out for the empiricist. Despite often-cited convergence theorems, subjective Bayesians cannot hold that any empirical hypothesis is ever objectively justified in the relevant sense. Rationalism is thus the only alternative to an implausible skepticism

Ideological innocence

Quine taught us the difference between a theory’s ontology and its ideology. Ontology is the things a theory’s quantifiers must range over if it is true, Ideology is the primitive concepts that must be used to state the theory. This allows us to split the theoretical virtue of parsimony into two kinds: ontological parsimony and ideological parsimony. My goal is help illuminate the virtue of ideological parsimony by giving a criterion for ideological innocence—a rule for when additional ideology does not count against parsimony. I propose the expressive power innocence criterion: if the ideology of theory one is expressively equivalent to that of theory two, then neither is ideologically simpler than the other. In its favor I offer the argument from accuracy, showing that any account of a theoretical virtue that is supposed to make theories that have it more likely to be true than theories that do not must respect it. Next I consider its ramifications, eliminating rival views and passing judgment on some arguments from parsimony that can be found in the literature. Finally, I consider two objections. First: I address an objection arising from the possibility of languages with a ‘primitive’ operator that allows us to list a theory’s primitives in the object-language. Second: I address an objection raised by Nelson Goodman against attempts to reckon simplicity by expressive power. Both objections fail

Confirmation and Evidence

The question how experience acts on our beliefs and how beliefs are changed in the light of experience is one of the oldest and most controversial questions in philosophy in general and epistemology in particular. Philosophy of science has replaced this question by the more specific enquiry how results of experiments act on scientific hypotheses and theories. Why do we maintain some theories while discarding others? Two general questions emerge: First, what is our reason to accept the justifying power of experience and more specifically, scientific experiments? Second, how can the relationship between theory and evidence be described and under which circumstances is a scientific theory confirmed by a piece of evidence? The book focuses on the second question, on explicating the relationship between theory and evidence and capturing the structure of a valid inductive argument. Special attention is paid to statistical applications that are prevalent in modern empirical science. After an introductory chapter about the link between confirmation and induction, the project starts with discussing qualitative accounts of confirmation in first-order predicate logic. Two major approaches, the Hempelian satisfaction criterion and the hypothetico-deductivist tradition, are contrasted to each other. This is subsequently extended to an account of the confirmation of entire theories as opposed to the confirmation of single hypothesis. Then the quantative Bayesian account of confirmation is explained and discussed on the basis of a theory of rational degrees of belief. After that, I present the various schools of statistical inference and explain the foundations of these competing schemes. Finally, I argue for a specific concept of statistical evidence, summarize the results, and sketch some open questions. </p

Evidentialism, Inertia, and Imprecise Probability

Evidentialists say that a necessary condition of sound epistemic reasoning is that our beliefs reflect only our evidence. This thesis arguably conflicts with standard Bayesianism, due to the importance of prior probabilities in the latter. Some evidentialists have responded by modelling belief-states using imprecise probabilities (Joyce 2005). However, Roger White (2010) and Aron Vallinder (2018) argue that this Imprecise Bayesianism is incompatible with evidentialism due to “inertia”, where Imprecise Bayesian agents become stuck in a state of ambivalence towards hypotheses. Additionally, escapes from inertia apparently only create further conflicts with evidentialism. This dilemma gives a reason for evidentialist imprecise probabilists to look for alternatives without inertia. I shall argue that Henry E. Kyburg’s approach offers an evidentialist-friendly imprecise probability theory without inertia, and that its relevant anti-inertia features are independently justified. I also connect the traditional epistemological debates concerning the “ethics of belief” more systematically with formal epistemology than has been hitherto done