Online research in philosophy

A number of authors have noted that vagueness engenders degrees of belief, but that these degrees of belief do not behave like subjective probabilities. So should we countenance two different kinds of degree of belief: the kind arising from vagueness, and the familiar kind arising from uncertainty, which obey the laws of probability? I argue that we cannot coherently countenance two different kinds of degree of belief. Instead, I present a framework in which there is a single notion of degree of belief, which in certain circumstances behaves like a subjective probability assignment and in other circumstances does not. The core idea is that one’s degree of belief in a proposition P is one’s expectation of P’s degree of truth.

What role, if any, does formal logic play in characterizing epistemically rational belief? Traditionally, belief is seen in a binary way - either one believes a proposition, or one doesn't. Given this picture, it is attractive to impose certain deductive constraints on rational belief: that one's beliefs be logically consistent, and that one believe the logical consequences of one's beliefs. A less popular picture sees belief as a graded phenomenon.

The concept of strength of belief, or degree of confidence, is central to a considerable amount of contemporary work in epistemology, decision theory, statistics, economics and artificial intelligence. This timely and impressive collection of essays brings together a number of important contributions to its philosophical study, authored by some of the most influential figures in the field. FRANZ HUBER’s introduction situates the various subsequent contributions within useful, concise overview of some of the main issues at stake, an overview that covers similar ground to his excellent recent SEP entry (‘Formal Representations of Belief’ (2008) in E.N. Zalta (ed.) Stanford Encyclopaedia of Philosophy). Like the remainder of the book, the scope of this chapter extends beyond coverage of the probabilist paradigm that is so ubiquitous in the mainstream philosophical literature. We find here a clear summary of various alternative models, such as DST belief functions, possibility/necessity measures and ranking functions. Huber’s enthusiasm for the latter transpires fairly clearly, although the claims made regarding its theoretical advantages will no doubt prove to be somewhat contentious. The remainder of the book is then divided into three sections— ‘Plain Belief and Degrees of Belief’, ‘What Laws Should Degrees of Belief Obey?’ and ‘Logical Approaches’—with Part II collecting the bulk of the contributions (seven of twelve articles). Although the issue is also touched upon further on in the volume, Part I is devoted to the relation between reports of degrees of confidence (e.g. ‘I am pretty certain that I locked the car’) and reports of plain belief or disbelief (e.g. ‘I believe I locked the car’). All three papers in this section seem to share a (possibly somewhat uncritical) commitment to the view that, to the extent that one’s degrees of confidence ought to determine the beliefs that one holds, the only plausible mapping is given by the so-called ‘Lockean Thesis’ (LT)..

It is a common view that the axioms of probability can be derived from the following assumptions: (a) probabilities reflect (rational) degrees of belief, (b) degrees of belief can be measured as betting quotients; and (c) a rational agent must select betting quotients that are coherent. In this paper, I argue that a consideration of reasonable betting behaviour, with respect to the alleged derivation of the first axiom of probability, suggests that (b) and (c) are incorrect. In particular, I show how a rational agent might assign a ‘probability’ of zero to an event which she is sure will occur.

Plantinga defines S's belief as ‘privately rational if and only if it is probable on S's evidence’, and ‘publicly rational if and only if it is probable with respect to public evidence’, and he claims that ‘it is an immediate consequence of these definitions that all my basic beliefs are privately rational’. I made it explicitly clear in my review that on my account of a person's evidence (quoted and used by Plantinga) as ‘the content of his basic beliefs (weighted by his degree of confidence in them)’, that is not the case. I emphasize ‘weighted by his degree of confidence in them’. I wrote explicitly: ‘for more or less any belief, however convinced you are of it initially, other evidence of which you are equally convinced could rend it overall improbable’. Put technically, in probabilistic terms, basic beliefs come to us with different degrees of prior probability varying with our degree of confidence in them, but a belief with a high prior probability can in the light of other beliefs of our current set have a lower posterior probability. If you continue to hold on to a basic belief when its probability on the total evidence is below half, that belief is not privately rational. Footnotes1 Note: This brief discussion arises out of Richard Swinburne's critical notice of Alvin Plantinga's Warranted Christian Belief (New York NY: Oxford University Press, 2000) and Plantinga's reply in Religious Studies, 37 (2001), 203–214, 215–222.

Rationalizations of deliberation often make reference to two kinds of mental state, which we call belief and desire. It is worth asking whether these kinds are necessarily distinct, or whether it might be possible to construe desire as belief of a certain sort — belief, say, about what would be good. An expected value theory formalizes our notions of belief and desire, treating each as a matter of degree. In this context the thesis that desire is belief might amount to the claim that the degree to which an agent desires any proposition A equals the degree to which the agent believes the proposition that A would be good. We shall write this latter proposition ‘A◦’ (pronounced ‘A halo’). The Desire-as-Belief Thesis states, then, that to each proposition A there corresponds another proposition A◦, where the probability of A◦ equals the expected value of A.

Degrees of belief are familiar to all of us. Our confidence in the truth of some propositions is higher than our confidence in the truth of other propositions. We are pretty confident that our computers will boot when we push their power button, but we are much more confident that the sun will rise tomorrow. Degrees of belief formally represent the strength with which we believe the truth of various propositions. The higher an agent’s degree of belief for a particular proposition, the higher her confidence in the truth of that proposition. For instance, Sophia’s degree of belief that it will be sunny in Vienna tomorrow might be .52, whereas her degree of belief that the train will leave on time might be .23. The precise meaning of these statements depends, of course, on the underlying theory of degrees of belief. These theories offer a formal tool to measure degrees of belief, to investigate the relations between various degrees of belief in different propositions, and to normatively evaluate degrees of belief.

There has been much discussion about whether traditional epistemology's doxastic attitudes are reducible to degrees of belief. In this paper I argue that what I call the Straightforward Reduction - the reduction of all three of believing p, disbelieving p, and suspending judgment about p, ~p to precise degrees of belief for p, ~p that ought to obey the standard axioms of the probability calculus - cannot succeed. By focusing on suspension of judgment (agnosticism) rather than belief, we can see why the Straightforward Reduction is bound to fail. I argue that, in general, suspending about p is not just a matter of having some specified standard credence for p, and in the end I suggest some ways to extend the arguments that will put pressure on other credence-theoretic accounts of belief and suspension of judgment as well.

What propositions are rational for one to believe? With what confidence is it rational for one to believe these propositions? Answering the first of these questions requires an epistemology of beliefs, answering the second an epistemology of degrees of belief.

John Locke proposed a straightforward relationship between qualitative and quantitative doxastic notions: belief corresponds to a sufficiently high degree of confidence. Richard Foley has further developed this Lockean thesis and applied it to an analysis of the preface and lottery paradoxes. Following Foley's lead, we exploit various versions of these paradoxes to chart a precise relationship between belief and probabilistic degrees of confidence. The resolutions of these paradoxes emphasize distinct but complementary features of coherent belief. These features suggest principles that tie together qualitative and quantitative doxastic notions. We show how these principles may be employed to construct a quantitative model - in terms of degrees of confidence - of an agent's qualitative doxastic state. This analysis fleshes out the Lockean thesis and provides the foundation for a logic of belief that is responsive to the logic of degrees of confidence.