In pure mathematics, the term "probability" simply refers to a non-negative, additive set function that takes a maximum value of 1. But this definition, while clear and precise, does not help us understand how the term is applied to the world, e.g. in empirical research, by insurance companies, or in everyday assertions like "Jane will probably catch her flight" (Kyburg & Smokler, 1980). There have been three main proposals for connecting the mathematical function with its use in statements about world: the empirical or frequentist, the logical, and the subjective (Kyburg & Smokler, 1980; see also Hájek, 2012).
The frequentist interpretation
In its first formulation, the empirical or frequentist conception identified probability with the limit of a relative frequency (Reichenbach, 1949; Venn, 1886; von Mises, 1951). The probability of a coin landing heads was held to be equal to the limit of the relative frequency of heads among the tosses, as the number tosses is increased towards infinity (Kyburg & Smokler, 1980). This interpretation runs into problems, for instance because it is difficult to explain the step from a sample frequency to a probability without a further conception of probability that is not itself based on frequencies (de Finetti, 1937/1980, p. 110; Hájek, 2012).
71 An alternative version of the frequentist interpretation is as a theoretical concept that receives its meaning through the rules or procedures through which it is applied (Braithwaite, 1953), in particular the rules for rejecting a statistical hypothesis given the evidence of a sample with specific characteristics (Fisher, 1956; Neyman, 1952).
Common to the above versions of the frequentist, empirical position is that a probability statement is an assertion about the world, like an assertion about length or weight, which can be true or false. The evidence for it is observational: in order to find out whether a probability statement is true or false, we must conduct an empirical investigation. This investigation will lead to one or more sample frequencies, e.g. a coin cannot be tossed indefinitely, and so will not usually terminate with certainty, but with a probability judgment, in some other sense, about the objective frequency (Kyburg & Smokler, 1980).
The logical interpretation
In contrast, the logical conception of probability holds that probabilities are not empirical but logical statements: statements about the logical relation between a hypothesis (one statement) and a body of knowledge or evidence (another statement or set of statements) (Kyburg & Smokler, 1980). Crucially, it argues that for a hypothesis and body of evidence, there is a single probability value that the hypothesis can take, given the evidence (Carnap, 1950/1962; Hintikka, 1965; Keynes, 1921). For example, Carnap (1950/1962) built an axiomatic system for probability theory based on this logical interpretation, and always hoped there could be a set of axioms that would rule out all but one acceptable probability function given a body of evidence. But he never found a set of intuitively acceptable axioms that led to this result (Kyburg & Smokler, 1980).
The subjectivist interpretation
The subjectivist conception probability is similar to the logical one, but differs from the latter in holding that there is no single probability function that is rationally acceptable, given some body of evidence. In the subjectivist view, probabilities represent degrees of belief. A hypothesis can be assigned any probability between 0 and 1, given a body of evidence, depending on the inclination of the person whose degrees of belief the probability represents. But this does not mean that there are no rational constraints on people's degrees of belief. The subjectivist theory of probability is a logical theory in the sense that only certain combinations of degrees of belief in related statements are admissible (Kyburg & Smokler, 1980). In line with this, the founders of subjective probability theory described it as the logic of partial belief
72 (Ramsey, 1926/1990), and as the logic of uncertainty (de Finetti, 1972), and contrasted it with classical logic as the logic of full belief or of certainty.
The constraints on the relations between people's degrees of belief in related statements are given by coherence (de Finetti, 1937/1980; Ramsey, 1926/1990), which is a generalisation of logical consistency to cover degrees of belief. The probabilities of two statements are coherent if and only if they respect the axioms of probability theory.4 For example, if we believe it's 80% likely to rain today, then to be coherent, we would also have to be willing to believe it's 20% likely not to rain today, otherwise the probabilities used to represent our degrees of belief would not sum to 1. The logical conception of probability also includes the constraint of coherence, but in the subjectivist interpretation coherence it is the only normative constraint (Kyburg & Smokler, 1980).
The relation between the three notions of probability can be illustrated with an urn example. Imagine there is an urn with a number of balls inside, some of which have a blue dot and others not. Imagine we draw a ball at random. What is the probability that this ball will have a blue dot? On the frequentist interpretation, we would first have to draw a series of balls before we could make an estimate based on the relative frequency of balls with blue dots in our sample. Questions about the probabilities of single events do not make sense unless they can be related to similar events (or chains of events) for which frequency information is available. On the logical interpretation, we could argue as follows. Given that the only evidence we have is that some of the balls have a blue dot and others do not, it is rational to apply the principle of indifference and say, as our best guess, that the probability of drawing a ball with a blue dot is .5. On the subjectivist interpretation, we could use the same principle and also respond .5. But our neighbour could object, pointing out that the balls with no blue dot may have a green, red, or orange dot instead. Applying the principle of indifference to this new partition of the parameter space, it would be reasonable to respond that the probability of drawing a ball with a blue dot is .25. Our neighbour could add that the original partition is not more or less correct than the new partition, as both depend on the subjective question of how the domain of the problem is interpreted. However, on a subjectivist interpretation our neighbour and ourselves would nonetheless converge in our probability judgments as we sampled more and more balls from the urn, as long as we both conformed to probability theory and so made coherent judgments (Howson & Urbach, 2006).
The present thesis follows the subjectivist interpretation of probability, studying people's degrees of belief and the extent to which these are subject to the constraints of coherence. But the subject matter and results of the thesis do not presuppose a subjectivist interpretation of
4 There are different axiomatic systems for probability theory even within the subjectivist view (Adams,
1998, Appendix 1; De Finetti, 1937/1980, pp. 60-61), but unless otherwise specified in this thesis, the axioms can be taken as those of Kolmogorov: (1) non-negativity: P(p) 0; (2) normalisation: P(tautologies) = 1; (3) finite additivity: P(p or q) = P(p) + P(q) for all p and q that are logically incompatible, i.e. that are disjunct (Hájek, 2012).
73 probability in order to be meaningful. Within a frequentist or logical perspective, people can still have degrees of belief, and the relation between degrees of belief and coherence constraints may be of interest independently of the relation between degrees of belief and probabilities.
HOW CAN WE MEASURE DEGREES OF BELIEF, AND WHY WOULD WE WANT