The logical view: probability as objective degree of belief
The view of probability as an objective degree of belief was developed in the early 20th century by people such as Harold Jeffreys and the the young John Maynard Keynes
Probability theory is in this view seen as an extension of logic: using deductive logic we can tell what logically follows from the premises. If our premises are that “all cats are mammals”, and “Tibbles is a cat”, then it follows logically that “Tibbles is a mammal”.
When there is no logical certainty, we can at least, using probability theory, assess the probability of something, following from the premises. This is sometimes called “inductive logic”. If our premises are merely that “Tibbles is a cat”, and “all cats we have seen so far have been mammals”, then it only follows that “Tibbles is probably a mammal” (since it is always possible that the next cat we observe turns out to be a robot). The statement is only partially entailed by the premises.
The probability of an event is then defined as the degree of partial entailment, because the premises partially entail it. This also means that a probability is always a probability given some premises. This is written as [math]P(h|e)[/math], the probability of [math]h[/math] given [math]e[/math], or, in the cat example, the probability of the hypothesis (all cats are mammals), given the evidence (all cats we have seen so far have been mammals).
We can use this definition to work out how we can learn from experience, as in the cat example. To work out the probability we should assign to the hypothesis, [math]P(h|e)[/math], given the evidence we have for it (and our background assumptions), we need to know the probability of [math]h[/math] (given only our background assumptions), the probability of [math]e[/math] (again, given the background assumptions) and the probability that [math]P(e|h)[/math], i.e. the probability that if [math]h[/math] were true, [math]e[/math] would follow (in the cat example that's simply 1: if the hypothesis were true, then [math]e[/math] would follow). We can then work out [math]P(h|e)[/math]. Then we can use a formula named after the 18th century mathematician Thomas Bayes. The logical interpretation is therefore also often called “objective Bayesianism”.
The actual working out of the probability may take some work, but at least according to the logical interpretation it is at least possible to arrive at an objective (i.e. logically entailed) probability in which it is rational to believe (here we run into a possible confusion with the words objective and subjective: probability under this interpretation is subjective because it is a person's degree of belief rather than a feature of the world. It is however objective in the sense that there is only one degree of belief which may be rationally held).
The problems of the scheme arise in the determining the other probabilities. In particular, when there is absolutely no other evidence which we can use to determine a probability, we have to resort to finding a reasonable alternative. In the classical definition of probability it is assumed that, if there is no information to suggest otherwise, the different outcomes are given equal weight. Thus a die has 6 sides, therefore the probability of getting any one side is simply 1/6. Similarly, the logical interpretation stipulates that if there is no other information, then we need to give a range of outcomes equal probability. Keynes called this the “principle of indifference”.
This principle does, however, gives rise to some paradoxes, which for many philosophers is reason enough to abandon the logical interpretation, as none of the proposed solutions seem completely convincing. Suppose we blindly take a book from a library, but we don't know its colour. We have as much reason to suppose it is red than it is black, so we assign the probability that it is red as ½. However, we also have as much reason to suppose it is yellow than that it is black, and therefore the probability of it being yellow is also ½. This is clearly not right, as the probabilities should all add up to one, yet the probabilities we assigned for red, black and yellow (and any other colour using similar reasoning) add up to more than one.
So, we assign equal probabilities to all the possible colours. Say, if there are five possible colours, then the probability of the random book being black is 1/5.
This solution works well when there are discreet colours and their number is known, say when we draw smarties or lego bricks at random. In the example of the books however, this is not known: the colour spectrum is not discrete, and there can be arguments for example over whether Prussian Blue is a shade or a colour in its own right. But if there is supposed to be only one rationally possible assessment of a probability it shouldn't be affected by such arguments over how to divide the spectrum. However there are worse problems with the principle of indifference. Sometimes to assess a probability there are several different areas where we can assume indifference, and these lead to different results.
The “book paradox” is described by Keynes himself; Keynes 1921 chapter 4, and covered in Gillies 2000