Bayesian probability theory

From a book review:

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The mathematical theory of probability allows us to derive more probabilities once we have some probabilities. To give a simple example, given a probability of x for proposition P, the theory tells us that the probability of ¬P equals 1 – x; but it does not provide the probability of P, or ¬P, or any other proposition ex nihilo. These probabilities we have to start with, and that are not provided by probability theory itself, are usually called the prior probabilities or just priors. How do we come to our priors? This question is differently answered by subjectivists and those hoping for some kind of logical interpretation of probability. According to the former, our priors are just our subjective degrees of confidence. These, of course, may be vastly different for different individuals, but so be it. Those in favour of logical probability hold that there is some objective, or at least more objective, way of determining prior probabilities. It has been suggested, for instance, that the prior probability of a given proposition can be determined on the basis of the syntactical structure of the sentence expressing that proposition, a suggestion that, however, led already to insurmountable difficulties for very simple artificial languages. Other suggestions have proven equally problematic.

http://www.arsdisputandi.org/publish/articles/000041/article.pdf