Interpretations of Probability: There have been many interpretations of what probability is. I will sketch five, namely two more or less old-fashioned ones; two currently fashionable ones; and my own theory of cardinal probability, that is compatible with the last two.
Again, to treat the subject of the interpretations of probability well, one needs at least a book, and so I will limit myself to the four interpretations that make some sense, and refer the reader to Weatherford for a book-length survey of the field, followed by my own theory, which will be considered in some more detail later on in this paper.
Logical interpretation: The logical interpretation seems to be the oldest interpretation if what probability is, and is often rendered as "probability is the ratio of selected cases to possible cases". Thus, the probability of throwing a 4 with an ordinary die is 1/6, and the probability of throwing an even number with an ordinary die is 3/6.
As will be seen, there is an underlying assumption that all possibilities that are distinguished count for the same and as one, which much simplifies the treatment of problems involving probability, but cannot easily or at all deal with weighted dice or different and unpredictable lengths of life.
Empirical interpretation: The empirical interpretation soon followed the logical interpretation, and tends to look at the actual frequencies with which distinguished possibilities in fact happen as information for what the probability of an event is.
This works well in practice with subjects where one can easily establish frequencies or samples, but this also makes it difficult to say what probability really is, both for such things as have frequencies, since these may change and anyway are partial information, and for such things as have no frequencies, like unique events and future events.
Both the logical and the empirical interpretation are somewhat old-fashioned, though one still meets the empirical interpretation in social statistics.
Objective interpretation: This can be best rendered in the form of two claims, namely (1) there is real chance in the world, in the form of chance processes and chance events in physics, and real contingency in life and free choice and (2) probability theory provides the tools to represent its basic properties.
This can be seen as motivated by physics: According to quantum-mechanics there are real chance processes in nature. Until the rise of quantum-mechanics, all physical theories were deterministic, and probabilities only entered because one nearly always has incomplete knowledge and samples of populations. With the rise of quantum-mechanics this supposed determinism of nature had to be given up.
Subjective interpretation: There are various subjective or personal theories of probability. One way of rendering their intuitive basis is in terms of two claims:
(1) Persons have their own personal estimates of probabilities, which, if they are consistent, indeed behave according to probability-theory, and (2) these probabilities can be used for Bayesian confirmation.
This does justice to the fact that different persons may have different estimates of what is the probability of something, and enables each person to recalculate his original probabilities when given new evidence.
The first claim can be spelled out in quite a few different ways, based on different considerations, but these will not occupy us here since we assume its conclusion anyway.
It is especially the second claim which makes subjective interpretations useful. The reason is that while Bayes' Theorem is a rather elementary theorem of formal probability theory, applying Bayes' Theorem requires that one has p(T), i.e. the probability of a theory of which one desires to recalculate the probability of given new evidence about the predictions of the theory, and this one does not have on the standard non-subjective interpretations, for whatever theories represent, these things cannot be counted like cherries, and anyway will at least start to be largely unknown for new theories.
The reason one does not have this on the standard non-subjective interpretations is a fundamental lack of knowledge about the hypothesis T. But since one does have this on subjective interpretations, one may make any guess about the probability of any statement, provided only it is consistent with one's further assumptions. The set-back of this is that if this is wholly subjective, one can in principle fix it so that almost any evidence will have hardly any effect on it. Thus, not only need subjective probabilities not be based on the evidence, but they also can be chosen so extreme as to make almost any evidence have almost no effect.
Cardinal interpretation: The interpretation of probability I propose I call the cardinal interpretation, because it rests on the existence of cardinal numbers, which are guaranteed by non-probabilistic assumptions, namely, those given for extension and number and which exist anyway.
Hence there always will be some probability for any statement, and this probability will exist objectively because it derives from the cardinal numbers of the sets that are involved.
One set-back is that normally one does not know the cardinal probability, though one can normally establish evidence for such statements as represent things that can be counted empirically, rather as in the empirical interpretation of probability.
Another set-back is that one cannot count the things that are represented by a theory. The way to solve that problem is to make an assumption about the probability of a theory that is consistent with the rest of probability theory, and does not depend on personal whim but on logic. It is what I called the abductive condition: The probability of a theory T on background knowledge K is the probability of its least probable known proper consequence on K.
This amounts to a strengthening of the following theorem of any standard formal probability theory:
(T)(Q) (T |= Q --> p(T)Rp(Q) )
That is: For any statements T and Q, if T does entail (explain) Q, then the probability of T is not larger than the probability of Q.
The abductive condition strengthens this inequality to an equality in the case that Q is the least probable of T's known consequences, and it does so to obtain a probability for T - that then can be changed by any incoming evidence by using Bayes' Theorem.
The cardinal interpretation of probability is compatible with the objective interpretation, and is like the subjective interpretation in enabling the use of Bayesian confirmation, but it does not make this subjective, though it does make this dependent on such evidence as one has, including such consequences of the theory one has established.
Note it also has the interesting consequence that wherever we have a domain of sets we have implied probabilities for the sets, which exist as much as do the cardinal numbers of these sets - but that very often we don't have enough information to determine these cardinal numbers, and accordingly the best we can do is to make a guess about it, and try to confirm or infirm that guess by evidence.