Philosophy of Probability

What it asks

A fair coin has a probability of one half of landing heads. A patient has a probability of eighty per cent of surviving five years. There is a sixty per cent chance of rain tomorrow. Those three sentences use one word and it is not clear they use it in one sense.

The coin can be tossed repeatedly. The patient will do one thing once. Tomorrow will happen once and either be wet or not. So the question is what a probability statement is about - a frequency, a physical tendency, a logical relation between evidence and hypothesis, or the speaker's confidence.

This branch has no parent among the core divisions. It sits between logic and epistemology and belongs wholly to neither.

The recurring problems

The reference class problem. A frequency is always a frequency within a class, and any individual belongs to indefinitely many. A patient is a fifty-year-old, a smoker, a Scot, a marathon runner. Each class gives a different number and nothing in the theory says which is correct.

The single case. Frequentism handles repeatable trials well and says nothing about events that happen once, which includes most events anyone cares about.

Where the priors come from. Bayesian probability updates a starting confidence on evidence, and the mathematics is unimpeachable. Where the starting confidence comes from, and whether two rational people may begin anywhere they like, is the standing objection.

Connecting chance to belief. If a coin has an objective chance of one half, your confidence should be one half. Everyone agrees; stating the principle without circularity is hard, and Lewis's attempt to do it in 1980 generated a literature of its own.

How it developed

The mathematics began in the seventeenth century with correspondence about gambling problems, and by Laplace's time there was a full classical theory resting on counting equally possible cases - which works for dice and gives no guidance about anything else.

The nineteenth century replaced it with frequency. A probability is the limiting proportion in a long run, which is objective, measurable and silent about single cases.

The 1920s and 1930s produced the subjective alternative. Ramsey and de Finetti independently argued that a probability is a degree of belief, and showed that anyone whose degrees of belief violate the axioms can be offered a set of bets they are guaranteed to lose - which turns coherence into a requirement of rationality rather than a convention. Kolmogorov axiomatised the mathematics in 1933, deliberately leaving the interpretation open.

Popper proposed propensities, physical tendencies attaching to single cases, largely to make sense of quantum mechanics. Carnap pursued a logical interpretation. Neither became standard, and the field has been three-cornered since.

The positions within it

Frequentism, subjective Bayesianism, objective Bayesianism, propensity theory and logical probability are the standing interpretations, all covered in the Schools section.

The dispute is unusual in philosophy for having direct institutional consequences: which interpretation is assumed determines how clinical trials are designed and how evidence is presented in court.

Where it stands now

Bayesianism has won in practice across statistics, machine learning and much of science, without winning the philosophical argument about priors.

The pressure now comes from cases where it visibly fails: courtroom reasoning, where base rates are routinely mishandled with serious consequences, and machine learning systems whose confidence outputs are treated as probabilities without anyone being clear what they are probabilities of.