Logical relation theory of probability

Theory due especially to English economist John Maynard Keynes (1883-1946) in his Treatise on Probability (1921), Chapter 1.

It says that the probability of a hypothesis is a logical relation (rather like logical entailment, only weaker) between a hypothesis and a body of evidence for it. Probability is thus made relative to evidence.

This could be avoided by considering all the evidence (requirement of total evidence), but there are difficulties in specifying this.

The relation in question is hard to specify, and would not give an analysis of ‘probably’ anyway, for if we have some evidence which entails a conclusion we can assert the conclusion; but if it only makes it probable (‘probabilifies’ it) we can only say ‘the conclusion is probable’, without saying what this means (unless we are saying merely that the evidence exists, without saying what it does, that is without using it).

The theory is also subject to various paradoxes.

Source:
H E Kyburg, Probability and Inductive Logic (1970), ch. 5

Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of these outcomes is called an event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes, which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion. Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability theory describing such behaviour are the law of large numbers and the central limit theorem.

As a mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of data.^[1] Methods of probability theory also apply to descriptions of complex systems given only partial knowledge of their state, as in statistical mechanics. A great discovery of twentieth-century physics was the probabilistic nature of physical phenomena at atomic scales, described in quantum mechanics.