A Treatise on Probability was published by John Maynard Keynes while at Cambridge University in 1921. The Treatise attacked the classical theory of probability and proposed a "logical-relationist" theory instead.^[1] In a 1948 review, Bertrand Russell, the co-author of Principia Mathematica, called it "undoubtedly the most important work on probability that has appeared for a very long time," and a "book as a whole is one which it is impossible to praise too highly."^[2]

The Treatise is fundamentally philosophical in nature despite extensive mathematical formulations.^[3] The Treatise presented an approach to probability that was more subjective than the highly quantified classical version. Keynes saw numerical probabilities as special cases of probability, which did not have to be quantifiable or even comparable.^[4]

Keynes used the example of taking an umbrella in case of rain to express the idea of "irreducible uncertainty," a type of probability that is not only non-numerical but also not cardinal – incapable of being compared.^[5]

Is our expectation of rain, when we start out for a walk, always more likely than not, or less likely than not, or as likely as not? I am prepared to argue that on some occasions none of these alternatives hold, and that it will be an arbitrary matter to decide for or against the umbrella. If the barometer is high, but the clouds are black, it is not always rational that one should prevail over the other in our minds, or even that we should balance them, though it will be rational to allow caprice to determine us and to waste no time on the debate.^[6]

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