A Treatise on Probability by Keynes is a very important book, one of its kind, in setting the philosophical and logical foundations of probabilistic reasoning.
Firstly, Keynes address key philosophical questions about the nature of probability, its interpretation and its measurement. Keynes’s conception of probability is that it is a strictly logical relation between evidence and hypothesis, a degree of partial implication. Furthermore, probability is not necessarily numerical and often it is impossible to compare degrees of probability. Numerical probability which allows precise quantification and comparison is a special case.
Secondly, he establishes a rigorous logical framework, along the lines of Russel and Whitehead’s approach to the foundations of mathematics in the Principia Mathematica.
Thirdly, inductive reasoning and the role of analogy are dissected in the established new perspective.
Fourthly, some general semi-philosophical questions are addressed with the new probabilistic understanding.
Finally, the scene is ready for Keynes to delve into statistical inference and elucidates the flaws in the methods proposed so far. While he dismantles the “just compute” approaches, he also presents a constructive alternative, based on comparing multiple, and not just one, series of events.
I wish I had kept notes while reading the book, and wrote this review in a stage-wise process, so that I could comment more deeply about key passage. Nonetheless, I believe the Treatise will accompany me for many years to come, as it contains such lucid and insightful arguments about what we should mean when we talk about probabilistic reasoning.