Brady, Michael Emmett and Rogério Arthmar. 2012. “Keynes, Boole and the Interval Approach to Probability,” History of Economic Ideas 20.3: 65–84.An earlier version of the paper is also available:
Brady, Michael Emmett and Rogério Arthmar. 2010. “Keynes’ Lower-Upper Bound Interval Approach to Probability,” SSRN Working Paper Series, February 2If correct, this is an important paper, which confirms Keynes’s genius, not only in the neglected Treatise on Probability (1921), but also in decision making theory, in that Keynes is the founder of modern non-additive, nonlinear probability and decision making theory, and provided it with a mathematical foundation (Brady and Arthmar 2012: 65).
I will not attempt to do anything more than sketch what appear to be the main points, as follows:
(1) Keynes thought only a certain class of probabilities were capable of objective numeric estimates, such as those with equiprobable outcomes with use of the principle of indifference (Brady and Arthmar 2012: 70). Nor did Keynes reject the relative frequency approach as long as it was confined to instances where it was appropriate. The laws of the probability calculus hold only in cases where probabilities are linear and have additive, precise values (Brady and Arthmar 2012: 80).BIBLIOGRAPHY
My additional reading is that, in Keynes’s view, when a probability is not objective in the sense that it is (1) a priori or (2) derived from a relative frequencies, if it can given at all it is non-numeric or represented as an approximate interval. I would assume that such non-numeric and interval probabilities apply to what are now called epistemic probabilities which we attach to the conclusions of inductive arguments.
(2) Keynes did provide a mathematical structure for his approach to probability, including his interval estimates, adopted from the work of Boole (Brady and Arthmar 2012: 69–70). When such probabilities are non-linear and non-additive, they do not follow the laws of the probability calculus (Brady and Arthmar 2012: 80). Some have incorrectly thought Keynes’s non-numeric probabilities were only capable of ordinal ranking, and then only some of the time.
(3) While Keynes briefly sketched his general approach to probability in Chapter 3 of the Treatise, the important and neglected treatment in detail occurred in Part II from Chapters 15 to 29 (Brady and Arthmar 2012: 71).
(4) Brady and Arthmar (2012: 76) point to this passage in Chapter 15 of the Treatise on Probability that summarises Keynes’s view:“5. It is evident that the cases in which exact numerical measurement is possible are a very limited class, generally dependent on evidence which warrants a judgment of equiprobability by an application of the Principle of Indifference. The fuller the evidence upon which we rely, the less likely is it to be perfectly symmetrical in its bearing on the various alternatives, and the more likely is it to contain some piece of relevant information favouring one of them. In actual reasoning, therefore, perfectly equal probabilities, and hence exact numerical measures, will occur comparatively seldom.(5) Ramsey’s reviews of Keynes’s Treatise on Probability badly misunderstood the latter’s approach to probability, and that misunderstanding was passed on to many subsequent commentators (Brady and Arthmar 2012: 79).
The sphere of inexact numerical comparison is not, however, quite so limited. Many probabilities, which are incapable of numerical measurement, can be placed nevertheless between numerical limits. And by taking particular non-numerical probabilities as standards a great number of comparisons or approximate measurements become possible. If we can place a probability in an order of magnitude with some standard probability, we can obtain its approximate measure by comparison. (Keynes 1921: 160).
Brady, Michael Emmett and Rogério Arthmar. 2010. “Keynes’ Lower-Upper Bound Interval Approach to Probability,” SSRN Working Paper Series, February 2
Brady, Michael Emmett and Rogério Arthmar. 2012. “Keynes, Boole and the Interval Approach to Probability,” History of Economic Ideas 20.3: 65–84.
Keynes, John Maynard. 1921. A Treatise on Probability. Macmillan, London.