This paper was originally written in 1987 in response to Cohen’s 1986 paper,” Twelve Questions ab... more This paper was originally written in 1987 in response to Cohen’s 1986 paper,” Twelve Questions about Keynes’s Concept of Weight. “It is the only one of approximately 200 papers written by me between 1979 and 1992 that I did not destroy in early 2025.
This paper is the foundation for some six years of continuous correspondence between myself and Cohen over his claim that Keynes
“… does not suggest any method by which weights might be measured and in fact admits that often one cannot even compare the weights of different arguments.” (Cohen,1986, p.264).
between 1988 and 1993.Cohen is correct that Keynes provided no way of measuring the evidential weight of the argument in chapter VI. Cohen’s conclusion is superior to that of, for just one instance, Isaac Levi in his 1967 Gambling with Truth, which was that Keynes had provided NO index defined on the interval [0,1] and measured V by K ,the total amount of evidence. Cohen’s conclusion was also maintained by Henry E Kyburg and Rudolf Carnap.
All of these illustrious logicians, however, overlooked that Keynes DID define a measure of evidential weight in chapter 26 of Keynes’s A treatise on Probability.
Specifically, Keynes stated the following on page 315 of chapter 26 of his A Treatise on Probability:
“We could, if we liked, define a conventional coefficient c of weight and risk, such as c = 2pw/ (1 +q) (1 +w) , where w measures the ‘weight,’ which is equal to unity when p = 1 and w = 1, and to zero when p = 0 or w =0 and has an intermediate value in other cases.∗”
The reader should note that Keynes does exactly what Cohen(and Levi) claim Keynes did not do, which was to define an index to measure the evidential weight of the argument on the unit interval [0,1].
The star footnote that Keynes appended shows how one applies the conventional coefficient of weight and risk, c, which is a substitute for Keynes’s interval valued analysis in chapters III, XV, XVII, XX and XXII of the A Treatise on Probability. Keynes states:
“∗If pA = p’A’, w >w’, and q =q’, then cA>c’A’; if pA=p'A’, w =w’, and
q <q’, then cA >c’A’; if pA=p’A’, w >w’, and q <q’ then cA>c’A’; but if
pA =p’A’, w= (this should read as w>w’) w’, and q>q’, we cannot compare cA and c’A’.”(Keynes,1921, p.315, ft.2)
Cohen was not convinced by my letters, however, even though they contained the material presented above .However, he did intervene, indirectly ,“behind the scenes “,in order to make sure that the papers I had submitted, in early 1990 and early 1991, were eventually published in mid-1993 and mid- 1994 in The British Journal for the Philosophy of Science(BJPS) and International Studies for the Philosophy of Science(ISPS) ,respectively. Both papers dealt with Keynes’s c coefficient, the measurement of V by w, where w is an index defined on the interval [0,1], and numerous applications. Numerous additional submissions on the index for w and applications using the c coefficient, made to the BJPS and ISPS after 1994 ,were unanimously rejected by all referees; however, three of the rejected submissions were successfully published later on in 2000 and 2001 in The Journal of Applied Economics and Econometrics, due to the intervention of Dr. K Puttaswamaiah, editor in chief.
I am only going to publish it now due to the appearance of an error-filled paper published in September ,2025 by J.Brekel in Erkinntnis,in which he lists Cohen’s paper in his references, but never actually cites Cohen’s paper anywhere in his paper. His 2025 paper is a revised version of his error- filled 2022 MA Thesis. Brekel makes the claim, first made by Baylis (1935) and many others, that Keynes had provided contradictory measurements for V, even though it is impossible to do this, as V is a logical relation. Levi correctly pointed out that NO CORRECT MEASUREMENT OF V IS POSSIBLE unless an index to measure evidential weight has been provided for it.
Moscati (2025), in a review of Zappia’s 2025 Uncertainty in economics, makes statements at the en... more Moscati (2025), in a review of Zappia’s 2025 Uncertainty in economics, makes statements at the end of his review which are pretty close to being correct about the lack of Keynes’s logical theory of probability having any influence, but those statements are not new, novel, original, unique, innovative or creative. Hishiyama made the same ,nearly identical ,observation 56 years ago. Moscati’s statement, that
“… I find the Keynesian interpretative framework adopted by Zappia problematic. One major issue is that Keynes and his ideas on uncertainty have, in my view, played only a marginal role in the development of the economic theory of decision-making under risk and uncertainty. In the decision theory literature since the 1950s, references to Keynes are very rare.” (Moscati,2025, p.4), is very close to being correct. We need to replace one word in the second sentence of his statement above. That one word is “marginal”. We replace it with the word, “no”, to obtain the following amended version: “…, played no role in the development of the economic theory of decision-making under risk and uncertainty.” This conclusion follows directly from the fact that no economist was able to read Keynes’s A Treatise on Probability, especially chapter I of Part I, and Parts II ,III and V.
It was I. Hishiyama who provided that correct answer about Keynes’s lack of influence in his 1969 paper long before Moscati’s observation:
“It may be said that the reason why a Treatise on Probability has been disregarded by those who have made a study of Keynes is, not because it was immature or that it has been replaced by his own thought in later days, but because it was an achievement in a field having nothing to do with economics, or because it had no concern whatsoever with the formation of his economic thought. Maybe it was assumed that the line of thought in terms of a Treatise on Probability and that which has penetrated through The General Theory by integrating the thoughts in the field of his economics were, so to speak, two parallel lines, neither of which would ever cross the other. Or rather, the truth might lie in the fact that a Treatise on Probability has never been read, and that those specialists interested in this field who perused the theory so happened to show no interest in Keynes' economics.”(Hishiyama,1969, p.24; italics and underline added). Hishiyama is correct. However, he left out one explanation, which is probably the main explanation. Consider the following correct as far as it goes, but incomplete, discussion by Hishiyama:
“For the purpose of making a thorough study of the meanings of J. S. Mill's thought on economics it would probably be essential to make a close examination of his system of logic. The same should be held true with regard to the case of Keynes. That is, for the purpose of clarifying the theoretical meaning of his thought on economics developed in his General Theory it would be necessary to examine his system of logic. In the meanwhile,speaking of the system of logic of Keynes, there is no such book except his Treatise on Probability, and it can be seen that Keynes sacrificed his whole youthful life, when his intellectual energies were brought into full play, only to complete this book in which he wanted to accomplish the task of creating a new type of logic which could well be set up against Mill's system of logic6,7. This book is not anything like a layman's piece-work, but is literally one of his life works8, as commented on by R. F. Harrod. It appears to me, as far as my view is concerned, that Keynes' philosophical thought, being different from the case of economic thought, grew to maturity in his youthful days and that his systematical thought of logic which bloomed in his youth never came to be revised in all his life.”(Hishiyama,1969, p.25; italic and underline added).
The oversight on Hishiyama’s part, which has been duplicated by every single economist, philosopher, social scientist, behavioral scientist and historian, who has written on Keynes’s theory of probability, in the 20th and 21st centuries, is the failure to realize that Keynes NEVER would be able “…to accomplish the task of creating a new type of logic.”, Because just SUCH A NEW LOGIC HAD ALREADY BEEN CREATED BY GEORGE BOOLE in The Laws of Thought in 1854.Keynes BUILT on Boole and ONLY BOOLE. This conclusion is fundamental. The failure to study Boole first before attempting to read Keynes can only lead to a complete failure to understand Keynes’s approach. One need only mention the same problem currently existing in Thomas Aquinas “studies”, which has overlooked Aquinas’s general qualitative(but quantitative when using actual and expected given market prices as numbers), logical theory of probability for nearly 8 centuries. No economist or academician has any chance of grasping Keynes’s analysis because that analysis IS BUILT ON THE LOGICAL ANALYSIS OF GEORGE BOOLE. Moscati’s further claims about “…see, for example, Dardi’s classic 1991 paper on the subject…”, ignore that, just like Hishiyama, Dardi overlooked that it is Boole, not Keynes, who developed a full scale logical theory of probability. Dardi ‘s paper is intellectually equivalent to H E Kyburg’s 1995,1999,2003 and 2010 papers. Dardi had NO KNOWLEDGE of the Boolean foundations for the A Treatise on Probability, the A Treatise on Money(vol.I) or the General Theory For instance, NO ECONOMIST ever read chapters 6,7, and 8 of Volume I of the TM (1930), which are required prerequisites in order to be able to understand how Keynes would OPERATIONALIZE the Fundamental Equations of chapter 9 as being interval valued with lower and upper bounds.
It is extremely important to recognize that von Neumann clearly and concisely recognized that Key... more It is extremely important to recognize that von Neumann clearly and concisely recognized that Keynes’ theory of logical probability was a nonstandard logic, while the two valued, “Boolean logic “of Jevons, Peirce and Schroder was a standard logic that was in conflict with the original logic and algebra of George Boole when it came to applications to probability. The Jevons-Peirce -Schroder(J-P-S) two valued “Boolean logic” REQUIRES linearity and additivity when applied to probability. This means that it can’t deal with insufficiencies/deficiencies in the availability/completeness of the data or what Boole called indeterminate probabilities .Thus, the (J-P-S)”Boolean logic “ is a standard logic ,while Boole’s original The Laws of Thought (1854)logic is a non-standard logic .Keynes’s A Treatise on Probability was directly based on Boole’s The Laws of Thought ,and hence is a nonstandard logic. Thus, von Neumann is giving Keynes too much credit, as Keynes’s nonstandard logic of 1921 is built directly on Boole’s nonstandard logic of 1854.Keynes was also well aware of Boole’s later, much easier, applications of indeterminate probabilities using Henry Wilbraham’s approach (Keynes,1921, p.161). Theodore Hailperin (1986,1996) had already grasped what Boole’s nonstandard approach was, which Boole modified shortly after The Laws of Thought was published in 1854 by switching to Henry Wilbraham’s littoral’s approach, which allowed one to bypass Boole’s four valued logic, while obtaining the same results in a much easier manner. Given Keynes’s application of Boole’s relational, propositional, Boolean logic, which was the foundation for his logical theory of probability, von Neumann’s understanding of Keynes’s work is complete and correct, while Ramsey’s understanding of Keynes’s work is nil (equal to 0).
How Ramsey’s pathetic interpretation came to be universally accepted among economists, philosophers, historians, as well as by all other academicians working on Keynes’s A treatise on Probability, will be examined in this paper. This will require a comparison-contrast with von Neumann’s unknown assessment, which was based on von Neumann’s very deep understanding of either (a) Keynes’s graphical lattice illustration on page 39 of his book or (b) Keynes’s worked out interval valued probability problems contained on pp.161-163 and 186-194 of chapters XV and XVII of the A Treatise on Probability(TP,1921;1973,CWJMK version,p.42).
von Neumann’s deep understanding of Keynes’s analysis in his A Treatise on Probability resulted i... more von Neumann’s deep understanding of Keynes’s analysis in his A Treatise on Probability resulted in von Neumann realizing that Keynes had generalized precise, numerical mathematical probability, which von Neumann called standard probability logic, to be a special case of imprecise, non-numerical, non-standard probability logic. This means that all economists, historians and philosophers “interpretations “of Keynes’s logical theory of probability, as being an ordinal theory based on the Platonic Intuitionism of G E Moore, combined with Plato’s forms, ,completely collapses. The basic foundation for this idiotic “interpretation “of Keynes’s logical theory of probability by economists, philosophers and historians is, of course, F P Ramsey.
F P Ramsey made up a bunch of claims about axioms that do not exist in Keynes’s A Treatise on Probability and Boolean logical relations that he also claimed did not exist. F P Ramsey’s claims about fictitious nonexistent axioms and logical relations were then combined with his complete ignorance concerning the main role played by Boole’s The Laws of thought (1854) in Keynes’s A treatise on Probability .Keynes’s A treatise on Probability is built completely on Boole’s relational, propositional logic ,Boole’s objective ,logical, probability relations among propositions ,Boole’s interval valued, indeterminate probabilities, as well as Boole’s severe strictures against the use of the POI. Thus Ramsey’s claim that the POI is required for the application of Keynes’s ‘new logic of uncertainty ‘is utter nonsense .
I am not going to try to explain the wild, wooly, weird and wacko ideas that Ramsey made up about Keynes’s A Treatise on Probability. I leave such a study to the psychoanalysts, psychiatrists and psychologists, who have the training to investigate the mental disease or problems afflicting FP Ramsey in the 1920’s .
I will note that von Neumann’s 1937 assessment completely undermines any and all economist “interpretations” of Keynes’s A Treatise on Probability being about some theory of ordinal probability:
“Von Neumann then makes the following declaration:
“We prefer, therefore, to disclaim any intention to interpret the relations P (a, b) =θ (0<θ<1) in terms of strict logics. In other words, we admit:
Probability logics cannot be reduced to strict logics, but constitute an essentially wider system than the latter, and statements of the form P (a, b) =θ (0<θ<1) are perfectly new and sui generis aspects of physical reality.
So probability logic appears as an essential extension of strict logics. This view, the so-called ‘logical theory of probability’ is the foundation of J. N. [sic] Keynes’s work on the subject.”
In short, the later von Neumann interprets quantum probabilities as logical probabilities. Moreover, he explicitly identifies this view with that worked out by Keynes. “(Stacey ,2016, p.4).
It is important to realize that von Neumann is using the standard approach to defining an interval. Von Neumann is using interval notation when he writes (a, b).
Von Neumann’s definition, that P (a, b) =θ (0<θ<1), states that the probability of the interval (a, b) is equal to θ, which is a probability between a and b, where a is the lower bound and b is the upper bound, which is then between 0 and 1. This is NOT a numerical probability .It is NOT an ordinal probability. It is not a marginal probability, joint probability or union of probabilities. This IS an interval probability. It is a great, unsolved mystery as to why there were NO social scientists, statisticians, behavioral scientists ,economists, historians, or philosophers ,who had recognized that Keynes’s non numerical or logical probabilities were interval probabilities ,in the 20th or 21st centuries.
Thus, it appears that only F Y Edgeworth, John von Neumann, Bertrand Russell and Theodore Hailperin clearly recognized the mathematical and technical basis for the interval valued nature of Keynesian probabilities.
A careful reading of von Neumann’s 1937 discussion of his decision to adapt Keynes’s logical appr... more A careful reading of von Neumann’s 1937 discussion of his decision to adapt Keynes’s logical approach to probability in order to discuss quantum probability shows that von Neumann had carefully read Keynes’s Parts I and II. von Neumann grasped that Keynes had correctly generalized the standard mathematical laws of the calculus of probability in order to incorporate interval valued probability ,such as his non numerical probabilities p1= [.47,.55] and p2=[.54,.60] ,such that ,as Keynes stated in his A Treatise on Probability,
“… I maintain, then, in what follows, that there are some
pairs of probabilities between the members of which no comparison
of magnitude is possible; that we can say, nevertheless, of some pairs
of relations of probability that the one is greater and the other less,
although it is not possible to measure the difference between them;
and that in a very special type of case, to be dealt with later, a
meaning can be given to a numerical comparison of magnitude. I
think that the results of observation, of which examples have been
given earlier in this chapter, are consistent with this account.
By saying that not all probabilities are measurable, I mean that
it is not possible to say of every pair of conclusions, about which we
have some knowledge, that the degree of our rational belief in one
bears any numerical relation to the degree of our rational belief in
the other; and by saying that not all probabilities are comparable in
respect of more and less, I mean that it is not always possible to say
that the degree of our rational belief in one conclusion is either equal
to, greater than, or less than the degree of our belief in another.”(Keynes,1921,p.34;boldface,italics,and underline added).
Of course ,by “numerical” ,Keynes means by a single number.Keynes will show that “ non numerical” probabilities ,intervals like p1 and p2,can provide support for rational belief
Note that there are an infinite number of examples as the one I gave above. Therefore,following Keynes, it is not possible to say that p1>p2 ,p1<p2 or p1=p2.p 1 and p2 are both indeterminate probabilities that overlap one another.
Of course, Keynes,who mainly used the terminology approximation, inexact measurement and non numerical probability , is talking about what are today called imprecise or non-additive probability ,just as Boole’s indeterminate probabilities are interval probabilities that are partially ordered .Keynes is defining ,just as Boole did ,a partial ordering of probabilities in the probability space by his constant use of the word “between “ in chapter III of the A Treatise on Probability, which corresponds to an ordering based on the inequality ,≤ ,and leads to the illustration by Keynes of a mathematical lattice structure on p.39 of chapter III in the A Treatise on Probabiity. H. E. Kyburg demonstrated this FOUR times, in 1994,1999,2003, and 2010, without having any knowledge about Keynes’s use of Boole’s mathematical apparatus to specify interval valued probability, as discussed by Keynes in Part II in the appendix to chapter XIV, chapter XV, chapter XVI, and chapter XVII of the A Treatise on Probability. Following Boole, Keynes is using propositions about events and not the events themselves. Second, Keynes is using Boole’s relational, propositional logic that uses the logical connectives “and”, “not” and “or” to analyze sets of conjunctions of propositions and sets of disjunctions of propositions. Third, Keynes is calculating least upper bounds(l.u.b’s) and greatest lower bounds(g.l.b.’s) for the sets of conjunctions and disjunctions, just as Boole did on pp.287-325 of The Laws of Thought, that, in modern terminology specify a mathematical lattice structure in Euclidean space.
This approach is termed by von Neumann “a non-standard probability approach.” This non-standard probability approach is illustrated by Keynes with the interval probability paths U,V,W,X,Y, and Z,where 0VA is an explicit interval probability,and the standard probability path is given by 0AI.Keynes’s illustration is of a mathematical lattice structure ,which is comprised of the non linear,non-additive interval probabilities U,V,W,X,Y, and Z plus the additive and linear probabilities given by 0AI,where A is an additive and linear probability.
It will be straightforward to show that von Neumann understood this and was then able to generalize to a Hilbert space ,where Keynes’s paths are replaced by subspaces in Hilbert space in order to model a propositional logic leading to a mathematical lattice structure based on lub’s and glb’s .
It is important to note ,as was first pointed out by Hailperin (1986,1996) that the Boolean approach being used by Keynes is NOT the Jevons, Peirce and Schroder interpretation based on a two valued(1,0) logic .Boole’s approach was based on a four valued logic that allowed Boole to deal with vagueness ,ambiguity, uncertainty ,and unavailable ,incomplete and missing evidence
Feynman’s discussions of the two slit electron experiment support von Neumann’s decision to make ... more Feynman’s discussions of the two slit electron experiment support von Neumann’s decision to make the standard calculus of ,or logic of, probability ,where all probabilities are linear and additive ,as a special case of an approach based on Keynes’s logical theory of probability, based on Boole’s indeterminate approach to probability ,where the probabilities are intervals, which means that some probabilities have nonlinear and non-additive representations, which is a nonstandard logic of probability . This leads to Boole’s discussions of the existence of least upper bounds and greatest lower bounds, based on Boole’s relational, propositional logic, which is based on a partial ordering of the space of probabilities (partially ordered set) using ≤ as the ordering principle. Boole’s system, based on the logical connectives of “and(intersection)”,” or(union)” and “not”, allows one to analyze the
set of all propositions which form conjunctions and the set of all propositions which form disjunctions. This then leads to the existence of the Meet (greatest lower bound ) and the Join (least upper bound).Keynes followed Boole’s approach most of the way. However, instead of using ≤, Keynes used the word “between” in his ordering of the illustration on p.39 of the A Treatise on Probability. Kyburg demonstrated this analysis, which was carried out on Keynes’s part in chapter III of the A Treatise on Probability, in four articles that appeared in 1994,1999,2003 and 2010. Keynes thus had illustrated his nonstandard ,Boolean approach ,by constructing a more general ,nonstandard probability logic in Euclidean space ,with his diagram on p.39 ,which incorporates the standard, additive probability approach by the one dimensional line OAI, where A is a standard probability while U,V,W, X,Y,Z represent an interval valued set of probabilities that are nonlinear ,non-additive and multidimensional. Keynes’s diagram is a mathematical lattice structure .von Neumann generalized Keynes’s Euclidean space analysis by using a Hilbert space analysis ,where the sub spaces of the Hilbert space form a lattice structure ,which would replace Keynes’s U,V,W,X,Y, and Z.A standard, classical probabilistic representation of quantum probabilities ,based on the standard probability logic, becomes a special case similar to the specification by Keynes of 0AI. Feynman’s analysis shows why one must go beyond the standard linear and additive approach to probability, as the standard approach breaks down when confronted with waves .
It has been shown that Feynman’s results are in agreement with von Neumann’s logical interpretation of wave -particle duality ,establishing that the probabilities for the two-slit experiment have a nonstandard logical structure , since they violate the additivity requirement, which is one of the purely mathematical laws upon which classical probability theory is based.
How did Richard Feynman's two slit electron experiment demonstrate that the probabilities are not additive? Richard Feynman's two-slit experiment demonstrates that the probabilities of quantum particles do not add up to 1, as required by standard, linear, classical probabilities. Instead, the probabilities are described by a probability amplitude that can interfere with itself, leading to regions in which the probability of finding a particle is zero [regions where the probability |ψ|2 (the square of the absolute value of the complex probability amplitude) of finding a quantum particle has a value of zero.]. This interference pattern is the hallmark of quantum mechanics and shows that the probabilities are not additive but rather depend on the interference generated by the probability amplitudes. The experiment also illustrates how the concept of wave-particle duality, where the particles can exhibit both wave-like and particle-like behavior, requires a nonlinear and non-additive characterization of probability.
It was F P Ramsey, in a 1923 Apostle’s paper, who first claimed that he saw a very strong analog... more It was F P Ramsey, in a 1923 Apostle’s paper, who first claimed that he saw a very strong analogy between Moore’s Platonic Intuitionism, which Moore applied to ethical and moral questions involving his concept of “the Good “ and Keynes’s logical, objective probability relation that Keynes applied in his concept of” the Probable”. In fact, there is no analogy at all that exists between Moore’s Platonic Intuitionism and Keynes’s logical theory of probability. Ramsey’s claims ,about a very strong analogy connecting Moore’s work in morals and ethics with Keynes’s work in probability and statistics, is actually a hallucination on Ramsey’s part that Ramsey dreamed up out of nothing. At no time in his life did Ramsey cite any page or pages where such a Moorean view can be discovered underlying Keynes’s foundation for his objective, logical, probability relation. Keynes’s objective, logical probability relation was a relation that held,just like Boole’s objective,logical probability relation, between related propositions, and had nothing to do with F P Ramsey’s queer claims about unrelated propositions. Keynes’s foundation can easily be found in Boole’s 1854 The Laws of Thought on
pp.7-8, as stated by Keynes in a footnote on p.5 of chapter I of Keynes’s A Treatise on Probability. In short, a complete and total refutation of Ramsey could have been recognized to exist by any reader of chapter 1 of the A Treatise on Probability. The fact that this has not happened was explained by Hishiyama in 1969-NO ONE had read Keynes’s A Treatise on Probability ,not even pp.3-5. We will cover five papers, that are highly representative from the thousands of such papers that could have been selected since 1921 ,showing how this immense, intellectual fraud has been successfully perpetrated and continues to spread throughout the world of academia in the year 2025.We examine Correa(2000), Mourouzi (2017) , Guerra-Pujols(2020) , Gerrard(2023) and Coates(2025).
Consider the following analysis supplied by Stacey that examines why von Neumann rejected his ini... more Consider the following analysis supplied by Stacey that examines why von Neumann rejected his initial 1928-1932 approach to basing quantum probability on a limiting frequentist approach to probability. von Neumann’s decision, to base his treatment of quantum probability on the logical probability approach of Keynes, which, in fact, is based on Boole’s non classical logic, results in incorporating Boole’s specification of interval valued probability, whenever no complete ordering of the probability space is possible. Note that Boole’s logic and algebra is NOT the standard Boolean logic interpretation of Peirce and Jevons, which can’t deal with Boole’s integration of a method to deal with uncertainty or ambiguity (decision situations of partial knowledge and partial ignorance) and ignorance. Consider the following statement in Stacey (2016):
“To see how von Neumann’s thinking on the foundations of probability changed, we turn next to an unfinished manuscript from about 1937, which is included in his Collected Works [44] … Von Neumann then makes the following declaration:
“We prefer, therefore, to disclaim any intention to interpret the relations P (a, b) =θ (0<θ<1) in terms of strict logics. In other words, we admit: Probability logics cannot be reduced to strict logics, but constitute an essentially wider system than the latter, and statements of the form P(a,b)=θ (0<θ<1) are perfectly new and sui generis aspects of physical reality. So probability logic appears as an essential extension of strict logics. This view, the so-called ‘logical theory of probability’, is the foundation of J. N. [sic] Keynes’s work on the subject. “
In short, the later von Neumann interprets quantum probabilities as logical probabilities. Moreover, he explicitly identifies this view with that worked out by Keynes. “(Stacey,2018, pp.3-4, underline added).
Keynes’s mathematical analysis is actually worked out in chapters XII, XV, XVI and XVII of the A Treatise on Probability. Keynes’s diagram on p.39 in chapter III of the A Treatise on Probability was provided as a graphical representation that illustrates Keynes’s use of the Boolean representation of indeterminant probability as interval valued probability, that Boole then later explicated as a mathematical lattice structure (Boole,1854, pp.287-325).
There are a number of points that will be required to be filled in, as Stacey’s summary of what von Neumann did, as regards Keynes and Boole, while correct, is too short.
H.E. Kyburg correctly analyzed Keynes’s page 39 illustration in a diagram presented in chapter II... more H.E. Kyburg correctly analyzed Keynes’s page 39 illustration in a diagram presented in chapter III after having read pp.22-33 of Keynes’s A Treatise on Probability involving discussions of some applications of interval probability to conduct. Kyburg carefully analyzed Keynes’s analysis on pp.33-36, showing that Keynes was describing a mathematical lattice structure consisting of interval probability. The Heterodox economists, on the other hand, did not read what Keynes had written, but, instead, decided to “ interpret” Keynes through the foggy bifocals of F P Ramsey’s wretched and intellectually worthless 1922 and 1926 reviews, where Ramsey claimed that Keynes’s “mysterious non numerical probabilities “were scientifically useless, ordinal probabilities. The following list provides a short list of the main heterodox economists advocating such an ordinal “interpretation”. The rest of the members of all heterodox schools also all come to the same bizarre conclusion- that Keynes’s theory of logical probability was an ordinal theory. The short list follows below: • Gay Meeks • Robert Skidelsky • Donald Moggridge • Rod O’Donnell • Anna Carabelli • Jochen Runde • John Davis • Athol Fitzgibbons • Basili and Zappia • Charles McCann • Bradley Bateman • Carlo Zappia
Despite Kyburg basically publishing the same material on Keynes’s diagram four times, in 1994,1999,2003 and 2010, all of which provided overwhelming analysis that the conclusion that Keynes’s theory was based on ordinal probability made no sense technically, Kyburg’s position made absolutely no headway. The reason it made no headway is the existence of very severe ,mathematical ,logical, probabilistic and statistical deficiencies that exist in all schools of heterodox economics involved in studying Keynes’s A Treatise on Probability, A Treatise on Money ,General Theory ,as well as all of his major post 1936 articles ,such as the 1937 ER and QJE, the 1937-38 exchanges with H. Townshend, and the 1938 -40 EJ articles dealing with Tinbergen’s limiting frequency approach to macroeconomic Investment and business cycles .
Consider the following statements taken from Chapter One of Keynes’s A Treatise on Probability t... more Consider the following statements taken from Chapter One of Keynes’s A Treatise on Probability that explains perfectly why Keynes, following in the footsteps of only one other scholar, George Boole, called his theory a logical theory of probability:
First, Keynes explains why the use of propositions is relevant and important ,for both philosophical reasons and the application of probability to conduct:
“With the term “event,” which has taken hitherto so important a place in the phraseology of the subject, I shall dispense altogether. † Writers on Probability have generally dealt with what they term the “happening” of “events.” In the problems which they first studied this did not involve much departure from common usage. But these expressions are now used in a way which is vague and ambiguous; and it will be more than a verbal improvement to discuss the truth
and the probability of propositions instead of the occurrence and the probability of events. ‡ 5. These general ideas are not likely to provoke much criticism. In the ordinary course of thought and argument, we are constantly assuming that knowledge of one statement, while not proving the truth of a second, yields nevertheless some ground for believing it. We assert that we ought on the evidence to prefer such and such a belief. We claim rational grounds for assertions which are not conclusively demonstrated”. (Keynes,1921, p.5).
Secondly, Keynes then explains what a relational, propositional logic is to the reader. Keynes is assuming that the reader has carefully read Keynes’s second footnote on page 5 discussing the innovator in this approach, George Boole:
“Between two sets of propositions, therefore, there exists a relation, in virtue of which, if we know the first, we can attach to the latter some degree of rational belief. This relation is the subject-matter of the logic of probability.
A great deal of confusion and error has arisen out of a failure to take due account of this relational aspect of probability. From the premisses “a implies b” and “a is true,” we can conclude something about b—namely that b is true—which does not involve a. But, if a is so related to b, that a knowledge of it renders a probable belief in b rational, we cannot conclude anything whatever about b which has not reference to a; and it is not true that every set of self-consistent premisses which includes a has this same relation to b. It is as useless, therefore, to say “b is probable” as it would be to say “b is equal,” or “b is greater than,” and as unwarranted to conclude that, because a makes b probable, therefore a and c together make b probable, as to argue that because a is less than b, therefore a and c together are less than b.” (Keynes,1921, pp.6-7; underline added ).
Finally, Keynes then provides the reader with his final conclusion concerning the introductory material covered in chapter I on pp.3-9 of his A treatise on Probability. It is important to realize that if this introductory, prerequisite material is NOT understood, then it will be highly likely that the reader will fail to grasp the later aspects of Keynes’s theory as they are developed systematically in Keynes’s book, especially in Part II. The failure to grasp
Part II will result in the failure to Grasp Part III. The failure to grasp Part III will result in the final failure, which is the failure to grasp Part V:
“This chapter has served briefly to indicate, though not to define, the subject matter of the book. Its object has been to emphasize the existence of a logical relation between two sets of propositions in cases where it is not possible to argue demonstratively from one to the other. This is a contention of a most fundamental character. It is not entirely novel(author’s note-Keynes is referring to Boole), but has seldom received due emphasis, is often overlooked, and sometimes denied. The view, that probability arises out of the existence of a specific relation between premiss and conclusion, depends for its acceptance upon a reflective judgment on the true character of the concept. It will be our object to discuss, under the title of Probability, the principal properties of this relation. First, however, we must digress in order to consider briefly what we mean by knowledge, rational belief, and argument.” (Keynes,1921, p.9; underline added).
Zappia, in his footnote 11 to chapter 3 of Zappia(2025) on the comparison of Keynes and Knight on the issue of uncertainty , makes the following decision about how he intends to go about dealing with the logical foundations of Keynes’s argument form, where the term “argument form” is Keynes’s short hand term for the application of Boole’s relational, propositional logic, which permeates all five Parts of Keynes’s A Treatise on Probability:
“It should be noted that, in Keynes’s approach, probability does not apply to events, but to propositions that express a degree of probability that a hypothesis is true, given some evidence. According to Keynes, it is necessary to speak of the probability of a proposition a, rather than ‘Dinosaurs disappeared from the Earth because of the fall of a meteorite,’ which can be judged to be true (or false) with some degree of probability, but not to the event of disappearance itself. Despite the philosophical importance of the question, we will ignore this distinction in what follows in order to be able to compare Keynes’s argument more easily with those of later authors.”(Zappia,2025, ch.3,ft.11 ;underline added). We first need to make some corrections in Zappia’s footnote, as he has confused Carnap’s very similar definition of logical probability, p1, with Keynes’s definition of α,a partial degree of rational belief. Carnap’s definition of degree of confirmation, C, is that C(h/e) =p1,
where h = hypothesis, e=evidence and p1=logical probability, as opposed to p2, which equals statistical frequency. Carnap’s p1 is fundamental, as p2 is derived from the application of p1. Keynes’s definition is that P(a/h) =α, which incorporates the argument form, (a/h),where a is the conclusion of an argument form, h contains the premises of the argument form, α is a partial degree of rational belief and P denoted the objective, logical, probability relation which MUST connect the a and h propositions internally.(See Keynes,1921, p.119). Zappia’s example is also confused, as it contains two propositions, not one. Zappia’s ‘Dinosaurs disappeared from the Earth because of the fall of a meteorite’ is composed of an a proposition, ‘Dinosaurs disappeared from the Earth”, and an h proposition ,” the fall of a meteorite.” It has some partial degree of rational belief, α1. Using Keynes’s p.119 definition in his A Treatise on Probability, we have P(a/h) =α1.
It is quite obvious from the above that Zappia never actually ever read Keynes’s A Treatise on Probability. He certainly never read Chapters I or II. What Zappia attempted to read was chapters III,IV,VI, and XXVI of Keynes’s A Treatise on Probability, which he combined with the huge literature based on F P Ramsey’s 1922 and 1926 reviews ,plus some 75 articles which have appeared in the Cambridge Journal of Economics since 1977. Zappia apparently does not understand that it is quite impossible “...to compare Keynes’s argument more easily with those of later authors. “if one ignores the distinction made by Keynes, as the distinction Keynes is making goes far beyond “… the philosophical importance of the question…” For example, It is quite impossible to compare Keynes’s degree of rational belief with Ramsey’s degree of subjective(actual) belief, which can be a) rational, b)irrational or c) non rational. Keynes’s 1931 point, never understood by Zappia, was that Ramsey’s approach had failed completely to deal with rational degrees of belief: “But in attempting to distinguish “rational” degrees of belief from belief in general he was not yet, I think, quite successful. It is not getting to the bottom of the principle of induction merely to say that it is a useful mental habit.” (Keynes,1931,Oct.3).
During the years 1937-1942 (See Stacey,2016; Mansson, 2007), Von Neumann decided to reject his ea... more During the years 1937-1942 (See Stacey,2016; Mansson, 2007), Von Neumann decided to reject his earlier 1928-32 frequentist approach to modelling and explaining quantum probability. He decided, instead, to base his foundation on Keynes’s logical theory of probability. This entailed using propositions to represent statements about quantum probabilities and using the logical connectives of Keynes’s(Boole’s) relational, propositional logic, “or”, “and”, and “not”, to build a mathematical lattice structure ,representing the sub spaces of a Hilbert space that ,by analogy, follows from the mathematical lattice structure specified in Euclidian space by Keynes in the diagram on page 39(Keynes,1921;1973,p.42) of Keynes’s A Treatise on Probability. Thus, least upper bounds and greatest lower bounds must be specified in order to incorporate non comparable or non-measurable probabilities resulting from the great uncertainties involved in modelling quantum states.
Following Keynes’s lead in 1921, von Neumann interpreted quantum probabilities as logical probabilities, viewing quantum mechanics as a form of generalized, formal logic. He followed the approach of Keynes, who worked with sets of propositions in Euclidian space, by showing that propositions dealing with quantum systems could be represented by subspaces of Hilbert space, where the relationships between these propositions followed a non-classical probability logic, which is now called quantum logic. This approach builds on Keynes’s uncertainty concept ,but applies it to the states of quantum probabilities, so that quantum probabilities arise from the inherent uncertainty in the logical structure of quantum mechanics, which differs from classical, mathematical probabilities.
Essentially, von Neumann's approach presents a logical analysis such that quantum mechanics, with its unique probabilistic rules, can be understood as a logical system where probabilities are derived from the structure of sets of propositions within a generalized, non-classical logic. This is, by analogy, practically the same type of generalization as arrived at by Keynes ,who ,in his mathematical lattice structure diagram, presented on p.39 of his A Treatise on Probability, specifies a set of non-numerical ,logical probabilities U,V,W,X,Y, and Z in Euclidian space ,which generalized the linear and additive classical mathematical probabilities given by the horizontal line segment 0AI,where A is a linear and additive classical ,mathematical probability. The contrast provided by Keynes is striking, as it allows a reader to grasp that Keynes’s analysis involves nonlinearity and non-additivity, which is a necessary prerequisite to model non comparability and non -measurability and hence Keynesian uncertainty.
Ramsey’s theory, on the other hand, was represented by the horizontal, linear line,OAI.H.E. Kyburg, in four articles published in 1994,1999,2003 and 2010, established that Keynes’s logical theory of probability is correctly represented only by a mathematical lattice structure, which incorporates non -measurability and non-comparability. It is interesting that all economists, historians, philosophers, etc., writing on Keynes’s logical theory of probability in the 20th and 21st centuries, rejected Kyburg’s hard analysis in favor of an idiotic, bizarre, absurd, moronic and foolish assertion that Keynes’s diagram on p.39 is an example of ordinal probability. This is quite impossible ,as the ordinal theory concept being deployed by economists is linear and additive, not nonlinear and non-additive .Thus, if p1>p2 or if p3<p4,then p1,p2,p3 and p4 are all raised to the first power ,meaning that they are linear and additive so that p1+p2=1 or p3+p4=1.It is impossible to represent them by Keynes’s nonlinear and non-additive logical probabilities U,V,W,X,Y and Z as shown in the diagram on page 39 .
The following claim made by McCann (see McCann,1994,pp.41-42),which is representative of the work of all economists, philosophers and historians analyzing the diagram on p.39(Keynes,p.42,1973),was wrong in 1994 and wrong in 2025.McCann’s claim is that Keynes’s OAI line represents numerical probability, which must be one dimensional, while Keynes’s logical probabilities(intervals) U,V,W,X,Y ,and Z represent ordinal probability ,which must be multidimensional. As demonstrated above, this is simply mathematically impossible. Keynes’s diagram is actually a representation of what Boole was doing on pp.287-325 ,which was working out what modern theory calls a least upper bounds(lub) and greatest lower bound(glb). The existence of l.u.b.’s and g.l.b.’s automatically defines a mathematical lattice structure.
Keynes’s page 39 diagram has been very severely misinterpreted for over 100 years by every single historian, economist, and philosopher, who has written on Chapter III of Keynes’s A Treatise on Probability, as being a representation of ordinal probability. This is impossible because the ordinal concept of probability is additive, which means that ordinal probabilities sum to 1 so that there can be no uncertainty. This error, like so many of the errors that permeates the work done on Keynes’s logical theory of probability by economists, follows from the wild, wooly, weird and wacko claims made by F P Ramsey in 1921,1922,1923 and 1926, which are still, mysteriously, universally accepted as of 2025 in the Liberal Arts and Social and Behavioral Sciences.
Add the following book to the references-
McCann, Charles. (1994). Probability Foundations of Economic Theory. London; Routledge.
Chapter XII contains Keynes’ s all -important axiom (i).A careful consideration of Axiom (i) ,esp... more Chapter XII contains Keynes’ s all -important axiom (i).A careful consideration of Axiom (i) ,especially given Keynes’ s emphasis in this axiom on the specifications of sets of all conjunctions and disjunctions of propositions, given that this axiom has been combined with Keynes’s earlier assumption of a partially ordered set(all probabilities are not measurable by a single numeral between 0 and 1) and Keynes’s diagram on page 39,leads to one ,and only one, conclusion - Keynes has specified a mathematical lattice structure to represent his non numerical(interval valued ) probabilities ,which MUST be presented as being non linear and non -additive .Keynes thus allows the reader to compare his nonlinear ,non-additive , lattice structure with the degenerate ,or pseudo, lattice structure representing a complete order ,OAI, which MUST ,of course, be linear and additive .The fact that only ONE individual ,Henry E Kyburg, was able to arrive at this conclusion in both the 20th and 21st centuries ,calls into serious question the scientific credentials and credibility of the so called social sciences ,behavioral sciences and Liberal arts.
Keynes’s Axiom (i) has nothing to do with Ramsey’s talk about an axiom I that does not exist. This is easily established by comparing Keynes’s axiom (i) with both the 1922 version of Ramsey’s axiom I and his 1926 version. First, we will present Keynes’s axiom (i) and then both of Ramsey’s versions of his Axiom I:
“5. Preliminary Axioms:
We shall assume that there is included in every premiss with
which we are concerned the formal implications which allow us to
assert the following axioms:
(i.) Provided that a and h are propositions or conjunctions of
propositions or disjunctions of propositions, and that h is not an
inconsistent conjunction, there exists one and only one relation of
probability P between a as conclusion and h as premiss. Thus any
conclusion a bears to any consistent premiss h one and only one
relation of probability.” (Keynes, p.135).
Ramsey’s two versions of what he claims was Keynes’s axiom I are given below:
“First, he thinks that between any two non-self-contradictory propositions there holds a probability relation (Axiom I), for example between 'My carpet is blue' and 'Napoleon was a great general.’(Ramsey,1922, p.3)
and
“Mr. Keynes accounts for this by supposing that between any two propositions, taken as premiss and conclusion, there holds one and only one relation of a certain sort called probability relations; and that if, in any given case, the relation is that of degree α, from full belief in the premiss, we should, if we were rational, proceed to a belief of degree α in the conclusion. “(Ramsey,1926; In Kyburg and Smokler ,1980,2nd ed., pp.26-27).
Nowhere in either the A Treatise on Probability or in any of the CWJMK does Keynes ever state
“…that between any two non-self-contradictory propositions there holds a probability relation (Axiom I)..”(Ramsey ,1922,p.3)
or
“Mr. Keynes accounts for this by supposing that between any two propositions, taken as premiss and conclusion, there holds one and only one relation of a certain sort called probability relations…”( Ramsey,1926; In Kyburg and Smokler ,1980,2nd ed., pp.26-27).
It is quite shocking that there was NO logician,statistician,mathematician,economist,philosopher,historian,poli-tical scientist, psychologist, sociologist, anthropologist or behavioral scientist, in either the 20th or 21st centuries, who pointed out the obvious contradictions in Ramsey’s 1922 and 1926 claims, especially given Bertrand Russell’s complete refutation of Ramsey’s assertions in the July,1922 issue of the Mathematical Gazette in his star footnote on page 120,which showed that all of Ramsey’s examples of axiom I are composed of UNRELATED OR IRRELEVANT pairs of propositions ,which were ruled out by Keynes on pp.4-6 of chapter I of the A Treatise on Probability:
“2+2=4;Napoleon disliked poodles…” (Russell,1922,p.120,*footnote).
It is very straightforward to conclude that Keynes’s logical theory of probability does NOT deal with only two propositions and that it is simply false that Keynes stated that “…between any two non-contradictory propositions” or “…between any two propositions…” there was a logical relation.
Keynes basically used chapter XI of the A Treatise on Probability to reiterate and reemphasize wh... more Keynes basically used chapter XI of the A Treatise on Probability to reiterate and reemphasize what he had originally covered in chapter X, which was his theory of groups. Keynes starts out by repeating materials originally presented in earlier chapters that dealt with his argument form, where the argument form is Keynes’s Boolean, relational, propositional logic, which incorporates ,as it MUST ,an objective, logical, probability relation connecting the premises and the conclusion:
“The Theory of Probability deals with the relation between two sets of propositions(author’s note-note that this has nothing to do with Ramsey’s two propositions only), such that, if the first set is known to be true, the second can be known with the appropriate degree of probability by argument from the first.∗ The relation, however, also exists when the first set is not known to be true and is hypothetical. In a symbolical treatment of the subject it is important that we should be free to consider hypothetical premisses, and to take account of relations of probability as existing between any pair of sets of propositions, whether or not the premiss is actually part of knowledge. But in acting thus we must be careful to avoid two possible sources of error. . . . The first is that which is liable to arise wherever variables are concerned. This was mentioned in passing in § 18 of Chapter IV[author’s note-section 18 of chapter IV depends on section 16 of chapter IV where Keynes introduces variables into the analysis .This requires the use of a 1st order (predicate) logic, which Keynes uses heavily in chapter XXIII]. We must remember that whenever we substitute for a variable some particular value of it, this may so affect the relevant evidence as to modify the probability. This danger is always present except where, as in the first half of Chapter XIII., the conclusions respecting the variable are certain. 3. The second difficulty is of a different character. Our premisses may be hypothetical and not actually the subject of knowledge. But must they not be possible subjects of knowledge? How are we to deal with hypothetical premisses which are self-contradictory or formally inconsistent with themselves, and which cannot be the subject of rational belief of any degree?” (Keynes,1921, p.123; underline and boldface added)
Keynes concludes that, as was the case considered in chapters I and II, that any unrelated propositions will be excluded from any logical analysis, so will inconsistent propositions and all propositions that are not the subject of knowledge, which also applies to Ramsey’s examples:
“Whether or not a relation of probability can be held to exist between a conclusion and a self-inconsistent premiss, it will be convenient to exclude such relations from our scheme, so as to avoid having to provide for anomalies which can have no interest in an account of the actual processes of valid reasoning. Where a premiss is inconsistent with itself it cannot be required.” (Keynes,1921, p.124; underline and boldface added)
Keynes’s theoretical development of a logical theory of probability based on Boole’s original mat... more Keynes’s theoretical development of a logical theory of probability based on Boole’s original mathematical logic begins with Part II of the A Treatise on Probability. Part I was the Introduction, which incorporated a coverage that, although non-technical, would have allowed a careful reader to have been able to make an accurate assessment of what Keynes was doing. However, there is only one individual who appears to have been able to successfully make such an assessment-Francis Ysidro Edgeworth. Edgeworth was very straightforward in his correspondence with Edwin Bidell Wilson, who was also straightforward with Edgeworth. Both of them admitted to the other that they had no idea what it was that Keynes was doing in Part II. However, Edgeworth, by making a very careful reading of Part I, figured out what Keynes was doing -presenting an interval valued theory of probability. No other academic economist succeeded in doing what Edgeworth did in the 20th and 21st centuries.
-Keynes presents a very brief summary of Part I that is a valuable restatement ONLY if the reader... more -Keynes presents a very brief summary of Part I that is a valuable restatement ONLY if the reader of chapter IX has, in fact, read and understood all of the previous eight chapters. This is precisely what DID NOT OCCUR. An easy example of this failure is the universal interpretation of the diagram on page 39 of chapter III as being a representation of ordinal probability. Someone, who intends to get his assessment of what Keynes had written in Part I by skipping chapters I -VIII and relying on chapter IX , will completely fail to grasp Keynes’s systematic reliance on • Boole’s relational, propositional logic in chapters I and II • Boole’s critique of the POI’s applicability in chapters IV, V,VI and VII • Boole’s use of interval valued probability in chapter III based on applications of the word “between” • Boole’s use of partially ordered sets[(existence of non-numerical probability)Boole’s indeterminate probability],plus the emphasis placed on the sets of all disjunctions (or,ᴠ) and conjunctions(and,ᴧ) defined in chapter XII, which leads to the mathematical lattice structure presented by Keynes on page 39 ,as a simplification for the reader of Part I, of Part II’s future discussion of interval valued probability in chapters XV,XVI and XVII. Thus, in retrospect, Keynes probably needed to seriously considered if he needed to also include the final paragraph from chapter X as becoming the final paragraph in chapter IX also. Of course, Keynes could not have foreseen that no academicians in either the 20th or 21st centuries, especially economists, historians and
philosophers, WOULD read his Part II and that all academicians WOULD ,instead, substitute Ramsey’s idiotic and absurd assessments, which Ramsey claimed to represent Keynes’s logical theory ,for a reading of Keynes’s A Treatise on Probability:
“7. In Chapter XV. I bring the non-numerical theory of probability developed in the preceding chapters into connection with the usual numerical conception of it, and demonstrate how and in what class of cases a meaning can be given to a numerical measure of a relation of probability. This leads on to what may be termed numerical approximation, that is to say, the relating of probabilities, which are not themselves numerical, to probabilities, which are numerical, by means of greater and less, by which in some cases numerical limits may be ascribed to probabilities which are not capable of numerical measures.” (Keynes ,1921, pp.121-122).
This does the job perfectly. Unfortunately, it is in Part II, not Part I,of the A Treatise on Probability.
: At the time Keynes published his A Treatise on Probability, all frequentists required all proba... more : At the time Keynes published his A Treatise on Probability, all frequentists required all probabilities to be precise and exact numbers. Therefore, it appears that Keynes’s improved imprecise, logical, frequentist theory of 1921, which Keynes linked to Alfred North Whitehead’s previous approach, would have to have been rejected from any serious consideration due to the following boldfaced sentences that deal with Keynes’s theoretical exposition: “10. A proposition can be a member of many distinct classes of propositions, the classes being merely constituted by the existence of particular resemblances between their members or in some such way. We may know of a given proposition that it is one of a particular class of propositions, and we may also know, precisely or within defined limits, what proportion of this class are true, without our being aware whether or not the given proposition is true. Let us, therefore, call the actual proportion of true propositions in a class the truth-frequency† of the class, and define the measure of the probability of a proposition relative to a class, of which it is a member, as being equal to the truth-frequency of the class
The fundamental tenet of a frequency theory of probability is,then, that the probability of a proposition always depends upon referring it to some class whose truth-frequency is known within wide or narrow limits. Such a theory possesses most of the advantages of Venn’s, but escapes his narrowness. There is nothing in it so far which could not be easily expressed with complete precision in the terms of ordinary logic. Nor is it necessarily confined to probabilities which are numerical. In some cases we may know the exact number which expresses the truth-frequency of our class; but a less precise knowledge is not without value, and we may say that one probability is greater than another, without knowing how much greater, and that it is large or small or negligible, if we have knowledge of corresponding accuracy about the truth-frequencies of the classes to which the probabilities refer.” (Keynes,1921, p.101;boldface and underline added) Keynes has thus introduced a Boolean , relational, propositional logic ,as well as imprecise probability(Keynes used the verbiage approximate measures and/or inexact measurement or non numerical probability),into his logical, frequentist theory. From Chapter VI, Keynes had already defined a measure about how the evidential weight of the argument would be measured in a frequentist approach:
“The same distinction may be explained in the language of the frequency theory.∗ We should then say that the weight is increased if we are able to employ as the class of reference a class which is contained in the original class of reference.” (Keynes,1921, p.76-Keynes’s star footnote refers explicitly to chapter VIII).
Since the frequentists of Keynes’s time , following Venn, Peirce, Reichenbach or von Mises, rejected any other decision theoretic measure except for a precise probability concept, it follows that they could not accept Keynes’s construction of incorporating imprecise probability along with a second variable to measure evidential weight in a decision theoretic context, which ,for instance ,Carnap later called degree of firmness and Savage called sureness(sure probabilities versus not sure probabilities). This means that Keynes’s theoretical exposition would have been rejected, given that all frequentists of the time accepted only some strict version of a limiting frequency approach to probability, which rejects any type of interval valued probabilities or evidential weight .(There are three papers that are relevant ,which appeared long after 1921.The first is Peter Walley and Terrence L. Fine’s(1982) “ Towards a frequentist theory of upper and lower probability” ,published in the Ann. Stat. 10 (3) (1982) 741–761. Two recent papers which attempted to put forth constructs similar in some ways to what Keynes tried to do in 1921 are Hubert’s (2021),” Reviving Frequentism”, published in Synthese and Fröhlich, Derr, and Williamson(2024), “Strictly frequentist imprecise probability”, published in The Journal of Approximate Reasoning. These papers will not be dealt with in this paper).
This chapter is a historical overview of the role played by the Principle of Non-Sufficient Reas... more This chapter is a historical overview of the role played by the Principle of Non-Sufficient Reason (PNSR), which Keynes reformulated and restructured as the Principle of Indifference (POI). The Bernoulli -Laplace PNSR was basically applied when decision makers needed to come up with the initial probabilities to implement the application of Bayes Law. They needed a starting point. Thus, PNSR operated based on an assertion of equally balanced ignorance among the possible alternatives so that equally likely probabilities could be used as a starting point. This allowed the decision maker to come up with the initial a priori probabilities, which would then be combined with new evidence in order to specify the a posteriori probability. Keynes discusses academicians from the past who have questioned, either wholly or in part, the legitimacy of the application of the POI. Thus, he covers thinkers such as Locke, Hume, Poincare, Venn, von Kries and Boole. Keynes missed a great opportunity to delve closely into Boole’s criticisms of the POI here. Unfortunately, and inexplicably, Keynes decided NOT to do so. Keynes waits until Part II is reached in order to cover Boole in the appendix to chapter XIV, and chapters XV, XVI and XVII of his A Treatise on Probability, which present a detailed exposition of Boole’s The Laws of Thought. The great problem here ,of course, is that, just as no academician ,excepting Hailperin ,read Boole’s chapters XVI-XXI ,no academician read Part II, chapters X to XVII, of the A Treatise on Probability .Hailperin did not read much of Part II ,given the elementary mistakes he makes about (a) Keynes’s Carnapian like definition of probability, Carnap’s p1, (b) the meaning of Keynes’s “non numerical” probabilities (See Hailperin,1996,pp.15,137,143) and (c) his failure to grasp the mathematical lattice structure in chapter XII that must result from Keynes’s poset assumptions and non-linear and non additive interval valued probabilities. More serious is that these errors led to Hailperin’s inexplicable failure to recognize that Keynes was committed to using Boole’s logical approach to probability ,which is based on a four valued logic ,and not the standard two valued logic, which is a very special case ,as no two valued logic, based strictly on (1,0) values ,can possibly deal with probability, which is why Boole rejected the Peirce-Jevons version, offered to Boole in correspondence with Jevons in 1863, based on a two valued logic. This is inexplicable, given Hailperin’s own clear cut understanding that Boolean logic is NOT a two valued logic and that Keynes was following Boole. Given that no other academician, in either the 20 or 21st centuries, knows what Boole is doing, then it is not possible for any academician in the 20th or 21st centuries to know what Keynes is doing either, since to understand what Keynes is doing requires that one understands what Boole is doing. The problem grows exponentially from this point on, as Keynes’s A Treatise on Money is based on Keynes’s A Treatise on Probability and Keynes’s General Theory is based on both Keynes’s A Treatise on Money and A Treatise on Probability. There was only one economist who clearly suspected that this was the case-Hugh Townshend. Unfortunately, he gave up his correspondence with Keynes just as he was on the verge of successfully grasping what Keynes had explained to him. I. Hishiyama also suspected that Keynes had created a ‘new logic of uncertainty, ‘but never realized that the Boole -Keynes connections were pervasive. The following statements, taken from The Laws of Thought, should have been given as long footnotes to one of the pages in chapter VII where Keynes briefly mentions Boole. That would have been sufficient to demonstrate Boole’s severe criticisms of the POI:
“The above solution is usually founded upon a supposed analogy of the problem with that of the drawing of balls from an urn containing a mixture of black and white balls, between which all possible numerical ratios are assumed to be equally probable. And it is remarkable, that there are two or three distinct hypotheses which lead to the same final result. For instance, if the balls are finite in number, and those which are drawn are not replaced, or if they are infinite in number, whether those drawn are replaced or not, then, supposing that m successive drawings have yielded only white balls, the probability of the issue of a white ball at the m+1 th drawing is m+1/ m+2 It has been said, that the principle involved in the above and in similar applications is that of the equal distribution of our knowledge, or rather of our ignorance—the assigning to different states of things of which we know nothing,and upon the very ground that we know nothing, equal degrees of probability. I apprehend, however, that this is an arbitrary method of procedure. Instances may occur, and one such has been adduced, in which different hypotheses lead to the same final conclusion. But those instances are exceptional. With reference to the particular problem in question, it is shown in the memoir cited, that there is one hypothesis, viz., when the balls are finite in number and not replaced, which leads to a different conclusion, and it is easy to see that there are other hypotheses, as strictly involving the principle of the “equal distribution of knowledge or ignorance,” which would also conduct to conflicting results…”(Boole,1854,pp.369-370; underline added) and
“…These results only illustrate the fact, that when the defect of data is supplied by hypothesis, the solutions will, in general, vary with the nature of the hypotheses assumed; so that the question still remains, only more definite in form, whether the principles of the theory of probabilities serve to guide us in the selection of such hypotheses. I have already expressed my conviction that they do not—a conviction strengthened by other reasons than those above stated. …Still it is with diffidence that I express my dissent on these points from mathematicians generally, from one who, of English writers, has most fully entered into the spirit and the methods of Laplace; and I venture to hope, that a question, second to none other in the Theory of Probabilities in importance, will receive the careful attention which it deserves.”(Boole,1854,p. 375 ;underline added). More than enough evidence is presented here that the POI has absolutely NOTHING to do with Boole’s logic and hence NOTHING to do with Keynes’s Boolean logic approach.
Keynes, in chapter VI of his A Treatise on Probability, applies Boole’s logical, relational, prop... more Keynes, in chapter VI of his A Treatise on Probability, applies Boole’s logical, relational, propositional logic to the problem that is today called analyzing evidentiary support, represented by the underlying data, evidence, knowledge or information supporting the probability assessment being made by a decision maker. Following Boole’s dictum ,that a logical analysis must always take place first before any mathematical analysis takes place second , Keynes devotes Chapter VI of his A Treatise on Probability to providing a logical analysis of his logical symbol, V =V (a/h),designated as the “evidential weight of the argument”, where the argument form is identical to that introduce in chapter I of his book. Thus, h will represent the premises stated in the argument and a will be the conclusion reached. Like the probability argument discussed in chapters I and II, Keynes will eventually add a mathematical analysis in chapter XXVI of the A Treatise on Probability; however, the reader of chapter VI is fully informed of this by a footnote explicitly linking chapter VI to Chapter XXVI. Now Keynes decided, when introducing his logical relation of probability in chapters I and II, to also incorporate an initial, introductory, mathematical analysis, so that one could arrive at P(a/h) =α, where 0≤α≤1, and P is a symbol representing a logical, objective, probability relation .α is the partial degree of rational belief resulting from the argument. However, Keynes only presents V(a/h) in chapter VI. So, no measurement takes place, although he does discuss comparisons between different logical V relations.
Keynes waits to define a mathematical analysis, that will define a measure of V, until he arrives at Chapter XXVI. The answer Keynes provides in chapter XXVI is that V(a/h) = w, where Keynes defines w as “…the degree of completeness of the information upon which a probability is based…”, where 0≤w≤1 and w=[K/(K+I)], given that K+I=1,where K =Absolute Knowledge and I =Absolute Ignorance (Keynes,1921,p.71) .This is identical to the normalization process for probabilities that define the sum of all probabilities to equal 1,i.e,p+q=1 as defined by Keynes in a footnote on page 315: “We could, if we liked, define a conventional coefficient c of weight and risk, such as c =2pw/ (1 +q) (1 +w), where w measures the ‘weight,’ which is equal to unity when p = 1 and w = 1, and to zero when p = 0 or w =0 and has an intermediate value in other cases. ∗” (Keynes,1921, p.315). This is easily rewritten as c=p[(1/(1+q)] [2w/(1+w)]. In this form, it is easy to see that Keynes has added two weights to the standard additive, linear probability model.
Keynes’s decision weight coefficient, c, allows one to reach the same conclusions, as reached by the use of the more difficult to handle interval valued approach to probability, without having to deal with the difficult, mathematical, technical issues of interval probability. Thus, intervals that have a small(narrow) range between the lower and upper bound have a greater w than interval probabilities that have a large (wide) range between the lower and upper bound. Keynes is very clear on this in his initial, introductory discussions in chapter III of the A Treatise on Probability:
“Whilst he may be able to make sure of a profit, on the principles of the bookmaker, yet the individual figures that make up the book are, within certain limits, arbitrary.”(Keynes,1921,p.22)
and
“In fact underwriters themselves distinguish between risks which are properly insurable, either because their probability can be estimated between comparatively narrow numerical limits…” (Keynes,1921,p.23)
and
“A distinction, interesting for our present purpose, between probabilities, which can be estimated within somewhat narrow limits, and those which cannot, has arisen in a series of judicial decisions respecting damages.” (Keynes ,1921, p.24)
and
“A relation of probability does not yield us, as a rule, information of much value, unless it invests the conclusion with a probability which lies between narrow numerical limits.” (Keynes,1921, p.31)
and
“We frame two ideal arguments, that is to say, in which the general character of the evidence largely resembles what is actually within our knowledge, but which is so constituted as to yield a numerical value, and we judge that the probability of the actual argument lies between these two. Since our standards, therefore, are referred to numerical measures in many cases where actual measurement is impossible, and since the probability lies between two numerical measures…” (Keynes,1921, p.32).
The reader should note that Keynes is incorporating chapter VI and chapter XXVI analysis in his section 5 of chapter III:
“The plaintiff had evidently suffered damage, and justice required that she should be compensated. But it was equally evident, that, relative to the completest information available and account being taken of the arbiter’s personal taste, the probability could be by no means estimated with numerical precision. Further, it was impossible to say how much weight ought to be attached to the fact that the plaintiff had been head of her district (there were fewer than 50 districts); yet it was plain that it made her chance better than the chances of those of the 5 left in, who were not head of their districts.” (Keynes,1921, p.27; underline added).
Chapter V of Keynes’s A Treatise on Probability has never been read and/or understood by any econ... more Chapter V of Keynes’s A Treatise on Probability has never been read and/or understood by any economist, philosopher, historian, social scientist or behavioral scientist. The complete failure to grasp this chapter explains why Chapter XV of the A Treatise on Probability was not read or understood either, as chapter V is the introduction to and prerequisite for chapter XV:
“Our previous conclusion that numerical measurement is often impossible agrees very well, therefore, with the argument of the preceding chapter that the rules, in virtue of which we can assert equiprobability, are somewhat limited in their field of application… But the recognition of this same fact makes it more necessary to discuss the principles which will justify comparisons of more and less between probabilities, where numerical measurement is theoretically, as well as practically, impossible.” (Keynes,1921 , p.65)
Keynes also appends an extremely important footnote which was overlooked by every reader attempting to cover Keynes’s theory in his A Treatise on Probability: “∗Parts of Chap. XV. are closely connected with the topics of the following paragraphs, and the discussion which is commenced here is concluded there.” (Keynes,1921, p.65).
Keynes next makes it clear that Part II must be covered (Keynes,1921, pp.66,68) for a clear understanding of (i) and (ii) discussed below:
“We are able, I think, always to compare a pair of probabilities which are (i.) of the type ab/h and a/h or (ii.) of the type a/hh1 and a/h provided the additional evidence h1 contains only one independent piece of relevant information. (i.) The propositions of Part II. will enable us to prove that ab/h < a/h unless b/ ah = 1; that is to say, the probability of our conclusion is diminished by the
addition to it of something, which on the hypothesis of our argument cannot be inferred from it. This proposition will be self-evident to the reader. The rule, that the probability of two propositions jointly is, in general, less than that of either of them separately, includes the rule that the attribution of a more specialised concept is less probable than the attribution of a less specialized concept.” (Keynes,1921, p.66).
The conclusion is obviously clear. Chapter XV and Part II is where Keynes finishes/ends the discussion begun in chapters IV and V on applying the POI. Unfortunately, no academician in the 20th or 21st century carried out Keynes’s requirement.
A very severe problem afflicting all philosophers, economists, historians, social scientists and ... more A very severe problem afflicting all philosophers, economists, historians, social scientists and behavioral scientists working on J M Keynes is their misbelief that the POI served in any manner at all as a foundation for his logical theory of probability as presented in his A Treatise on Probability regarding either initial a priori or revised a posteriori probability. Extremely severe errors have been made by D. Gillies, Howson, Urbach, Howson and Urbach, Childers, and Runde. Their argument is that the POI lies at the heart of Keynes’s logical theory of probability and collapses without it .Contrary to D. Gillies, Howson ,Urbach ,Howson and Urbach ,Childers, and Runde, Keynes does not rely on the POI at any place in the 33 chapters of the A Treatise on Probability, as Keynes’s main foundation is a revised and improved version of Boole’s interval valued probability approach contained in Boole’s 1854 The Laws of Thought, which was used for BOTH the estimation of either initial a priori or revised a posteriori probabilities. What has happened involves two severe deficiencies in the assessments made by D. Gillies, Howson ,Urbach ,Howson and Urbach ,Childers, and Runde of Keynes’s work regarding (a) their ignorance of Boole’s estimation approach to both initial, a priori probability and a posteriori ,revised probability and (b) their reading into chapter IV of Keynes’s A Treatise on Probability the Ramsey claim that Keynes’s substantial efforts to successfully modify and tighten up the conditions needed to reliably apply the PNSR ,now designated by Keynes as the POI, meant that Keynes was doing this so that he could make use of the result as the foundation for his theory. Nothing could be further from the truth.
Keynes was very clear about this in Part II, chapter XV, which D. Gillies, Howson, Urbach, Howson and Urbach, Childers, and Runde, never read:
“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.” (Keynes,1921, p.160; underline and boldface added).
Keynes makes it very clear what he is relying on in the very next paragraph, again not read by academicians. D. Gillies, Howson, Urbach, Howson and Urbach, Childers, and Runde have read into Keynes’s works ideas which are hallucinations on their part:
“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. This method is frequently adopted in common discourse.” (Keynes,1921, p.160; boldface and underline added)
The wild, wooly, weird and wacko claims of Runde, based on idiotic assertions made by F P Ramsey, that Keynes is relying on comparative probability, is a reflection of Runde’s reliance on the gobbledygook of F P Ramsey when he was reading some small parts of the TP.
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This paper is the foundation for some six years of continuous correspondence between myself and Cohen over his claim that Keynes
“… does not suggest any method by which weights might be measured and in fact admits that often one cannot even compare the weights of different arguments.” (Cohen,1986, p.264).
between 1988 and 1993.Cohen is correct that Keynes provided no way of measuring the evidential weight of the argument in chapter VI. Cohen’s conclusion is superior to that of, for just one instance, Isaac Levi in his 1967 Gambling with Truth, which was that Keynes had provided NO index defined on the interval [0,1] and measured V by K ,the total amount of evidence. Cohen’s conclusion was also maintained by Henry E Kyburg and Rudolf Carnap.
All of these illustrious logicians, however, overlooked that Keynes DID define a measure of evidential weight in chapter 26 of Keynes’s A treatise on Probability.
Specifically, Keynes stated the following on page 315 of chapter 26 of his A Treatise on Probability:
“We could, if we liked, define a conventional coefficient c of weight and risk, such as c = 2pw/ (1 +q) (1 +w) , where w measures the ‘weight,’ which is equal to unity when p = 1 and w = 1, and to zero when p = 0 or w =0 and has an intermediate value in other cases.∗”
The reader should note that Keynes does exactly what Cohen(and Levi) claim Keynes did not do, which was to define an index to measure the evidential weight of the argument on the unit interval [0,1].
The star footnote that Keynes appended shows how one applies the conventional coefficient of weight and risk, c, which is a substitute for Keynes’s interval valued analysis in chapters III, XV, XVII, XX and XXII of the A Treatise on Probability. Keynes states:
“∗If pA = p’A’, w >w’, and q =q’, then cA>c’A’; if pA=p'A’, w =w’, and
q <q’, then cA >c’A’; if pA=p’A’, w >w’, and q <q’ then cA>c’A’; but if
pA =p’A’, w= (this should read as w>w’) w’, and q>q’, we cannot compare cA and c’A’.”(Keynes,1921, p.315, ft.2)
Cohen was not convinced by my letters, however, even though they contained the material presented above .However, he did intervene, indirectly ,“behind the scenes “,in order to make sure that the papers I had submitted, in early 1990 and early 1991, were eventually published in mid-1993 and mid- 1994 in The British Journal for the Philosophy of Science(BJPS) and International Studies for the Philosophy of Science(ISPS) ,respectively. Both papers dealt with Keynes’s c coefficient, the measurement of V by w, where w is an index defined on the interval [0,1], and numerous applications. Numerous additional submissions on the index for w and applications using the c coefficient, made to the BJPS and ISPS after 1994 ,were unanimously rejected by all referees; however, three of the rejected submissions were successfully published later on in 2000 and 2001 in The Journal of Applied Economics and Econometrics, due to the intervention of Dr. K Puttaswamaiah, editor in chief.
I am only going to publish it now due to the appearance of an error-filled paper published in September ,2025 by J.Brekel in Erkinntnis,in which he lists Cohen’s paper in his references, but never actually cites Cohen’s paper anywhere in his paper. His 2025 paper is a revised version of his error- filled 2022 MA Thesis. Brekel makes the claim, first made by Baylis (1935) and many others, that Keynes had provided contradictory measurements for V, even though it is impossible to do this, as V is a logical relation. Levi correctly pointed out that NO CORRECT MEASUREMENT OF V IS POSSIBLE unless an index to measure evidential weight has been provided for it.
Moscati’s statement, that
“… I find the Keynesian interpretative framework adopted by Zappia problematic. One major issue is that Keynes and his ideas on uncertainty have, in my view, played only a marginal role in the development of the economic theory of decision-making under risk and uncertainty. In the decision theory literature since the 1950s, references to Keynes are very rare.” (Moscati,2025, p.4),
is very close to being correct. We need to replace one word in the second sentence of his statement above. That one word is “marginal”. We replace it with the word, “no”, to obtain the following amended version:
“…, played no role in the development of the economic theory of decision-making under risk and uncertainty.”
This conclusion follows directly from the fact that no economist was able to read Keynes’s A Treatise on Probability, especially chapter I of Part I, and Parts II ,III and V.
It was I. Hishiyama who provided that correct answer about Keynes’s lack of influence in his 1969 paper long before Moscati’s observation:
“It may be said that the reason why a Treatise on Probability has been
disregarded by those who have made a study of Keynes is, not because
it was immature or that it has been replaced by his own thought in
later days, but because it was an achievement in a field having nothing
to do with economics, or because it had no concern whatsoever with
the formation of his economic thought. Maybe it was assumed that the
line of thought in terms of a Treatise on Probability and that which has
penetrated through The General Theory by integrating the thoughts in
the field of his economics were, so to speak, two parallel lines, neither
of which would ever cross the other. Or rather, the truth might lie in the
fact that a Treatise on Probability has never been read, and that those
specialists interested in this field who perused the theory so happened
to show no interest in Keynes' economics.”(Hishiyama,1969, p.24; italics
and underline added).
Hishiyama is correct. However, he left out one explanation, which is probably the main explanation. Consider the following correct as far as it goes, but incomplete, discussion by Hishiyama:
“For the purpose of making a thorough study of the meanings of J.
S. Mill's thought on economics it would probably be essential to make
a close examination of his system of logic.
The same should be held true with regard to the case of Keynes. That is,
for the purpose of clarifying the theoretical meaning of his thought on
economics developed in his General Theory it would be necessary to
examine his system of logic. In the meanwhile,speaking of the system
of logic of Keynes, there is no such book except his Treatise on
Probability, and it can be seen that Keynes sacrificed his whole youthful
life, when his intellectual energies were brought into full play, only to
complete this book in which he wanted to accomplish the task of
creating a new type of logic
which could well be set up against Mill's system of logic6,7. This book
is not anything like a layman's piece-work, but is literally one of his
life works8, as commented on by R. F. Harrod. It appears to me, as
far as my view is concerned, that Keynes' philosophical thought, being
different from the case of economic thought, grew to maturity in his
youthful days and that his systematical thought of logic which bloomed
in his youth never came to be revised in all his life.”(Hishiyama,1969, p.25; italic and underline added).
The oversight on Hishiyama’s part, which has been duplicated by every single economist, philosopher, social scientist, behavioral scientist and historian, who has written on Keynes’s theory of probability, in the 20th and 21st centuries, is the failure to realize that Keynes NEVER would be able
“…to accomplish the task of creating a new type of logic.”,
Because just SUCH A NEW LOGIC HAD ALREADY BEEN CREATED BY GEORGE BOOLE in The Laws of Thought in 1854.Keynes BUILT on Boole and ONLY BOOLE. This conclusion is fundamental. The failure to study Boole first before attempting to read Keynes can only lead to a complete failure to understand Keynes’s approach.
One need only mention the same problem currently existing in Thomas Aquinas “studies”, which has overlooked Aquinas’s general qualitative(but quantitative when using actual and expected given market prices as numbers), logical theory of probability for nearly 8 centuries.
No economist or academician has any chance of grasping Keynes’s analysis because that analysis IS BUILT ON THE LOGICAL ANALYSIS OF GEORGE BOOLE.
Moscati’s further claims about “…see, for example, Dardi’s classic 1991 paper on the subject…”, ignore that, just like Hishiyama, Dardi overlooked that it is Boole, not Keynes, who developed a full scale logical theory of probability. Dardi ‘s paper is intellectually equivalent to H E Kyburg’s 1995,1999,2003 and 2010 papers. Dardi had NO KNOWLEDGE of the Boolean foundations for the A Treatise on Probability, the A Treatise on Money(vol.I) or the General Theory
For instance, NO ECONOMIST ever read chapters 6,7, and 8 of Volume I of the TM (1930), which are required prerequisites in order to be able to understand how Keynes would OPERATIONALIZE the Fundamental Equations of chapter 9 as being interval valued with lower and upper bounds.
Theodore Hailperin (1986,1996) had already grasped what Boole’s nonstandard approach was, which Boole modified shortly after The Laws of Thought was published in 1854 by switching to Henry Wilbraham’s littoral’s approach, which allowed one to bypass Boole’s four valued logic, while obtaining the same results in a much easier manner.
Given Keynes’s application of Boole’s relational, propositional, Boolean logic, which was the foundation for his logical theory of probability, von Neumann’s understanding of Keynes’s work is complete and correct, while Ramsey’s understanding of Keynes’s work is nil (equal to 0).
How Ramsey’s pathetic interpretation came to be universally accepted among economists, philosophers, historians, as well as by all other academicians working on Keynes’s A treatise on Probability, will be examined in this paper. This will require a comparison-contrast with von Neumann’s unknown assessment, which was based on von Neumann’s very deep understanding of either (a) Keynes’s graphical lattice illustration on page 39 of his book or (b) Keynes’s worked out interval valued probability problems contained on pp.161-163 and 186-194 of chapters XV and XVII of the A Treatise on Probability(TP,1921;1973,CWJMK version,p.42).
F P Ramsey made up a bunch of claims about axioms that do not exist in Keynes’s A Treatise on Probability and Boolean logical relations that he also claimed did not exist. F P Ramsey’s claims about fictitious nonexistent axioms and logical relations were then combined with his complete ignorance concerning the main role played by Boole’s The Laws of thought (1854) in Keynes’s A treatise on Probability .Keynes’s A treatise on Probability is built completely on Boole’s relational, propositional logic ,Boole’s objective ,logical, probability relations among propositions ,Boole’s interval valued, indeterminate probabilities, as well as Boole’s severe strictures against the use of the POI. Thus Ramsey’s claim that the POI is required for the application of Keynes’s ‘new logic of uncertainty ‘is utter nonsense .
I am not going to try to explain the wild, wooly, weird and wacko ideas that Ramsey made up about Keynes’s A Treatise on Probability. I leave such a study to the psychoanalysts, psychiatrists and psychologists, who have the training to investigate the mental disease or problems afflicting FP Ramsey in the 1920’s .
I will note that von Neumann’s 1937 assessment completely undermines any and all economist “interpretations” of Keynes’s A Treatise on Probability being about some theory of ordinal probability:
“Von Neumann then makes the following declaration:
“We prefer, therefore, to disclaim any intention to interpret the relations P (a, b) =θ (0<θ<1) in terms of strict logics. In other words, we admit:
Probability logics cannot be reduced to strict logics, but constitute an essentially wider system than the latter, and statements of the form P (a, b) =θ (0<θ<1) are perfectly new and sui generis aspects of physical reality.
So probability logic appears as an essential extension of strict logics. This view, the so-called ‘logical theory of probability’ is the foundation of J. N. [sic] Keynes’s work on the subject.”
In short, the later von Neumann interprets quantum probabilities as logical probabilities. Moreover, he explicitly identifies this view with that worked out by Keynes. “(Stacey ,2016, p.4).
It is important to realize that von Neumann is using the standard approach to defining an interval. Von Neumann is using interval notation when he writes (a, b).
Von Neumann’s definition, that P (a, b) =θ (0<θ<1), states that the probability of the interval (a, b) is equal to θ, which is a probability between a and b, where a is the lower bound and b is the upper bound, which is then between 0 and 1. This is NOT a numerical probability .It is NOT an ordinal probability. It is not a marginal probability, joint probability or union of probabilities. This IS an interval probability. It is a great, unsolved mystery as to why there were NO social scientists, statisticians, behavioral scientists ,economists, historians, or philosophers ,who had recognized that Keynes’s non numerical or logical probabilities were interval probabilities ,in the 20th or 21st centuries.
Thus, it appears that only F Y Edgeworth, John von Neumann, Bertrand Russell and Theodore Hailperin clearly recognized the mathematical and technical basis for the interval valued nature of Keynesian probabilities.
“… I maintain, then, in what follows, that there are some
pairs of probabilities between the members of which no comparison
of magnitude is possible; that we can say, nevertheless, of some pairs
of relations of probability that the one is greater and the other less,
although it is not possible to measure the difference between them;
and that in a very special type of case, to be dealt with later, a
meaning can be given to a numerical comparison of magnitude. I
think that the results of observation, of which examples have been
given earlier in this chapter, are consistent with this account.
By saying that not all probabilities are measurable, I mean that
it is not possible to say of every pair of conclusions, about which we
have some knowledge, that the degree of our rational belief in one
bears any numerical relation to the degree of our rational belief in
the other; and by saying that not all probabilities are comparable in
respect of more and less, I mean that it is not always possible to say
that the degree of our rational belief in one conclusion is either equal
to, greater than, or less than the degree of our belief in another.”(Keynes,1921,p.34;boldface,italics,and underline added).
Of course ,by “numerical” ,Keynes means by a single number.Keynes will show that “ non numerical” probabilities ,intervals like p1 and p2,can provide support for rational belief
Note that there are an infinite number of examples as the one I gave above. Therefore,following Keynes, it is not possible to say that p1>p2 ,p1<p2 or p1=p2.p 1 and p2 are both indeterminate probabilities that overlap one another.
Of course, Keynes,who mainly used the terminology approximation, inexact measurement and non numerical probability , is talking about what are today called imprecise or non-additive probability ,just as Boole’s indeterminate probabilities are interval probabilities that are partially ordered .Keynes is defining ,just as Boole did ,a partial ordering of probabilities in the probability space by his constant use of the word “between “ in chapter III of the A Treatise on Probability, which corresponds to an ordering based on the inequality ,≤ ,and leads to the illustration by Keynes of a mathematical lattice structure on p.39 of chapter III in the A Treatise on Probabiity. H. E. Kyburg demonstrated this FOUR times, in 1994,1999,2003, and 2010, without having any knowledge about Keynes’s use of Boole’s mathematical apparatus to specify interval valued probability, as discussed by Keynes in Part II in the appendix to chapter XIV, chapter XV, chapter XVI, and chapter XVII of the A Treatise on Probability. Following Boole, Keynes is using propositions about events and not the events themselves. Second, Keynes is using Boole’s relational, propositional logic that uses the logical connectives “and”, “not” and “or” to analyze sets of conjunctions of propositions and sets of disjunctions of propositions. Third, Keynes is calculating least upper bounds(l.u.b’s) and greatest lower bounds(g.l.b.’s) for the sets of conjunctions and disjunctions, just as Boole did on pp.287-325 of The Laws of Thought, that, in modern terminology specify a mathematical lattice structure in Euclidean space.
This approach is termed by von Neumann “a non-standard probability approach.” This non-standard probability approach is illustrated by Keynes with the interval probability paths U,V,W,X,Y, and Z,where 0VA is an explicit interval probability,and the standard probability path is given by 0AI.Keynes’s illustration is of a mathematical lattice structure ,which is comprised of the non linear,non-additive interval probabilities U,V,W,X,Y, and Z plus the additive and linear probabilities given by 0AI,where A is an additive and linear probability.
It will be straightforward to show that von Neumann understood this and was then able to generalize to a Hilbert space ,where Keynes’s paths are replaced by subspaces in Hilbert space in order to model a propositional logic leading to a mathematical lattice structure based on lub’s and glb’s .
It is important to note ,as was first pointed out by Hailperin (1986,1996) that the Boolean approach being used by Keynes is NOT the Jevons, Peirce and Schroder interpretation based on a two valued(1,0) logic .Boole’s approach was based on a four valued logic that allowed Boole to deal with vagueness ,ambiguity, uncertainty ,and unavailable ,incomplete and missing evidence
set of all propositions which form conjunctions and the set of all propositions which form disjunctions. This then leads to the existence of the Meet (greatest lower bound ) and the Join (least upper bound).Keynes followed Boole’s approach most of the way. However, instead of using ≤, Keynes used the word “between” in his ordering of the illustration on p.39 of the A Treatise on Probability. Kyburg demonstrated this analysis, which was carried out on Keynes’s part in chapter III of the A Treatise on Probability, in four articles that appeared in 1994,1999,2003 and 2010. Keynes thus had illustrated his nonstandard ,Boolean approach ,by constructing a more general ,nonstandard probability logic in Euclidean space ,with his diagram on p.39 ,which incorporates the standard, additive probability approach by the one dimensional line OAI, where A is a standard probability while U,V,W, X,Y,Z represent an interval valued set of probabilities that are nonlinear ,non-additive and multidimensional. Keynes’s diagram is a mathematical lattice structure .von Neumann generalized Keynes’s Euclidean space analysis by using a Hilbert space analysis ,where the sub spaces of the Hilbert space form a lattice structure ,which would replace Keynes’s U,V,W,X,Y, and Z.A standard, classical probabilistic representation of quantum probabilities ,based on the standard probability logic, becomes a special case similar to the specification by Keynes of 0AI.
Feynman’s analysis shows why one must go beyond the standard linear and additive approach to probability, as the standard approach breaks down when confronted with waves .
It has been shown that Feynman’s results are in agreement with von Neumann’s logical interpretation of wave -particle duality ,establishing that the probabilities for the two-slit experiment have a nonstandard logical structure , since they violate the additivity requirement, which is one of the purely mathematical laws upon which classical probability theory is based.
How did Richard Feynman's two slit electron experiment demonstrate that the probabilities are not additive? Richard Feynman's two-slit experiment demonstrates that the probabilities of quantum particles do not add up to 1, as required by standard, linear, classical probabilities. Instead, the probabilities are described by a probability amplitude that can interfere with itself, leading to regions in which the probability of finding a particle is zero [regions where the probability |ψ|2 (the square of the absolute value of the complex probability amplitude) of finding a quantum particle has a value of zero.]. This interference pattern is the hallmark of quantum mechanics and shows that the probabilities are not additive but rather depend on the interference generated by the probability amplitudes. The experiment also illustrates how the concept of wave-particle duality, where the particles can exhibit both wave-like and particle-like behavior, requires a nonlinear and non-additive characterization of probability.
In fact, there is no analogy at all that exists between Moore’s Platonic Intuitionism and Keynes’s logical theory of probability. Ramsey’s claims ,about a very strong analogy connecting Moore’s work in morals and ethics with Keynes’s work in probability and statistics, is actually a hallucination on Ramsey’s part that Ramsey dreamed up out of nothing. At no time in his life did Ramsey cite any page or pages where such a Moorean view can be discovered underlying Keynes’s foundation for his objective, logical, probability relation. Keynes’s objective, logical probability relation was a relation that held,just like Boole’s objective,logical probability relation, between related propositions, and had nothing to do with F P Ramsey’s queer claims about unrelated propositions. Keynes’s foundation can easily be found in Boole’s 1854 The Laws of Thought on
pp.7-8, as stated by Keynes in a footnote on p.5 of chapter I of Keynes’s A Treatise on Probability.
In short, a complete and total refutation of Ramsey could have been recognized to exist by any reader of chapter 1 of the A Treatise on Probability. The fact that this has not happened was explained by Hishiyama in 1969-NO ONE had read Keynes’s A Treatise on Probability ,not even pp.3-5.
We will cover five papers, that are highly representative from the thousands of such papers that could have been selected since 1921 ,showing how this immense, intellectual fraud has been successfully perpetrated and continues to spread throughout the world of academia in the year 2025.We examine Correa(2000), Mourouzi (2017) , Guerra-Pujols(2020) , Gerrard(2023) and Coates(2025).
“To see how von Neumann’s thinking on the foundations of probability changed, we turn next to an unfinished manuscript from about 1937, which is included in his Collected Works [44] … Von Neumann then makes the following declaration:
“We prefer, therefore, to disclaim any intention to interpret the relations P (a, b) =θ (0<θ<1) in terms of strict logics. In other words, we admit:
Probability logics cannot be reduced to strict logics, but constitute an essentially wider system than the latter, and statements of the form P(a,b)=θ (0<θ<1) are perfectly new and sui generis aspects of physical reality.
So probability logic appears as an essential extension of strict logics. This view, the so-called ‘logical theory of probability’, is the foundation of J. N. [sic] Keynes’s work on the subject. “
In short, the later von Neumann interprets quantum probabilities as logical probabilities. Moreover, he explicitly identifies this view with that worked out by Keynes. “(Stacey,2018, pp.3-4, underline added).
Keynes’s mathematical analysis is actually worked out in chapters XII, XV, XVI and XVII of the A Treatise on Probability. Keynes’s diagram on p.39 in chapter III of the A Treatise on Probability was provided as a graphical representation that illustrates Keynes’s use of the Boolean representation of indeterminant probability as interval valued probability, that Boole then later explicated as a mathematical lattice structure (Boole,1854, pp.287-325).
There are a number of points that will be required to be filled in, as Stacey’s summary of what von Neumann did, as regards Keynes and Boole, while correct, is too short.
The Heterodox economists, on the other hand, did not read what Keynes had written, but, instead, decided to “ interpret” Keynes through the foggy bifocals of F P Ramsey’s wretched and intellectually worthless 1922 and 1926 reviews, where Ramsey claimed that Keynes’s “mysterious non numerical probabilities “were scientifically useless, ordinal probabilities. The following list provides a short list of the main heterodox economists advocating such an ordinal “interpretation”. The rest of the members of all heterodox schools also all come to the same bizarre conclusion- that Keynes’s theory of logical probability was an ordinal theory. The short list follows below:
• Gay Meeks
• Robert Skidelsky
• Donald Moggridge
• Rod O’Donnell
• Anna Carabelli
• Jochen Runde
• John Davis
• Athol Fitzgibbons
• Basili and Zappia
• Charles McCann
• Bradley Bateman
• Carlo Zappia
Despite Kyburg basically publishing the same material on Keynes’s diagram four times, in 1994,1999,2003 and 2010, all of which provided overwhelming analysis that the conclusion that Keynes’s theory was based on ordinal probability made no sense technically, Kyburg’s position made absolutely no headway. The reason it made no headway is the existence of very severe ,mathematical ,logical, probabilistic and statistical deficiencies that exist in all schools of heterodox economics involved in studying Keynes’s A Treatise on Probability, A Treatise on Money ,General Theory ,as well as all of his major post 1936 articles ,such as the 1937 ER and QJE, the 1937-38 exchanges with H. Townshend, and the 1938 -40 EJ articles dealing with Tinbergen’s limiting frequency approach to macroeconomic Investment and business cycles .
First, Keynes explains why the use of propositions is relevant and important ,for both philosophical reasons and the application of probability to conduct:
“With the term “event,” which has taken hitherto so important
a place in the phraseology of the subject, I shall dispense altogether. †
Writers on Probability have generally dealt with what they term the
“happening” of “events.” In the problems which they first studied
this did not involve much departure from common usage. But these
expressions are now used in a way which is vague and ambiguous;
and it will be more than a verbal improvement to discuss the truth
and the probability of propositions instead of the occurrence and the
probability of events. ‡
5. These general ideas are not likely to provoke much criticism.
In the ordinary course of thought and argument, we are constantly
assuming that knowledge of one statement, while not proving the
truth of a second, yields nevertheless some ground for believing it.
We assert that we ought on the evidence to prefer such and such
a belief. We claim rational grounds for assertions which are not
conclusively demonstrated”. (Keynes,1921, p.5).
Secondly, Keynes then explains what a relational, propositional logic is to the reader. Keynes is assuming that the reader has carefully read Keynes’s second footnote on page 5 discussing the innovator in this approach, George Boole:
“Between two sets of propositions, therefore, there exists
a relation, in virtue of which, if we know the first, we can attach
to the latter some degree of rational belief. This relation is the
subject-matter of the logic of probability.
A great deal of confusion and error has arisen out of a failure to
take due account of this relational aspect of probability. From the
premisses “a implies b” and “a is true,” we can conclude something
about b—namely that b is true—which does not involve a. But, if a
is so related to b, that a knowledge of it renders a probable belief in b
rational, we cannot conclude anything whatever about b which has
not reference to a; and it is not true that every set of self-consistent
premisses which includes a has this same relation to b. It is as useless,
therefore, to say “b is probable” as it would be to say “b is equal,”
or “b is greater than,” and as unwarranted to conclude that, because
a makes b probable, therefore a and c together make b probable, as
to argue that because a is less than b, therefore a and c together are
less than b.” (Keynes,1921, pp.6-7; underline added ).
Finally, Keynes then provides the reader with his final conclusion concerning the introductory material covered in chapter I on pp.3-9 of his A treatise on Probability. It is important to realize that if this introductory, prerequisite material is NOT understood, then it will be highly likely that the reader will fail to grasp the later aspects of Keynes’s theory as they are developed systematically in Keynes’s book, especially in Part II. The failure to grasp
Part II will result in the failure to Grasp Part III. The failure to grasp Part III will result in the final failure, which is the failure to grasp Part V:
“This chapter has served briefly to indicate, though not
to define, the subject matter of the book. Its object has been
to emphasize the existence of a logical relation between two sets of
propositions in cases where it is not possible to argue demonstratively
from one to the other. This is a contention of a most fundamental
character. It is not entirely novel(author’s note-Keynes is referring to Boole), but has seldom received due
emphasis, is often overlooked, and sometimes denied. The view, that
probability arises out of the existence of a specific relation between
premiss and conclusion, depends for its acceptance upon a reflective
judgment on the true character of the concept. It will be our object
to discuss, under the title of Probability, the principal properties of
this relation. First, however, we must digress in order to consider
briefly what we mean by knowledge, rational belief, and argument.” (Keynes,1921, p.9; underline added).
Zappia, in his footnote 11 to chapter 3 of Zappia(2025) on the comparison of Keynes and Knight on the issue of uncertainty , makes the following decision about how he intends to go about dealing with the logical foundations of Keynes’s argument form, where the term “argument form” is Keynes’s short hand term for the application of Boole’s relational, propositional logic, which permeates all five Parts of Keynes’s A Treatise on Probability:
“It should be noted that, in Keynes’s approach, probability does not apply to events, but to propositions that express a degree of probability that a hypothesis is true, given some evidence. According to Keynes, it is necessary to speak of the probability of a proposition a, rather than ‘Dinosaurs disappeared from the Earth because of the fall of a meteorite,’ which can be judged to be true (or false) with some degree of probability, but not to the event of disappearance itself. Despite the philosophical importance of the question, we will ignore this distinction in what follows in order to be able to compare Keynes’s argument more easily with those of later authors.”(Zappia,2025, ch.3,ft.11 ;underline added).
We first need to make some corrections in Zappia’s footnote, as he has confused Carnap’s very similar definition of logical probability, p1, with Keynes’s definition of α,a partial degree of rational belief.
Carnap’s definition of degree of confirmation, C, is that
C(h/e) =p1,
where h = hypothesis, e=evidence and p1=logical probability, as opposed to p2, which equals statistical frequency. Carnap’s p1 is fundamental, as p2 is derived from the application of p1.
Keynes’s definition is that
P(a/h) =α,
which incorporates the argument form, (a/h),where a is the conclusion of an argument form, h contains the premises of the argument form, α is a partial degree of rational belief and P denoted the objective, logical, probability relation which MUST connect the a and h propositions internally.(See Keynes,1921, p.119).
Zappia’s example is also confused, as it contains two propositions, not one. Zappia’s
‘Dinosaurs disappeared from the Earth because of the fall of a meteorite’
is composed of an a proposition, ‘Dinosaurs disappeared from the Earth”, and an h proposition ,” the fall of a meteorite.”
It has some partial degree of rational belief, α1.
Using Keynes’s p.119 definition in his A Treatise on Probability, we have
P(a/h) =α1.
It is quite obvious from the above that Zappia never actually ever read Keynes’s A Treatise on Probability. He certainly never read Chapters I or II. What Zappia attempted to read was chapters III,IV,VI, and XXVI of Keynes’s A Treatise on Probability, which he combined with the huge literature based on F P Ramsey’s 1922 and 1926 reviews ,plus some 75 articles which have appeared in the Cambridge Journal of Economics since 1977.
Zappia apparently does not understand that it is quite impossible “...to compare Keynes’s argument more easily with those of later authors. “if one ignores the distinction made by Keynes, as the distinction Keynes is making goes far beyond “… the philosophical importance of the question…”
For example, It is quite impossible to compare Keynes’s degree of rational belief with Ramsey’s degree of subjective(actual) belief, which can be a) rational, b)irrational or c) non rational.
Keynes’s 1931 point, never understood by Zappia, was that Ramsey’s approach had failed completely to deal with rational degrees of belief:
“But in attempting to distinguish “rational” degrees of belief from belief in general he was not yet, I think, quite successful. It is not getting to the bottom of the principle of induction merely to say that it is a useful mental habit.” (Keynes,1931,Oct.3).
Following Keynes’s lead in 1921, von Neumann interpreted quantum probabilities as logical probabilities, viewing quantum mechanics as a form of generalized, formal logic. He followed the approach of Keynes, who worked with sets of propositions in Euclidian space, by showing that propositions dealing with quantum systems could be represented by subspaces of Hilbert space, where the relationships between these propositions followed a non-classical probability logic, which is now called quantum logic. This approach builds on Keynes’s uncertainty concept ,but applies it to the states of quantum probabilities, so that quantum probabilities arise from the inherent uncertainty in the logical structure of quantum mechanics, which differs from classical, mathematical probabilities.
Essentially, von Neumann's approach presents a logical analysis such that quantum mechanics, with its unique probabilistic rules, can be understood as a logical system where probabilities are derived from the structure of sets of propositions within a generalized, non-classical logic. This is, by analogy, practically the same type of generalization as arrived at by Keynes ,who ,in his mathematical lattice structure diagram, presented on p.39 of his A Treatise on Probability, specifies a set of non-numerical ,logical probabilities U,V,W,X,Y, and Z in Euclidian space ,which generalized the linear and additive classical mathematical probabilities given by the horizontal line segment 0AI,where A is a linear and additive classical ,mathematical probability. The contrast provided by Keynes is striking, as it allows a reader to grasp that Keynes’s analysis involves nonlinearity and non-additivity, which is a necessary prerequisite to model non comparability and non -measurability and hence Keynesian uncertainty.
Ramsey’s theory, on the other hand, was represented by the horizontal, linear line,OAI.H.E. Kyburg, in four articles published in 1994,1999,2003 and 2010, established that Keynes’s logical theory of probability is correctly represented only by a mathematical lattice structure, which incorporates non -measurability and non-comparability. It is interesting that all economists, historians, philosophers, etc., writing on Keynes’s logical theory of probability in the 20th and 21st centuries, rejected Kyburg’s hard analysis in favor of an idiotic, bizarre, absurd, moronic and foolish assertion that Keynes’s diagram on p.39 is an example of ordinal probability. This is quite impossible ,as the ordinal theory concept being deployed by economists is linear and additive, not nonlinear and non-additive .Thus, if p1>p2 or if p3<p4,then p1,p2,p3 and p4 are all raised to the first power ,meaning that they are linear and additive so that p1+p2=1 or p3+p4=1.It is impossible to represent them by Keynes’s nonlinear and non-additive logical probabilities U,V,W,X,Y and Z as shown in the diagram on page 39 .
The following claim made by McCann (see McCann,1994,pp.41-42),which is representative of the work of all economists, philosophers and historians analyzing the diagram on p.39(Keynes,p.42,1973),was wrong in 1994 and wrong in 2025.McCann’s claim is that Keynes’s OAI line represents numerical probability, which must be one dimensional, while Keynes’s logical probabilities(intervals) U,V,W,X,Y ,and Z represent ordinal probability ,which must be multidimensional. As demonstrated above, this is simply mathematically impossible. Keynes’s diagram is actually a representation of what Boole was doing on pp.287-325 ,which was working out what modern theory calls a least upper bounds(lub) and greatest lower bound(glb). The existence of l.u.b.’s and g.l.b.’s automatically defines a mathematical lattice structure.
Keynes’s page 39 diagram has been very severely misinterpreted for over 100 years by every single historian, economist, and philosopher, who has written on Chapter III of Keynes’s A Treatise on Probability, as being a representation of ordinal probability. This is impossible because the ordinal concept of probability is additive, which means that ordinal probabilities sum to 1 so that there can be no uncertainty. This error, like so many of the errors that permeates the work done on Keynes’s logical theory of probability by economists, follows from the wild, wooly, weird and wacko claims made by F P Ramsey in 1921,1922,1923 and 1926, which are still, mysteriously, universally accepted as of 2025 in the Liberal Arts and Social and Behavioral Sciences.
Add the following book to the references-
McCann, Charles. (1994). Probability Foundations of Economic Theory. London; Routledge.
Keynes’s Axiom (i) has nothing to do with Ramsey’s talk about an axiom I that does not exist. This is easily established by comparing Keynes’s axiom (i) with both the 1922 version of Ramsey’s axiom I and his 1926 version. First, we will present Keynes’s axiom (i) and then both of Ramsey’s versions of his Axiom I:
“5. Preliminary Axioms:
We shall assume that there is included in every premiss with
which we are concerned the formal implications which allow us to
assert the following axioms:
(i.) Provided that a and h are propositions or conjunctions of
propositions or disjunctions of propositions, and that h is not an
inconsistent conjunction, there exists one and only one relation of
probability P between a as conclusion and h as premiss. Thus any
conclusion a bears to any consistent premiss h one and only one
relation of probability.” (Keynes, p.135).
Ramsey’s two versions of what he claims was Keynes’s axiom I are given below:
“First, he thinks that between any two non-self-contradictory propositions there holds a probability relation (Axiom I), for example between 'My carpet is blue' and 'Napoleon was a great general.’(Ramsey,1922, p.3)
and
“Mr. Keynes accounts for this by supposing that between any two propositions, taken as premiss and conclusion, there holds one and only one relation of a certain sort called probability relations; and that if, in any given case, the relation is that of degree α, from full belief in the premiss, we should, if we were rational, proceed to a belief of degree α in the conclusion. “(Ramsey,1926; In Kyburg and Smokler ,1980,2nd ed., pp.26-27).
Nowhere in either the A Treatise on Probability or in any of the CWJMK does Keynes ever state
“…that between any two non-self-contradictory propositions there holds a probability relation (Axiom I)..”(Ramsey ,1922,p.3)
or
“Mr. Keynes accounts for this by supposing that between any two propositions, taken as premiss and conclusion, there holds one and only one relation of a certain sort called probability relations…”( Ramsey,1926; In Kyburg and Smokler ,1980,2nd ed., pp.26-27).
It is quite shocking that there was NO logician,statistician,mathematician,economist,philosopher,historian,poli-tical scientist, psychologist, sociologist, anthropologist or behavioral scientist, in either the 20th or 21st centuries, who pointed out the obvious contradictions in Ramsey’s 1922 and 1926 claims, especially given Bertrand Russell’s complete refutation of Ramsey’s assertions in the July,1922 issue of the Mathematical Gazette in his star footnote on page 120,which showed that all of Ramsey’s examples of axiom I are composed of UNRELATED OR IRRELEVANT pairs of propositions ,which were ruled out by Keynes on pp.4-6 of chapter I of the A Treatise on Probability:
“2+2=4;Napoleon disliked poodles…” (Russell,1922,p.120,*footnote).
It is very straightforward to conclude that Keynes’s logical theory of probability does NOT deal with only two propositions and that it is simply false that Keynes stated that “…between any two non-contradictory propositions” or “…between any two propositions…” there was a logical relation.
Keynes starts out by repeating materials originally presented in earlier chapters that dealt with his argument form, where the argument form is Keynes’s Boolean, relational, propositional logic, which incorporates ,as it MUST ,an objective, logical, probability relation connecting the premises and the conclusion:
“The Theory of Probability deals with the relation between
two sets of propositions(author’s note-note that this has nothing to do with Ramsey’s two propositions only), such that, if the first set is known to be true,
the second can be known with the appropriate degree of probability
by argument from the first.∗ The relation, however, also exists when
the first set is not known to be true and is hypothetical.
In a symbolical treatment of the subject it is important that we
should be free to consider hypothetical premisses, and to take account
of relations of probability as existing between any pair of sets of
propositions, whether or not the premiss is actually part of knowledge.
But in acting thus we must be careful to avoid two possible sources
of error.
.
.
.
The first is that which is liable to arise wherever variables
are concerned. This was mentioned in passing in § 18 of Chapter IV[author’s note-section 18 of chapter IV depends on section 16 of chapter IV where Keynes introduces variables into the analysis .This requires the use of a 1st order (predicate) logic, which Keynes uses heavily in chapter XXIII].
We must remember that whenever we substitute for a variable some
particular value of it, this may so affect the relevant evidence as to
modify the probability. This danger is always present except where,
as in the first half of Chapter XIII., the conclusions respecting the
variable are certain.
3. The second difficulty is of a different character. Our premisses
may be hypothetical and not actually the subject of knowledge. But
must they not be possible subjects of knowledge? How are we to deal
with hypothetical premisses which are self-contradictory or formally
inconsistent with themselves, and which cannot be the subject of
rational belief of any degree?” (Keynes,1921, p.123; underline and boldface added)
Keynes concludes that, as was the case considered in chapters I and II, that any unrelated propositions will be excluded from any logical analysis, so will inconsistent propositions and all propositions that are not the subject of knowledge, which also applies to Ramsey’s examples:
“Whether or not a relation of probability can be held to exist
between a conclusion and a self-inconsistent premiss, it will be
convenient to exclude such relations from our scheme, so as to avoid
having to provide for anomalies which can have no interest in an
account of the actual processes of valid reasoning. Where a premiss
is inconsistent with itself it cannot be required.”
(Keynes,1921, p.124; underline and boldface added)
Part I was the Introduction, which incorporated a coverage that, although non-technical, would have allowed a careful reader to have been able to make an accurate assessment of what Keynes was doing. However, there is only one individual who appears to have been able to successfully make such an assessment-Francis Ysidro Edgeworth. Edgeworth was very straightforward in his correspondence with Edwin Bidell Wilson, who was also straightforward with Edgeworth. Both of them admitted to the other that they had no idea what it was that Keynes was doing in Part II. However, Edgeworth, by making a very careful reading of Part I, figured out what Keynes was doing -presenting an interval valued theory of probability. No other academic economist succeeded in doing what Edgeworth did in the 20th and 21st centuries.
• Boole’s relational, propositional logic in chapters I and II
• Boole’s critique of the POI’s applicability in chapters IV, V,VI and VII
• Boole’s use of interval valued probability in chapter III based on applications of the word “between”
• Boole’s use of partially ordered sets[(existence of non-numerical probability)Boole’s indeterminate probability],plus the emphasis placed on the sets of all disjunctions (or,ᴠ) and conjunctions(and,ᴧ) defined in chapter XII, which leads to the mathematical lattice structure presented by Keynes on page 39 ,as a simplification for the reader of Part I, of Part II’s future discussion of interval valued probability in chapters XV,XVI and XVII.
Thus, in retrospect, Keynes probably needed to seriously considered if he needed to also include the final paragraph from chapter X as becoming the final paragraph in chapter IX also.
Of course, Keynes could not have foreseen that no academicians in either the 20th or 21st centuries, especially economists, historians and
philosophers, WOULD read his Part II and that all academicians WOULD ,instead, substitute Ramsey’s idiotic and absurd assessments, which Ramsey claimed to represent Keynes’s logical theory ,for a reading of Keynes’s A Treatise on Probability:
“7. In Chapter XV. I bring the non-numerical theory of probability
developed in the preceding chapters into connection with the usual
numerical conception of it, and demonstrate how and in what class of
cases a meaning can be given to a numerical measure of a relation
of probability. This leads on to what may be termed numerical
approximation, that is to say, the relating of probabilities, which are
not themselves numerical, to probabilities, which are numerical, by
means of greater and less, by which in some cases numerical limits
may be ascribed to probabilities which are not capable of numerical
measures.” (Keynes ,1921, pp.121-122).
This does the job perfectly. Unfortunately, it is in Part II, not Part I,of the A Treatise on Probability.
“10. A proposition can be a member of many distinct classes of
propositions, the classes being merely constituted by the existence of
particular resemblances between their members or in some such way.
We may know of a given proposition that it is one of a particular class
of propositions, and we may also know, precisely or within defined
limits, what proportion of this class are true, without our being aware
whether or not the given proposition is true. Let us, therefore, call the
actual proportion of true propositions in a class the truth-frequency†
of the class, and define the measure of the probability of a proposition
relative to a class, of which it is a member, as being equal to the
truth-frequency of the class
The fundamental tenet of a frequency theory of probability is,then, that the probability of a proposition always depends upon
referring it to some class whose truth-frequency is known within wide
or narrow limits. Such a theory possesses most of the advantages of
Venn’s, but escapes his narrowness. There is nothing in it so far which
could not be easily expressed with complete precision in the terms of
ordinary logic. Nor is it necessarily confined to probabilities which
are numerical. In some cases we may know the exact number which
expresses the truth-frequency of our class; but a less precise
knowledge is not without value, and we may say that one probability
is greater than another, without knowing how much greater, and that
it is large or small or negligible, if we have knowledge of
corresponding accuracy about the truth-frequencies of the classes to
which the probabilities refer.” (Keynes,1921, p.101;boldface and underline added)
Keynes has thus introduced a Boolean , relational, propositional logic ,as well as imprecise probability(Keynes used the verbiage approximate measures and/or inexact measurement or non numerical probability),into his logical, frequentist theory. From Chapter VI, Keynes had already defined a measure about how the evidential weight of the argument would be measured in a frequentist approach:
“The same distinction may be explained in the language of the
frequency theory.∗ We should then say that the weight is increased
if we are able to employ as the class of reference a class which is
contained in the original class of reference.” (Keynes,1921, p.76-Keynes’s star footnote refers explicitly to chapter VIII).
Since the frequentists of Keynes’s time , following Venn, Peirce, Reichenbach or von Mises, rejected any other decision theoretic measure except for a precise probability concept, it follows that they could not accept Keynes’s construction of incorporating imprecise probability along with a second variable to measure evidential weight in a decision theoretic context, which ,for instance ,Carnap later called degree of firmness and Savage called sureness(sure probabilities versus not sure probabilities). This means that Keynes’s theoretical exposition would have been rejected, given that all frequentists of the time accepted only some strict version of a limiting frequency approach to probability, which rejects any type of interval valued probabilities or evidential weight .(There are three papers that are relevant ,which appeared long after 1921.The first is Peter Walley and Terrence L. Fine’s(1982) “ Towards a frequentist theory of upper and lower probability” ,published in the Ann. Stat. 10 (3) (1982) 741–761.
Two recent papers which attempted to put forth constructs similar in some ways to what Keynes tried to do in 1921 are Hubert’s (2021),” Reviving Frequentism”, published in Synthese and Fröhlich, Derr, and Williamson(2024), “Strictly frequentist imprecise probability”, published in The Journal of Approximate Reasoning. These papers will not be dealt with in this paper).
Keynes discusses academicians from the past who have questioned, either wholly or in part, the legitimacy of the application of the POI. Thus, he covers thinkers such as Locke, Hume, Poincare, Venn, von Kries and Boole.
Keynes missed a great opportunity to delve closely into Boole’s criticisms of the POI here. Unfortunately, and inexplicably, Keynes decided NOT to do so. Keynes waits until Part II is reached in order to cover Boole in the appendix to chapter XIV, and chapters XV, XVI and XVII of his A Treatise on Probability, which present a detailed exposition of Boole’s The Laws of Thought.
The great problem here ,of course, is that, just as no academician ,excepting Hailperin ,read Boole’s chapters XVI-XXI ,no academician read Part II, chapters X to XVII, of the A Treatise on Probability .Hailperin did not read much of Part II ,given the elementary mistakes he makes about (a) Keynes’s Carnapian like definition of probability, Carnap’s p1, (b) the meaning of Keynes’s “non numerical” probabilities (See Hailperin,1996,pp.15,137,143) and (c) his failure to grasp the mathematical lattice structure in chapter XII that must result from Keynes’s poset assumptions and non-linear and non additive interval valued probabilities. More serious is that these errors led to Hailperin’s inexplicable failure to recognize that Keynes was committed to using Boole’s logical approach to probability ,which is based on a four valued logic ,and not the standard two valued logic, which is a very special case ,as no two valued logic, based strictly on (1,0) values ,can possibly deal with probability, which is why Boole rejected the Peirce-Jevons version, offered to Boole in correspondence with Jevons in 1863, based on a two valued logic. This is inexplicable, given Hailperin’s own clear cut understanding that Boolean logic is NOT a two valued logic and that Keynes was following Boole.
Given that no other academician, in either the 20 or 21st centuries, knows what Boole is doing, then it is not possible for any academician in the 20th or 21st centuries to know what Keynes is doing either, since to understand what Keynes is doing requires that one understands what Boole is doing. The problem grows exponentially from this point on, as Keynes’s A Treatise on Money is based on Keynes’s A Treatise on Probability and Keynes’s General Theory is based on both Keynes’s A Treatise on Money and A Treatise on Probability. There was only one economist who clearly suspected that this was the case-Hugh Townshend. Unfortunately, he gave up his correspondence with Keynes just as he was on the verge of successfully grasping what Keynes had explained to him. I. Hishiyama also suspected that Keynes had created a ‘new logic of uncertainty, ‘but never realized that the Boole -Keynes connections were pervasive.
The following statements, taken from The Laws of Thought, should have been given as long footnotes to one of the pages in chapter VII where Keynes briefly mentions Boole. That would have been sufficient to demonstrate Boole’s severe criticisms of the POI:
“The above solution is usually founded upon a supposed analogy of the problem with that of the drawing of balls from an urn containing a mixture of black and white balls, between which all possible numerical ratios are assumed to be equally probable. And it is remarkable, that there are two or three distinct hypotheses which lead to the same final result. For instance, if the balls are finite in number, and those which are drawn are not replaced, or if they are infinite in number, whether those drawn are replaced or not, then, supposing that m successive drawings have yielded only white balls, the probability of the issue of a white ball at the m+1 th drawing is
m+1/ m+2
It has been said, that the principle involved in the above and in similar applications is that of the equal distribution of our knowledge, or rather of our ignorance—the assigning to different states of things of which we know nothing,and upon the very ground that we know nothing, equal degrees of probability.
I apprehend, however, that this is an arbitrary method of procedure. Instances may occur, and one such has been adduced, in which different hypotheses lead to the same final conclusion. But those instances are exceptional. With reference to the particular problem in question, it is shown in the memoir cited, that there is one hypothesis, viz., when the balls are finite in number and not replaced, which leads to a different conclusion, and it is easy to see that there are other hypotheses, as strictly involving the principle of the “equal distribution of knowledge or ignorance,” which would also conduct to conflicting results…”(Boole,1854,pp.369-370; underline added)
and
“…These results only illustrate the fact, that when the defect of data is
supplied by hypothesis, the solutions will, in general, vary with the nature of the hypotheses assumed; so that the question still remains, only more definite in form, whether the principles of the theory of probabilities serve to guide us in the selection of such hypotheses. I have already expressed my conviction that they do not—a conviction strengthened by other reasons than those above stated. …Still it is with diffidence that I express my dissent on these points from mathematicians generally, from one who, of English writers, has most fully entered into the spirit and the methods of Laplace; and I venture to hope, that a question, second to none other in the Theory of Probabilities in importance, will receive the careful attention which it deserves.”(Boole,1854,p. 375 ;underline added).
More than enough evidence is presented here that the POI has absolutely NOTHING to do with Boole’s logic and hence NOTHING to do with Keynes’s Boolean logic approach.
Following Boole’s dictum ,that a logical analysis must always take place first before any mathematical analysis takes place second , Keynes devotes Chapter VI of his A Treatise on Probability to providing a logical analysis of his logical symbol, V =V (a/h),designated as the “evidential weight of the argument”, where the argument form is identical to that introduce in chapter I of his book. Thus, h will represent the premises stated in the argument and a will be the conclusion reached. Like the probability argument discussed in chapters I and II, Keynes will eventually add a mathematical analysis in chapter XXVI of the A Treatise on Probability; however, the reader of chapter VI is fully informed of this by a footnote explicitly linking chapter VI to Chapter XXVI.
Now Keynes decided, when introducing his logical relation of probability in chapters I and II, to also incorporate an initial, introductory, mathematical analysis, so that one could arrive at
P(a/h) =α, where 0≤α≤1, and
P is a symbol representing a logical, objective, probability relation .α is the partial degree of rational belief resulting from the argument.
However, Keynes only presents
V(a/h) in chapter VI. So, no measurement takes place, although he does discuss comparisons between different logical V relations.
Keynes waits to define a mathematical analysis, that will define a measure of V, until he arrives at Chapter XXVI.
The answer Keynes provides in chapter XXVI is that
V(a/h) = w, where Keynes defines w as
“…the degree of
completeness of the information upon which a probability is based…”,
where 0≤w≤1 and w=[K/(K+I)], given that K+I=1,where K =Absolute Knowledge and I =Absolute Ignorance (Keynes,1921,p.71) .This is identical to the normalization process for probabilities that define the sum of all probabilities to equal 1,i.e,p+q=1 as defined by Keynes in a footnote on page 315:
“We could, if we liked, define a conventional coefficient c of weight
and risk, such as c =2pw/ (1 +q) (1 +w), where w measures the ‘weight,’
which is equal to unity when p = 1 and w = 1, and to zero when p = 0
or w =0 and has an intermediate value in other cases. ∗” (Keynes,1921, p.315).
This is easily rewritten as
c=p[(1/(1+q)] [2w/(1+w)].
In this form, it is easy to see that Keynes has added two weights to the standard additive, linear probability model.
Keynes’s decision weight coefficient, c, allows one to reach the same conclusions, as reached by the use of the more difficult to handle interval valued approach to probability, without having to deal with the difficult, mathematical, technical issues of interval probability. Thus, intervals that have a small(narrow) range between the lower and upper bound have a greater w than interval probabilities that have a large (wide) range between the lower and upper bound. Keynes is very clear on this in his initial, introductory discussions in chapter III of the A Treatise on Probability:
“Whilst he may be able to make sure of a profit, on the principles of
the bookmaker, yet the individual figures that make up the book are,
within certain limits, arbitrary.”(Keynes,1921,p.22)
and
“In fact underwriters themselves
distinguish between risks which are properly insurable, either because
their probability can be estimated between comparatively narrow
numerical limits…” (Keynes,1921,p.23)
and
“A distinction,
interesting for our present purpose, between probabilities, which can
be estimated within somewhat narrow limits, and those which cannot,
has arisen in a series of judicial decisions respecting damages.” (Keynes ,1921, p.24)
and
“A relation of probability does not yield us, as a rule, information of
much value, unless it invests the conclusion with a probability which
lies between narrow numerical limits.” (Keynes,1921, p.31)
and
“We frame two ideal arguments, that is to say, in which the general
character of the evidence largely resembles what is actually within
our knowledge, but which is so constituted as to yield a numerical value,
and we judge that the probability of the actual argument lies between
these two. Since our standards, therefore, are referred to numerical
measures in many cases where actual measurement is impossible, and
since the probability lies between two numerical measures…” (Keynes,1921, p.32).
The reader should note that Keynes is incorporating chapter VI and chapter XXVI analysis in his section 5 of chapter III:
“The plaintiff had
evidently suffered damage, and justice required that she should
be compensated. But it was equally evident, that, relative to
the completest information available and account being taken of
the arbiter’s personal taste, the probability could be by no means
estimated with numerical precision. Further, it was impossible to say
how much weight ought to be attached to the fact that the plaintiff
had been head of her district (there were fewer than 50 districts); yet
it was plain that it made her chance better than the chances of those
of the 5 left in, who were not head of their districts.” (Keynes,1921, p.27; underline added).
“Our previous conclusion that numerical measurement is often
impossible agrees very well, therefore, with the argument of the
preceding chapter that the rules, in virtue of which we can assert
equiprobability, are somewhat limited in their field of application… But the recognition of this same fact makes it more necessary to
discuss the principles which will justify comparisons of more and less
between probabilities, where numerical measurement is theoretically,
as well as practically, impossible.” (Keynes,1921
, p.65)
Keynes also appends an extremely important footnote which was overlooked by every reader attempting to cover Keynes’s theory in his A Treatise on Probability:
“∗Parts of Chap. XV. are closely connected with the topics of the
following paragraphs, and the discussion which is commenced here is
concluded there.” (Keynes,1921, p.65).
Keynes next makes it clear that Part II must be covered (Keynes,1921, pp.66,68) for a clear understanding of (i) and (ii) discussed below:
“We are able, I think,
always to compare a pair of probabilities which are
(i.) of the type ab/h and a/h
or
(ii.) of the type a/hh1 and a/h
provided the additional evidence h1 contains only one independent
piece of relevant information.
(i.) The propositions of Part II. will enable us to prove that
ab/h < a/h unless b/ ah = 1;
that is to say, the probability of our conclusion is diminished by the
addition to it of something, which on the hypothesis of our argument
cannot be inferred from it. This proposition will be self-evident to
the reader. The rule, that the probability of two propositions jointly
is, in general, less than that of either of them separately, includes the
rule that the attribution of a more specialised concept is less probable
than the attribution of a less specialized concept.” (Keynes,1921, p.66).
The conclusion is obviously clear. Chapter XV and Part II is where Keynes finishes/ends the discussion begun in chapters IV and V on applying the POI. Unfortunately, no academician in the 20th or 21st century carried out Keynes’s requirement.
What has happened involves two severe deficiencies in the assessments made by D. Gillies, Howson ,Urbach ,Howson and Urbach ,Childers, and Runde of Keynes’s work regarding (a) their ignorance of Boole’s estimation approach to both initial, a priori probability and a posteriori ,revised probability and (b) their reading into chapter IV of Keynes’s A Treatise on Probability the Ramsey claim that Keynes’s substantial efforts to successfully modify and tighten up the conditions needed to reliably apply the PNSR ,now designated by Keynes as the POI, meant that Keynes was doing this so that he could make use of the result as the foundation for his theory. Nothing could be further from the truth.
Keynes was very clear about this in Part II, chapter XV, which D. Gillies, Howson, Urbach, Howson and Urbach, Childers, and Runde, never read:
“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.” (Keynes,1921, p.160; underline and boldface added).
Keynes makes it very clear what he is relying on in the very next paragraph, again not read by academicians. D. Gillies, Howson, Urbach, Howson and Urbach, Childers, and Runde have read into Keynes’s works ideas which are hallucinations on their part:
“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.
This method is frequently adopted in common discourse.” (Keynes,1921, p.160; boldface and underline added)
The wild, wooly, weird and wacko claims of Runde, based on idiotic assertions made by F P Ramsey, that Keynes is relying on comparative probability, is a reflection of Runde’s reliance on the gobbledygook of F P Ramsey when he was reading some small parts of the TP.