


































Advances in Politics and Economics 
ISSN 2576-1382 (Print) ISSN 2576-1390 (Online) 

Vol. 6, No. 1, 2023 

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11 
 

Original Paper 

Contrary to T. Hirai, Ramsey’s Critiques of Keynes in 1922 and 

1926 Were Completely Wrong 

Michael Emmett Brady
1
 

1
 Adjunct Lecturer, California State University, Dominguez Hills, College of Business Administration 

and Public Policy, Department of Operations Management, 1000 East Victoria St arson, California, 

90747 USA  

 

Received: December 14, 2022    Accepted: January 22, 2023   Online Published: January 29, 2023 

doi:10.22158/ape.v6n1p11          URL: http://dx.doi.org/10.22158/ape.v6n1p11 

 

Abstract  

This paper covers Harai’s analysis, contained in his section, titled “Keynes as a philosopher”, of 

Keynes’s logical theory of probability. Harai spends too much of his time repeating Ramsey’s claims 

about having uncovered serious errors in the structure of Keynes’s relational propositional theory.  

.For 100 years, Keynes’s logical theory has been interpreted and misevaluated through the eyes of an 

ignorant 18 year old teenager. The result has been the proliferation and spread of what can be called 

the “Ramsey myth”. 

The “Ramsey myth” is that an 18 year old teenager appeared at Cambridge University in 1921.This 18 

year old teenager was a genius who wrote a three page review in the Jan., 1922 issue of Cambridge 

Magazine, which supposedly destroyed, devastated and demolished the logical foundations of Keynes’s 

A Treatise on Probability, which was his relational, propositional logic founded on Boole’s relational, 

propositional logic. Russell countered this in his review, but was ignored (See Brady, 2016a). In 1931, 

it is further supposed that Keynes then capitulated to Ramsey and repudiated his own logical theory of 

probability, accepting some version of Ramsey’s subjectivist theory. This myth is what Hirai’s paper is 

based on. It was false in 1921 and it is false today in 2022. 

 

1. Introduction 

The paper will be organized in the following manner. Section Two will present a very brief overview of 

Keynes’s logical theory of probability. Section Three will show how Edgeworth’s two vastly superior 

reviews could have saved Hirai from making the many errors that he made in his paper, where he 

simply repeats all of Ramsey’s claims. Section Four will show that all of Hirai’s analysis concerning 

Keynes’s A Treatise on Probability (TP, 192) is wrong. Section Five will conclude the paper. 



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T. Harai’s recent paper in the Review of Keynesian Studies, 2021,contains a section called “sketch 2 

Keynes as a philosopher” (pp. 192-198). All seven pages are completely wrong as they are based on 

what Ramsey claimed, without any textual support, Keynes’s logical theory of probability entailed (See 

Brady 1990, Brady, 2004a, Brady, 2004b, Brady, 2012, Brady, 2021a, Brady, 2021b, Brady, 2015, 

Brady and Arthmar, 2012, Arthmar & Brady, 2010, 2017, 2018). All of Ramsey’s definitions of 

Keynes’s relational, propositional logic, made in 1922 and 1926, appear nowhere in anything written 

by Keynes in his Collected Writings of John Maynard Keynes or in his lifetime. Ramsey basically 

made things up as he went along in the years 1922-1926. Ramsey grossly distorted Keynes’s analysis 

until you end up with the very distorted assessments made by C. Misak (2020), nearly 100 years later, 

about Keynes purportedly making the following claims : 

“Ramsey peppered the Treatise with problems. He objected to the attempt to provide a logical 

foundation for the Principle of Indifference. He objected to the idea of an unmeasurable, non-numerical 

probability and would later, in the 1926 “Truth and Probability”, offer an account of how all 

probabilities are measurable. And he objected to the very idea of Keynes’s objectively fixed probability 

relations—the idea that all statements stand in logical relations to each other. As Ramsey put it, there is 

no such probability as the probability that “my carpet is blue” given that ‘Napoleon was a great 

general’.” (Misak, 2020, p. 114). 

First of all, Keynes never attempted “… to provide a logical foundation for the Principle of 

Indifference.” (Misak, 2020). Second, Keynes’s supposed “idea of an unmeasurable, non-numerical 

probability.” (Misak, 2020) is actually Keynes’s interval valued, imprecise probabilities that involve 

inexact measurement and approximation, as covered in great mathematical detail by Keynes in chapters 

15 and 17 in Keynes’s Part II of the A Treatise on Probability. Neither Ramsey or Braithwaite or Misak 

or Hirai ever read Part II. Third, Keynes never stated Misak’s claim concerning “the idea that all 

statements stand in logical relations to each other. As Ramsey put it, there is no such probability as the 

probability that ‘my carpet is blue’ given that ‘Napoleon was a great general’” (Misak, 2020) at any 

time during his life. 

Hirai, much like Misak, has absolutely no idea about what he is  

 

2. Method: Keynes’s Probability (P) Relation and the Weight (V) Relation. A Decision (D) Is a 

Function of Both, so That D = f (P,V) 

Keynes’s actual decision function included the time constraint, T. The Time constraint concerned how 

much time the decision maker had available in order to make a decision. If the decision must be made 

within a fixed, short run period of time, as in a rated chess tournament involving over the board chess 

games, then this constraint would become binding. However, if the chess game was a correspondence 

(postal) game, then the time constraint would not be binding Only Savage also considered such a 

binding time constraint in his subjectivist approach because he recognized the severe problem of 

vagueness that impacted a decision maker’s assessment of his personal probability. In other words, 



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Savage recognized Keynes’s concern with the importance of weight, but felt that it could not be 

formally integrated into his theory, although he hinted at seriously taking a long look at interval valued 

probability. These concerns made his subjective approach to decision making vastly superior to 

Ramsey’s very inferior subjective approach. On the other hand, if the decision can be postponed until 

the long run, as it can be in a rated tournament correspondence (Postal) chess game, then the time 

constraint, as noted above, is not binding and Keynes’s concern with weight becomes much less 

relevant. Savage recognized the binding nature of this constraint when he restricted the application of 

his theory to the short run and micro level only. 

Therefore, suppressing the T variable yields Keynes’s two logical relations, P and V, where P(a/h)= 

α,0≤α≤1, where α is a rational degree of belief and V(a/h) =w, 0≤w≤1, where w measures the degree of 

the completeness of the relevant information (See Keynes, 1921, TP, p. 313, pp. 314-315; p. 315) 

supporting Keynes’s argument form. This logical relation, V, was severely misinterpreted in Runde 

(1990, pp. 278-283) as being a mathematical variable which was being used to measure weight. Runde 

relies completely on chapter 6 alone for his evaluation. This had been done previously by IJ Good and I 

Levi. Keynes made it very clear on p. 315 of the TP, which Runde overlooked, that weight, V, is 

measured by w. It should be clear from V(a/h)=w, 0≤w≤1,that V can’t measure anything, as it is the 

logical relation of weight between the a and h propositions that connects them(See Brady,2017c,d),. All 

heterodox, Post Keynesian, neo -Keynesian and Institutionalist economists, working in this area in the 

20th and 21
st
 centuries, continue to cite Runde’s 1990 article (just under 300 citations according to 

Google Scholar). 

The result is that it is impossible for them to make use of Keynes’s conventional coefficient, c, of 

weight and risk, as it is asserted that it must be V that is used to measure weight and not Keynes’s w. It 

is also asserted that Keynes erroneously presented three different definitions of weight in the TP when 

Keynes actually only presented one, which was that V(a/h)=w,0≤w≤1. 

Runde’s errors were first introduced into the academic literature by the philosopher I J Good in the 

early 1950’s and then emphasized by I. Levi in the late 1960’s.Both Good and Levi overlooked, as did 

Runde, Keynes’s clear definition on p.315 of the TP that only w measures V, weight (See Brady, 2018). 

Runde’s error, of confusing and conflating V with w, was repeated one year later in Runde (1991, pp. 

128-129), where Runde confuses and conflates P with α,which is a mathematical variable that can take 

on a numerical value from the interval (0, 1):  

“The probability relation may be written as a/h=P. This relation is objective, on Keynes’s account, and 

expresses the degree of belief P is rational to hold in a in the circumstances h.If a is implied by h, 

P=1,if the relation between a and h is contradictory, P=0.In all other cases, where a is only partially 

entailed by h, P will lie somewhere somewhere between these two extremes…” (Runde, 1991, pp. 

128-129). 

It is mathematically impossible for a logical relation to equal 0, 1 or a number between 0 and 1.It is α 

that is equal to 0, 1 or a number between 0 and 1 once the probability relation has been normalized on 



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the unit interval [0,1].The probability relation has to be written as a/h, and not as a/h=P.  

Runde errs again on the same page when he writes the following: “This can be seen if we again write 

a/h=P (1) where a is the ‘primary proposition ‘and relation (1) the secondary proposition.” (Runde, 

1991, p. 129). 

One can suppress the notation, denoting “relation”, as Keynes did on p.4, where he writes (a/h) =α or 

on p.40 of the TP, where he writes (a/h)=P. However, then P is defined as the probability P, not the 

relation P. 

Runde has confused and conflated P with α. See above. This is identical to his conflation of V with w. 

Vercelli spotted Runde’s errors in 2011. Vercelli spots the fundamental error that it is obvious is 

contained in the technical analysis of Runde He fixes the flaw, but, again, does not see the contradiction 

when he talks about Keynes’s c coefficient. Consider the following : 

“Therefore, most interpreters believe that a satisfactory measure of the weight of argument may be 

given simply by 

V(x/h) = K (1) 

Runde (1990) suggests the following measure: V(x/h)=K/I (2) 

We may overcome these shortcomings by introducing the following measure, which is derived from the 

third definition [author’s note-Vercelli accepts Runde’s illusion that Keynes had three separate 

definitions of weight in his TP]: 

V(x/h)= K/(K+I) “ (3)” 

(Vercelli, 2011, pp. 153-154). 

Therefore, Vercelli was able to see the nature and scope of the mathematical problems that existed with 

the Runde, Cr articles. He attempts to remedy this in his footnote 1: 

By introducing the usual criterion of normalization of probability measures K+I=1 (a criterion that 

Keynes himself has utilized in the TP, for example, on page 348)….(Vercelli, 2011, p. 168). 

This fixes one problem, but simply creates another problem that takes the place of the failure of Runde 

to normalize on the unit interval [0, 1].  

Vercelli’s requirement, that K +I =1, must actually be written as  

0≤V≤1, 

which now states that the logical relation, V, is restricted to applications between 0 and 1,which makes 

no sense at all because you can’t restrict a logical relation in this fashion. 

Keynes’s position on belief thus depends on both P and V, not P alone, as was argued by Ramsey. 

Ramsey further confused the issue by claiming that there was something called a non numerical 

probability relation. This so called non numerical probability relation does not exist anywhere in 

Keynes’s A Treatise on Probability. 

Keynes’s P and V relations are independent, since α =p/(p+q) and w=K/(K+I),where p is the probability 

of success, q is the probability of failure, K is the absolute amount of Knowledge and I the absolute 

amount of Ignorance.  



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However, V constrains the type of probability that α is, either numerical or non numerical. If w=1, so 

that all of the α values are numerical and additivity holds, then it is numerical. It does not hold in the 

general case
1
, where w<1,where the α values are interval valued probabilities so that non additivity 

holds. A decision maker can accept, and have a different opinion about, any value within the boundary 

set up by the upper and lower probabilities defining the interval.  A w<1 creates very complicated, 

complex, severe and intricate problems concerning/about one’s beliefs because of the non- linearities 

introduced by w in Keynes’s decision theory, which is a function of both α(probability) and w(weight) 

or the relations P and V. This can best be seen by using the Mathematica program or MathLab to 

generate three dimensional contours of Keynes’s conventional coefficient of risk and weight, c, 

involving c, p and w. Everything is substantially simplified if w=1. 

Ramsey does not have any idea about what V represents. Hence, he has no inkling about the role that 

confidence plays in belief. There is no simple, direct, linear connection between probability and belief 

for Keynes as there is for Ramsey because Ramsey has conflated confidence with probability. The one 

exception
1
 is if w=1, so that V drops out and one is left with α.Only in this very special case would 

Ramsey be correct. However, Keynes never disputed this outcome, correctly seeing that it would be 

rare in the real world of actual decision making, as opposed to Ramsey’s imaginary academic world 

where everyone uses only the purely mathematical theory of the calculus of probability to base their 

decisions on. 

[Place Footnote at bottom of this page]
 
 

The exception to the general case occurs, as discussed by Keynes on p. 160 (Keynes, TP, 1921, p. 160), 

if the principle of Indifference can be applied, where the existing evidence is “…perfectly symmetrical 

in its bearing on the various alternatives”. This would be the case in Keynes’s discussion of urn models 

on pp. 75-76 of chapter 6 of the TP. The uncertain urn has numerical probabilities of ½, which are 

identical to the risky urn. The important point made by Keynes, which was that the risky urn has more 

and stronger evidence than the uncertain urn, has nothing to do with risk allowing quantitative 

estimates and uncertainty requiring non quantitative estimates of probability. 

Keynes specified an exact argument form that all of the propositions needed to satisfy in his, 

propositional logic. Keynes’s fundamental requirement is that the propositions must be related, 

connected or similar: “Let our premisses consist of any set of propositions h, and our conclusion 

consist of any set of propositions a, then, if a knowledge of h justifies a rational belief in a of degree α, 

(then-author’s insert) we say that there is a probability-relation of degree α between a and h.” (Keynes, 

1921, p. 4). 

Only if a knowledge of h justifies a can there be a “…probability-relation of degree α between a and h.” 

Otherwise, there is NO SUCH LOGICAL RELATION between a and h. Ramsey’s fanciful belief that 

there can be only one h proposition and one a proposition is nonsense, given Keynes’s very clear 

statement throughout his TP that his relational propositional logic works with sets of h and a 

propositions.  



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Keynes gives an excellent example of his argument form on pp.5-6 of chapter II of the TP: 

“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. We allow, in fact, that statements may be unproved, 

without, for that reason, being unfounded. And it does not seem on reflection that the information we 

convey by these expressions is wholly subjective. When we argue that Darwin gives valid grounds for 

our accepting his theory of natural selection, we do not simply mean that we are psychologically 

inclined to agree with him; it is certain that we also intend to convey our belief that we are acting 

rationally in regarding his theory as probable. We believe that there is some real objective relation 

between Darwin’s evidence and his conclusions, which is independent of the mere fact of our belief, 

and which is just as real and objective, though of a different degree, as that which would exist if the 

argument were as demonstrative as a syllogism. We are claiming, in fact, to cognise correctly a logical 

connection between one set of propositions which we call our evidence and which we suppose 

ourselves to know, and another set which we call our conclusions, and to which we attach more or less 

weight (author’s insert-the word “weight” refers to chapters 6 and 26 of the TP analysis of Keynes’s 

evidential weight of the argument) according to the grounds supplied by the first. It is this type of 

objective relation between sets of propositions—the type which we claim to be correctly perceiving 

when we make such assertions as these—to which the reader’s attention must be directed.” (Keynes, TP, 

1921, pp. 5-6; italics added; this is expanded upon on by Keynes on pp. 160-161). 

Ramsey’s 1922 example, “…for example between ‘My carpet is blue’ and ‘Napoleon was a great 

general’…”, that was given on page 3 of his Cambridge Magazine review in 1922, is false because 

knowledge of the color of a carpet can’t ever supply a connection (or evidence ) to justify the 

conclusion that Napoleon was a great general, where one proposition is regarded as providing the 

information or evidence upon which to accept the other proposition provisionally. The same conclusion 

holds for Ramsey’s “this is red; that is round” and his “this is red; that is blue “examples. There is no 

connection between the propositions. 

I believe that Ramsey had to have been on some type of medication, narcotic or drug treatment or been 

suffering from some unknown mental disorder at the time he was writing his review as his entire 

paragraph (Ramsey, 1922, p. 3) represents a complete contradiction of what Keynes presented very 

clearly on pp. 4-6 and pp. 54-56 of the TP. The same holds for his very similar characterization of 

Keynes’s theory made in 1926. 

Finally, Keynes would not conclude that “…the probability is not numerical.” (Ramsey, 1922, p. 3). 

Keynes would conclude that Ramsey’s example violated his argument form and was irrelevant.  

Therefore, there is no conditional probability because neither of the propositions provides any relevant 

evidence for the other proposition. It is extraordinary that so many philosophers and economists have 



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described Ramsey’s 1922 article as brilliant. In fact, it makes absolutely no sense whatever. 

How could have Hirai have avoided the intellectual mess that he created in his article? Hirai needed to 

have read chapters I, XI and XII of Boole’s 1854 The Laws of Thought while simultaneously reading 

chapters I and II of Keynes’s TP. Doing this would have provided him with hard evidence that 

Ramsey’s many, many attacks on Keynes’s logical theory of probability were also attacks on Boole’s 

logical theory of probability. It is easy to then conclude that Ramsey’s claims needed to be evaluated 

from a critical standpoint as no serious scholar had impugned Boole’s theory 

 

3. Discussion I: A Brief View of Edgeworth’s Superior Understanding of Keynes’s A Treatise on 

Probability 

Consider Edgeworth’s very brief summary of the TP: “Suffice it that many probabilities which are 

incapable of numerical measurement can be placed “between two numerical measures” (p. 32). We thus 

obtain the idea of a “finite probability” as one which exceeds some numerical probability (p. 257). 

Such measurements play a leading part in induction. To establish a generalization it is necessary that 

“with the experience we have” actually had there are finite probabilities, however small, derived “from 

some other source, in favour of the generalization” (p. 238). Round this nucleus of finite probability, 

through the operation of repetition and likeness, science grows. “An argument from” induction must 

always involve some element of analogy, and, ‘on the other hand, few arguments from analogy can 

afford to “ignore altogether the strengthening influence of pure induction “[repetition]” (p. 255).” 

(Edgeworth, JRS, 1922b, pp. 107-108; also see Edgeworth, 1922a, Brady, 2016b, c, 2017a). 

This very powerful summary by Edgeworth is still the best short summary by far that has ever been 

made of Keynes’s A Treatise on Probability (TP, 1921), even though it is now one hundred years old. 

Edgeworth’s summary is the perfect antidote to the extremely poor coverage made of Keynes’s 

approach by R. B. Braithwaite in his editorial foreword to the TP that was placed at the front of Volume 

8 of the 1973 Collected Writings of John Maynard Keynes’s TP by Donald Moggridge, as well as the 

many false claims made by the 18 year old “boy genius”, Frank Ramsey. 

Reading Edgeworth would also inoculate readers of the TP from the extremely poor assessments made 

of Keynes’s work by economists, who simply do not know what an interval valued probability is, like 

R. Dimand, R. Skidelsky, A. Carabelli, J. Runde, B. Davis, T. Lawson, G Meeks, R. O’ Donnell and 

many others. Their unanimous acceptance of Ramsey’s false claim that Keynes’s theory was primarily 

an ordinal theory that would be useless in science is probably their greatest error. 

Hirai could have saved himself from his errors if he had carefully read Edgeworth’s two reviews; 

however, there is no evidence from his references that he had ever read Edgeworth’s reviews. Hirai thus 

made the fundamental mistake of putting all of his intellectual eggs into one basket, a very inferior 

basket based on Ramsey comprising the first page (p. 115, 1921) of Chapter 10 of Part II and several 

pages taken from Keynes’s Introductory Chapter 18 (pp. 117-121, 1921) of Part III of the TP. He relied 

completely on his belief that the Ramsey’s reviews were the only one’s worth reading because only 



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Ramsey had supposedly spotted alleged flaws in Keynes’ s logical structure about propositions and the 

objective, logical, probability relation connecting them. In fact, Ramsey never correctly identified even 

one, single flaw in Keynes’s TP in either of his reviews. Ramsey’s understanding about Keynes’s 

propositional logic, which follows directly from Boole’s approach, is nil.  

The two crucial pieces of information that Hirai could have gleaned from a reading of Edgeworth were 

(a) Keynes’s theory of probability is an interval valued theory of imprecise probability and (b) the very 

important role played by finite probabilities in Keynes’s mathematical theory of induction and analogy 

explained by Keynes in chapters 20 and 22 on pp.233-238 and pp. 253-27, respectively,where a finite 

probability was defined by Keynes as being greater than both a numerical probability and/or a non 

numerical probability, where a non numerical probability is an interval valued probability. 

 

4. Discussion II: Hirai’s Mistaken Reliance on Ramsey for His Understanding of the Foundations 

of Keynes’s Logical Theory of Probability 

Consider the following claims made by Ramsey that is the foundation for Hirai’s 2021 work on 

Keynes’s logical theory of probability in the Review of Keynesian Studies: see  

“Mr. Keynes starts from the supposition that we make probable inferences for which we claim objective 

validity; we proceed from full belief in one proposition to partial belief in another, and we claim that 

this procedure is objectively right, so that if another man in similar circumstances entertained a 

different degree of belief, he would be wrong in doing so. 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 & Smokler, 1980 (2nd ed.), pp. 26-27; italics added; see 

also Bateman, 1987, p. 106). 

The italics emphasis involve Ramsey in severe error; the second to last italics emphasis holds only for 

the special case where Keynes’s Principle of Indifference (POI) is applicable or one is dealing with a 

least upper bound or a greatest lower bound for interval valued probability. 

First, nowhere in the TP does Keynes claim objective validity, which can only occur in a deductive 

logic, as opposed to practical certainty (Keynes, TP, p. 160). Second, nowhere does Keynes claim that 

his procedure is objectively right. Keynes claims that it is rational and reasonable because it is based on 

all of the relevant, available evidence which appears in the h propositions. Third, the circumstances 

must be identical, not similar. Fourth, Keynes never claimed that a decision maker would be wrong. 

Keynes stated that such a decision maker would be non rational, as opposed to being irrational. Thus, 

for Keynes, the only rational answer for both of the Ellsberg two urn ball (red and black ball) problems 

is that the probability of a red or black is ½, based on a correct application of Keynes’s POI, as based 

on Keynes’s summary on pp. 54-56 of the A Treatise on Probability (TP; 1921), and not on Keynes’s 

restatement of the erroneous Laplace-Bernoulli version on p. 42. Fifth, Keynes’s relational, 



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propositional logic does not hold’ between any two propositions’, but only between sets of propositions 

that are related or connected by a degree of similarity (or likeness or resemblance). Ramsey’s claim is 

simply a regurgitation of the false claim he first made four years earlier in 1922 in his Jan., Cambridge 

Magazine review on p. 3 that “…between any two non contradictory propositions there holds a logical 

relation…”, which Ramsey then illustrated with his completely irrelevant ‘my carpet is blue; Napoleon 

was a great general ‘example. Neither of these propositions are related or connected. All of Ramsey’s 

1922 and 1926 examples involve unconnected or unrelated propositions. Sixth, there holds one and 

only one relation of logical probability for exact, precise, numerically connected probabilities based on 

the POI or for the calculation of a least upper bound (LUB) and /or Greatest Lower Bound (GLB) for 

Keynes’s interval valued probabilities. Seventh, Keynes’s approach does not hold ‘in any given case’; it 

holds in some cases. Finally, Ramsey is very confused in his belief that Keynes’s approach only holds 

for two propositions, i.e. for one h proposition and one a proposition. In fact, Keynes works with sets of 

h and a propositions throughout the TP. The best example of this would be his Darwin example on pp. 

5-6 and its extension on p. 160-161 (TP, 1921). 

Hirai takes an approach to reading the TP which can only be described as reading “little bits and 

pieces” of Keynes’s TP. Hirai’s coverage of Part II of Keynes’s TP is a mirage. The two pages he cites 

from, pp.125-126 (Hirai, pp. 192-194) from the 1973 CWJMK version (p. 115, TP, 1921) have nothing 

to do with Keynes’s main conclusions in Part II, which were that (a) additivity is a special 

case,(b)interval valued probability is the general case, and (c) the normal distribution can’t be simply 

assumed, but must be shown to hold by the examination of preliminary data (Chapter 17, TP). The 

researcher must consider the use of the geometric and harmonic means, the median, and the mode and 

not just assume the arithmetic mean.  

Hirai’s discussions of Part III on induction and analogy on pp.241,243 and 245 (Hirai, pp. 195-196) 

from the 1973 CWJMK edition of the TP (pp. 217, 219 and 221, TP, 1921) ignore Keynes’s crucial 

discussions of finite probabilities on pp.233-238 and 254-257,as well as Keynes’s qualification that 

inductive(practical) validity is NOT the same as deductive(universal)validity (pp. 244-245, TP, 1921). 

I have already pointed out that a reading of Edgeworth’s two reviews in 1922 would have allowed Hirai 

to actually have come to grips with what Keynes was presenting in Parts II and III of the TP. 

However, it is on pp.196-198 of Hirai, where Hirai’s assessments of Ramsey’s empty criticisms of 

Keynes’s A Treatise on Probability are simply accepted as being true, that Hirai’s failure to absorb Parts 

II and III of the TP become obvious. 

Hirai’s belief, that he can take one small paragraph out of context from p. 245 (p. 221, TP, 1921) of Part 

III’s introductory chapter 18 and present it as the major conclusion of Part III of the TP, is comparable 

to Ramsey’s total confusions about Keynes’s Boolean, relational, propositional logic. 

We will consider Hirai’s confusions paragraph by paragraph. 

Consider the following:  

“Here “probability” is defined as the degree of rational belief between propositions, and, moreover, 



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“inductive arguments” are defined, being drawn by this probability. Furthermore, it is argued that 

induction should be a problem of formal logic (the existence of probability relation) rather than a 

factual problem.” (Hirai, 2021, p. 196). 

If the a and h propositions are related and connected, then there exists an objective, logical, relation of 

probability (similarity between them such that there is a degree of rational belief α. However, it is quite 

impossible that induction “….should be a problem of formal logic (the existence of probability relation) 

rather than a factual problem.” since the best outcome for an inductive argument is “practical certainty 

“only, which is NOT the deductive result of universal certainty (the conclusion is always true. See 

pp.160-161 and 244-245, TP, 1921). 

Consider now Hirai’s next confusion: “To sum up, Keynes argues that (i) induction is to be seen as the 

most important element in the argument on probability, (ii) although induction is composed of 

“analogy” and “pure induction”, analogy is more important, (iii) prior probability can be obtained 

through analogy.” (Hirai, 2021, p. 196). Of course, this leaves out of the discussion Keynes’s “finite 

probabilities”π and η as emphasized by Edgeworth. Without these finite probabilities, pure induction 

can’t be used to support a generalization. 

It is when Hirai explicitly considers Ramsey’s extraordinarily weak critique of Keynes’s theory that 

Hirai goes astray: “And in the process, he criticized Keynes’s Treatise on Probability (Ramsey (1926) 

[Ramsey [1931] pp. 156-98]). His criticism runs as follows. (1) There does not exist anything like a 

probabilistic relation between propositions. (Rejection of the definition of “probability”). (2) 

Consistency is not maintained in the arguments regarding the main principles. (Ambiguity of the 

objectivity and subjectivity of “probability” found in the TP). (3) The attempt to absorb the world of 

induction into the world of deduction is nonsense. Ramsey argues (3) as follows. The logical relation 

which justifies the inference is that the sense or import of the conclusion is contained in that of the 

premisses. But in the case of an inductive argument this does not happen in the least; it is impossible to 

represent it as resembling a deductive argument and merely weaker in degree; it is absurd to say that 

the sense of the conclusion is partially contained in that of the premisses. (Ramsey [1931] p. 186) 

Keynes responded to it with acceptance to a large extent as follows (Keynes, 1931). He states that the 

development of formal logic by Russell and [early] Wittgenstein (1922) gradually emptied out the 

content, eventually eliminating not only any experience, but the most rational thinking principles. 

Ramsey tried to counter this with a kind of pragmatism and turned to human logic. “Thus, he was led to 

consider “human logic” as distinguished from “formal logic”. Formal logic is concerned with nothing 

but the rules of consistent thought. But in addition to this we have certain “useful mental habits” for 

handling the material with which we are supplied by our perceptions and by our memory and perhaps 

in other ways, and so arriving at or towards truth; and the analysis of such habits is also a sort of logic. 

The application of these ideas to the logic of probability is very fruitful. “(JMK.10, p. 338) Keynes 

went on to admit, “So far I yield to Ramsey—I think he is right” (JMK. 10, p. 339). This is a brief 

commentary, but the matter is weighty and deep. Keynes agrees with Ramsey’s theory of probability. 



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We may reasonably conclude that he is critical of his own theory of probability, which belongs to a 

stream of formal logic.” (Hirai, 2021, pp. 196-198). 

Hirai’s (1) fails completely, since Keynes’s relational, propositional logic used in the TP comes directly 

from G. Boole’s relational, propositional logic as contained in chapters XVI-XXI of the Laws of 

Thought (see TP, 1921, p. 5, ft. 2). 

Hirai’s (2) fails completely, as he never establishes any inconsistency in any part of the TP. Hirai’s (3) 

makes no sense, because Keynes never tried to absorb induction into deduction. Where this is supposed 

to have occurred in the TP is never cited by Hirai. I await Hirai’s citing of the exact page and paragraph 

where Keynes made such an attempt in the TP. Hirai’s reliance on Ramsey’s assessment of the value of 

induction is completely erroneous. Let us consider Ramsey’s assessment: “The logical relation which 

justifies the inference is that the sense or import of the conclusion is contained in that of the premisses. 

But in the case of an inductive argument this does not happen in the least; it is impossible to represent 

it as resembling a deductive argument and merely weaker in degree; it is absurd to say that the sense of 

the conclusion is partially contained in that of the premises. (Ramsey, 1931, p. 186; p. 44, Kyburg & 

Smokler, 1980, (2
nd

 ed); italics added). 

Of course, Ramsey’s assessment of induction relative to deduction is identical to J.S. Mill’s assessment 

as pointed out by Keynes on pp.267-268: “An induction has no validity….unless it is absolutely 

certain” (Keynes, 1921, p. 267). 

Keynes had already countered Ramsey’s attack on induction with his discussion of “practical certainty” 

on p.161 of the TP, where such “practical certainty” (Darwin’s theory or Einstein’s theories or the work 

of Linus Pauling)is based on approximation. See also Keynes’s discussions of Darwin’s theory on 

pp.5-6 of the TP (1921). Ramsey’s expressions, “this does not happen in the least”, “it is impossible to 

represent it”, and “it is absurd to say”, as far as physics, biology, chemistry and engineering go, would 

be rejected by practically all physical, life and biological scientists. 

Hirai severely misquotes Keynes’s discussion of Ramsey in The New Statesman and Nation in 1931 

(Vol. 10, 1973, CWJMK, pp. 337-338; 1931, p. 407): 

“Ramsey argues, as against the theory that I had put forward, that probability is concerned not with 

objective relations between propositions but (in some sense) with degrees of belief, and he succeeds in 

showing that the calculus of probabilities simply amounts to a set of rules for ensuring that the system 

of degrees of belief which we hold shall be a consistent system. Thus the calculus of probabilities 

belongs to formal logic. But the basis for our degrees of belief—or the a priori probabilities, as they 

used to be called—is part of our human outfit, perhaps given us by natural selection, analogous to our 

perceptions and our memories rather than to formal logic. So far I yield to Ramsey—I think he is right. 

But….” (Keynes, Vol. 10, 1973, CWJMK, pp. 337-338; 1931, New Statesman and Nation, p. 407). 

The only part of this paragraph where Keynes “yields” to Ramsey is the following part: 

“…and he succeeds in showing that the calculus of probabilities simply amounts to a set of rules for 

ensuring that the system of degrees of belief which we hold shall be a consistent system. Thus the 



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calculus of probabilities belongs to formal logic.” 

However, the vast majority of Keynes’s probabilities are interval valued, which Keynes described as 

being non numerical because they required two numerals, not one, the laws of the purely mathematical 

theory of probability do not apply. Since Keynes had always agreed that instances where numerical 

probabilities held could be analyzed using the calculus of probability, Keynes’s “yielding “to Ramsey 

has no import for the theory presented in Keynes’s A Treatise on Probability. Finally, Hirai’s summary 

makes no sense: “Keynes went on to admit, “So far I yield to Ramsey—I think he is right” (JMK. 10, p. 

339). This is a brief commentary, but the matter is weighty and deep. Keynes agrees with Ramsey’s 

theory of probability. We may reasonably conclude that he is critical of his own theory of probability, 

which belongs to a stream of formal logic (Hirai, 2021, p. 198). 

Keynes does not agree with Ramsey’s theory of probability unless the evidential weight of the evidence, 

w,=1 or the lower limit = the upper limit of an interval valued probability. Keynes was not critical of 

his own theory as demonstrated in the Keynes-Townshend (see references on Townshend, 1937-1938) 

correspondence in 1937-38.Finally, Keynes’s theory does not belong to a stream of formal logic. It is 

Ramsey’s theory that belongs to formal logic (the purely mathematical theory of  

Hirai’s article directly conflicts with the first article published in the 2021 issue of the RKS by Sakai. 

Sakai argued that Keynes’s theory was an interval probability approach. Harai, as the editor of this 

journal, had to have read Sakai’s paper as he would have been responsible for sending the paper out for 

review. As Professor James S Earley told me in December, 1980,before agreeing to sit on my 

dissertation committee, that “if, and it is a big if, Edgeworth is right about Keynes’s probabilities being 

between two numbers, then Ramsey is wrong.” 

Nowhere in Harai’s article is Sakai’s result concerning interval probability Nor does Hirai used, listed 

or cited. Nor does Hirai realize that Keynes’s logic is the logic of George Boole’s The Laws of 

Thought. 

 

5. Conclusions 

Hirai bases his analysis of Keynes’s work in Part II on one irrelevant page (p. 115, TP, 1921) taken 

from Part II of the TP, three introductory pages (pp. 217, 219, 221) taken from Part III and one page 

taken from the first page of chapter 33 of Part V of the TP. The vast majority of his assessment of 

Keynes’s TP is based only on Ramsey’s 1926 review that I have demonstrated to be intellectually 

worthless and without any merit at all (See Brady, 2021a; 2021b; 2004a; 2004b). 

I will now again provide the Edgeworth quotation from his 1922 review in the Journal of the Royal 

Statistical Society so that the reader can see how inferior Hirai’s assessment of Keynes is compared to a 

100 year old reviews by Edgeworth that have never been used by any heterodox or orthodox economist 

in the 20th or 21
st
 centuries. Both of Edgeworth’s 100 year old reviews combined completely refute all 

heterodox claims made about Keynes’s logical theory of probability. The work of Bateman (1987, 1989, 

1990, 2016, 2021a, b), Davis (1994), Dimand (2021), Faulkner, Fed-uzi, McCann and Runde (CJE, 



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2021), Meeks (2003), Lawson (2012), O’Donnell (2021a, b), Skidelsky (1992) and Weatherson (2002) 

is completely refuted because all of these articles rely on Ramsey’s flawed claims. It should be noted 

that both Edgeworth reviews appear in the references of the Faulkner, Feduzi, McCann and Runde 

paper, but none of Edgeworth’s analysis is ever used or quoted anywhere in their paper. 

It is interesting to see that Bateman, Skidelsky, Meeks, Davis, O’Donnell, Runde and Weatherson all 

rely, either completely or partially, on Ramsey’s claims about Keynes’s relational, propositional logic 

being wrong, while ignoring everything actually written about it by Keynes in the TP. 

 

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