Conventional Coefficient Of Weight Research Articles

Using Keynes’s Conventional Coefficient of Weight and Risk, C, From Chapter 26 of His a Treatise on Probability, to Help Explain Keynes’s Three Equations in Chapter 17 of the General Theory That Integrate (A) Expectations (Probability, Linear Risk), (B) Liquidity (Uncertainty, Weight of the Evidence, W) And (C) The Risk of Appreciation/Depreciation, a (Nonlinear Risk), One at a Time Sequentially

Using Keynes’s Conventional Coefficient of Weight and Risk, C, From Chapter 26 of His a Treatise on Probability, to Help Explain Keynes’s Three Equations in Chapter 17 of the General Theory That Integrate (A) Expectations (Probability, Linear Risk), (B) Liquidity (Uncertainty, Weight of the Evidence, W) And (C) The Risk of Appreciation/Depreciation, a (Nonlinear Risk), One at a Time Sequentially

J M Keynes’s Theory of Evidential weight can only be expressed in a formal manner as V=V(a/h) =V(a/h1, h2, h3, h4……hn, hn+1….) =w, 0≤w≤1, where w=K/[K+I] and 1-w=I/[K+I] when K=knowledge and I =Ignorance.The erroneous Heterodox , Post Keynesian version is V(a/h)=V = K/(K+I),0≤V≤1.

In 1990,J. Runde presented a formal, mathematical representation of Keynes’s theory of evidential weight, as presented in chapters 6 and 26 of Keynes’s A Treatise on Probability ,as V(a/h)=V = K/(K+I).In 1999,A. Vercelli revised this to read as V =V(a/h) = K/(K+I), 0≤V≤1.We can identify this as the Post Keynesian – Heterodox formulation of what they thought Keynes was supposed to have meant .We will abbreviate this as PK-H. The problem with the PK-H formulation is that it does not make any sense .It involves a basic ,fundamental error in conflating and confusing two separate fields, logic and mathematics,as Keynes’s V is a logical relation and not a mathematical variable. Keynes correctly transforms his logical relation ,V,into a mathematical variable ,on p.315 of the A Treatise on Probability in the following manner : “….where w measures the ‘weight,’(sic)…”(Keynes,1921,p.315). Now Keynes had already properly discussed w as being “….the degree of completeness of information upon which a probability is based…”(Keynes,1921,p.313) where Keynes had already done the necessary and sufficient normalization to the unit interval [0,1] needed to define a measure of weight ,so that w comes in degrees as 0≤w≤1. Keynes thus finishes what he started in the first paragraph of chapter VI on p.71 of the A Treatise on Probability,where ,instead of the balance being between the favorable and unfavorable evidence,which is given by the concept of probability ,where P is the probability relation ,so that one obtains the following: P(a/h)=α,0≤α≤1,where α=p/(p+q) and 1-α=q/(p+q),where p+q =1.(Keynes,1921,p.315). One obtains the other balance ,which is now a balance between the absolute amounts of relevant knowledge and of relevant ignorance ,respectively ,as V(a/h)=w, 0≤w≤1,where w=K/(K+I) and 1-w=I/(K+I),where K is ,obviously ,relevant Knowledge and I is, just as obviously,relevant ignorance,so that K+I=1. Only after this has been recognized can one understand Keynes’s conventional coefficient of weight and risk c. Unfortunately,no academic philosopher,economist,historian or decision theorist was ever able to grasp Keynes’s mathematical analysis underling the c coefficient of p.315 of the A Treatise on Probability in either the 20th or 21st century except Edgeworth. We will examine one out of the hundreds of erroneous papers which have incorporated Runde’s mathematical errors about V by examining a paper by Crocco (2006).

J M Keynes’s Contribution to Solving the Certainty Effect Problem: How Some Philosophers Overlooked Keynes’s Conventional Coefficient of Weight and Risk, C

J M Keynes solved the problems of the certainty, reflection, translation, and preference reversal effects long before these effects were specified in the post world war II literature by psychologists. Keynes recognized in chapter 26 of the A Treatise on Probability (1921; p.313) that all of these effects were a result of non linear probability preferences on the part of the decision maker. An understanding of Keynes’s contribution would have helped philosophers, such as I. Levi and B. Weatherson, to deal with this problem.

Can Shiozawa’s, Morioka’s and Taniuchi’s Microfoundations for Evolutionary Economics (2019) Serve As the Microfoundations for “… Post-Keynesian Economics “ (2019, p.vii)? The Answer Is Definitely Yes if Post –Keynesians Can Break Away From Joan Robinson’s Anti-Mathematical, Anti-Formalist Views

Can Shiozawa’s, Morioka’s and Taniuchi’s Microfoundations for Evolutionary Economics (2019) Serve As the Microfoundations for “… Post-Keynesian Economics “ (2019, p.vii)? The Answer Is Definitely Yes if Post –Keynesians Can Break Away From Joan Robinson’s Anti-Mathematical, Anti-Formalist Views

On the Unfortunate Ignorance of Academics regarding the Technical and Mathematical Tools for Dealing with Uncertainty Developed by J M Keynes in the A Treatise on Probability in 1921 and in His Earlier Fellowship Dissertations in 1907 and 1908: Interval Valued Probability, Decision Weights, and Safety First Analysis

Keynes developed a number of technical, mathematical tools for dealing with the problems of uncertainty (ambiguity, vagueness, indeterminate probabilities, imprecise probabilities) in his A Treatise on Probability in 1921 (and in his earlier Fellowship Dissertations of 1907 and 1908) that continue to be overlooked by practically all academics in the nearly 100 years since its publication in 1921. Keynes’s technical and mathematical modeling developments took place in chapters 15-17 of Part II of the A Treatise on Probability, chapters 20 and 22 of Part III of the A Treatise on Probability, Chapter 26 of Part IV of the A Treatise on Probability, and chapters 29 and 30 of Part V of the A Treatise on Probability. Part II of the A Treatise on Probability included Keynes’s original work dealing with non –additive probabilities that was based on the original work of George Boole in his 1854 The Laws of Thought. Keynes’s Boolean upper and lower bounded probabilities, which Keynes categorized as inexact measurement and approximation, established his complete grasp of the concept of interval valued probability. The American Theodore Hailperin showed in 1986 that both Boole and Keynes were using early versions of linear programming techniques to solve both linear and non linear systems of equations and inequations. Keynes used the name “non –numerical probability“ when discussing his no additive, non linear interval valued probability approach. Keynes’s approach has been erroneously identified as an ordinal probability approach by all Post Keynesians, Institutional and heterodox economists, who were greatly influenced by the 1975 paper on Keynes and uncertainty in HOPE by E R Weintraub, which misinterpreted Keynes in terms of G L S Shackle’s theory of possibility. Part III of the A Treatise on Probability, building on Part II’s interval valued probability concept, developed the concept of finite probability, applicable to both numerical and non numerical (interval valued) probabilities, in order to deal with applications of probability to Keynes’s concepts of induction, based on degrees of analogy, involving degrees of similarity and dissimilarity using human memory ,intuition and pattern recognition. Part IV of the A Treatise on Probability provided the first decision weight approach to decision making, Keynes’s conventional coefficient of weight and risk, c, that incorporated both non linear probability preferences and non additivity by the use of decision weights .These decision weights incorporated Keynes’s logical analysis, contained in chapter 6 of the A Treatise on Probability of his Evidential weight of the argument approach, V(a/h),which was a logical relation like Keynes P(a/h)=α, 0≤α≤1,but which Keynes waited to provide the comparable mathematical analysis for in chapter 26.In chapter 26,Keynes set V(a/h) equal to w,where w was normalized on the unit interval, so that V(a/h) =w, 0≤w≤1, holds. Keynes then defined uncertainty in the General Theory as an inverse function of V in chapter 12 on page 148 in footnote 1. Keynes then defined his conventional coefficient of weight and risk, c, to equal the standard EMV and SEU rules multiplied by the decision weights [(2w/1+w)] for non additivity and [(1/(1+q)] for non linearity. F Y Edgeworth recognized Keynes’s approach as breaking new ground. However, his plea for reader assistance from subscribers to Mind was ignored. Part V of the A Treatise on Probability provided the first “safety first” approach to decision making. As acknowledged by Edgeworth, Keynes used Chebyshev’s Inequality to establish lower bounds for decision problems in order to minimize risk (Keynes, 1921, pp. 353-358). Samuelson, while analyzing Keynes‘s simplified Least Risk analysis on p. 315 of the A Treatise on Probability, unfortunately overlooked Keynes’s generalized approach using Chebyshev’s Inequality in chapter 29. Samuelson also overlooked on this same page Keynes’s development of the logical and mathematical tools needed to specify the investment multiplier in the General Theory, as well as Kahn’s employment multiplier on p.183 of his June, 1931 article in the Economic Journal.

Given Keynes's Definitions of Logical Probability and Evidential Weight, It Is Impossible for Keynes’s Approach to Measurement to Be an Ordinal Theory; His 'Non Numerical' Probabilities Must Be Based on Inexact and Imprecise Measurement Using Approximation Involving Boolean Upper and Lower Probability Bounds and Bounded Outcomes

Keynes spent all of his introductory Chapter One of the A Treatise on Probability emphasizing that his logical relation of probability (a relation of similarity – dissimilarity based on analogy) P(a/h)=α, where α came in degrees and 0≤α≤1. In chapter 6, Keynes introduced his logical relation of the evidential weight of the argument, V(a/h), but did not specify what it equaled, although he clearly stated that he would integrate both probability and weight together later. In chapter 26, Keynes specified that V(a/h)= w, where we came in degrees and 0≤w≤1. Keynes showed how to integrate both probability and weight together in his conventional coefficient of weight and risk, c, in chapter 26 of the A Treatise on Probability. Probability and weight were also integrated together by Keynes using the theory of inexact measurement or approximation based on intervals with upper and lower bounds or limits. Ordinal probability can do none of these things. Advocates of the claim that Keynes’s theory was an ordinal theory of probability concentrate exclusively on a misinterpreted diagram that appears near the end of chapter 3 of the A Treatise on Probability. Thus, while Keynes’s theory can easily deal with ordinal probability with the aid of the Principle of Indifference, it is not part of Keynes’s main or major analysis in the A Treatise on Probability, which was interval probability. The same conclusion holds for numerical probability and rational expectations. Rational expectations are a very special case that occurs if w=1. The heterodox, institutionalist, or Post Keynesian approaches, involving irreducible, fundamental, or radical uncertainty, are a very, very special case that occurs only if w=0 in the very long run. Their argument, that measurement is not possible at all, be it exact or inexact, is rejected by Keynes in chapter 4 of the General Theory. Thus, while Keynes’s theory can easily deal with ordinal probability, it is impossible for the ordinal probability to deal with inexact and imprecise interval-valued probability measurement. Keynes’s theory of measurement involves interval-valued probability in the A Treatise on Probability with the numerical and ordinal probability being very special cases of his general theory of interval-valued probability. Keynes’s approach to measurement in the General Theory also involved inexact and imprecise measurement as deployed by Keynes in chapter 4 of the General Theory. No ordinal probability approach to measurement is used by Keynes in the General Theory. The great value of the 1936-1938 Keynes-Townshend exchanges is that they show that Keynes presented an approach to measurement that was imprecise and inexact in both the A Treatise on Probability and the General Theory.

A Demonstration of the Impossibility That Keynes’s Logical Theory of Probability Was Primarily an Ordinal Theory: Keynes’s C Coefficient of the a Treatise on Probability Is a Mathematical Translation of Interval Valued Probability into a Decision Weight Approach That Modifies the Additive, Linear Approach to Mathematical Probability

It is straightforward to demonstrate that Keynes’s logical theory of probability could not primarily be an ordinal theory of probability. The demonstration is based on three main points made by Keynes. Keynes’s first major point in the A Treatise on Probability covered Keynes’s extensive discussion of non additivity (the general inapplicability of the addition rule of the calculus of probabilities) in Part II of the A Treatise on Probability (1921), as well as his many worked out problems using interval valued probability in Part II of the A Treatise on Probability (1921). Any attempt to juxtapose ordinal probability and non additivity is an oxymoron because ordinal probability can’t be added or multiplied. Part II of the A Treatise on Probability has nothing to do with ordinal probability. Keynes’s second major point was that the difficult topic of developing an interval valued approach to probability can be avoided by using decision weights that translate additive probability into non linear and non additive decision weights. Keynes’s conventional coefficient of weight and risk,c, was designed to allow his interval valued probability concept to be translated into non linear ,non additive coefficients or decision weights while starting the analysis using additive probability.It is mathematically and statistically impossible to do this with ordinal probability. Keynes’s third major point in the A Treatise on Probability shows that Keynes’s discussion of statistical frequencies involves his use of interval valued probability through the use of upper and lower limits or bounds,an approach identical to that used by Keynes in Part II. None of Keynes’s results from Parts II ,III, IV, and V of the A Treatise on Probability using either interval valued probability or decision weights, first introduced by Keynes to deal with the problems of non linearity and non additivity are alluded to by any Post Keynesian or Fundamentalist Cambridge Keynesian in any article or book written by them over the last 40 years. O’Donnell (2019) and Dow (2019) are have simply overlooked all of Keynes’s analysis in Part II, IV, and V of the TP dealing with measurement. O’Donnell (2019) and Dow (2019) rely completely on a very severe misinterpretation of pages 38-40 of the TP, as well as the diagram on page 39. The diagram on page 39 of the A Treatise on Probability illustrates the linear and additive probabilities of OAI with the nonlinear (quadratic) and non additive nature of the probabilities of all of the other paths. There is no theory of ordinal probability used, developed, deployed, or applied on pp.38-40 of the TP.

Confusing Metaphors With Mathematics in Chapter 6 of the A Treatise on Probability, When Analyzing Keynes’s Modeling of the Evidential Weight of the Argument V(a/H), Leads to confusion: If V(a/H)=W, As Defined in Chapter 26, Where 0≤w≤1, Then It Is Mathematically Impossible That Keynes’s Weight Is Monotonically Increasing

Philosophers and economists continue to erroneously claim that Keynes’s Evidential Weight of the Argument, V(a/h) and/or his degree of the weight of the evidence, w, is always increasing monotonically over time based on a reading of a metaphor Keynes used in chapter 6 to help explain why the Evidential Weight of the Argument is different from probability. The Evidential Weight of the Argument will increase if there is additional relevant evidence, data, information or knowledge. It is also the case that we know more if we later discover additional relevant evidence that establishes that we, in fact, did not know what we thought we knew .What is required to actually measure weight mathematically ,as opposed to logical considerations, is an index that is normalized on the unit interval between 0 and 1 that is identical to the normalization of probability on the unit interval between 0 and 1 so that one can talk about degrees. Any attempt to discuss Keynes’s concept of weight is an exercise in futility unless the correct mathematical structure has been provided first so that one can talk about degrees of probability and degrees of knowledge and ignorance. Such a structure was not provided by Keynes in Chapter 6 of the A Treatise on Probability. However, Keynes did create such a structure in chapter 26 of the A Treatise on Probability. First, Keynes had to define V(a/h) to be equal to some type of degree. Keynes’s designated this variable as w, where w is normalized on the unit interval and represents the degree of completeness of the relevant evidence. Therefore, V(a/h) = w performs the same task for Keynes as α does for his other logical relation, P(a/h) =α ,0≤α≤1. W measures the degree of the completeness of the relevant evidence, so that V(a/h)= w .Keynes then combines P and V in chapter 26 into his “conventional coefficient of weight and risk, c”, where c is also normalized on the unit interval between 0 and 1 for the specific form chosen by Keynes that integrates both non additivity, sub additivity, and non-linearity in the decision weights chosen by Keynes, which were 2w/(1 plus w) and 1/(1 plus q),respectively. It is mathematically impossible for w to be monotonically increasing in the decision weight, w, chosen by Keynes to illustrate his concepts on non-additivity and non-linearity, where w=K/(K plus I),where K denotes knowledge and I denotes Ignorance. Thus, α =p/(p plus q) and 1-α =q/(p plus q) while w= K/(K plus I) and 1-w=I/(K plus I). The major error committed in all journal articles and books published over the last sixty years that discuss Keynesian weight is the complete misinterpretation of the following quote from page 77 of the A Treatise on Probability: “The fundamental distinction of this chapter may be briefly repeated. One argument has more weight than another if it is based upon a greater amount of relevant evidence; but it is not always, or even generally, possible to say of two sets of propositions that one set embodies more evidence than the other. It has a greater probability than another if the balance in its favor, of what evidence there is, is greater than the balance in favor of the argument with which we compare it; but it is not always, or even generally, possible to say that the balance in the one case is greater than the balance in the other. The weight, to speak metaphorically, measures the sum of the favorable and unfavorable evidence, the probability measures the difference.” Keynes is talking, as he himself so clearly stated, metaphorically. However, just as Adam Smith’s purely metaphorical statements about an Invisible Hand, as discussed by Gavin Kennedy for over 20 years, have been grossly misunderstood, so were Keynes’s statements. Current assessment of Keynes’s weight concept is erroneously based on a metaphor Keynes used before he presented the mathematical analysis in chapter 26 of the TP on page 77 of chapter 6. Keynes’s chapter 26 is the omega while chapter 6 is the alpha. It is impossible to fully grasp Keynes’s weight of the evidence concept without a detailed study of chapter 26 of the TP.

On J M Keynes's Original Contributions to Decision Making Under Uncertainty: Indeterminate, Interval Valued Probabilities in Part II of the a Treatise on Probability and Imprecise, Interval Valued Probabilities in Part V of the a Treatise on Probability

All of the logical, statistical and mathematical foundations for Keynes’s work on uncertainty in the General Theory and his 1937 Quarterly Journal of Economics reply article can be found in Parts II and V of Keynes’s A Treatise on Probability. Keynes, building on the original, mathematically developed, logical theory of probability first put forth by George Boole in his The Laws of Thought in 1854, demonstrated how to construct upper and lower probabilities that would define an interval valued probability. Keynes dealt with indeterminate probabilities first. Indeterminate probabilities are interval valued probabilities where the range between the upper and lower bounds or limits will not diminish as new additional data or evidence is examined because there will always be important relevant evidence that will never be attained and is permanently missing or unavailable when a decision or assessment must be made. The great American mathematician, Edwin Bidwell Wilson, acknowledged this begrudgingly in 1934 in his JASA article titled, “A Problem of Boole’s.” Keynes also developed the first approach to “Safety First” by showing how to use Chebyshev’s Inequality to provide a lower bound or limit. This imprecise lower bound or limit would be based on the initial, small amount of data or information that was available to the decision maker at the start of his assessment of a problem. However, over time, as more and more relevant data and information became available, the decision maker would be able to make use of more accurate and reliable statistical methods related to the Normal distribution. The Normal distribution estimate would then serve as an upper bound. The range between the lower and upper bounds would then diminish as larger and larger samples were used. Eventually, a precise probability estimate would result. The lower bound would be relevant at the initial stages of a estimation process. Keynes’s views on imprecise probability estimates, in cases where significant, additional data and information can be obtained over time and a decision to act can be postponed indefinitely, imply that eventually a precise probability would be able to be calculated that was stable. When Keynes defined uncertainty to be an inverse function of the weight of the evidence, he automatically supported his GT with a foundation consisting of the mathematical and logical analysis contained in Parts II and V of the TP on indeterminate probabilities, imprecise probabilities, the weight of the evidence, and the conventional coefficient of weight and risk. Unfortunately, 81 years after Keynes wrote the GT and 96 years after Keynes published his TP, economists are still unable to read Parts II and V of the TP. Economists, like the French mathematician, Emile Borel, and the American mathematician, Edwin Wilson, found that Parts II and V of the TP were just too overwhelming for them.

R. Lucas's Research Program: Far Too Narrow with Too Many Special Assumptions

R. Lucas's Research Program: Far Too Narrow with Too Many Special Assumptions

J M Keynes's Evidential Weight of the Argument, V(A/H),Which Applies to Arguments, is Different from the Weight of the Evidence, W, Which is a Measure of the Completeness of the Information, Data, Evidence, or Knowledge in the Premises, H

Keynes’s V(a/h) relation, first presented on p.72 of chapter 6 of the A Treatise on Probability (1921), represents the Evidential Weight of the Argument from h to a. H and a are related to each other and can’t be separated. V relates two sets of propositions, a and h. H is the premises and a is the conclusion. It is not possible to talk about h separately from a or a separately from h when using V. Keynes introduced w, the weight of the evidence, in Chapter 26 of the A Treatise on Probability in preparation for the construction of his conventional coefficient of weight and risk, c. C is not a probability. C is a decision weight. It represents the first use of and creation of a decision weight in history. W is defined by Keynes as a measure of the completeness of the relevant evidence, knowledge or data that is contained in the premises, h. It is a measure of the completeness of the evidence contained in the premises. It has nothing to do with V(a/h). It was introduced in the first paragraph of chapter 6 on p.71, before Keynes introduced his V(a/h) relation on p.72. The first paragraph deals exclusively with the premises containing the evidence. It does not deal with V. It is impossible to substitute V for w since they are different. V and w can’t be compared because V represents the logical relation between h and a, while w is a mathematical variable, normalized on the unit interval [0,1], that represents the completeness of the amount of available evidence, data, or knowledge, represented by h. Keynes never normalized V on the unit interval [0,1]. Keynes only normalized w on the unit interval [0,1]. The failure to understand the differences between V(a/h) and w, which is a measure of the strength of h, leads to confusion and error among heterodox economists, Post Keynesians, and economists associated with Cambridge University, England.

The Formal, Normative Theories of Savage, De Finetti, and Savage and De Finetti Require Additivity. They Never Introduced Sub or Non Additivity (Interval Valued Probability, Imprecise Probability, Indeterminate Probability) into Their Formal, Normative Theories of Subjective Probability or of Maximizing Subjective Expected Utility

George Boole was the first scholar to present a rigorous, mathematically derived, concept of upper and lower probability with a large number of applications in 1854 in The Laws of Thought. Boole made it clear that this was due to the existence of missing or insufficient information, evidence or knowledge. Boole mentioned a number of times that the problems under consideration would always have incomplete or insufficient data. He termed these kinds of problems as being logically indeterminate. If, however, there was no missing or unavailable information, then the upper and the lower probabilities will become equal to each other. There will no longer be any interval valued, or upper and lower, probabilities. There will be a determinate, exact or precise probability. Additivity will always hold. Keynes introduced his weight of the argument,V,a logical relation, in chapter 6 of the A Treatise on Probability. Its purpose was to grade single pieces of data or evidence. Thus V1(h/x1 x2 x3)>V2(h/ x1 x2) means that the weight of the argument, V1, is greater than V2. The comparable mathematical representation of V was accomplished by Keynes with his weight of the evidence variable, w. w was defined on p.315 of the A Treatise on Probability in chapter 26. W was defined on the unit interval [0,1], just like probability is defined on the unit interval between 0 and 1. A w=1 meant that there was no missing or unavailable evidence or knowledge. The evidence is complete. If w=1, then there can’t be any upper and lower probability estimates. Only if w<1 can interval valued probabilities be defined. If w=0, then no probability can be defined. This is G L S Shackle’s argument. Keynes integrated w into his decision weight formula, c. Keynes called c a “conventional coefficient of weight and risk”. C is defined as being equal to p [1/(1 q)][2w/(1 w)]. One obtains a decision rule when c is multiplied by the outcome A or the utility of the outcome, U(A ), and the adjective ‘Maximize’ is appended. [1/(1 q)] models probability preferences, which are assumed to be linear by Subjectivists. [2w/(1 w)] models weight, w, non linearly. W introduces sub and non additivity. Ellsberg, in a manner similar to Keynes, in 1961 modeled what he defined as the ambiguity of the evidence or data by a variable he defined as rho, ρ, where rho measures the degree of ambiguity. ρ, like Keynes’s w, is defined on the interval [0,1]. A rho equal to 0 means that there is no ambiguity. A rho equal to 1 means that complete ambiguity exists. Interval valued probabilities, based on Ellsberg’s analysis of the work of Smith and Good, is recommended by Ellsberg (Ellsberg,pp.111-125), in cases where 0<ρ<1. Feduzi, Runde and Zappia (2012, 2014, 2017) have claimed repeatedly that de Finetti and Savage formally allowed imprecise and indeterminate probabilities to be used by decision makers in their normative theory of decision making. The only way in which this can be formally incorporated into a decision theory is by the use of a variable similar to Keynes’s w or Ellsberg’s ρ. Nowhere in anything written by de Finetti and/or Savage in their lifetimes does any such variable appear in their formal theoretical analysis or in any of their supporting axioms. Savage stated that he did not know how to integrate such a variable, to take account of vagueness, into the additive probability measure that represented his formal, normative theory. Feduzi, Runde and Zappia (2012, 2014, 2017) have misinterpreted the de Finetti and Savage concerns and sympathy for vacillating, indecisive decision makers, confronted with vague and ambiguous evidence, with their recognition that defining one’s personal probabilities can be difficult, with the erroneous and mistaken belief that de Finetti and Savage incorporated a role for imprecise and indeterminate probabilities within their formal, subjectivist, normative theory of probability and their SEU decision theory that rests completely on additive probability. Feduzi, Runde and Zappia need to cite and show exactly where, in any written article, book, or co-authored paper, that Savage, de Finetti, or Savage and de Finetti introduced sub or non additivity into their formal, normative model. If they are unable to do so, then they need to publicly withdraw their claims in the journals in which they published their claims.

Hicks's IS LM Interpretation of the IS-LM Model in J M Keynes's Chapter 15 of the General Theory: Partly Wrong in 1937 and Partly Wrong in 1981

Hicks’s 1937 interpretation of Keynes’s IS-LM model in chapter 15 of the General Theory was inferior to Champernowne’s 1936 interpretation which correctly incorporated variables, denoted by Champernowne as Q and Q’, that stood for “The State of the News”, investor nervousness, uncertainty, or the confidence a decision maker had in his expectations. These variables appeared in both the IS and LM functions as independent variables that could shift both curves simultaneously. Only Champernowne’s work in 1936 incorporated these variables. No economist in the time period 1936 -2017 has acknowledged Champernowne’s unique and far sighted accomplishment although his article has been cited repeatedly since 1936. Hicks deliberately removed these variables from his own 1937 paper in Econometrica because it would have prevented him from creating what he called the general, general theory. In 1981, after wasting much of his readers time talking about different time periods and fix prices (money wages) versus flex prices (money wages), Hicks finally admits, at the very end of his 1981 recantation-retraction of his 1937 IS LM paper in an article published in the Journal of Post Keynesian Economics, that the problem with IS LM is that it does not incorporate Keynes’s concerns with uncertainty into the LM curve. Nowhere is the reader told that it was Hicks himself who had deliberately removed the uncertainty variable explicitly when presenting his interpretation of Keynes’s IS – LM curves from chapter 15 of the General Theory. Hicks’s 1981 paper’s citation of Shackle on uncertainty demonstrates that he was still confused and did not understand that Keynes’s definition of uncertainty, on p.148 of the General Theory, was that uncertainty was a unique function of Keynes’s weight of the evidence variable only. Keynes integrated uncertainty explicitly into decision theory with his indeterminate, interval valued probability approach, based on Boole, in chapters 3, 5, 10, 15, 16, 17, 20 and 22 of the A Treatise on Probability, and in his conventional coefficient of weight and risk, c, from chapter 26 of the A Treatise on Probability. Keynes dealt with imprecise probability in chapter 29 with his suggested use of Chebyshev’s Inequality as a lower bound. Since the Liquidity Preference LM curve is built on Keynes’s decision theory from the A Treatise on Probability, Hicks needed to have stated that he had no idea about what Keynes meant by uncertainty in either 1937 or 1981, but that uncertainty, as a function of the weight of the evidence, needed to be reintegrated into the IS LM model in both the IS curve and the LM curve.

The Unbridgeable Gulf Forever Separating A. Smith and J. M. Keynes from Bentham, Classical Economists, Neoclassical Economists, And Modernn Economists (Friedman, Becker, Stigler, Lucas, Sargent, Wallace, Muth, Kydland, Prescott, etc.): The Formal and Mathematical Concepts of Uncertainty, Weight of the Evidence, and Confidence

It is logically impossible for Classical Economists, Neoclassical Economists, and “Modern” Economists (Irving Fisher, Friedman, Becker, Stigler, Lucas, Sargent, Wallace, Muth, Kydland, Prescott, etc.) to grasp Smithian and Keynesian economics because both Smith and Keynes explicitly incorporated a specific term, weight of the evidence, into their decision theories to deal with uncertainty besides an additive concept of probability. F. Knight’s distinction between risk and uncertainty is flawed because, although he tried, he failed to explicitly integrate such a term as the “weight of the evidence” into his theory of decision making in order to differentiate uncertainty from risk mathematically and logically. Knight needed an understanding of non-additivity. Integrating such a term into decision theory automatically creates a violation of the purely mathematical laws of the probability calculus because the probabilities can no longer be precise (additive), but must be modeled mathematically as interval valued probabilities (non additive). A concept of weight of evidence thus violates the additivity, complementarity, and linearity requirements needed for the application of either the frequency approach to probability, using objective, precise probabilities or the Subjective, Bayesian approach to probability of Ramsey, de Finetti, Savage, and M Friedman upon which Subjective Expected Utility is build, which requires precise, subjective probabilities. George Boole explicitly provided the first explicit logical and mathematical foundation for the concept of upper and lower probabilities, or interval valued probability, in 1854 with the publication of The Laws of Thought in which he put forth the first explicit logical theory of probability in history. Keynes built on Boole’s foundation and based his logical theory of probability and decision theory on interval valued probability. However, this non-additive and nonlinear approach directly conflicts with the linearity and additivity constructs that underlie the subjective and frequency approaches to probability of I. Fisher, Friedman, Becker, Stigler, Lucas, Sargent, Wallace, Muth, Kydland, Prescott, etc. It is not possible for subjectivists to deal with weight because it would automatically make the subjective theory of probability and SEU theories very special theories that could never be more than a limiting case of the work of Adam Smith, George Boole, and J M Keynes. Thus, the intellectual conflict over the role of the weight of the evidence and uncertainty variables in decision theory, which completely divided Smith from Cantillon and Bentham in the eighteenth century, also completely divided Keynes from the Classical and Neoclassical economists of the 20th century. Practically all of the “What did Keynes mean in the General Theory?” literature can be traced back to economists’ gross ignorance of the Keynesian concept of the weight of the evidence and its role in the General Theory. Once Keynes introduced the weight of the evidence variable into the General Theory on p.148, the probabilities(expectations) automatically become interval valued. Non-additivity and non-linearity become the general case, while classical and neoclassical theories based on Benthamite utilitarianism’s use of additive and linear precise probabilities, becomes a very special case of Keynes’s. Keynes’s concept of liquidity preference is directly founded on the logical and mathematical analysis of the weight of the evidence that he incorporated into his conventional coefficient of weight and risk, c, in chapter 26 of the A Treatise on Probability. The conventional coefficient , which substitutes a mathematically more tractable analysis for the much more difficult Boolean interval valued approach, adapted with revisions by Keynes for use in Part II of the A Treatise on Probability, allows a reader to see the explicit role that weight plays in Keynes’s theory .Part II of the A underlies Part III and Part III underlies Part V of the A Treatise on Probability. Liquidity preference automatically brings in interval valued probability if w, the weight of the evidence, is less than 1. Keynes’s analysis of imprecise probability, which occurs in chapter 29 the A Treatise on Probability, is based, as acknowledged by Edgeworth in 1922, on using Chebyshev’s inequality to form a lower bound with the upper bound being given by probability estimates using other known probability distributions, such as the Normal distribution. As more evidence is accumulated, improved estimates will start to approach the upper bound and become precise. Keynes’s analysis of indeterminate probabilities takes place in chapters 3, 5, 10, 15, 16, 17, 20, 22, and 26 of the A Treatise on Probability. “Modern” Economists (Irving Fisher, Friedman Becker, Stigler, Lucas, Sargent, Wallace, Muth, Kydland, Prescott, etc.) can’t deal with Keynes’s general theory of decision making, which underpins the General Theory, because all of their theoretical approaches must use precise probabilities that are additive.