I should like to thank for valuable suggestions and comments on my earlier draft, Yuji Aruka, Eiichi Asano, Mitsuharu Ito, Kunitake Ito, Myoung-Kyu Kang, Atsushige Matsushima, David McMurray and Satoshi Sechiyama.

  • 1. Indeed, Keynes avoided an empirical interpretation of probability, but it is often overlooked that he sidestepped the empiricist philosophy of John Locke and David Hume by following two distinct routes: (a) insofar as the category of knowledge is concerned, he distinguished between knowledge and probability, which means between absolute, certain knowledge and doubtful, uncertain, probable knowledge based on experience; and (b) he also distinguished two kinds of uncertainty, that is, uncertainty in the object itself and uncertainty in human judgement. In this respect, see J. Locke (1975 [1690]), Chapter 15 ('Of Probability'), and particularly, D. Hume (1978 [1739]), Part III ('Of Knowledge and Probability'). See also K. Ito's (2001), 'Hume no Kakuritsu-ron' (Hume's Theory of Probability) which includes a discussion of the relevance of Hume's theory to Keynes's interpretation of probability.
  • 2. The term 'objective' is polysemic and its meaning depends upon which writer is referring to it. In this regard, one can call attention to a number of well-known treatments: M. Weber, 'Die "Objectivitat" sozialwissenschaftli- cher und sozialpolitischer Erkenntnis' (1971 [1904]), Emile Durkheim, Les regies de la methode sociologique (1999 [1895]), and V. Pareto, Trattato di socio- logia generale (1916).

A certain probability relation (between a premise and a conclusion) is taken to be 'objective' because several other (not just a few) thoughtful and prudent people who qualify as 'rational spectators' consider it to be 'reasonable' and 'objective'. If so, Keynes's view of 'objective' seems to be rather similar to the meaning of 'logique' in Pareto's expression 'action logique', which means an individual's rational action suited (subjectively or objectively) to its purpose (see Pareto, 1917, p. 67, and Busino, 1987, p. 803). In that connection, Keynes wrote: 'the theory of probability is logical, therefore, because it is concerned with the degree of belief in which it is rational [emphasis in original] to entertain in given conditions, and not merely with the actual beliefs of particular individuals, which may or may not be rational' (TP, p. 4; emphasis added). It may be said that Keynes is concerned not with 'actual beliefs' but with 'ideal beliefs'. But this assertion could not elude Ramsey's severe criticism. See his critique beginning with the passage: 'But we are concerned with the relation which actually holds between two propositions' (Ramsey, 1999 [1922], p. 253; emphasis added).

3. Rudolf Carnap (1950, p. 43; 1966, p. 29), although he is in a sense a critical successor of Keynes's TP, calls Keynesian probability 'a logical probability'.

But Carnap's conception is generally recognized as 'a rationalist theory' (cf. Hacking, 1987, p. 2). According to Hacking, the label 'subjective theory' should be applied not to Keynes's probability but to the tradition stemming from Frank Ramsey (Ramsey, 1926). This is of course not wrong, but I am mostly interested in the classification of probability concepts proposed by Gilles-Gaston Granger (1988, pp. 282 ff.). According to Granger, the theories of probability may be classified in the following terms: (a) 'l'interpretation objectiviste (including Kolmogorof and von Mises) in which 'les frequences de realisation des evenements' are considered to be 'les probabilites', and (b) ^'interpretation subjectiviste, which is further subdivided into two subclasses: (i) the stream of Wittgenstein-Keynes-Carnap, in which 'la probabil- ite' is regarded as 'une mesure d'un degre de liaison logique entre deux propositions’, and (ii) the stream of F. Ramsey, von Neumann and Morgenstern, and Savage, in which 'la probabilite' is 'une mesure du degre de credibilite d'un evenement’. Incidentally, it would seem that my own interpretation has much in common with that of Granger (see Hishiyama, 1983 [1969]).

  • 4. According to the atomic hypothesis, or logical atomism, it should be possible to reduce the 'world' to individual units or 'atoms'. According to Bertrand Russell and Wittgenstein, a general proposition must ultimately be reducible to elementary or atomic propositions, the former may be completely construed from the latter. To this effect, Wittgenstein (1971 [1918]) presents his basic standpoint as follows: '[i]f all true elementary propositions are given, the result is a complete description of the world' (Wittgenstein, 1971 [1918], s. 4. 231) and '[t]he world is completely described by giving all elementary propositions, and adding which of them are true and which false' (Wittgenstein, 1971 [1918], s. 4. 232). The most typical example of logical atomism in the social sciences is neoclassical economic theory, especially the Walrasian scheme of general equilibrium theory.
  • 5. In this regard, see Kunitake Ito (1999, p. 150 ff.).
  • 6. It is interesting to note here that Shinichiro Tomonaga was also puzzled by the infinite quantities appearing in his quantum electrodynamics model. This kind of infinity concerned 'the infinities appearing in the field reaction, particularly in the scattering process'. He would break through this difficulty by creating 'the theory of renormalization' or, more precisely, 'the theory of renormalization of mass and charge'. This 'theory' serves, it would seem, as deus ex machina for him because it helped him to solve the difficulty of infinity and to make his model determinate. See his Nobel lecture ('Development of Quantum Electrodynamics. Personal Recollections'), 6 May 1966, reprinted in Tomonaga (1997, Appendix, pp. 10-33).
  • 7. Let us denote the concept of 'risk' by R and recall its definition. Let A be the amount of good which may result from some course of ethical behaviour, p its probability (assume p + q = 1), and E the value of 'mathematical expectation'. As a result, E = pA. Then 'risk' R is defined according to the following equations: R = p(A - E) = p(A - pA) = p(1 - p)A = pqA = qE. The value of 'mathematical expectation' E, that is, pA, represents 'the net immediate sacrifice which should be made in the hope of obtaining A' (Keynes, TP, p. 348) - in other words, opportunity cost associated with hope. Since q is 'the probability that this sacrifice will be made in vain' (TP, p. 348), qE measures 'risk'. Keynes adds: '[t]he ordinary theory supposes that the ethical value of an expectation is a function of E only and is entirely independent of R' (TP, p. 348).
  • 8. There is a need to refer to the two cases mentioned in TP, as these are somewhat similar to the one considered here: (a) a court case brought by a breeder of racehorses to recover damages for breach of contract (taken from the Times Law Reports), and (b) the case of an offer of a beauty prize (taken from the Daily Express). Keynes sums up as follows: '[w]hether or not such a thing is theoretically conceivable, no exercise of practical judgment is possible, by which a numerical value can actually be given to the probability of every argument. So far from our being able to measure them, it is not even clear that we are always able to place them in an order of magnitude. Nor has any theoretical rule for their evaluation ever been suggested' (TP, p. 29).
  • 9. This 'first classical fundamental postulate' is not admitted here because, as a result of the controversy in the 1960s over the theory of capital, the general applicability of the neoclassical production function, whether it concerns the industries as a whole or a single industry, was definitively disproved. See Pasinetti (2000, pp. 408-12) for the reconstruction of the process whereby this result was suppressed by influential economic theorists and finally disappeared as if the controversy had never taken place.
  • 10. As for the assumption of Marshallian increasing costs, Sraffa rejects Marshall's device by which 'the whole industry [is considered] as a single firm which employs the "whole of the constant factor" and employs successive doses of the other factors' (Sraffa, 1998 [1925], pp. 341-2; Japanese translation by Hishiyama, 1956, p. 41). Of course 'the whole industry' (tutta l'industria) here means a particular industry, so that we should reasonably leave out Keynes's peculiar device of 'an economy-wide huge industry', whose marginal cost increases would correspond to an increase in the scale of output as a whole.
  • 11. One of the main topics of Sraffa's critique of Marshall's cost analysis (Sraffa, 1925), in a sense, is its emphasis upon its logical incompatibility with the so-called partial equilibrium framework. While Keynes must have been aware of the implications in Sraffa's critique, he simply includes in GT the Marshallian (increasing-costs) supply curve as a cornerstone of his theoretical scheme. This is somewhat surprising, but also a rather interesting sign of the prevailing intellectual atmosphere among his fellow economists. Eiichi Asano (1987, p. 174 ff.) conjectures that Keynes adopted the Marshallian short-run supply curve probably because Richard Kahn advised him to. But it should also be noted that in his 'Notes on Ohlin's Final Section' (April, 1937), Keynes clearly states: 'I have always regarded decreasing physical returns in the short period as one of the very few incontrovertible propositions of our miserable subject!' (Keynes, J. M. 1973c [1937], p. 190).
  • 12. There is a sense in which, unlike in the physical sciences, in the moral sciences, and particularly for investment decisions in economics, future exert an influence upon the present. The following statements by Keynes are worth recalling: 'The schedule of the marginal efficiency of capital is of fundamental importance because it is mainly through this factor (much more than through the rate of interest) that the expectation of the future influences the present1 (GT p. 145; emphasis added). But the type of entrepreneurial behaviour behind that schedule is, it would seem, essentially similar to the ethical behaviour of human beings facing choices among alternative courses of action, as considered in TP. Of course the 'indefinite future' in the case of an ethical act is transformed into a series of discrete periods (t1,t2,., tn) and, correspondingly, the 'goodness or advantage' of the particular chosen course is changed into a series of values for the 'prospective yield' of the chosen investment (Qp Q2,..., Qn).

We may conjecture that the values of prospective yields should probably be based on 'probable beliefs' arrived at by virtue of probabilistic inference proceeding from actual or expected knowledge (or information). The kernel of this approach, somewhat differently from the case of TP, would be the method of capitalization by which one is able to roughly reduce a 'future' value to a 'present' one. Moreover, while 'the state of confidence' is considered by Keynes to be one of the major factors determining the investment demand schedule (see GT, pp. 148-9), this is in a certain sense a reformed formulation of the 'probable beliefs' of TP, which is obtained by explicitly taking into account both 'risk' and 'weights' of arguments. In the case of investment, 'risk' means 'the likelihood of our best forecast turning out quite wrong' (GT, p. 148). On the other hand, 'weights', that is, the amount of evidence, strengthens, as it were, the credibility of probable belief independently of its degree. In short, we might say that, though the state of confidence may be a decreasing function of 'risk', the increase in 'weight', that is, the amount of evidence associated with probable arguments, does not always strengthen the state of confidence because the latter may be either weakened or strengthened according to whether the increase of evidence strengthens the unfavourable case or the favourable one. The emphasis upon 'the state of confidence' in GT might in part reflect F. P. Ramsey's criticism of Keynes's TP (Ramsey, 1931, pp. 62-8).

  • 13. Pasinetti's views of the 'principle' of effective demand can be traced to his book Growth and Income Distribution (Pasinetti, 1974, pp. 31-3). But the original formulation seems to have been made much earlier. In fact, I learned at Cambridge University in 1969-70 that Pasinetti had displayed the graphic representation of the 'principle' in his lectures there.
  • 14. See on this issue this statement by Keynes:

We are merely reminding ourselves that human decisions affecting the future, whether personal or political or economic, cannot depend on strict mathematical expectation, since the basis for making such calculations does not exist; and that it is our innate urge to activity which makes the wheels go round, our rational selves choosing between alternatives as best we are able, calculating where we can, but often falling back for our motive on whim or sentiment or chance.

  • (GT, pp. 162-3; emphasis added)
  • 15. As far as the interpretation of Keynes's philosophical thought is concerned, I have always assumed the following simple classification of the sciences (excluding metaphysics):

While empirical sciences in general, and especially the reconstruction of their fundamental logic, are the main theme of TP, the moral sciences, and particularly the logic of economics as a branch of the moral sciences, are considered in GT. Keynes emphasizes 'the point about economics being a moral science', not a natural science, in letters to Roy Harrod after GT. This does not, it would appear, require any change in our interpretation of GT. (See Keynes' s letters to Harrod, dated 4 and 16 July 1938, in Keynes, 1973d [1939], pp. 295-7, 299-301.) However, I wonder with an uneasy feeling why Harrod does not, in his reply, pay attention to the 'point' mentioned above.

  • 16. It seems clear that when Keynes published this article in 1926 he undoubtedly believed that 'the atomic hypothesis [had] worked so splendidly in physics' (Keynes, 1972 [1933], p. 262). In TP, he takes for granted that 'atomic uniformity' is generally applicable in the material universe.
  • 17. As Bertrand Russell writes: 'there are two very different classes of wholes, of which the first will be called aggregates, while the second will be called unities.... Each class of wholes consists of terms not simply equivalent to all their parts; but in the case of unities, the whole is not even specified by its parts' (Russell, 1903, pp. 140-1). Aggregates are quite different: Russell had also written: '[s]uch a whole [the aggregate] is completely specified when all its simple constituents are specified; its parts have no direct connection inter se, but only the indirect connection involved in being parts of one and same whole' (Russell, 1903, p. 140).

In this respect, see R. M. O'Donnell's noteworthy book Keynes: Philosophy, Economics and Politics (1989). O'Donnell is interested in the 'parts-wholes' issue, and explicitly mentions the locus classicus in Russell's Principles of Mathematics (O'Donnell, 1989, pp. 177-8). He writes: 'His [Keynes's] philosophical position on this issue may be described as "pluralist"'. He adds that 'in economics, Keynes' position was, I suggest, similarly pluralist', because 'the principle of atomism or methodological individualism was applicable in some situations, and the principle of organic unity in others' (O'Donnell, 1989, p. 177). Indeed, O'Donnell's general point of view may not be wrong, but there is need, I think, to further investigate the issue so as to arrive at the ultimate ground. In that sense, my point of view might be somewhat different from O'Donnell's.

  • 18. In this respect, see Tarde's article 'La croyance et le desir' in Tarde (1895, pp. 235-308). Tarde summed up his thoughts with regard to the ultimate elements of 'phenomenes internes' into 'trois termes irreductibles, la croy- ance, le desir, et leur point d'application, le sentir pur' [Three irreducible terms, belief, desire and their point of application, mere feeling]. And he adds: 'les deux premiers termes sont les formes ou forces innees et constitutives du sujet, les moules oh il refoit les materiaux bruts de la sensation [The two former terms are the innate and constitutive forms of the subject, the springs from which he receives the rough materials of sensibility]’ (Tarde, 1895, p. 240). See also 'Monadologie et sociologie’, in Tarde (1895, pp. 371-2).
  • 19. It is worth noting what K. Maruyama (1985) writes about Durkheim:

Durkheim endeavours to derive the notion of 'society’ as an objective fact depending on a rigorous inductive method. According to Durkheim 'society’ cannot be reduced to the psychological and physiological characteristics of individuals. The notion of 'society’ specified in such a manner is not obtained from adding up individuals’ voluntary intentions. On the contrary, it is a system of constraints that moulds those individuals from outside.

  • (Maruyama, 1985, p. 135)
  • 20. We find a similar statement in Saussure’s Les sources manuscrites du cours de linguistique generale edited by R. Godel:

Des que nous disons: terme au lieu de mot, l’idee de systeme est evoquee.... Mais de plus, ne pas commencer par le mot ou le terme pour en deduire le systeme. Ce serait croire que les termes ont d’avance une valeur absolue; au contraire, c’est du systeme qu’il faut partir, du tout solidaire. Ce dernier se decompose entre certains termes qui ne sont pas si faciles a degager qu’il peut sembler’ [As soon as we say 'term' instead of 'word', the idea of system is invoked ... Furthermore, never start with the term or the word in order to deduce the system. That would entail thinking that terms have since the very beginning an absolute value. On the contrary, we should start with the system, with the organic whole. This latter must be decomposed into terms that are not so easy to disentangle as it may seem.]

  • (Godel, 1969, p. 228).
  • 21. Sraffa’s 'system’, unlike Durkheim’s, consists not of institutions, laws, customs and morals, and so on but of the economic structure based on technological interrelationships among various industries. In Marxian terms, Sraffa identifies the starting point of his theory with the material base, not the superstructure of society. Nevertheless, it would seem that Sraffa’s system of production is a system of constraints similar to Durkheim’s system: individuals’ economic behaviour ultimately adapts to the operation of an objective system of production. Moreover, there is a sense in which it shares the methodological standpoint of Saussure, which tends to a complete denial of atomism. The following statement is, I think, precisely applicable to Sraffa’s view: 'c’est du systeme, du tout solidaire qu’il faut partir’. Whereas Durkheim makes a direct attack on 'utilitarian individualism’, that point of view would also be unacceptable to Sraffa and perhaps also to Keynes.
  • 22. Pierangelo Garegnani explicitly accepts the hypothesis of classical free competition when he writes: '[a]gain, hasn’t that assumption [that of the uniform rate of profits for all industries] been generally held as the necessary implication of the hypothesis of free competition?’ (Garegnani, 1990, p. 350).
  • 23. For a multi-sector production system, the 'macroeconomic condition’ is determined by the production coefficients per unit of output and by the per capita demand coefficients associated with each sector (or commodity) in the system. It is a necessary condition for full employment, or the full expenditure of income. In other words, the condition basically depends upon the configuration of technology and the composition of demand in terms of sectors or commodities (on this point, see Pasinetti, 1993; 1997).
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