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Weight of argument and modalities of uncertainty

In TP Keynes often illustrates general concepts by examining specific instances considered to be particularly important or emblematic. This method of exposition has the advantage of favouring constant intuitive control of the meaning of arguments, but it may jeopardize the rigorous definition of concepts. In the preceding section we have seen a significant example of this kind of difficulty. In order to clarify the distinction between probability and weight of argument, Keynes insists on the particular case of new evidence that may increase or decrease the probability of an event while increasing at the same time the weight of argument. This happens, we may add, only when the new piece of evidence reduces the relevant ignorance of the decision maker (henceforth DM). This emblematic example illustrates in an intuitive way the semantic difference between probability and weight of argument, but it may mislead the reader if he is induced to believe that an increase in relevant evidence necessarily implies an increase in the weight of argument. This is not necessarily true, because an increase in relevant knowledge may increase the awareness that relevant ignorance is greater than previously believed. As Socrates, Plato and many other eminent philosophers often maintained, 'wisest is he who knows he knows not'.

In order to go deeper into the problem, we have to clarify the concept of uncertainty. We may start from a generic definition that I believe to be consistent with Keynes's epistemic approach: uncertainty is in its essence 'rational awareness of ignorance'. We have to distinguish between ignorance relative to a conclusion x of an argument A (given the premises h) expressed by the probability, and ignorance relative to the argument as expressed by the weight of argument V(x|h). The weight of argument is expressed through a proposition having as its object the argument A. This establishes a hierarchical relation between probability and weight of argument that may be expressed in the following way: the probability of a proposition expressing the conclusion of an argument given the premises is a first-order measure of uncertainty while the weight of argument is a second-order measure having as its object the reliability of the first-order measure.

The concept of weight of argument, as defined and measured here, permits a clear definition of different modalities of (first-order) uncertainty, which may be ordered on the basis of an homogeneous criterion. The range of values of V(x/h), as defined in measure (3), goes from 0 to 1 and allows a distinction between three modalities of (first-order) uncertainty that play a different role in Keynes's analysis in TP and then in GT. Uncertainty may be defined as 'radical' when the weight of argument is nil. In other words, the DM is aware that he does not know anything relevant about the occurrence of a certain event. For example: '[T]he prospect of a European war is uncertain, or the price of copper and the rate of interest twenty years hence ... about these matters there is no scientific basis on which to form any calculable probability whatever. We simply do not know' (Keynes, 1973c [1937], pp. 113-14). Conversely, uncertainty may be defined as 'soft' (or weak) when the weight of argument is 1. In other words, in this case the DM is uncertain only in the weak sense that he does not know which of a set of possible events will occur but believes that he knows their 'true' probability distribution. This is the typical case in a game of chance, as the emblematic case of roulette well illustrates. If the roulette is fair, the DM knows exactly the complete list of possible events and knows the 'objective' or 'true' probability of each of the possible events. Traditionally only these two extreme cases have been considered. The weight of argument clarifies that between the two extremes - the white of soft uncertainty and the black of complete relevant ignorance - there is a wide grey zone characterized by the DM's awareness that his relevant knowledge is incomplete but not nil. It is thus rational to exploit all relevant knowledge. In other words, the weight of argument allows a measure of the degree of incompleteness of relevant knowledge and provides a guide for its rational exploitation.

The threefold classification of uncertainty that we represent in graphical terms in Figure 7.1 emerges naturally - we believe - from the interpretation of weight of argument suggested here, but it is not universally accepted. Many interpreters of Keynes believe that what Keynes had in mind was a simple dichotomy between weak uncertainty (which may be expressed by probability) and radical uncertainty (or 'uncertainty' in its strict sense): see , for example, Davidson (1988; 1991). There is no doubt that a dichotomy of this kind often appears in Keynes's economic arguments, but in our opinion its role is to emphasize the hierarchy between first-order and second-order uncertainty. In any case, we want to show that the interpretation that focuses on a simple dichotomy raises a series of textual and contextual difficulties.

The first observation refers to radical uncertainty and takes into account a qualification by Keynes. In this case the knowledge relevant for probability is altogether absent, and this prevents the use of probability. Keynes maintains that this is already true for a value of weight of argument inferior to e. which defines the minimum degree of relevant knowledge that makes probability meaningful. As Keynes emphasizes (TP, p. 78):

Weight of argument and uncertainty

Figure 7.1 Weight of argument and uncertainty

A proposition cannot be the subject of an argument, unless we at least attach some meaning to it, and this meaning, even if it only relates to the form of the proposition, may be relevant in some arguments relating to it. But there may be no other relevant evidence ... in this case the weight of the argument is at its lowest.

According to the dichotomous view of uncertainty modalities, when the weight of argument is maximum (1 in our interpretation) the probability is either 1 or 0. This assertion seems at first sight inescapable as the completeness of relevant knowledge seems to imply the convergence of probability towards one of its extreme values (0 or 1; on this point see in particular O'Donnell, 1989). This thesis, however, is misleading. The knowledge that intervenes in the definition of weight of argument is, according to Keynes, the relevant knowledge that may be acquired by an epistemic subject characterized by bounded rationality. In general, as the weight of argument increases, the probability converges towards a more reliable value, which may be any value between 0 and 1 (extremes included). This conclusion should be obvious as soon as we refer to games of chance. If the DM knows that a dice is fair, the probability of any of the numbers written on its faces is assumed to be equal to 16 and this assertion has the maximum weight (equal to 1). Even if we believe that the outcome of the throw of the dice ultimately depends on deterministic factors, and we agree with Laplace that a demon knowing all the relevant initial conditions would be able to forecast the exact outcome, this is patently beyond human reach. It would be meaningless in a case like this to maintain that an argument based on the probability 16 for each number on the dice has a weight inferior to the maximum, 1. We have thus to conclude that the probability converges towards its extreme values (0 or 1) only in a deterministic argument. In addition, we have to emphasize that the weight of argument has a significant role in decision theory only when uncertainty is hard. As a matter of fact, if uncertainty is radical, probabilities are groundless, while when uncertainty is weak probabilities are seen as fully reliable. Only when uncertainty is hard may a change in the weight of argument affect economic decisions (see ss. 7.4 and 7.5). In GT Keynes refers to the weight of argument within a conceptual framework based on the distinction between probability (when the weight of argument is at the maximum) and genuine 'uncertainty' in the other cases. He wants to stress how demanding the hypothesis of soft uncertainty underlying classical economics is and how fragile the approach based on such an extreme assumption is: a small deviation from the assumption that agents have complete relevant knowledge is enough to produce deep modifications in financial and real choices and in the theories that account for them. The prevailing interpretation identifies with radical uncertainty what Keynes calls simply uncertainty in contraposition to probability. This seems justified by a few passages where Keynes refers to uncertainty as complete relevant ignorance (as the famous passage, too well-known to be quoted here, qualifying his reply in 1937 to the early critiques of GT (Keynes, 1973c [1937], pp. 113-14). However, this interpretation does not work in different crucial passages of GT where Keynes focuses on the effects of changes in one or more crucial variables on the degree of uncertainty perceived by economic agents. The interpretation of these variations as a jump between extreme values of the weight of argument would be misleading. The weight of argument can play an active role in causal analysis only in the hypothesis of hard uncertainty, where a change in the weight may bring about different behaviour.

To avoid confusion we believe that the distinction between probability and uncertainty should be interpreted not as a dichotomy between two extreme modalities of first-order uncertainty, but as a distinction between two levels of a hierarchy: probability is a first-order uncertainty measure while 'uncertainty' in its strict sense refers to second-order uncertainty as measured by the weight of argument. We may understand in this way why Keynes relates the weight of argument to uncertainty in the strict sense and why the classical economists, by neglecting altogether this dimension of the analysis, limit themselves to considering probability in the hypothesis of soft uncertainty.

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