EQUATING IMPERFECT KNOWLEDGE WITH RATIONAL BELIEF IS AN ERROR
8. Here I wish to refute mainly the equation of rational degrees of belief with imperfect knowledge, and incidentally current or accepted scientific opinion with imperfect knowledge.
What is needed for this refutation is one lemma, concerning the betting quotient of certitude or perfect knowledge : we bet, if we do at all, against any odds for a certainty and never against it. That is to say, no odds are fair against certainty. It is therefore no risk to bet on a certainty; it is no sport to do so: gentlemen do not bet on a certainty.Assume this lemma (the italicized bits in the last paragraph) to be true. Consider now the case of imperfect knowledge in this light. No matter how high is the probability that one’s information is true, unless it equals unity it may be possible to calculate the odds for a fair betting on it. Moreover, however high the probability may be that one’s information is true, unless it equals unity, the refutation of that information is not the refutation of the claim that it was probable: refuting one’s expectation does not amount to the finding of the injustice of one’s claim for high probability: only the frequent refutation of such expectations amounts to this. Indeed, we have ample (though relatively rare) realistic examples of forecasts which are highly probable yet turn out to be unrealized. (A good forecaster who forecasts with 90% accuracy may predict that 90% of the members of a given class A will have a property B. He may be right even when a high number of ^4’s turn out to be non-^’s. For this may be one of his 10% erroneous forecasts.)
Conversely, however often an expectation comes out correctly, this is no proof of its probability being the highest. This is a point which Russell has emphasized in his criticism of Reichenbach: only infinite sequences provide the accurate assessment of any probability.
Thus, whether philosophically or on the basis of common sense, when we identify certainty with demonstrability with the highest probability, or with the probability one, we can be sure that we never are certain: we know with probability one that we can never assign utter certainty to any forecast. Therefore, the maxim, a gentleman never bets on a certainty, is not applicable ever.This conclusion is false: we have imperfect knowledge of cases in which it is ungentlemanly to bet.We may rescue the maxim by saying, a very high probability makes the fair odds so stiff that there is no chance of the odds offered in reality to be fair. This rescue operation, however, will be rejected by most experts as not getting the point of the maxim. Of course a gentleman may be offered unfair odds and sometimes honestly agree to bet on them; what he cannot honestly do is bet on a certainty - under any circumstances. (If he bets on certainty under expediency, he views the expediency as one which allows dishonesty - love or war.)
10. To clinch matters, we should show that the certainty on which a gentleman may not bet is not a case of perfect knowledge of probability one, but of ordinary imperfect knowledge. This seems impossible because only perfect knowledge has probability one. And the last sentence implies the equation of imperfect knowledge with, at most, a very high probability. Hence, we cannot show what we should be able to show if our case were correct. Hence our case is incorrect.
But this argument confuses necessary and sufficient conditions for knowledge: perfect and imperfect knowledge do not share sufficient conditions : yet this is what the previous argument utilizes. Perfect and imperfect knowledge share a necessary condition which is fairly obvious. We cannot claim any knowledge of a refuted proposition.2 We sometimes do claim perfect knowledge of a false proposition, as well as imperfect knowledge of a false proposition: there exist ample instances of either in history.
But after a proposition is refuted, either the claim is withdrawn or the claimant is declared unreasonable.This similarity, however, should not be exaggerated. Though in both cases withdrawal of the claim is mandatory, the assessment of the claim as it had been made in the past is different. It is the case that we do declare all claims for perfect knowledge of refuted propositions to have been unjustified even before the discovery of their refutations; yet this does not usually hold for claims for imperfect knowledge!3
11. Consider the claim that a certain statistical hypothesis is known to be true. Regardless of how one claims to have attained such knowledge, ordinarily one’s claim will be dismissed if the ordinarily observed frequency is repeatedly different. This was noticed by Hans Reichenbach and Rudolf Carnap, who, however, chose to put it differently. In their opinion it is rational to gamble with truth according to some fair betting criterion, where the fair betting criterion is properly defined so that the computed fair betting quotient is always rational. Carnap adds that when an entrepreneur acts according to this criterion of rationality, then he acts rationally even on the occasion on which he subsequently loses, because in the long run this rational method will make the entrepreneur as well-off as possible. If all this is applicable, then there is a simple test for statistical imperfect knowledge: in the long run he who possesses more of it, and more properly utilizes it, must be better off than his neighbor. Hence, he who claims to possess more of it and use it more properly yet does not get better off is making a false claim.
12. The regrettable fact with the above criterion, however, is that even if it were applicable, the above test for a claim for knowledge will not work because seldom are claims for knowledge made concerning elements of random series. (This criticism is Popper’s.) On the contrary, when referring to elements of random series one speaks not of elements of the series but of (imperfect) knowledge of the relative frequency of the series in question.
Thus, a capable gambler in an honest house has an imperfect knowledge of the relative frequency and of the fair betting quotients of the gambles that are going on, not of any outcome of any given gamble. Of a given gamble he has an imperfect knowledge of the likelihood of its outcome, not of its outcome itself.4 Moreover, most ordinary claims for imperfect knowledge concern not easily discernible random sequences but either unrepeatable events or elements of unknown random series. Consider the question of the success of a certain marriage or of a certain business venture, or even of an investment in the stock exchange. Furthermore, very often claims for imperfect knowledge of unique cases are tested, and it is a simple empirical business to describe the way these are tested. Thus, Reichenbach’s or Carnap’s system is seldom helpful in practice, yet imperfect knowledge is often a matter of practical significance. Moreover, se wee that imperfect knowledge is not the same as likelihood, though we may have an imperfect knowledge of a likelihood.13. Sir Francis Bacon has made a suggestion which has nowadays become very popular, namely that we consider unjust claims for knowledge as including false promises. Thus, the alchemist makes promises to deliver a recipe for making gold; which he cannot deliver. This is very obvious in all claims for perfect knowledge: when a proposition claimed to be known with certainty turns out to be false the claim must be declared unjust. When we wish to extend this to imperfect knowledge we must be cautious - not only because of the difference between perfect and imperfect knowledge, but also because of the nature of promises under ordinary circumstances. Ordinarily, the interference of an act of God exempts one from the charge of having failed to deliver what one has promised: all promises are nullified by acts of God which render their execution impossible. A claim for perfect knowledge is a claim for a share in the divine (Plato, Bacon, Spinoza) in that it promises no act of God (short, perhaps, of a miracle) to nullify it.
With this difference taken care of we can safely equate perfect and imperfect knowledge: imperfect knowledge that x becomes perfect knowledge that barring an act of God x. Also, the same holds for the imperfect knowledge that x is probable or likely (Shimony).
14. This opinion, finally, enables us to find the error in the argument, since perfect knowledge that x is probability one, imperfect knowledge that x must have a lower probability. Imperfect knowledge that x too is probability one, but not of the same proposition x. Hence, when we say I know that he will propose to her within a fortnight, meaning ‘know’ in the sense of having imperfect knowledge (soft sense, so-called), then we mean, I know that if they are alive and well for another fortnight, if he is not soon inducted into the armed forces, and if she does not all of a sudden decide to visit her aunt in New Orleans, etc., etc., then, etc. Now, it seems cheating to declare imperfect knowledge of x to be the perfect knowledge of a quite different proposition - namely of if y then x. However, this exercise in clarifying claims for imperfect knowledge has a perfect precedent in the case of clarifying claims for perfect knowledge. Bertrand Russell’s analysis of the claim for perfect knowledge of the Pythagoras theoremp is making it depend on Euclid’s axioms e: so that in his view the perfect knowledge is not of p, as claimed, but of if e then p, I need not say this analysis is now common knowledge.5
III.