popper-logic-scientific-discovery

270
some structural components of a theory of experience
Among those who argue in this way is Keynes who uses the expression
‘a priori probability’ for what I call ‘logical probability’. (See note 1
to section 34.) He makes the following perfectly accurate remark 1
regarding a ‘generalization’ g (i.e. a hypothesis) with the ‘condition’ or
antecedent or protasis φ and the ‘conclusion’ or consequent or
apodosis f: ‘The more comprehensive the condition φ and the less
comprehensive the conclusion f, the greater à priori* 4 probability do we
attribute to the generalization g. With every increase in φ this probability
increases, and with every increase in f it will diminish.’ This, as I
said, is perfectly accurate, even though Keynes does not draw a sharp
distinction* 5 between what he calls the ‘probability of a
generalization’—corresponding to what is here called the ‘probability
of a hypothesis’—and its ‘a priori probability’. Thus in contrast to my
degree of corroboration, Keynes’s probability of a hypothesis increases with its a
priori logical probability. That Keynes nevertheless intends by his ‘probability’
the same as I do by my ‘corroboration’ may be seen from the
fact that his ‘probability’ rises with the number of corroborating
instances, and also (most important) with the increase of diversity
This is the crucial result. My later remarks in the text merely draw the conclusion from
it: if you value high probability, you must say very little—or better still, nothing at all:
tautologies will always retain the highest probability.
1 Keynes, A Treatise on Probability, 1921, pp. 224 f. Keynes’s condition φ and conclusion f
correspond (cf. note 6 to section 14) to our conditioning statement function φ and our
consequence statement function f; cf. also section 36. It should be noticed that Keynes
called the condition or the conclusion more comprehensive if its content, or its intension, rather
than its extension, is the greater. (I am alluding to the inverse relationship holding
between the intension and the extension of a term.)
* 4 Keynes follows some eminent Cambridge logicians in writing ‘à priori’ and ‘à posteriori’;
one can only say, à propos de rien—unless, perhaps, apropos of ‘à propos’.
* 5 Keynes does, in fact, allow for the distinction between the a priori (or ‘absolute
logical’, as I now call it) probability of the ‘generalization’ g and its probability with
respect to a given piece of evidence h, and to this extent, my statement in the text needs
correction. (He makes the distinction by assuming, correctly though perhaps only
implicitly—see p. 225 of the Treatise—that if φ = φ1φ2, and f = f1f2, then the a priori probabilities
of the various g are: g(φ, f1) � g(φ, f) � g(φ1, f).) And he correctly proves that the
a posteriori probabilities of these hypotheses g (relative to any given piece of evidence h)
change in the same way as their a priori probabilities. Thus while his probabilities change
like (absolute) logical probabilities, it is my cardinal point that degrees of corroborability
(and of corroboration) change in the opposite way.