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Published online by Cambridge University Press: 01 September 1963
In The Logic of Scientific Discovery, K. R. Popper says that he has A. provided “a mathematical refutation of all those theories of induction which identify the degree to which a statement is supported or confirmed or corroborated by empirical tests with its degree of probability in the sense of the calculus of probability.” He tells us that “those who identify, explicitly or implicitly, degree of corroboration, or of confirmation, or of acceptability, with probability [are] Keynes, Jeffreys, Reichenbach, Kaila, Hosiasson, and, more recently, Carnap.”
1 K. R. Popper, The Logic of Scientific Discovery, New York, 1959, p. 390 All page references hereinafter will be to this book.
2 Writers on the theory of probability sometimes require that the statements x and y in ‘p(x, y)’ satisfy certain logical or epistemic conditions. Popper ignores this requirement, and so shall I. Popper uses the expression ‘p(x)’ in his argument against (I). ‘p(x)’ is short for ‘p(x, t)’, where t is any logical truth, and may be read ‘the degree of absolute probability of x’ or ‘the degree of probability of x relative to logical truth or logical information only’.
3 The following is one of Popper's examples, (p. 398) Let a be a coloured counter. Suppose that the colours red, green, blue, and yellow are exclusive and, as Popper puts it, ‘equally probable’. Given just this information, Popper says that the absolute probability of ‘a is blue’ is 1/4, and the probability of ‘a is blue’ relative to ‘a is not yellow’ is 1/3. This example clearly suggests that ‘p(x, y)’ is being interpreted according to either the classical or the logical interpretation. Incidentally, this example of Popper's is defective. He should have stipulated that the four colours mentioned be exhaustive as well as exclusive within the class of coloured things. Strictly speaking, the absolute probability of ‘a is blue’ is not 1/4. The value 1/4 is the value of the probability of ‘a is blue’ relative to ‘a is coloured’. And further, Popper offers no explanation of ‘equally probable’ as applied to properties.
In connection with the classical and logical interpretation of ‘p(x, y)’, I think it fair to say that under neither of these interpretations does ‘p(x, y)’ turn out to be a normative expression.
4 (Ia) is a patent falsehood. Since ‘p(x, y)’ expresses a probability function, if x is a logical truth, p(x, y) = 1 for any logically contingent statement y. But in this case the left hand side of (Ia) reduces to O.
5 I find it difficult to say anything very helpful about Popper's ‘C(x, y)’. My trouble is not that Popper himself says nothing about ‘C(x, y)’. On the contrary, he says many things. But what he says on one occasion differs—and not merely verbally—from what he says on other occasions. And on no occasion does he give a detailed explanation of ‘C(x, y)’; he tends to content himself with three- or four-line comments.
6 Popper gives two examples to back (2). One concerns a coloured counter; see my footnote 3. The other will be set forth here. Examples are needed to back (2) because (2) is neither an axiom nor a theorem of the cajculus of probability.
7 This is called a criterion of qualitative confirmation because is confirmatory with respect to is a qualitative, and neither a comparative nor a quantitative (metrical) term. Although Popper's argument does contain (3) as a premise, Popper does not accept (3) as it stands. He would, I think, accept (3) with the following proviso added: “provided that if y expresses observational data and x is a logically contingent statement, p(x) ≡ O.”
8 Which thesis is, incidentally, false. For if a statement x logically implies an observation statement y that, as it happens, is true, then it is clear that x has passed a test and that y results from a test of x that x passed. For if y were false then x would be false too; hence by ascertaining the truth-value of y, x is exposed to falsification. Now let x be a conjunction of logically contingent statements, one of which logically implies the statement y. Hence x logically implies y. But who would say that the information y is confirmatory with respect to x no matter what the other conjuncts of x may be?
9 An attempt to undermine this thesis is made in my ‘A Note on Some Quantitative Theories of Confirmation’, Philosophical Studies, Vol. XII, December, 1961, pp. 91–92.
