popper-logic-scientific-discovery

382
new appendices
inferences from observed to unobserved objects, then, Hume says, ‘I
wou’d renew my question, why from this experience we form any conclusion
beyond those past instances, of which we have had experience’. In other words,
Hume points out that we get involved in an infinite regress if we appeal
to experience in order to justify any conclusion concerning unobserved
instances—even mere probable conclusions, as he adds in his Abstract. For there
we read: ‘It is evident that Adam, with all his science, would never have
been able to demonstrate that the course of nature must continue uniformly
the same. ... Nay, I will go farther, and assert that he could not
so much as prove by any probable arguments that the future must be
conformable to the past. All probable arguments are built on the supposition
that there is conformity betwixt the future and the past, and
therefore can never prove it.’ 6 Thus ( + ) is not justifiable by experience;
yet in order to be logically valid, it would have to be of the character of
a tautology, valid in every logically possible universe. But this is clearly
not the case.
Thus ( + ), if true, would have the logical character of a synthetic a priori
principle of induction, rather than of an analytic or logical assertion. But
it does not quite suffice even as a principle of induction. For ( + ) may
be true, and p(a) = 0 may be valid none the less. (An example of a
theory which accepts ( + ) as a priori valid—though, as we have seen, ( + )
must be synthetic—and which at the same time accepts p(a) = 0, is
Carnap’s. 7 )
An effective probabilistic principle of induction would have to be
even stronger than ( + ). It would have to allow us, at least, to conclude
that for some fitting singular evidence b, we may obtain p(a, b) > 1/2,
or in words, that a may be made, by accumulating evidence in its
6 Cf. An Abstract of a Book lately published entitled A Treatise of Human Nature, 1740, ed. by J. M.
Keynes and P. Sraffa, 1938, p. 15. Cf. note 2 to section 81. (The italics are Hume’s.)
7 Carnap’s requirement that his ‘lambda’ (which I have shown to be the reciprocal of a
dependence measure) must be finite entails our ( + ); cf. his Continuum of Inductive Methods,
1952. Nevertheless, Carnap accepts p(a) = 0, which according to Jeffreys would entail the
impossibility of learning from experience. And yet, Carnap bases his demand that his
‘lambda’ must be finite, and thus that ( + ) is valid, on precisely the same transcendental
argument to which Jeffreys appeals—that without it, we could not learn from experience.
See his Logical Foundations of Probability, 1950, p. 565, and my contribution to the
Carnap volume of the Library of Living Philosophers, ed. by P. A. Schilpp, especially note 87.
This is now also in my Conjectures and Refutations, 1963.