popper-logic-scientific-discovery

probability 185
property ‘1’, and also an element z with the property ‘o’. A statement of
this form (‘for every x there is a y with the observable, or extensionally
testable, property β’) is both non-falsifiable—because it has no falsifiable
consequences—and non-verifiable—because of the ‘all’ or ‘for
every’ which made it hypothetical.* 2 Nevertheless, it can be better, or
less well ‘confirmed’—in the sense that we may succeed in verifying
many, few, or none of its existential consequences; thus it stands to the
basic statement in the relation which appears to be characteristic of
probability statements. Statements of the above form may be called
‘universalized existential statements’ or (universalized) ‘existential
hypotheses’.
My contention is that the relation of probability estimates to basic
statements, and the possibility of their being more, or less, well
‘confirmed’, can be understood by considering the fact that from all
probability estimates, existential hypotheses are logically deducible. This
suggests the question whether the probability statements themselves
may not, perhaps, have the form of existential hypotheses.
Every (hypothetical) probability estimate entails the conjecture that
the empirical sequence in question is, approximately, chance-like or
random. That is to say, it entails the (approximate) applicability, and
the truth, of the axioms of the calculus of probability. Our question is,
* 2 Of course, I never intended to suggest that every statement of the form ‘for every x,
there is a y with the observable property β’ is non-falsifiable and thus non-testable:
obviously, the statement ‘for every toss with a penny resulting in 1, there is an immediate
successor resulting in 0’ is both falsifiable and in fact falsified. What creates nonfalsifiability
is not just the form ‘for every x there is a y such that . . . ’ but the fact that the
‘there is’ is unbounded—that the occurrence of the y may be delayed beyond all bounds: in
the probabilistic case, y may, as it were, occur as late as it pleases. An element ‘0’ may occur at
once, or after a thousand tosses, or after any number of tosses: it is this fact that is
responsible for non-falsifiability. If, on the other hand, the distance of the place of
occurrence of y from the place of occurrence of x is bounded, then the statement ‘for every
x there is a y such that . . .’ may be falsifiable.
My somewhat unguarded statement in the text (which tacitly presupposed section 15)
has led, to my surprise, in some quarters to the belief that all statements—or ‘most’
statements, whatever this may mean—of the form ‘for every x there is a y such that . . .’
are non-falsifiable; and this has then been repeatedly used as a criticism of the falsifiability
criterion. See, for example, Mind 54, 1945, pp. 119 f. The whole problem of these
‘all-and-some statements’ (this term is due to J. W. N. Watkins) is discussed more fully in
my Postscript; see especially sections *24 f.