popper-logic-scientific-discovery

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some structural components of a theory of experience
opposition to such views I believe that the relations in question can be
fully analysed in terms of the ‘classical’ logical relations of deducibility
and contradiction.* 1
From the non-falsifiability and non-verifiability of probability
statements it can be inferred that they have no falsifiable consequences,
and that they cannot themselves be consequences of verifiable statements.
But the converse possibilities are not excluded. For it may be (a)
that they have unilaterally verifiable consequences (purely existential
consequences, or there-is-consequences) or (b) that they are themselves
consequences of unilaterally falsifiable universal statements (allstatements).
Possibility (b) will scarcely help to clarify the logical relation
between probability statements and basic statements: it is only too
obvious that a non-falsifiable statement, i.e. one which says very little,
can belong to the consequence class of one which is falsifiable, and
which thus says more.
What is of greater interest for us is possibility (a) which is by no
means trivial, and which in fact turns out to be fundamental for our
analysis of the relation between probability statements and basic statements.
For we find that from every probability statement, an infinite
class of existential statements can be deduced, but not vice versa. (Thus
the probability statement asserts more than does any of these existential
statements.) For example, let p be a probability which has been estimated,
hypothetically, for a certain alternative (and let 0 ≠ p ≠ 1); then
we can deduce from this estimate, for instance, the existential consequence
that both ones and zeros will occur in the sequence. (Of
course many far less simple consequences also follow—for example,
that segments will occur which deviate from p only by a very small
amount.)
But we can deduce much more from this estimate; for example that
there will ‘over and over again’ be an element with the property ‘1’ and
another element with the property ‘o’; that is to say, that after any
element x there will occur in the sequence an element y with the
* 1 Although I do not disagree with this, I now believe that the probabilistic concepts
‘almost deducible’ and ‘almost contradictory’ are extremely useful in connection with
our problem; see appendix *ix, and chapter *iii of the Postscript.