popper-logic-scientific-discovery

probability 203
class of occurrences;* 1 for example, ‘the probability of throwing five
with the next throw of this die is 1/6′ or ‘the probability of throwing
five with any single throw (of this die) is 1/6′. From the standpoint of
the frequency theory such statements are as a rule regarded as not quite
correct in their formulation, since probabilities cannot be ascribed to
single occurrences, but only to infinite sequences of occurrences or
events. It is easy, however, to interpret these statements as correct, by
appropriately defining formally singular probabilities with the help of
the concept of objective probability or relative frequency. I use ‘ αP k(β)’
to denote the formally singular probability that a certain occurrence k
has the property β, in its capacity as an element of a sequence α—in
symbols: 1 k εα—and I then define the formally singular probability as
follows:
αP k(β) = αF(β) (k εα) (Definition)
This can be expressed in words as: The formally singular probability
that the event k has the property β—given that k is an element of the
sequence α—is, by definition, equal to the probability of the property
β within the reference sequence α.
This simple, almost obvious, definition proves to be surprisingly
useful. It can even help us to clarify some intricate problems of modern
quantum theory. (Cf. sections 75–76.)
As the definition shows, a formally singular probability statement
would be incomplete if it did not explicitly state a reference-class. But
although α is often not explicitly mentioned, we usually know in such
cases which α is meant. Thus the first example given above does not
specify any reference sequence α, but it is nevertheless fairly clear that
it relates to all sequences of throws with true dice.
In many cases there may be several different reference sequences for
an event k. In these cases it may be only too obvious that different
formally singular probability statements can be made about the same
* 1 The term ‘formalistisch’ in the German text was intended to convey the idea of a
statement which is singular in form (or ‘formally singular’) although its meaning can in
fact be defined by statistical statements.
1 The sign ‘. . . ε . . .’, called the copula, means ‘. . . is an element of the class . . .’; or
else, ‘. . . is an element of the sequence . . .’.