popper-logic-scientific-discovery

284
appendices
it is connected with the fact that a theory always prohibits some ‘homotypic’
events, as we have called them (i.e. a class of occurrences which
differ only in their spatio-temporal co-ordinates; cf. sections 23 and
31). For this reason, spatio-temporal co-ordinates will, in general,
appear in the schema which generates the field of application, and
consequently the field of the relatively atomic statements will, in
general, show a topological and even a metrical order.
The proposed definition says: A theory t is called ‘d-dimensional
with respect to the field of application F’ if and only if the following
relation holds between t and F: there is a number d such that (a) the
theory does not clash with any d-tuple of the field and (b) any given
d-tuple in conjunction with the theory divides all the remaining
relatively atomic statements uniquely into two infinite sub-classes A
and B, such that the following conditions are satisfied: (α) every
statement of the class A forms, when conjoined with the given dtuple,
a ‘falsifying d + 1-tuple’ i.e. a potential falsifier of the theory;
(β) the class B on the other hand is the sum of one or more, but
always a finite number, of infinite sub-classes [B i] such that the conjunction
of any number of statements belonging to any one of these
subclasses [B i] is compatible with the conjunction of the given
d-tuple and the theory.
This definition is intended to exclude the possibility of a theory’s
having two fields of application such that the relatively atomic statements
of one field result from the conjunction of the relatively
atomic statements of the other (this must be prevented if the field of
application is to be identifiable with that of its graphic representation;
cf. section 39). I may add that by means of this definition the
problem of atomic statements (cf. note 2 to section 38) is solved in a
manner which might be described as ‘deductivist’, since the theory
itself determines which singular statements are relatively atomic (with
respect to the theory). For it is the theory itself through which the
field of application is defined—and with it the statements which
because of their logical form have equal status with respect to the
theory. Thus the problem of the atomic statements is not solved by
the discovery of statements of some elementary form out of which
the other, more composite, statements are built up inductively, or
composed by the method of truth-functions. On the contrary, the