popper-logic-scientific-discovery

33 DEGREES OF FALSIFIABILITY COMPARED BY
MEANS OF THE SUBCLASS RELATION
The following definitions are introduced provisionally, to be
improved later in the course of our discussion of the dimensions of
theories.* 1
(1) A statement x is said to be ‘falsifiable in a higher degree’ or
‘better testable’ than a statement y, or in symbols: Fsb(x) >Fsb(y), if and
only if the class of potential falsifiers of x includes the class of the
potential falsifiers of y as a proper subclass.
(2) If the classes of potential falsifiers of the two statements x and y
are identical, then they have the same degree of falsifiability, i.e.
Fsb(x) = Fsb(y).
(3) If neither of the classes of potential falsifiers of the two statements
includes the other as a proper subclass, then the two statements
have non-comparable degrees of falsifiability (Fsb(x) ||Fsb(y)).
If (1) applies, there will always be a non-empty complement class.
In the case of universal statements, this complement class must be
infinite. It is not possible, therefore, for the two (strictly universal)
theories to differ in that one of them forbids a finite number of single
occurrences permitted by the other.
The classes of potential falsifiers of all tautological and metaphysical
statements are empty. In accordance with (2) they are, therefore, identical.
(For empty classes are subclasses of all classes, and hence also of
empty classes, so that all empty classes are identical; which may be
expressed by saying that there exists only one empty class.) If we denote
an empirical statement by ‘e’, and a tautology or a metaphysical statement
(e.g. a purely existential statement) by ‘t’ or ‘m’ respectively, then
we may ascribe to tautological and metaphysical statements a zero
degree of falsifiability and we can write: Fsb(t) = Fsb(m) = 0, and
Fsb(e) >0.
A self-contradictory statement (which we may denote by ‘c’) may be
said to have the class of all logically possible basic statements as its class
of potential falsifiers. This means that any statements whatsoever is
comparable with a self-contradictory statement as to its degree of
* 1 See section 38, and the appendices i, *vii, and *viii.
degrees of testability 99