popper-logic-scientific-discovery

degrees of testability 107
statement function of q (which may be denoted by ‘φ qx’); or in other
words, if ‘(x) (φ qx → φ px)’ is tautological (or logically true). Similarly we
shall say that p has greater precision than q if ‘(x) (f px → f qx)’ is tautological,
that is if the predicate (or the consequent statement function)
of p is narrower than that of q, which means that the predicate of
p entails that of q.* 1
This definition may be extended to statement functions with more
than one variable. Elementary logical transformations lead from it to
the derivability relations which we have asserted, and which may be
expressed by the following rule: 1 If of two statements both their universality
and their precision are comparable, then the less universal or
less precise is derivable from the more universal or more precise;
unless, of course, the one is more universal and the other more precise
(as in the case of q and r in my diagram). 2
We could now say that our methodological decision—sometimes
metaphysically interpreted as the principle of causality—is to leave
nothing unexplained, i.e. always to try to deduce statements from
others of higher universality. This decision is derived from the demand
for the highest attainable degree of universality and precision, and it
can be reduced to the demand, or rule, that preference should be given
to those theories which can be most severely tested.* 2
* 1 It will be seen that in the present section (in contrast to sections 18 and 35), the arrow
is used to express a conditional rather than the entailment relation; cf. also note *1 to
section 18.
1 We can write: [(�qx → � px).(f px → f qx)] → [(� px → f px) → (� qx) → f qx)] or for short:
[(� q → � p).(f p → f q] → (p → q). *The elementary character of this formula, asserted in
the text, becomes clear if we write: ‘[(a → b).(c → d)] → [(b → c) → (a → d)]’. We then
put, in accordance with the text, ‘p’ for ‘b → c’ and ‘q’ for ‘a → d’, etc.
2 What I call higher universality in a statement corresponds roughly to what classical
logic might call the greater ‘extension of the subject’; and what I call greater precision
corresponds to the smaller extension, or the ‘restriction of the predicate’. The rule
concerning the derivability relation, which we have just discussed, can be regarded as
clarifying and combining the classical ‘dictum de omni et nullo’ and the ‘nota-notae’ principle,
the ‘fundamental principle of mediate predication’. Cf. Bolzano, Wissenschaftslehre II,
1837, §263, Nos. 1 and 4; Külpe, Vorlesungen über Logik (edited by Selz, 1923), §34, 5,
and 7.
* 2 See now also section *15 and chapter *iv of my Postscript, especially section *76, text to
note 5.