popper-logic-scientific-discovery

appendix *vii 389
and ‘ ’.) The use of these symbols can be explained by the following
rules:
(1) ‘C(a) C(b)’ and thus its equivalent ‘p(b) p(a)’ may be used
to state that the content of a is greater than that of b—at least in the sense
of the fine structure of content. We shall thus assume that C(a) C(b)
entails C(a) C(b), and that this in turn entails C(a) � C(b), that is to
say, the falsity of C(a) C(b) always entails C(a) C(b).
(4) Corresponding rules will hold for p(a) p(b), etc.
The problem now arises of determining the cases in which we may
say that C(a) C(b) holds even though we have C(a) = C(b). A number
of cases are fairly clear; for example, unilateral entailment of b by a.
More generally, I suggest the following rule:
If for all sufficiently large finite universes (that is, for all universes
with more than N members, for some sufficiently large N), we have
C(a) >C(b), and thus, in accordance with rule (3), C(a) C(b), we
retain C(a) C(b) for an infinite universe even if, for an infinite
universe, we obtain C(a) = C(b).
This rule seems to cover most cases of interest, although perhaps not
all. 15
The problem of a1 = ‘All planets move in circles’ and a2 = ‘All planets
move in ellipses’ is clearly covered by our rule, and so is even the case
of comparing a1 and a3 = ‘All planets move in ellipses with an eccentricity
other than zero’; for p(a3)>p(a1) will hold in all sufficiently
large finite universes (of possible observations, say) in the simple sense
that there are more possibilities compatible with a3 than with a1. *
15 Related problems are discussed in considerable detail in John Kemeny’s very stimulating
paper ‘A Logical Measure Function’, Journal of Symb. Logic 18, 1953, pp. 289 ff.
Kemeny’s model language is the second of three to which I allude on p. xxiv of my
Preface, 1958. It is, in my opinion, by far the most interesting of the three. Yet as he
shows on p. 294, his language is such that infinitistic theorems—such as the principle
that every number has a successor—must not be demonstrable in it. It thus cannot
contain the usual system of arithmetic.