popper-logic-scientific-discovery

(D)
a→ N b is true if, and only if, (a → b) ε N.
appendix *x 455
In words, perhaps: ‘If a then necessarily b’ holds if, and only if, ‘If a
then b’ is necessarily true. Here ‘a → b’ is, of course, the name of an
ordinary conditional with the antecedent a and the consequent b. If it
were our intention to define logical entailment or ‘strict implication’,
then we could also use (D), but we should have to interpret ‘N’ as
‘logically necessary’ (rather than as ‘naturally or physically necessary’).
Owing to the definition (D), we can say of ‘a→ N b’ that it is the name
of a statement with the following properties.
(A) a→ N b is not always true if a is false, in contradistinction to a → b.
(B) a→ N b is not always true if b is true, in contradistinction to a → b.
(A′) a→ N b is always true if a is impossible or necessarily false, or if its
negation, ā, is necessarily true whether by logical or by physical
necessity. (Cf. the last three pages of the present appendix, and
note 26, below.)
(B′) a→ N b is always true if b is necessarily true (whether by logical or
physical necessity).
Here a and b may be either statements or statement functions.
a→ N b may be called a ‘necessary conditional’ or a ‘nomic conditional’.
It expresses, I believe, what some authors have called
‘subjunctive conditionals’, or ‘counterfactual conditionals’. (It seems,
however, that other authors—for example Kneale—meant something
else by a ‘counterfactual conditional’: they took this name to imply
that a is, in fact, false. 17 I do not think that this usage is to be
recommended.)
17 In my ‘Note on Natural Laws and so-called Contrary-to-Fact Conditionals’ (Mind 58,
N.S., 1949, pp. 62–66) I used the term ‘subjunctive conditional’ for what I here call
‘necessary’ or ‘nomic conditional’; and I explained repeatedly that these subjunctive
conditionals must be deducible from natural laws. It is therefore difficult to understand
how Kneale (Analysis 10, 1950, p. 122) could attribute to me even tentatively the
view that a subjunctive conditional or a ‘contrary to fact conditional’ was of the form