popper-logic-scientific-discovery

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new appendices
indirect proof. Indeed, an indirect proof may well be put thus: ‘Conceive
that a is true. Then we should have to admit that b is true. But we
know that b is absurd. Thus it is inconceivable that a is true.’ It is clear
that although this use of ‘conceivable’ and ‘inconceivable’ is a little
vague and ambiguous, it would be misleading to say that this way of
arguing must be invalid since the truth of a cannot be inconceivable,
considering that we did start by conceiving, precisely, the truth of a.
Thus ‘inconceivable’ in logic and mathematics is simply another
word for ‘leading to an obvious contradiction’. Logically possible or
‘conceivable’ is everything that does not lead to an obvious contradiction,
and logically impossible or ‘inconceivable’ is everything that
does. When Kneale says that the contradictory of a theorem may be
‘conceivable’, he uses the word in another sense—and in a very good
sense too.
(9) Thus an assumption is logically possible if it is not selfcontradictory;
it is physically possible if it does not contradict the laws
of nature. The two meanings of ‘possible’ have enough in common to
explain why we use the same word; but to gloss over their difference
can only lead to confusion.
Compared with logical tautologies, laws of nature have a contingent,
an accidental character. This is clearly recognized by Leibniz who
teaches (cf. Philos. Schriften, Gerhardt, 7, p. 390) that the world is the work
of God, in a sense somewhat similar to that in which a sonnet, or a
rondeau, or a sonata, or a fugue, is the work of an artist. The artist may
freely choose a certain form, voluntarily restricting his freedom by this
choice: he imposes certain principles of impossibility upon his creation,
for example upon his rhythm, and, to a lesser extent, his words
which, as compared to the rhythm, may appear contingent, accidental.
But this does not mean that his choice of form, or of rhythm, was not
contingent also. For another form or rhythm could have been chosen.
Similarly with natural laws. They restrict the (logically) possible
choice of singular facts. They are thus principles of impossibility with
respect to these singular facts; and the singular facts seem highly contingent
as compared with the natural laws. But the natural laws, though
necessary as compared with singular facts, are contingent as compared
with logical tautologies. For there may be structurally different worlds—
worlds with different natural laws.