"Logical priority may be defined" down to "although conversely, not."
The two passages which I contrast with these are (d) p. 46, "Much of mathematics" down to the words on the same page "mechanics and physics must be false"; and (e) on p. 205, the sentence beginning "Thus, à propos" and ending "pure mathematics (arithmetic)." See also pp. 220 and 225, which suggest an empirical basis for the doctrine of (d) and (e).
Passages (a) and (c) state the doctrine of Logical Priority,[1] and point out its application, in perfect agreement with each other, and consistently with the traditional rules of Formal Logic. Here is passage (a): " By Logical Priority is meant that relation which holds between a proposition and its necessary condition. Thus if A implies B, but B does not imply A, then B is the necessary condition of A; for A's truth depends upon B's truth. That is, should B prove to be false, A must be false; and though A be false, yet B may prove true; for we are saying merely that A's truth is a sufficient condition of B's truth, and are not maintaining that it is the only condition, or a necessary condition. For example, let us assume it to be true that if the tissues of a man's body absorb a certain amount of arsenic, he must die; that there is no preventing cause either known or unknown. Then evidently for it to be true that this man's
- ↑ The doctrine is one apparently obvious to commonsense, and the definition might be borrowed from Aristotle, M. δ 11, 1019 a 2. πρότερα ὅσα ἐδέχεται εἶναι ἄνευ ἄλλων, ἐκεῖνα δὲ ἄνευ ἐκείνων μή.
