Logic and foundations
1.1. Material implication
The material implication “If A, then B” (or “A implies B”) can be thought
of as the assertion “B is at least as true as A” (or equivalently, “A is at
most as true as B”). This perspective sheds light on several facts about the
(1) A falsehood implies anything (the principle of explosion). Indeed,
any statement B is at least as true as a falsehood. By the same
token, if the hypothesis of an implication fails, this reveals nothing
about the conclusion.
(2) Anything implies a truth. In particular, if the conclusion of an
implication is true, this reveals nothing about the hypothesis.
(3) Proofs by contradiction. If A is at most as true as a falsehood, then
it is false.
(4) Taking contrapositives. If B is at least as true as A, then A is at
least as false as B.
(5) “If and only if” is the same as logical equivalence. “A if and only
if B” means that A and B are equally true.
(6) Disjunction elimination. Given “If A, then C” and “If B, then C”,
we can deduce “If (A or B), then C”, since if C is at least as true
as A, and at least as true as B, then it is at least as true as either
A or B.
(7) The principle of mathematical induction. If P(0) is true, and each
P(n + 1) is at least as true as P(n), then all of the P(n) are true.
(Note, though, that if one is only 99% certain of each implication