Why is it that in formal logic, the phrase “If Bob drinks, then everyone drinks” is true if Bob doesn’t drink ?

Question

in other words, why are statements of the form “If A then B” always true if A is false?

In: 8

Answer 1

You used the math flair, so I will assume you understand basic logic. A conditional of the form “P → Q” is equivalent to “¬P V Q”, where ¬ is the negation operator and V is a stand-in for the proper OR operator (I’m on my phone right now).

If we then take your example:
P is “Bob drinks”,
¬P is “Bob doesn’t drink”,
and Q is “everyone drinks”.

Thus, “If Bob doesn’t drink, then everyone drinks” can be symbolically represented as “¬P → Q”, which becomes “¬(¬P) V Q”. ¬¬P = P, so we have P V Q, which is True V True