in other words, why are statements of the form “If A then B” always true if A is false?
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This sounds like the drinker’s paradox.
This statement remains true when you look at the two possibilities.
1) Bob is drinking:
As that statement says, if Bob drinks, everyone drinks. If A, then B. Bob is a part of everyone, so if Bob drinks, it is not contradictory that everyone might drink. So, that statement could be true; it is not necessarily wrong.
2) Bob is not drinking:
In logic, there is something called the contrapositive. It is basically the logical reverse of a statement. You find the contrapositive by reversing the order of the argument and reversing the positive or negative quality of the parts of the argument. So, the original state is “if A, then B”. The contrapositive is therefore “if not B, then not A.” Both these statements are logically identical.
The contrapositive of the original statement about Bob is “if not everyone is drinking, Bob isn’t drinking.” Again, this statement could be true. Since Bob is a part of everyone, Bob’s absence of drinking is a sign that not everyone is drinking.
So, we say that “If Bob drinks, then everyone drinks” is true if Bob doesn’t drink” because when you look at both sides of the coin, the logic does not contradict itself. It’s true in the sense that it could be true. This kind of truth is called vacuous truth. It’s “true” because it isn’t untrue.
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