in other words, why are statements of the form “If A then B” always true if A is false?
In: 8
“If Bob drinks, then everyone drinks.”
In formal logic, this can be alternatively worded as:
“In the event that Bob drinks, then I know for certain that everyone must also drink.”
So now you suppose, “So what happens in the event that Bob *does not* drink?” The answer is: we do not have enough info to know. Nothing about the initial phrase told us about what happens in that scenario.
It’s tempting to *infer* that the opposite of the phrase (in strict logic terms, the *inverse*) must also be true, i.e. “If Bob does not drink, not everyone drinks.” We do that a lot in common speech. Like, if I told you, “If it is Tuesday, I will be gone”, you may reasonably gather from context the inverse, “If it is not Tuesday, I will be here”. That works in day-to-day speech. But in formal logic, that doesn’t fly. I did not explicitly tell you what happens when it is not Tuesday. Maybe I will be here. Maybe I won’t. For all you know, my presence and the day of the week are not related at all on all days that are not Tuesday.
If you do want to explicitly say the inverse is also true without adding another statement, you can say “if *and only if*” instead of just “if”, e.g. “If *and only if* it is Tuesday, I will be gone.”
EDIT: Reread the question.
In general, “if A is true, then B is true” can still be a true statement even when A is not true. You never *said* “A is true”, you said, “*if* A is true”.
If I said, “If you turn right at the intersection, then you will go to the grocery store”, but you turn right instead, what I said doesn’t stop being true. *If you did* turn left, you *would have* gotten to the grocery store. But you didn’t that time.
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