Consider the sentence, “If pigs could fly, then Trump is Queen of England”. Both components are false and when this occurs, the if-statement is considered false.
However, this is purely a matter of convention. There’s no reason why we couldn’t consider these sentences where the condition is false to be false or undefined.
Sentences of the form “If P, then Q” are true when both P and Q are false. According to standard logical interpretation, at least. (After all, if P is false, then not-P is true, and you yourself said if statements are equivalent to B-or-not-A.) (Also, I think you mean “if pigs can fly” since using “could” makes it a counterfactual rather than logical claim, right?) And there are good reasons for this, if I recall correctly from my logic classes (though I don’t remember what they were, and I agree that ultimately it’s a matter of convention.)
Stipulate that we want to choose some boolean function → on two inputs to represent “if P, then Q”. Then we want T→T=T,T→F=F or what are we even doing with our lives.
So we have four choices for how to define F→T and F→F. The standard choice says both are true. What about other choices?
Suppose we have F→F=F. Then we’ve translated “if P, then Q” into something that asserts Q and maybe (depending on F→T) also asserts P. That seems like a bad translation. So let’s say F→F=T. (Notably, if we add “undefined”, then F→F=U has the same problem, when P and Q are defined it lets P→Q be true only if Q is true.) Without this, we can’t really translate “if P, then Q. Not Q. Therefore, not P,” because (P→Q)∧¬Q is a contradiction. (Which does still give us ¬P, but it also gives us P, so.)
All that’s left is F→T. If we say this is false, then our “if P, then Q” is translated into something that means ”P and Q are either both true, or both false”, or P=Q. That seems like a bad translation too.
Basically, “if P, then Q” just doesn’t translate very well into boolean logic, but all the other ways to translate it seem worse.
Sentences of the form “If P, then Q” are true when both P and Q are false. According to standard logical interpretation, at least. (After all, if P is false, then not-P is true, and you yourself said if statements are equivalent to B-or-not-A.) (Also, I think you mean “if pigs can fly” since using “could” makes it a counterfactual rather than logical claim, right?) And there are good reasons for this, if I recall correctly from my logic classes (though I don’t remember what they were, and I agree that ultimately it’s a matter of convention.)
Stipulate that we want to choose some boolean function → on two inputs to represent “if P, then Q”. Then we want T→T=T,T→F=F or what are we even doing with our lives.
So we have four choices for how to define F→T and F→F. The standard choice says both are true. What about other choices?
Suppose we have F→F=F. Then we’ve translated “if P, then Q” into something that asserts Q and maybe (depending on F→T) also asserts P. That seems like a bad translation. So let’s say F→F=T. (Notably, if we add “undefined”, then F→F=U has the same problem, when P and Q are defined it lets P→Q be true only if Q is true.) Without this, we can’t really translate “if P, then Q. Not Q. Therefore, not P,” because (P→Q)∧¬Q is a contradiction. (Which does still give us ¬P, but it also gives us P, so.)
All that’s left is F→T. If we say this is false, then our “if P, then Q” is translated into something that means ”P and Q are either both true, or both false”, or P=Q. That seems like a bad translation too.
Basically, “if P, then Q” just doesn’t translate very well into boolean logic, but all the other ways to translate it seem worse.
Thanks for posting this. I didn’t realise that there were such good reasons for this convention!
Fixed now. I really should have checked my post for mistakes like this.