Need help with truth table: I'm trying to understand WHY, in this table, that X→Z would be true when X is false.

[TheQuestor]

X | Z| X→Z| Z→~X
=| =| === | ====
T | T | T | F
T | F | F | T
F | T | T | T
F | F | T | T

I'm trying to understand WHY, in this table, that X→Z would be true when X is false. In my thinking, X can't imply anything if it's false.
Apparently, that's not the case. I want to understand WHY this truth table comes out as it does. Every place I've looked online
is coming up with the same answers, but I'd like a full explanation so that I can truly understand. Thanks!

 

[Dr.Peterson]

In my thinking, X can't imply anything if it's false.

That's almost the right idea, but missing one detail.

If I claim that X implies Z, then I'm wrong if X is true but Z is false; but if X is false, we have no evidence against the claim. So we can't say the claim is false; the most I can say is "I can't tell".

But in logic, we need every statement we make to be either true or false; so we have agreed to an "innocent until proven guilty" approach: If there's no evidence of falsity, we call it true. That's simply how we choose to define this connective.

In particular, though, we define it this way in large part because that's what we need in order to apply logic to testing logical arguments. A valid argument isn't invalidated just because it can be applied to cases where the premises are not true; we want to call it valid as long as true premises lead to true conclusions.

 

[eipi45]

Think of the implication 'eating carrots helps you see in the dark'
If it is True that you eat carrots, and you have good night-vision, we can say the implication is correct
If you eat carrots, and are night-blind, we can say the implication is incorrect
But if you dont eat carrots, and have good night-vision, the implication is not necessarily incorrect (eating radishes might have the same beneficial effect) and as Dr Peterson says, we say the implication is correct as a default position
Finally, if you don't eat carrots, and you can't see in the dark, the implication is obviously still correct

 

[Dr.Peterson]

Think of the implication 'eating carrots helps you see in the dark'
If it is True that you eat carrots, and you have good night-vision, we can say the implication is correct
If you eat carrots, and are night-blind, we can say the implication is incorrect
But if you dont eat carrots, and have good night-vision, the implication is not necessarily incorrect (eating radishes might have the same beneficial effect) and as Dr Peterson says, we say the implication is correct as a default position
Finally, if you don't eat carrots, and you can't see in the dark, the implication is obviously still correct

There's one thing I don't like about this example: A logical implication does not mean causation; in fact, thinking of it that way is one reason people are confused by this idea. So rather than say "helps" (which really doesn't even imply an absolute implication!), I'd rather say "if you eat carrots, you can see in the dark". Other than that, this is fine: If you don't eat carrots, then you can't disprove the claim.

I've discussed this in more depth here, including a similar example:

Why, in Logic, Does “False” Imply Anything? – The Math Doctors

www.themathdoctors.org

 

[eipi45]

There's one thing I don't like about this example: A logical implication does not mean causation; in fact, thinking of it that way is one reason people are confused by this idea. So rather than say "helps" (which really doesn't even imply an absolute implication!), I'd rather say "if you eat carrots, you can see in the dark". Other than that, this is fine: If you don't eat carrots, then you can't disprove the claim.

I've discussed this in more depth here, including a similar example:

Why, in Logic, Does “False” Imply Anything? – The Math Doctors

www.themathdoctors.org

Thanks for the comment! The arrow notation is unfortunate here; it really does look like causation is implied. But better to have said, "if you eat carrots, you can see in the dark better than if you did not eat carrots" then it is obvious that "not eating carrots" stops the implication having any discerning value.

 

[Steven G]

Your teacher tells you that if you study, then you will pass the exam.
How can the teacher be wrong? There is only one way for the teacher to be wrong and that is if you did study but you failed the exam.
If you didn't study, then you can't complain to your teacher that you failed the exam (you could have even passed the exam).
The only way to get a false implication is when T->F

 

[Dr.Peterson]

Your teacher tells you that if you study, then you will pass the exam.
How can the teacher be wrong? There is only one way for the teacher to be wrong and that is if you did study but you failed the exam.
If you didn't study, then you can't complain to your teacher that you failed the exam (you could have even passed the exam).
The only way to get a false implication is when T->F

Of course, even the best "real-life" examples we give tend to have something wrong about them. This one sounds causal; in addition, in real life, it would have to be "if you study effectively, you will probably pass the exam."

And the teacher could be wrong in many ways; whether you study does not determine whether the teacher is wrong, only whether you can prove them to be wrong! Maybe everyone else studied and failed, because the exam was a bad one. Or if you study and pass, maybe everyone else studied and failed, so T->T doesn't prove they're right; it's just compatible with truth, as are F->T and F->T, but not T->F.

But this is one of the clearer examples I've seen, especially since it fits a student's context, making it very relatable.