| if p is… | and q is… | then p → q is. |
|---|---|---|
| true | true | true |
| true | false | false |
| false | true | true |
| false | false | true |
What trips most people up about it is the last two lines: how can it be true if the premise is false? An example should illustrate why it's defined this way.
Let's say that at the beginning of the semester, the instructor goes over the syllabus and grading, then says "Oh, and if you get 100% on the final, you automatically pass the course." Here, p is "you ace the final" and q is "you pass the course". At the end of the semester, one of four things will happen:
1) You ace the final, and pass the course.
2) You ace the final, but fail the course.
3) You don't ace the final, but still pass the course. This can happen if, say, you got 90% on the midterm and the final.
4) You don't ace the final, and don't pass the course. This can happen if, say, you get 20% on both the miderm and the final.
In the third case, I think we can agree that it would be unreasonable to storm into the instructor's office and say, "Hey! I didn't ace the final, but I passed anyway! You lied at the beginning of the semester!". It would be even more unreasonable, in the fourth case, to storm in and say "Hey! I failed the tests, and you failed me for the course! You lied at the beginning of the semester!".
In fact, the only case in which you'd have a legitimate grievance is the second case: "Hey! I aced the final, but you failed me anyway! You lied at the beginning of the semester!". (Note that I'm using hyperbole. I don't actually advocate yelling at instructors. Gregory House can get away with it because he's portrayed as an indispensable genius. You are most likely not an indespensable genius.)
This also has implications (see what I did there? I slay me) for how we evaluate claims. Again, let's look at a concrete example: let's say that I tell you that I can make it rain by washing my car. That is, I'm making a statement of the form p → q, where p is "I wash my car" and q is "it rains".
You are, of course, dubious. So I rattle off a list of times when I washed my car, and it rained within the following 24 hours, backed up by photos and NOAA archives. What should you do to see whether my claim is true or not?
You might be tempted to look at times when I didn't wash my car, but it rained anyway. In other words, where p ("I wash my car") is false, but q ("it rains") is true. But as we've seen above, "false → true" is itself true, and even if you find a hundred rainstorms when I didn't wash my car, it doesn't mean I'm wrong. Nor would it help to find cases when I didn't wash my car, and it also didn't rain. It does, after all, rain for reasons other than my clean-car fetish.
If you look at the table above, you'll see that there's only one case where implication is false. In fact, "p → q" can be rewritten as "it is not the case that p is true and q is false". This is a cumbersome way to put it, but it provides us with the way to proceed.
What you need to do is come up with a list of cases when I did wash my car, but it didn't rain. If you can come up with such a list, then you can definitely say that I'm wrong.
I think the reason this is confusing is that in colloquial speech, when people say "if" (p → q), what they really mean is "if and only if" (p↔q). When parents say "if you don't eat your carrots, you can't have dessert", they also mean "if you do eat your carrots, you can have dessert". If your friend says "If I'm free on Thursday, I'll meet you for dinner", we tacitly understand that as meaning, "If I'm not free, I won't meet you". But if we want to be careful in separating fact from fiction, we need to understand what claims are being made, and how to test them.
