Reference request: How to understand $p\to q$?

Question

I have studied calculus, linear algebra, some naive set theory, some analysis. These were not satisfactory at all. I want to find a way to understand the mathematical system in an axiomatic way.

The most difficult thing I have is "$p\to q$". There was no book that gave satisfactory explanation about this.

For example

$1+1 = 2 \to \sqrt 7$is irrational.

This is a true statement, but does not make sense to me. How does $1+1 = 2$ imply that $\sqrt7$ is irrational?!

I can't accept the definition of $p \to q$'s truth table, especially "vacuous truth".

• Would you recommend me books regarding foundations of mathematics?
• Would you recommend me a book that explains $p \to q$ well?

Answer 1

A statement of the form $p \rightarrow q$ asserts "if $p$, then $q$".

To prove it, you are allowed to assume the truth of $p$, and then, using a sequence of valid logical statements, you try to prove $q$.

But there's no requirement to use $p$. It's just that you're allowed to use $p$, if it helps. Thus, for your example "$1 + 1 = 2$" implies "$\sqrt{7}$ is irrational", you have a statement of the form $p \rightarrow q$, where $p$ (the hypothesis) is the statement "$1+1=2$", and $q$ (the conclusion) is the statement "$\sqrt{7}$ is irrational".

If you don't need to use the hypothesis in the proof, that's fine, as long as you can prove the conclusion somehow.

But I agree, in English, it seems wrong, since the conclusion is not seen as a direct consequence of the hypothesis.

In math, it's allowed, since we only care about proving the conclusion, so the hypothesis can be used if it helps, but otherwise can be ignored.

Answer 2

Your question is really interesting and a sort of philosophical. Sometimes, I also feel confused about the truth table. It just needs more time for you to understand. In my opinion, the truth table is really helpful when you want to write a good proof or break a proof down. You can refer to Prof. Martin J. Osborne's "Mathematical methods for economic theory" 1.1 Logic on his website.

P \implies Q only says: if P is true, then Q is true, that's all! We say P is the sufficient condition of Q. Without the observation of "P is true", you cannot criticize this statement. But if you observe P happens but Q does not happen, undoubtedly you will deny the statement that "P implies Q" is true.

In your example, you think that "square root 7 is irrational" is always true without any conditions. In other words, we can actually say that everything in the world (including" 1+1=2","1+1=4","sun rises in the east", "snow is white") can be a sufficient condition for a "forever-true" statement, because there doesn't exist the case that "Q is false". Does it break the rules of truth table down? No, it doesn't. Therefore, you cannot criticize the logic system by using this example. It's weird just because you simply interpret P implies Q as a causal relationship.