how is $p\to{\sim} p$ not a contradiction??

Question

i understand that a contradiction is a proposition that Is always false, and in this case if p is false the implication is true, but in English this sentence sounds so contradictory: "if it's blue, then it's not blue" how does this even make sense?

Answer 1

The important thing is to realise that $q \to r$ is true already if $q$ is false. So also $p \to \lnot p$ is true if $p$ is false. This is contrary to the usage people expect in English or natural language.

Answer 2

A naive person learns to recite the alphabet and thinks that means he's a great genius.

Another naive person learns to recite the alphabet and realizes that's a bare beginning of erudition and he must do a great deal more. He continues his schooling and goes on to become the author of the general theory of relativity.

Of the first naive person, one says: "If he's genius only because of that, then he's not a genius."

Thus "If $p$ then not $p.$"

This would be a contradiction only if $p$ were actually true.

Answer 3

"if it's blue, then it's not blue" how does this even make sense?

Let me rephrase:

"IF it's blue, then it's not blue"

OK, so it cannot be blue, for IF it were blue, THEN it's be both blue and not blue, and we'd have a problem.

OK, so it is not blue ... and note: if it is not blue ... then there is no problem at all!

So: if you ever have $p \to \neg p$, then we can conclude $\neg p$ ... and there is no contradiction.

Indeed: $p \to \neg p \Leftrightarrow \neg p$

But yes, many beginning students of logic get that one wrong. They believe $p \to \neg p \Leftrightarrow \bot$. No, we have $p \land \neg p \Leftrightarrow \bot$, but that's different. So, you're far from the only one! It's a classic, really.