# What do we call $P \rightarrow \lnot Q$?

Let us assume that we have a statement $$P \rightarrow Q$$. In this case, what would $$P \rightarrow \lnot Q$$ be called?

The reason why I want to know is that I want to show that $$P$$ is true by contradiction, proving that both $$\lnot P \rightarrow \lnot Q$$ and $$\lnot P \rightarrow Q$$ are true to conclude that $$\lnot P$$ is false. I initially framed this by stating that $$\lnot P \rightarrow \lnot Q$$ is a contradiction to $$\lnot P \rightarrow Q$$, but my supervisor said that this was incorrect, since both theorems are technically true.

• $P\to\lnot Q$ is the same thing as $\lnot(P\wedge Q)$. – Don Thousand Dec 9 '19 at 6:11
• In computer science it is Nand(P,Q). That is, "not (P and Q)". – DanielWainfleet Dec 9 '19 at 6:23
• There are lots of names for statements related to implications, e.g. inverse, converse, contrapositive, but this one does not seem to have a name. – user856 Dec 9 '19 at 6:27
• "proving that both ¬P→Q and ¬P→Q are true to conclude that P is false" um.... huh? – fleablood Dec 9 '19 at 6:27
• "Sorry, I meant "both ¬P→Q and ¬P→¬Q"" Did you also mean "P is true". If $\lnot P \implies Q$ and $\lnot P\implies Q$ that (which is not a contradiction) would mean $\lnot P$ is false. – fleablood Dec 9 '19 at 6:32

There seems to be no name for $$P \rightarrow \lnot Q$$.
The statement "$$\neg P \to \neg Q$$ is a contradiction to $$\neg P \to Q$$" is indeed wrong, since both these implications are true, they don't contradict each other.
The correct way to state this was to use modus ponens. You have already proven $$\vdash \neg P \to \neg Q$$ and $$\vdash \neg P \to Q$$. Since you assume $$\neg P$$, you then get $$\vdash Q$$ and $$\vdash \neg Q$$, and this is the contradiction you've been looking for.