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

Question

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.

In computer science it is Nand(P,Q). That is, "not (P and Q)". — DanielWainfleet, Dec 09 '19 at 06:23
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. — , Dec 09 '19 at 06:27
"proving that both ¬P→Q and ¬P→Q are true to conclude that P is false" um.... huh? — fleablood, Dec 09 '19 at 06:27
Sorry, I meant "both $\lnot P \rightarrow Q$ and $\lnot P \rightarrow \lnot Q$". — J.M.W Turner, Dec 09 '19 at 06:29
" by stating that ¬P→¬Q is a contradiction to ¬P→Q" Well that not a contradition. They'll both be true if $P$ is true. So that isn't a contradiction. (But if both statements are true that would be a way of showing P is true). — fleablood, Dec 09 '19 at 06:30
"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 09 '19 at 06:32

score 1 · Answer 1 · answered Dec 09 '19 at 06:53

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.

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

1 Answers1