# Is there a term for the contradiction of the converse of a logical relationship?

new here (apologies for the formatting). I want to express the relationship between $$A \longrightarrow B$$ and $$B \longrightarrow \neg A.$$

Clearly this is neither converse, inverse, nor contrapositive; does it have a name at all?

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If the first one implies the second one, you get a contradiction ($A \rightarrow \neg A$). Same the other way around since the second statement is equivalent to $A \rightarrow \neg B$ which means that $A$ implies both $B$ and $\neg B$. In all, I don't know if there is a specific name for this. – David Kohler Apr 4 '11 at 15:39
I'm not sure you have the notation right in your title - the converse of the inverse (or inverse of the converse) is the contrapositive $\neg B \longrightarrow \neg A$. – Michael Chen Apr 4 '11 at 15:40
@Cheezmeister: What is the "contradiction" of a proposition? If you mean the negation, then note that the negation of the converse of $A\rightarrow B$ is $\neg(B\rightarrow A) = B\land \neg A$, which is not $B\rightarrow \neg A$. – Arturo Magidin Apr 4 '11 at 15:44
B->-A gives the same truth values as -(A and B) – yoyo Apr 4 '11 at 15:44
@David These are independent statements that I'm simply comparing. @Michael Right you are, sir. I've fixed the title. – Cheezmeister Apr 4 '11 at 15:45

Then there's endless ways you can string those steps together, among which are "the converse of the contradiction of the inverse of $A \rightarrow B$", or the shorter "the contradiction of the converse of $A \rightarrow B$", or the equally short "the contrapositive of the contradiction of $A \rightarrow B$".