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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

If you want to construct a name using the terms that you've mentioned in the comments:

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 $A \rightarrow B$".

My guess would be that there isn't much of a standard name for this since it doesn't appear to be common fallacious deduction, nor is it an equivalent statement, and the usefulness of its demonstration in an argument seems to be the same as that of "contradictions" (in the sense of the term).

I think among those above names, "contrapositive of the contradiction" is most suggestive for how we should be thinking about this. Knowing that contraposition gives an equivalent statement, and noting the potential role of deducing this statement when we have the other, it seems like we should just view this one as "contradiction" as well.

We might get lucky with a response from someone who studied rhetoric or logic more from the perspective of some area of philosophy, and my suggestion would be to look more toward those areas if you're not satisfied with the above.

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