# Are there multiple contrapositives?

Are there "multiple contrapositives"?

Normally a contrapositive from P implies Q changes to not Q implies not P.

Secondly, can a contrapositive be in the from of P in the antecedent and Q be the consequent?

I'm not going to post the question because I'm not looking for someone to do it for me, just looking for some info as to the form of contrapositives.

## 2 Answers

The contraposition law states that a conditional statement ($\phi \rightarrow \psi)$ is logically equivalent to the inverse ($\psi \rightarrow \phi$) of its converse ($\neg \phi \rightarrow \neg \psi$). Symbolically, the law states that the following holds for every interpretation:

$\vDash (\phi \rightarrow \psi) \leftrightarrow (\neg \psi \rightarrow \neg \phi)$

Now if what you mean by "multiple contrapositives" is the fact that contrapositives can be successivelly iterated, your answer is yes, so that the following holds:

$\vDash (\neg \psi \rightarrow \neg \phi) \leftrightarrow (\neg (\neg \phi) \rightarrow \neg (\neg \psi))$

Recall that the Greek letters $\phi$ and $\psi$ in the above statements are actually not part of the language, but just metavariables, as they stand for any well-formed formula of the language L of our statments (This also answers your second question with a yes). Then the successive iteration of contrapositive comes easily.

(Note however, that in classic logic they are superfluous: $\vDash \neg \neg \phi \leftrightarrow \phi$ so that the successive negation can be eliminated)

"Multiple contrapositives" is a vague thing to ask for; others have already explained the case where it is interpreted as repeatedly applying contrapositives in a context without double negation elimination. But I think, based on the question you posed on whether "a contrapositive can be in the from of P in the antecedent and Q be the consequent", that you might be inquiring whether you can derive $\neg P\to \neg Q$ from $P\to Q$, in addition to the contrapositive $\neg Q\to\neg P$. This is actually not true, because when $P$ fails and $Q$ holds then $P\to Q$ holds but $\neg P\to\neg Q$ fails.

An intuitive explanation is this: when you assert an if-then condition, all you care about is what happens when the antecedent holds. When it doesn't, there is nothing you can conclude. Doing so is called the fallacy of denying the antecedent, which is unfortunately often encountered in informal debates.

Another way to see it is to remember that $P\to Q$ is equivalent to $(\neg P)\lor Q$, and you simply need to introduce a double negation to $Q$ (one can do this even in intuitionistic logic where double negation elimination fails), apply commutativity of disjunction, and change it back to conditional form to get the contrapositive. One cannot do something like this to get $\neg P\to \neg Q$.