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(I think the title is appropriate) Given a statement S, "if there is a funnel cloud at a location, then the sirens are sounding." Can I use S to show that there is a funnel cloud? Can I use S to show that there is no funnel cloud?

If I suppose S is true, are either of these two statements true: a) If the sirens are not sounding, there must be no funnel cloud at that location. b) If there is no funnel cloud at that location, then the sirens are not sounding.

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(a) True, (b) False – zapkm Mar 1 '12 at 2:32
Relabel the proposition X, then define F = "there is a funnel cloud" and S = "sirens are sounding." In order to answer your questions, figure out which of these four are consistent with X and which aren't: $$F \wedge S \qquad F \wedge \neg S \qquad \neg F \wedge S \qquad \neg F \wedge \neg S$$ – anon Mar 1 '12 at 2:33
up vote 2 down vote accepted

Part (a) is certainly true; in fact, it is called the contrapositive. Given any statement $P \rightarrow Q$ that is true, $\neg Q \rightarrow \neg P$ is also true. Part (b) need not be true; it is the converse and looks like $\neg P \rightarrow \neg Q$.

Maybe this parallel example will help. We know the following is true: "If it is snowing, it is cold." If we say $P$ is "It is snowing," and $Q$ is "It is cold," this is of the form $P \rightarrow Q$.

Just by common sense we know that "If it is not cold, it is not snowing" is true, and this is in fact the contrapositive $\neg Q \rightarrow \neg P$. However, "If it is not snowing, it is not cold" need not be true (it can be cold but not snowing), and that is the converse ($\neg P \rightarrow \neg Q$).

Arguing by the converse is not valid, but arguing by the contrapositive is fine and is sometimes the most straightforward way to go about proving a hypothesis in the first place.

Consider the hypothesis "If $X$ is a square, $X$ is a rectangle." If we can show that if $X$ is not a rectangle, it cannot be a square, we have also proven the original hypothesis.

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Exactly what I wanted to know, thank you. – Austin Henley Mar 1 '12 at 2:55

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