Contrapositives and Converses

Suppose the conditional ‘If P, then Q’ is one of the premises of a mixed hypothetical syllogism.  If P was the other premise then you may validly conclude Q (by the rule of affirming the antecedent AKA modus ponens).  In other words, we may think of the conditional statement, ‘If P, then Q’ as issuing an inference ticket from P to Q.

By the same token, ‘If P then Q’ also allows you to infer not-P from not-Q by modus tollens (denying the consequent).  That is, it also issues an inference ticket from not-Q to not-P.

Now consider the conditional ‘If not-Q, then not-P.’  What inferences does it license?  By modus ponens (affirming the antecedent), it allows you to infer not-P from not-Q, and by modus tollens (denying the consequent) it allows you to infer Q from P.  The thing to notice about all this is that the conditional ‘If not-Q, then not-P’ issues the very same inference tickets as ‘If P, then Q.’  That is, logically speaking, those two conditionals are interchangeable; one might say that they mean the same thing.

This is an instance of the law of contraposition: take any conditional.  You may form an equivalent (logically interchangeable) conditional to it simply by interchanging its antecedent and consequent – or “flipping it around” – and then denying each side (deleting double negations as appropriate).

So here are the four schematic instances of the law of contraposition:

The contrapositive of…	(1) If P, then Q.	..is..	(1)* If not-Q, then not-P.
“  ”	(2) If not-P, then Q.	“	(2)* If not-Q, then P.
“  ”	(3) If P, then not-Q.	“	(3)* If Q, then not-P
And ‘’  “	(4) If not-P, then not-Q.	“	(4)* If Q, then P.

A word of warning: you must not confuse the contrapositive of a conditional with that conditional’s converse.  The converse of a conditional is formed simply by keeping the antecedent and consequent of a conditional in the same place and denying them both.  For example, the converse of ‘If P, then not-Q’ is ‘If not-P, then Q.’  Whereas a conditional is logically equivalent to its contrapositive, it is clearly not equivalent to its converse.  A conditional and its converse issue entirely different inference tickets.

Exercises:

I. Look again at the exercises at the end of the “Conditionals” webpage. For each of those, identify both of that conditional’s contrapositive and its converse.  To complete this exercise, you will first need to put the conditionals in standard form (which was what the original exercise asked you to do).

II. In a column in the Las Vegas Sun (April 11, 2007), J. Patrick Coolican tries to play “gotcha” with Democratic officials (accusing them, among other things, of launching ad hominen attacks against Gov. Jim Gibbons – he was right about that, BTW). However, consider the following:

Then there's Titus: "If he was in Elko," she said of the Journal reporter Wilke, "he didn't introduce himself to me."

Coolican then goes on to try to make fun of Titus:

Does that mean if he wasn't in Elko, he did introduce himself to her?

Answer Coolican’s (rhetorical) question. Does it accurately paraphrase what Titus said? Is his paraphrase the contrapositive or the converse of her original conditional?

III. Consider the following passage (especially the highlighted sections) from an LV Sun article entitled “Official: Water pipeline to be built if ‘absolutely necessary’”:

The Southern Nevada Water Authority is making a major effort to get rights to water in rural valleys that could be piped to Las Vegas, but a spokeswoman said Wednesday the pipeline project will be built only if "absolutely necessary."

…

We are not going to build the instate project unless it is absolutely necessary, unless there is absolutely nothing else we are going to do," Mulroy told the Assembly Natural Resources, Agriculture and Mining Committee. (LV Sun 12 Feb 2009)

1. First, translate the two highlighted conditionals into standard form:

(i) The pipeline will be built only if “absolutely necessary.

(ii) We are not going to build the instate project [the pipeline] unless it is absolutely necessary.

2. Are these two conditionals equivalent to one another? Why or why not?

3. Finally, consider the Sun’s title of the article (Official: Water pipeline to be built if ‘absolutely necessary’). Does it accurately convey what the official [Pat Mulroy] said? Why or why not?