The Contrapositive

This post is part of a four-part series on logical reversal. The truth may lie on the other side or in the other direction, but there is more than one way to reverse a sentence: obverting, converting, inverting, and contraposing are four ways.

VileBodies

I know very few young people, but it seems to me that they are all possessed with an almost fatal hunger for permanence. I think all these divorces show that. People aren’t content just to muddle along nowadays … And this word “bogus” they all use … They won’t make the best of a bad job nowadays. My private schoolmaster used to say, “If a thing’s worth doing at all, it’s worth doing well.” My Church has taught that in different words for several centuries. But these young people have got hold of another end of the stick, and for all we know it may be the right one. They say, “If a thing’s not worth doing well, it’s not worth doing at all.” It makes everything very difficult for them.

The contrapositive

In traditional categorical logic, an A-type (universal affirmative) sentence, “All A are B,” immediately implies its contrapositive, “All not-B are not-A.”

All roses are flowers. So all non-flowers are non-roses.

The contrapositive is the obverse of the converse of the obverse: (1) “All A are B” ⇒ (2) “No A are non-B” ⇒ (3) “No non-B are A” ⇒ (4) “All non-B are non-A.”

E.g., all roses are flowers, so no roses are non-flowers. And if no roses are non-flowers, then no non-flowers are roses. And if no non-flowers are roses, then all non-flowers are non-roses.

But the contrapositive is also the inverse of the converse, or the converse of the inverse, which means that the inverse and the converse of any A-type sentence are logically equivalent. I.e., it is not true that all flowers are roses, but the converse is true: all roses are flowers. And thus so is the inverse: all non-flowers are non-roses.

Contraposed conditionals

In contemporary logic, contraposition is more often applied to the conditional P → Q, yielding ¬Q → ¬P. This is also called transposition. Or in Evelyn Waugh’s words, “getting hold of another end of the stick.”

For example, when you’re runnning, your heart rate is elevated. So if your heart rate isn’t elevated, you’re not running.

It’s similar to a modus tollens argument:
P → Q
¬Q
∴ ¬P
E.g., if it’s a rose, it’s a flower. But it’s not a flower. Therefore it’s not a rose.

Two further examples

The twist involved in “getting hold of another end of the stick” explains the difference between a romantic and his realist critic.

The romantic endeavors to escape from the world of the actual into that of fancy, or more specifically into the realm of esthetic and speculative imagination. The critic, however, interprets this attitude as an unwillingness to submit to outward fact and accuses the romantic of a desire to withdraw into a world of his own creation.

The realist says to the romantic, “If you insist on using imagination, you must not be willing to limit yourself to what is actual.”

The romantic replies, “If you insist on limiting reality to what is actual, you must not be using your imagination.”

Raphael Demos

The indictment rests on a confusion between the actual and the real. The critic begins with a fallacious identification of the real with the actual and then goes on to describe any breaking loose from the latter into the world of imagination as a detachment from the objective and a retreat into the subjective. But it is only a vulgar preference or extreme naïveté that could lead one to limit reality to the actual; over and above the actual, there is the field of subsistence, of ideal entities, of forms, of possibilities; and the romantic imagination is not vain dreaming but an extension of the area of knowledge itself beyond perception into the realm of these ideal essences.

See Raphael Demos’ article, “Romanticism vs. the worship of fact” (1922).

As a final example, consider the difference between two kinds of environmentalism – which prompted a website to name itself contraposition.org

P → Q
If we’re going to maintain the current economy, then we’ll need alternative fuels that meet current energy needs.¬Q → ¬P
If we can’t get alternative fuels to meet current energy needs, then we won’t be able to maintain the current economy.

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