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.
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.
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
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 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.”
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
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.