The Converse

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.

downtherabbithole

Down, down, down. There was nothing else to do, so Alice soon began talking again. ‘Dinah’ll miss me very much to-night, I should think!’ (Dinah was the cat.) ‘I hope they’ll remember her saucer of milk at tea-time. Dinah my dear! I wish you were down here with me! There are no mice in the air, I’m afraid, but you might catch a bat, and that’s very like a mouse, you know. But do cats eat bats, I wonder?’ And here Alice began to get rather sleepy, and went on saying to herself, in a dreamy sort of way, ‘Do cats eat bats? Do cats eat bats?’ and sometimes, ‘Do bats eat cats?’ for, you see, as she couldn’t answer either question, it didn’t much matter which way she put it.

The Converse

Conversion is one traditional way to flip a sentence: it means exchanging the positions of subject and predicate. The converse of “All men are mortal” is “All mortals are men.” The converse of “Cats eat bats” is “Bats eat cats.”

The converse is not always either equivalent or opposite to a statement, but it does seem to arise quite naturally in the delirium of thought – as Alice found when tumbling down the rabbit hole “saying to herself, in a dreamy sort of way, ‘Do cats eat bats? Do cats eat bats?’ and sometimes, ‘Do bats eat cats?’” – as if every predicate will change places with the subject given half a chance.

Not merely a delirious reversal, it also figures in rigorous treatments of relationships.

For example, in Sigmund Freud’s explanation of dream displacement, he explains how the true meanings of a dream, the “dream-thoughts” which his method seeks to unearth, are doubly hidden from the apparent dream content that represents them. Not only do the most obvious elements of the actual dream point elsewhere than to this true basis, but the converse is also true: “That which is obviously the essential content of the dream-thoughts need not be represented at all in the dream” (The Interpretation of Dreams, VI B). Thus the central idea of the dream is not found in the dream thoughts, and conversely the central idea of the dream thoughts is not found in the dream.

(One might say the same of political representation.)

In what cases can both a statement and its converse be true?

In universal affirmative sentences (A-type propositions in the form “all A are B”), the converse is not generally equivalent. If the converse is also true, it means two sets (A, B) are coextensive: one is a definition of the other (A≡B) or its necessary and sufficient condition (A iff B). For example, when Émile Durkheim says that a social fact is “a manner of acting, thinking, and feeling external to the individual, which is invested with a coercive power by virtue of which it exercises control over him,” he is evidently crafting a definition. He means every social fact has these features, and conversely every such manner of acting, thinking, and feeling is a social fact.

But where an A-type sentence is not a definition, the converse is not immediately inferrable. All alligators bite. But not all bites are from alligators.

Among other categorical sentence types, there are two types where the converse is always equivalent, and one type where it isn’t.

  • In universal negative (E-type) sentences, the converse is an immediate inference. If no men are mothers, it follows immediately that no mothers are men.
  • The same is true of particular affirmative (I-type) sentences. If some mothers are teenagers, it can be immediately inferred that some teenagers are mothers.
  • In particular negative sentences (O-type), however, the converse is not equivalent. Some women aren’t mothers, but it’s not the same thing to say that some mothers aren’t women. If the converse is true it means that each set has members outside the other, but this need not be the case in a given sentence of this type.

When it comes to cause-and-effect sentences, a statement whose converse is also true makes either an infinite regress, as in the chicken ⇄ egg paradox, or a vicious cycle, as in the no experience ⇄ no job Catch-22.

What if neither a sentence nor its converse is true?

For universal affirmative (A-type) statements, this amounts to a refusal of the relation on both sides. When Gilles Deleuze, in his book on Spinoza, explains that Spinoza challenges the Cartesian superiority of mind over body, he feels compelled to point out that he is not therefore arguing the converse: “the superiority of the body over the mind,” he says, “would be no more intelligible” (Spinoza: practical philosophy, 18).

When getting sticky tape or a wet hair off your hand, you often have to get it off the other hand too before you’re done. In a similar way, you can’t deny a comparison that doesn’t make sense without taking another moment to deny it the other way around. When this deliriously necessary moment of thought ossifies, it takes the form of a false dichotomy that either a statement or its converse must be true.

“Guns don’t kill people; people do” for example, is a clever package of arguments bundled up in a simple statement. Apart from a straw version of the gun-control argument, painting gun control supporters as soft-on-crime liberals who make sociological excuses to avoid attributing individual responsibility for crime, it also hinges on the false dichotomy that in a given case of a person or people being killed by a gun, either the gun is more fundamentally responsible for the murder than the person who pulled the trigger, or the converse.

In this case, the correct response should be, as with Spinoza with respect to the Cartesian superiority of mind over body, to refuse it twice. The gun is not more responsible than the person, since responsibility doesn’t mean the same thing in each case. And the converse would be no more intelligible. The people who use guns to kill people and the guns that people use to kill people are separate problems that could be dealt with in parallel.

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