Sorting Through Sophistries
A Pair of Pseuds
By Gabriel Blanchard
To err is human; to forgive is divine; logically, then, it must be diabolical to subalternate.
Last week we discussed fallacies of ambiguity. These include equivocation, fallacious prosody, the motte-and-bailey, moving the goalposts—all sorts of things that sound like you’ve broken an obscure rule of croquet and have a maiden aunt yelling at you for it. They form a subcategory of what we called fallacies of lexis, in which fallacious-ness is due to a problem in the mechanics of what is said: the flaw may lie in the grammar, the syntax, even the stress.
We now pass to a second subcategory within fallacies of lexis. To explain it properly, we’ll need to talk a little bit first about the square of opposition.
The square of opposition describes the logical relationships between the four categorical statements. These are statements about kinds of things, and how they do, or do not, relate to each other. In our introduction to this series, we gave examples of syllogisms made up of categorical statements, such as the classic:
All cats have four legs.
My dog has four legs.
∴ Therefore, my dog is a cat.
For the purposes of making a syllogism, there are four kinds of categorical statements. They are traditionally named* as follows
- The A statement, or universal affirmative: “All X are Y”
- The I statement, or particular affirmative: “Some X are Y”
- The E statement, or universal negative: “No X are Y”
- The O statement, or particular negative: “Some X are Y”
The square of opposition is, more or less, a map of how these four statements relate to each other logically; universal statements go at the upper corners, affirmatives at the left-hand corners, the diagonals of the square represent simple contradictions, etc.; we don’t need to get into all that here. All we need to talk about is a type of inference the square of opposition allows us to make, known as subalternation.**
Alternations and Contradictions
That phrase may sound intimidating, but the way it works is fairly simple. It comes in two types: subalternation, and superalternation. The first word sounds a little less silly, so let’s begin there.
Subalternation means making deductions from universals about particulars. If an A statement is true, then we know—by definition—that an I statement about the same things is also true; if all cats are animals, then any specific cat must also be an animal (as long as no one sneakily equivocates about their terms). The same goes for the negatives: if the E statement is true, the O is necessarily true as well. If no dogs are cats, then this six-puppy litter over here, which consists in some of the dogs that exist, consists in six non-cats.
However, subalternation is a one-way street. You can’t (validly) go from an I statement like “some bacteria are beneficial” to the A statement “all bacteria are beneficial,” nor from an O statement like “some women do not speak Portuguese” to the E statement “no women speak Portuguese.”
By contrast, we can travel in either direction on a different pair of roads—the diagonals, the paths of contradiction, which we mentioned in passing above. As long as we’re working with a single pair of terms, the A statement and the O statement contradict each other, and the same goes for the E statement and the I. If some women, even just one, do not speak Portuguese, then the A-type claim “all women speak Portuguese” must be, or rather is, false; if some bacteria are beneficial, then the person who says “no bacteria are beneficial” has to be wrong.†
What has this got to do with fallacies, though?
The Fallacy of Composition
These two fallacies are basically invalid alternations—attempts to drive the wrong way down their one-way streets. They are known as the fallacy of composition and the fallacy of division, and have to do with properly relating a whole to its parts or vice versa.
Suppose you went to visit the great Inca citadel of Macchu Picchu. On observing that the towers and terraces of the Temple of the Sun were made of granite, you naturally conclude that the rest of the city is also granite: what seems to be grass is in fact finely carved and painted granite; the air, though ground to the finest texture possible, is a gaseous form of granite. After all you got a sample that really is from Macchu Picchu, as even your bitterest enemy will admit. Alas, you have committed the fallacy of composition: what is true about a part of something is not necessarily true about the whole. (The equivalent on the square of opposition would be affirming an I statement, and trying to deduce that the A statement must be true too—reversing the valid process of subalternation.)
The Fallacy of Division
Probably many readers would scoff at this obvious error. But most if not all of us, at one time or another, have committed its opposite. Consider: on the exact opposite side of the world from the Incas lie the jungles of Vietnam. Vietnam is one of the few remaining countries that is officially Communist.‡ On meeting a gentleman from that nation, perhaps we begin a conversation with him about politics, assuming that he is a Communist—only to find him denying every key doctrine of Communism. We are bewildered; but we are only bewildered because we had already committed the fallacy of division: what is true of a whole in general will not necessarily be true of every part.
The secret to this fallacy is a little more complex than the fallacy of composition, though. We cannot move from a particular statement to a universal one, so we might suppose that division must be about moving from universals to particulars in some invalid way. This is not actually what the fallacy of division is, though; division is moving from one I statement to another I statement, or from one O to another O. This is because the statement it starts with, while broad, is not truly a universal at all. “Vietnam is a Communist country” is (probably) a way of saying “Most Vietnamese are Communists”; and what does most mean? It may be consistent with “all,” but it doesn’t mean that of itself. In terms of categorical statements, most is only a synonym for some.
Fallacies of composition and division are thus both hasty generalizations. These are—well, I’m sure you know what I mean, we’ve all made them.
*Since the cathedral schools (which for some centuries were the only means by which logic was passed down at all) taught their pupils Latin first, these names come from two Latin words: affirmo, “I affirm,” for the affirmative statements, and nego, “I deny,” for the negatives.
**A related kind of inference, superalternation, also exists. However, its uses are slim to nil—any time it would be used, contradiction has almost certainly done its job already—so we can pass over it here.
†It’s worth taking a moment here to remember that which (if any) of these categorical statements are true is not the issue. That’s usually something you have to find out by investigation. What’s at stake is how these statements relate to each other logically, and what can and can’t be proven from them.
‡That is, in theory; like China and the former USSR, Vietnam loosened its grip beginning in the 1980s, and the ideological pretensions of the state have continued to weaken since.
Gabriel Blanchard has a degree in Classics from the University of Maryland, and is an uncle to seven nephews. He lives in Baltimore, MD, and serves as CLT’s editor at large.
If you enjoyed this piece, take a look at some of our installments from our previous Thursday series on the great ideas. We covered beauty, mathematics, opposition, prudence, and many more. Thank you for reading the Journal.
Published on 25th January, 2024. Page image of John Tenniel’s illustration of Alice meeting Tweedledum and Tweedledee in Lewis Carroll’s Through the Looking-Glass: And What Alice Found There (1871).