Can You Say Something True About a Thing That Doesn’t Exist?
Can a Sentence Be True If What It’s About Isn’t There?
Imagine you and a friend are arguing about unicorns. You say, “All unicorns are white.” Your friend says, “No unicorns are white.” Who is right? There are no unicorns at all. So can either sentence be true? If nothing exists for the words to point to, logic seems to wobble.
This wobble is at the heart of an ancient diagram called the Square of Opposition. For more than 2,000 years it was a trusted tool. It showed how four basic sentence forms — every, no, some, and some not — connect to one another. But in the last century many logicians decided the square was fundamentally broken. And then historians discovered something surprising: the square had never been broken at all. The story of this tiny diagram opens a window into what happens when language meets emptiness.
To understand it, step into a world where all terms point to real things: dogs, cats, animals. The four forms, labeled with the letters A, E, I, O, are:
- A: Every S is P. (Example: “Every dog is an animal.”)
- E: No S is P. (“No dog is a cat.”)
- I: Some S is P. (“Some dog is brown.”)
- O: Some S is not P. (“Some dog is not brown.”)
The square draws lines between these forms to show fixed logical relationships. Two statements are contradictories if they cannot both be true and cannot both be false. A and O are contradictories; so are E and I. Two are contraries if they can’t both be true but can both be false — that’s A and E. Two are subcontraries if they can’t both be false but can both be true — that’s I and O. Finally, an I statement is a subaltern of A: if every dog is an animal, then some dog is an animal. The same holds from E down to O. On the traditional square, all six of these links held firm.
The square also included a rule called simple conversion. “No S is P” always means the same as “No P is S.” And “Some S is P” always means the same as “Some P is S.” This made the diagram feel tight and complete.
The Square That Worked — Until It Didn’t
The trouble starts when we replace ordinary terms with one that is empty — a word that names nothing real, like “unicorn” or “chimera.” On the traditional view, an I statement like “Some unicorn is white” would be false, because there are no unicorns at all. Its contradictory, E, “No unicorn is white,” must then be true. But from E, by subalternation, we should get O: “Some unicorn is not white.” That seems bizarre. If there are no unicorns, how can any unicorn lack whiteness? The inference forces a positive claim about things that don’t exist. The square appears to collapse.
Modern logicians from the late 19th century onward came to a stark conclusion: the traditional square was inconsistent. They re-symbolized the four forms using quantifiers — ∀ and ∃ — and dropped subalternation. In the modern revised square, A and O are still contradictories, and E and I are contradictories, but the other relationships vanish. “Every dog is an animal” no longer implies “Some dog is an animal,” because if the subject class is empty, “Every” can be true (vacuously true) while “Some” is false. The resulting diagram is so sparse that hardly anyone bothers to draw it.
Aristotle’s Hidden Escape Hatch
But there is a twist. When you open Aristotle’s own text, De Interpretatione (4th century BCE), you find that he did not word the O form as “Some S is not P.” He wrote it as “Not every S is P.” The difference is tiny on the page, but enormous in logic.
Suppose again that there are no unicorns. The A form, “Every unicorn is white,” is false, because affirmatives like A were understood to have existential import: they assumed the subject really existed. So “Not every unicorn is white” — Aristotle’s O — is true. And that’s exactly what the square demands. The O form under this wording does not claim that some unicorn isn’t white; it only denies that every unicorn is white. Since the universal affirmative is false, its denial is true. No spooky non-existent creatures need to be summoned.
Thus Aristotle’s square was coherent from the start. The contradiction only appeared when later translators, beginning with Boethius in the early 6th century, rewrote O as “Some man is not just” and the tradition gradually lost sight of the existential-import difference. But the logical engine, when run on the original fuel, never stalled.
A wider principle emerged: affirmatives carried existential import and negatives did not. An A or I statement was false if its subject was empty; an E or O statement could be true even when nothing fell under the subject. That single rule kept every relation in the square firing correctly.
Mediaeval Logicians Faced the Dragon — and Sleighed It
By the Middle Ages, the square was taught everywhere using Boethius’s wording, but the deep understanding of emptiness never vanished entirely. Thinkers like John Buridan (c. 1300–1358) and Paul of Venice (1369–1429) knew perfectly well that some terms refer to nothing. They discussed chimeras, the void, and extinct things like roses in winter. They saw that the square required the O form to be true when the subject was empty, and they accepted that result because they treated O like the old “Not every” form.
They also ran into trouble with trickier inference rules. One was contraposition — the idea that “Every S is P” turns into “Every non-P is non-S.” If you start with “Every human is a being” (true), contraposition would yield “Every non-being is a non-human.” But “Being” is not empty, while “non-being” is empty; an A statement with an empty subject is false because it has existential import, so the contrapositive becomes false from a true premise. Buridan and others pointed out the failure and explained that extra premises were needed to make the rule safe. Similarly, obversion — changing “No S is P” into “Every S is non-P” — only worked reliably when the subject wasn’t empty.
By the 14th century there was a common understanding: negative propositions don’t carry existential import, and the square, when interpreted properly, was fully consistent. Paul of Venice calmly gave examples like “A chimera does not exist” and “Some man who is a donkey is not a donkey” as true statements, even though the subjects were empty or impossible. Far from being blind to the problem, medieval logicians designed their theories to handle it.
Why the Square Crashed in the Modern Era
After the medieval period, serious logical work faded. Popular textbooks from the 17th to the 19th century taught the square but fudged the details. Many used “Some S is not P” without clarifying whether it carried existential import. They also included the faulty versions of contraposition and obversion. The authors usually didn’t notice the hidden contradiction, or they simply assumed, without saying so, that no terms are empty. In 1847, Augustus De Morgan was unusual in openly forbidding universal and empty terms — an artificial rule that kept the old square standing, but only by pretending the problematic cases didn’t exist.
When modern logic arrived in the late 19th century, it swept the whole puzzle aside. The new symbolism made negation and existence precise, and the traditional relations broke. The square was replaced by a thin set of contradictories, and that has remained the standard in most logic classrooms ever since. The historical square became a curiosity, often presented as a dead end from a pre-scientific past.
What This Means for You and the Socks You Can’t Find
So why should you care about a dusty diagram? Because the square’s story is really about a question you face all the time: when you talk about things that might not exist, what are you actually saying?
Say you open your drawer and it’s empty. You announce, “All my socks are in the drawer.” Is that true? Your mind might say, “Well, there are no socks outside the drawer, so yes.” But a strict logic that gives A existential import would call it false, because there are no socks at all. The same puzzle appears when you talk about extinct animals (“All dodos are birds”), ideal objects in science (“All ideal gases obey this law”), or promises that were never made. The rules you choose about existence decide whether those sentences are true or false.
The Square of Opposition isn’t just a relic. It’s a training ground for noticing what we assume. The next time you hear someone say “No dragon is friendly,” you’ll know there’s a whole hidden story behind whether that counts as true — and you’ll be thinking like a logician who doesn’t duck the hard cases.
Think about it
- If a sign says “No bicycles are allowed in the park,” and there are zero bicycles in the entire town, is the sign lying? Why might someone say yes, and why might someone say no?
- You’re designing a video game that includes the statement “Some potions cure poison.” The player’s inventory currently holds no potions. Should the game treat that sentence as false, true, or something else? How would your choice change the way players talk about it?
- Imagine a scientist writes, “Every black hole in our galaxy has been mapped,” but we don’t know whether any black holes actually exist. Does the word “every” make an invisible promise about existence? If yes, what should the scientist have written instead?
