Can You Think a Round Square? The Philosopher Who Said Yes

The Impossible Shape on Your Desk

Try to picture a square that is also round. Really try. You can say the words “round square,” but can you actually hold the idea in your head? Most people feel a mental squirm — as if something in their thoughts just broke. Philosophers have wrestled with this squirm for over a century. If you can think about a round square, what exactly are you thinking about? It can’t exist, so is your thought about nothing?

Around 1900, the Austrian philosopher Alexius Meinong gave a bold answer. Every thought, he said, is directed at an object — even thoughts about things that don’t, or can’t, exist. When you imagine a unicorn, the fountain of youth, or a round square, your mind points to something. That something is an object, just not a real, physical one. Meinong called these nonexistent objects.

The trouble started right there. If a round square is an object, then it has properties. It is round. It is square. But that means it is round and not round at the same time — and that’s a flat-out contradiction. Logic seems to forbid it. The British philosopher Bertrand Russell pounced on this problem in 1905, calling it “intolerable.” A theory that lets round squares exist as objects appeared to be a theory that lets nonsense win.

Ernst Mally (1879–1944) was a student of Meinong’s, and he thought Meinong was mostly right — but the logic needed fixing. Mally’s repair job became one of the most elegant moves in 20th-century metaphysics.

Mally’s Two Ways of Having Properties

Mally looked at the round square and asked a simple question: what does “is” mean? If you say “the triangle is triangular,” that seems obvious. But what if you are talking about the concept of a triangle — not a specific wooden or drawn triangle, but the very idea of one? Is that concept itself triangular? Of course not. A concept doesn’t have angles or sides. You can’t hold it in your hand.

Mally drew a sharp line between two ways an object can connect to a property. The first way he called determination. A conceptual object is determined by a certain set of properties — those are the properties that define it. The concept of the triangle is determined by “triangular.” The concept of the round square is determined by “round” and “square.” Determination is like a label on the object’s name tag: it tells you which idea it is.

The second way is instantiation (Mally spoke of satisfying a property). This is what happens when an object actually has a property in reality. The concrete wooden triangle on the desk instantiates triangular. The conceptual triangle does not. Instead, it instantiates properties like “abstract” and “non-triangular.” It is not a shape at all.

Now the round square stops being a logical disaster. The conceptual object the round square is determined by roundness and squareness together. That’s fine — determination doesn’t force the object to instantiate those properties. It instantiates non-round and non-square, just as the conceptual triangle instantiates non-triangular. No contradiction. The object is complete and consistent when it comes to instantiation, and no law of logic is broken. Russell’s “intolerable” mess dissolves.

Mally’s distinction gave philosophers a safe way to talk about impossible, non-existent, and abstract objects. You can say “the fountain of youth is a fountain” without committing yourself to the real existence of magical water. The phrase names a conceptual object whose determination includes being a fountain, but which instantiates none of the ordinary fountain properties. The thought has content, even though the thing doesn’t exist. Your mind isn’t staring into blank nothing — it is gripping a structured, logical idea.

The Rule That Broke: Mally’s Ought-Logic

Mally wasn’t content with fixing impossible objects. He wanted to build a logic for ought — a formal system that could capture how we reason about duties and obligations. In 1926 he published The Basic Laws of Ought, introducing a special operator he wrote as !. A statement like !A meant “it ought to be the case that A.” The goal was to treat “ought” as precisely as mathematicians treat addition.

The system had axioms — basic rules Mally thought any reasonable person would accept. One rule said that if A is obligatory and B follows logically from A, then B is obligatory too. Another connected obligation to what is “willed” or demanded. Mally was working in the tradition of his teacher Meinong, who believed that values could be presented to the mind emotionally, almost like perceptions.

But something went badly wrong. The axioms entangled “ought” and “is” until they collapsed into each other. The system proved the astonishing claim: !A if and only if A — something is obligatory exactly when it is true in the actual world. If you are reading this sentence, then you ought to be reading this sentence. If someone tells a lie, the system says they ought to have lied. The mathematician Karl Menger spotted the collapse in 1939. The very first deontic logic had, in a technical sense, self-destructed.

Mally’s failure here is instructive, not humiliating. He was the first person to even try to build a formal logic of obligation. Before him, nobody had laid axioms for “ought” the way logicians lay axioms for “and” or “or.” His system crashed, but it started a field. Later logicians rebuilt deontic logic from the ground up, avoiding the collapse. Today, deontic logic is used in computer science, legal reasoning, and artificial intelligence — and it all traces back to Mally’s brave, broken first attempt.

A Fuzzy World of Tendencies

As Mally grew older, his thinking took a surprising turn. He began to doubt that the world is made of neatly finished individual things — trees, chairs, clouds — with sharp boundaries. Instead, he argued, reality is built out of tendencies or strivings. Objects are never fully determined. They are always slightly fuzzy, always in the process of becoming.

He pointed to a lake. You think of its surface as a plane, like a perfect sheet of geometry. But the actual water is never precisely flat. It ripples, curves, and shifts. The “plane” is only an approximation, a tendency the water never completely fulfills. Natural laws, Mally said, are not iron rules that lock the world into shape. They are more like descriptions of the directions in which reality tends to flow.

This dynamic ontology was a dramatic break from his mentor Meinong, who had treated objects as static. Mally now called Meinong’s view a “static conception” and criticized it for mistaking mental snapshots for the real, moving world. For Mally, everything is striving to become more fully itself — but the striving never finishes. Even you, in his view, are not a finished thing but a bundle of tendencies. That idea would echo in his ethics too.

The Philosopher and the Nazi Party

Now comes the hardest part of Mally’s story. During the 1930s, he became an active supporter of National Socialism — the Nazi movement. He joined the Nazi party in 1938, two months after Germany annexed Austria. He wrote pamphlets and a textbook that promoted racist ideas, anti-semitism, and the glorification of the German “people” as a kind of super-organism destined for dominance. Some of his philosophical language — about strivings, wholes, and essences — bent to fit that dark cause.

Philosophers today take this seriously. They examine Mally’s work in three categories. Grade one: purely philosophical writings from earlier years, such as his logic of objects and his deontic logic, which contain no Nazi content. Grade two: works like his 1935 book Erlebnis und Wirklichkeit (“Experience and Reality”), where some ideas might resonate with the ideology but aren’t outright propaganda. Grade three: openly Nazi texts where the philosophy becomes a mouthpiece for hate.

This sorting matters because Mally’s logical insights are genuine. They don’t automatically become false because he later embraced monstrous politics. At the same time, his political choices can’t be swept under the rug. The same mind that saw how to separate determination from instantiation also chose to promote lies about human inequality. Grappling with that tension is part of doing philosophy honestly. Ideas don’t float free from the people who make them — but they also aren’t doomed to be forever chained to their maker’s worst mistakes.

Why a Logic of Ought and Impossible Things Still Matters

You probably think about impossible objects more often than you realize. Every time you imagine a perfect friend, a flawless plan, or a completely fair world, you are playing with conceptual objects determined by properties that nothing in real life fully instantiates. Mally’s distinction between determination and instantiation gave philosophers — and in a way, all of us — a clear model for how such thinking can be meaningful without being magical.

Deontic logic, the field Mally launched with his flawed 1926 system, is now a living branch of philosophy and computer science. When engineers design a self-driving car, they need it to reason about obligations: it ought to stop for pedestrians, it ought to obey speed limits, yet sometimes obligations clash. Formal deontic systems help tackle these puzzles. They didn’t spring from nowhere — they began with a disabled, pain-ridden Austrian philosopher scribbling an exclamation mark as a logical symbol.

And Mally’s life? It remains a standing warning. Smart people can believe horrible things. A powerful logical mind can still build a cage of prejudice. That doesn’t mean we throw away every tool that mind forged — but it does mean we handle those tools with clear eyes and a sharp awareness of where they came from. Next time you struggle to draw that round square, you’ll be stepping into a problem Mally helped solve, even as his own story took a dark and cautionary turn.

Think about it

1. Can you think about something that is completely impossible, like a square circle? What does that tell you about how your mind works?
2. If a computer program can reason about what you “ought” to do, who gets to decide what “ought” means?
3. Should we study the ideas of a philosopher who held terrible political views, or does their whole body of work become off-limits?