When Math Tried to Prove Everything Is Right

A Philosopher Tries to Turn “Ought” Into Math

On a spring day in 1926, in the Austrian city of Graz, a philosopher named Ernst Mally (1879–1944) sat down to solve a puzzle that had nagged at him for years. He wanted a crystal-clear meaning for a word everyone uses: ought. People say “you ought to finish your homework,” or “we ought to be kind,” but what exactly does that word do? Mally thought the answer lay in logic—the study of good reasoning using symbols and strict rules, like mathematics. If we could write down the exact laws of “ought,” he believed, we could build an “exact system of pure ethics.”

He was the first person ever to propose a formal deontic logic, a logic of obligation and what is required. The title of his book, The Basic Laws of Ought, announced his ambition: to spell out the deepest principles behind every duty, every rule, every “should.”

Mally’s Toolbox: Five Simple Rules

Mally used a special symbol to mean “it ought to be that.” He wrote it as an exclamation mark: !A meant “A ought to be the case.” He also introduced a symbol for requires: A f B meant “A requires B”—that is, if A is true, then B ought to be true. He linked these two notions with a simple definition: A f B just means A → !B (if A, then B ought to happen). To get started, he wrote down five principles, or axioms, that seemed obviously reasonable:

1. Chaining requirements: If A requires B, and B leads to C, then A also requires C.
2. Combining requirements: If A requires B and A also requires C, then A requires both B and C together.
3. From requiring to ought: A requires B exactly when it is obligatory that “if A then B.”
4. Something is unconditionally obligatory: There is an U—the unconditionally obligatory—that always ought to be, no matter what. (He thought truth itself, V, worked that way.)
5. No built-in contradiction: The unconditionally obligatory does not require its own opposite.

From these five building blocks, Mally began to prove theorems—new statements that followed logically from the axioms. He thought he was uncovering the deep structure of all duties and correct willing.

When Logic Goes Wild: The Surprising Theorems

But as Mally cranked through his system, he started to get results that surprised him. One theorem said that if A requires B, then A requires everything that is true. Another claimed that if anything is required at all, then every fact is required. Even more bizarre, his rules implied that “the facts ought to be the case”—that whatever actually happens is automatically what should happen.

At first Mally called these outcomes “surprising” or even “paradoxical,” but he didn’t see just how badly they damaged his whole enterprise. Over and over, his axioms forced together two ideas that ordinary thought keeps strictly apart: is (what happens) and ought (what should happen). The hidden collapse became undeniable when the mathematician Karl Menger studied the system in 1939.

Menger proved that in Mally’s logic, the statement !A ↔ A was a theorem. In plain words: something is true if and only if it is obligatory. If you are reading this sentence, then you ought to be reading it; if you ought to do something, then you are already doing it. Menger pointed out that Mally himself had noted that ordinary language treats “p → (!q or !r)” and “p → !(q or r)” as different—yet in his system, because !A and A are equivalent, those two sentences collapse together. The logic of “ought” had eaten itself.

Why Is That a Disaster?

Almost all deontic logicians since 1939 have agreed with Menger’s verdict: Mally’s original system is broken. After all, if “ought” means exactly the same as “is,” then we cannot criticize anything that happens as wrong, and we cannot aim to make things better than they currently are. That is not how anyone actually uses the word. Mally had hoped to create an exact ethics, but his rules accidentally turned it into a rubber stamp for reality.

Yet Mally’s failure was not useless. It taught logicians a hard lesson: the logic of obligations must carefully protect the boundary between facts and duties. When you design a system of rules—whether for a legal code, a computer chip, or everyday morality—that distinction is everything.

Patching the System: The Birth of Modern Deontic Logic

Philosophers and logicians set out to fix Mally’s mistake. Two main strategies emerged.

Change the base logic. Mally built his system on classical propositional logic—the everyday “and,” “or,” “not” logic that allows certain oddities, like “from a contradiction, anything follows.” By moving to a relevance logic, where a conclusion must be properly connected to its premises, or to an intuitionistic logic, where the law of excluded middle is restrained, many of Mally’s surprising theorems vanish. In a relevance-based version, the dreaded !A ↔ A is no longer derivable. The unwanted collapse disappears.

Tweak the deontic rules themselves. Even staying inside classical logic, you can replace Mally’s definition of “requires” and one of his axioms with a slightly different recipe. The resulting system is nearly identical to standard deontic logic (often called KD), which is the version most widely studied today. It keeps the obligation operator, enforces that obligations don’t conflict with truths, and avoids making everything obligatory just because it happens. Mally’s own idea that there is an unconditionally obligatory U can still be added without breaking the system.

Neither fix is perfect—modern deontic logic still faces its own paradoxes—but each one shows that Mally’s original act of building a logic for “ought” was a pioneering step, not a dead end. It took only small adjustments to turn his failed attempt into a workable framework.

Why It Still Matters: Keeping “Is” and “Ought” Apart

Mally’s strange journey matters far beyond dusty logic books. Every time you make a to-do list, follow a rule, or argue that something is unfair, you rely on the difference between how things are and how they ought to be. A logical system that confuses those two things cannot help us think about right and wrong, about laws, or about improving our lives.

The attempt to build a math of morals did not, in the end, give us a perfect calculator for duties. But it did force philosophers to get precise about what “ought” means and to design logical tools that respect our deepest intuition: that the world doesn’t have to be the way it should be, and that we can work to change it. Mally’s brilliant failure is still teaching us that lesson.

Think about it

1. If a computer could calculate all the rules Mally proposed, would it ever be able to tell you what you really ought to do? Why or why not?
2. Can you imagine a world where whatever happens is automatically the right thing? What would that mean for apologies, promises, or learning from mistakes?
3. Why might people want a logic of “ought” that works like math? What would we gain, and what might we lose?