Can You Get an ‘Ought’ from an ‘Is’?
The Gap Between Facts and Rules
Suppose you are in a courtroom. The judge says, “The evidence shows the defendant took the money.” That is a fact about what is. Then the judge says, “Therefore, the defendant ought to go to prison.” A quiet, almost invisible leap has just happened — a leap from a description to a command, from a plain fact to a rule about what should be done.
David Hume (1711–1776), a Scottish philosopher with a sharp eye for hidden steps, spotted this leap in 1739. He pointed out that writers often move from sentences joined by is and is not to sentences joined by ought and ought not without ever explaining the connection. This shift, he argued, changes the whole subject — and nobody notices because it happens inside ordinary-sounding words. The question he raised became famous: can you ever derive an ought from an is?
Hume himself thought the answer was no. Most interpreters agree he believed such a derivation was impossible. But noticing the gap and proving it cannot be bridged are two different things. For centuries, philosophers have tried to figure out exactly what kind of connection — if any — could make the jump legitimate.
Adding a Missing Piece: The Bridge Principle
Imagine you have two disconnected islands. You can shout across the water, but you cannot walk from one to the other. That is the situation inside a formal logical system where “is”-statements and “ought”-statements live in different vocabularies. To get from one to the other, you need an explicit rule that links them. Philosophers call this a bridge principle.
A bridge principle is a logical statement that uses both descriptive language (about what is true) and deontic language (about what is permitted, forbidden, or obligatory — from the Greek deon, meaning duty). For example, the statement “If something is true, then it ought to be the case” is a bridge principle. So is “If you ought to do something, then it must be possible to do it” — a principle known as ought-implies-can, controversially linked to Immanuel Kant (1724–1804).
The philosopher Gerhard Schurz (writing in 1991) gave a precise definition: a bridge principle is any axiom that contains at least one symbol that appears both inside and outside the scope of the obligation operator O (the formal marker for “it ought to be that”). These principles do not solve the deep philosophical puzzle, but they do something almost as valuable: they force hidden assumptions out into the open, where everyone can see and argue about them.
Logics That Talk to Each Other
To treat Hume’s problem with real precision, you need two different kinds of logic working side by side. Alethic logic deals with necessity and possibility — what must be and what could be (from aletheia, Greek for truth). Deontic logic deals with obligation and permission — what ought to be and what may be. A system that uses both at once is called a bimodal logic, because it has two distinct modalities living in the same language.
This was not possible to study carefully until the 20th century, when general modal logic was developed. Before that, “ought” was often treated as a predicate — a property of actions, like “being obligatory.” But treating it as a modal operator (a little symbol that modifies a whole sentence, like the box □ for necessity or the diamond ◇ for possibility) turned out to be far more powerful. It allowed logicians to build precise semantic models — Kripke models, named after Saul Kripke (1940–2022) — where different worlds represent different possibilities, and obligation means something like “true in all morally acceptable worlds.”
Some philosophers, like A.N. Prior (1914–1969), used this new machinery to argue directly against Hume. Prior constructed a clever paradox. Take the sentence “Tea-drinking is common in England” — surely a descriptive fact. Now add: “Either tea-drinking is common in England, or all New Zealanders ought to learn Latin.” This mixed sentence looks partly normative. Depending on how you classify the mixed sentence, you can make a purely descriptive premise yield a normative conclusion using nothing but ordinary classical logic. Prior concluded that Hume’s thesis is simply false: you can derive ethical conclusions from non-ethical premises.
But the conclusion felt hollow even to Prior. The inferences are trivial and ethically irrelevant. Nobody would be convinced to learn Latin by a logic trick about tea. The real challenge became defining exactly what “ethical irrelevance” means — a problem that has not been fully settled even today.
A More Careful Humean Answer
Schurz took a different approach. Instead of trying to block all “is-to-ought” inferences, he asked what kind of conclusion could ever follow validly from purely descriptive premises if no bridge principle is allowed. His answer became known as the generalized Hume’s thesis.
The idea is this: if a mixed sentence (one containing both descriptive and deontic vocabulary) follows from purely descriptive premises, then the deontic part of that sentence must be completely irrelevant. You could swap the obligation operator’s content — replace “you ought to tell the truth” with “you ought to tell a lie” — and the logical structure would still hold. The normative part is, in a precise sense, hollow.
Schurz proved that this holds in a combined alethic-deontic first-order logic exactly when the logic can be built without any bridge principles. In other words, if you keep the two logical systems genuinely separate — fusing their languages but not adding secret connecting axioms — then no genuine normative content can sneak in from purely descriptive facts. Hume was right, in a carefully qualified way.
This result depends on the technique of fusion, a method for combining modal logics introduced by Richmond Thomason in 1984. In a fusion, you take two modal logics and let their operators coexist in a shared language, with separate accessibility relations in the Kripke models. The logics work side by side without interacting, unless you explicitly add bridge axioms to connect them. Fusion is one of the cleanest ways to keep “is” and “ought” from collapsing into each other.
Why the Pieces Matter Separately
Combining logics is not only about building bigger systems. It is also about decomposing them — taking a complex logical system and breaking it down into simpler fragments. This reverse direction, sometimes called splitting rather than splicing, can be just as revealing.
A method called possible-translations semantics, introduced by Walter Carnielli in 1990, works this way. Suppose you have a logic with a strange, non-classical connective — like a paraconsistent negation where a statement and its opposite can both be true. Instead of trying to give one big, messy meaning for that connective, you translate the whole formula into several simpler logics at once, each using a slightly different truth table. The original formula counts as valid only if every one of its translations is valid in the corresponding simpler system. A single, puzzling connective gets decomposed into a family of well-behaved ones.
This matters for Hume’s problem because it reveals something deep about mixed languages. When “is” and “ought” sit inside the same sentence, you are not dealing with one simple thing. You are dealing with a composite, built from logics that obey different rules. Recognizing that — and having the technical tools to separate them — is what allows philosophers to state clearly when a jump from fact to value is legitimate, and when it is just an illusion created by sloppy language.
What This Means for You
Every day, you move from is to ought without thinking about it. The ground is wet, so you ought to wear boots. Your friend is sad, so you ought to say something kind. These tiny leaps feel so natural that questioning them seems absurd. But Hume’s puzzle is worth sitting with precisely because it makes the invisible visible.
The technical machinery — bimodal logics, fusions, bridge principles — is not the point for you. The point is the habit of asking: What extra step did I just take? What hidden rule connects the fact I noticed to the action I am about to choose? Once you start seeing the gap, you cannot unsee it. And you become harder to manipulate with arguments that sneak values inside facts, pretending the leap was never there at all.
The debate is not settled. Some philosophers still argue, following Prior, that the gap is an illusion. Others, following Schurz, think it is real but carefully definable. What everyone agrees on is that the question is worth asking — and that combining logics, rather than keeping them in separate compartments, is the most honest way to face it.
Think about it
- If a scientist could perfectly predict every decision you will ever make using only facts about your brain, would it still make sense to say you “ought” to do anything?
- Think of a recent time when you told someone what they should do. What fact did you start from? What hidden bridge rule connected the fact to the “should”?
- Can you imagine a world where everyone agrees on all the facts but still disagrees on what they ought to do? What would that tell you about the gap between is and ought?
