When "If... Then..." Goes Crazy
A fight about “if”
You’re arguing with a friend. They say, “If I’m so bad at math, then my dog can fly.” You blink. Their statement sounds oddly confident, as if their bad math skills magically give their dog wings. In everyday life, we expect an “if… then…” to connect two ideas that really belong together. But in classical logic — the kind most people learn in school — that connection doesn’t matter at all.
A classical logician would say your friend’s sentence is true. Why? Because in classical logic, an “if… then…” (called material implication) is false only when the “if” part is true and the “then” part is false. If the “if” part is false — your friend isn’t actually bad at math? fine, it’s still false — then the whole statement counts as true, no matter how ridiculous the “then” part is. This leads to some deeply weird results.
When logic lets nonsense win
The problem isn’t just one weird sentence. Once you accept material implication, you can prove that anything follows from a contradiction. From “It is raining and it is not raining,” classical logic allows you to conclude “The president is a penguin.” That rule is called explosion, and most philosophers find it at least a little embarrassing.
Equally strange is that any truth implies a logical truth. For example, “If grass is green, then either it is raining or it is not raining” must be true. The second half is always true, so the “if… then…” is automatically true, even though grass’s color has nothing to do with rain. In the early 20th century, philosopher C.I. Lewis (1883–1964) tried to fix this with strict implication: “if A then B” means necessarily, in every possible world, either A is false or B is true. This ruled out some nonsense, but not enough. You still got “If grass is green, then if it is raining, then it is raining” — an irrelevant chain of reasoning that never uses the first idea at all.
Building a better “if” with three worlds
In the 1970s, logicians Richard Routley (1935–1996) and Robert K. Meyer (1931–1998) took a dramatic step. They said: to judge whether “if A then B” is true at a world, you need three worlds, not two. Their ternary relation says that A → B holds at a world a if and only if, for every pair of worlds b and c where the relation Rabc holds, whenever A is true at b, B is true at c.
What does that mean? Think of a as a communication channel, like a TV broadcast. Site b is where information starts, and site c is where it arrives. The conditional is a kind of “pipe” that guarantees information flows correctly. This idea — often called channel theory — is one way to make sense of the three-place relation. Another way, developed by Edwin Mares, treats a as a situation full of informational links (like the laws of physics). If b contains some facts, and a says those facts always bring others along, then c must contain the result.
Others see it differently. Philosopher J. Michael Dunn suggested that Rabc means that in context a, b is relevant to c — the information in a and b together allows you to derive what’s in c, but not from either alone. This connects relevance logic to the everyday idea that some clues are useful only when combined.
However you interpret it, the ternary relation blocks the old paradoxes. It lets there be worlds where “q → q” can fail, so “p → (q → q)” can fail too. Real connections become required.
Proofs that track what you really used
Relevance logicians didn’t stop at meaning. They also invented proof systems that enforce relevance. Imagine you’re building a logical argument step by step. In a classical proof, you can introduce a premise and then never use it, like tossing an extra fact into a bag just in case.
Alan Ross Anderson (1925–1973) and Nuel Belnap (b. 1930) created a special natural deduction system for relevance. In their system, every step is tagged with a set of numbers showing which premises were actually used to reach it. For example:
- Step 1: Assume A (tag {1})
- Step 2: Assume A → B (tag {2})
- Step 3: From 1 and 2, get B (tag {1,2})
If you later try to form an implication “A → (B → B)” but the proof shows you never used A to get B → B, the tag will expose that — and the system rejects the step. This keeps arguments honest.
The rule for “and” is especially strict: you can only combine two statements with “&” if they carry exactly the same set of dependency tags. That prevents you from smuggling in an unused premise and then cutting it away to create a fake connection.
When loops become dangerous
Even with these safeguards, relevance logics can run into trouble. Some versions include a rule called contraction: if A → (A → B), then A → B. This sounds harmless — you’re just using A once instead of twice. But contraction lets you build dangerous self-referential loops.
The most famous is the Curry paradox. It starts with a sentence that says of itself that if it is true, then any wild claim you like is also true. In a logic with contraction, you can actually prove that wild claim — anything from “pigs fly” to “0=1”. The problem is especially sharp if you want a naive set theory: a theory where for any property you can describe, there’s a set of all things with that property. Contraction makes such a theory trivial — it proves everything. Logicians like Ross Brady and Graham Priest (b. 1948) showed that if you drop contraction and a few other rules, you can have a safe naive set theory, one that doesn’t blow up.
In Anderson and Belnap’s proof system, contraction corresponds to allowing a premise to be used more than once. To block it, you need a proof system that counts how many times each premise is used — not just which ones. That leads to even more delicate systems that treat premises like resources you can spend.
Why it matters for real arguments and computers
You might wonder: is all this just a game with symbols? Not at all. Relevance logic has real applications.
In deontic logic — the logic of “ought” and “permitted” — classical systems face a problem. They say that if something is a logical truth, it ought to be the case. So “It ought to rain or not rain in Ecuador” is automatically true. But that seems silly. Relevance logicians avoid this by allowing non-normal worlds where logical truths can fail, so they don’t force silly obligations.
In computer science, linear logic — a close cousin of relevance logic — treats implications as recipes for using resources. “If you have two tokens, you get a sandwich” is different from “If you have one token, you get a sandwich.” Linear logic keeps track of token counts, just as relevance proof systems count premise uses. It’s used in designing programming languages and understanding how computer memory works.
Even in everyday argument, we sense that a good “if… then…” should be more than a trick of truth tables. When a parent says, “If you finish your homework, you can watch TV,” we understand there’s a real link — it’s not just true because the homework isn’t finished yet. Relevance logic gives us a vocabulary to explain why that link matters.
Think about it
- If a friend says, “If I have a million dollars, I’ll buy the moon,” and they don’t have a million dollars, would you call their statement true? Why or why not?
- Can you think of a time when someone gave a reason that didn’t actually support their conclusion, but everyone nodded as if it did? How would you explain what was wrong?
- Imagine a computer program that treats every unused instruction as still “true” for the final result. What could go wrong? Would you trust it to launch a spaceship?
