Why Some Logics Won’t Let You Add an Extra Fact
The Rules You Never Knew You Were Using
You and a friend are arguing about dessert. You say, “I finished my homework, so I should get cake.” Your friend adds, “You also brushed your teeth, and you wore a blue shirt.” Does piling on those extra facts make your argument stronger? In the logic you meet in school, you can toss in any extra premise without weakening an argument. That habit is so natural that you might not even notice you’re following a rule. But some logics say that rule is not always safe—and they back it up by studying the structural rules that control how premises can be combined.
A structural rule is a rule about the whole shape of an argument, not about any particular word like “if” or “not.” For example, the rule modus ponens says that from “If it’s raining, the ground is wet” and “It’s raining” you can conclude “The ground is wet.” That rule is about the conditional “if… then…,” so it is an operational rule—it cares about the logical form of the sentences. A structural rule, by contrast, doesn’t peek inside the sentences. One example is the Cut rule: if you can prove A from some premises, and you can prove B from A together with other premises, then you can prove B from all those premises put together. The Cut rule works no matter what A, B, or the premises are about.
Other structural rules tell you what you can do with the pile of premises itself. Can you always add an extra premise to a valid argument? Can you reuse a premise you already used? Can you swap the order of the premises? In ordinary logic, you can do all three. The logics that question one or more of these are called substructural logics. They were invented to model situations where resources run out, where order matters, or where extra facts really do spoil the dish.
What Happens When You Add a Silly Premise?
Imagine you reason, “It’s raining, so the ground is wet.” Now add “My socks are purple.” Does that new fact make the argument invalid? Most of us feel that the extra fact is just irrelevant—it shouldn’t hurt. And in classical logic, it doesn’t, because classical logic accepts the structural rule called weakening. Weakening says that if a conclusion follows from some premises, it also follows from those premises plus any extra ones.
But in the middle of the 20th century, several logicians grew uneasy. They argued that a truly valid argument must have a relevant connection between premises and conclusion. Wilhelm Ackermann (1896–1962) and Alonzo Church (1903–1995) were early voices for this idea. Later, Alan Ross Anderson (1925–1973) and Nuel Belnap (born 1930) built full systems of relevant logic in which weakening is rejected. In a relevant logic, if you have a proof of q from p, you cannot simply add r—because r might be completely irrelevant to q. So an argument like “From ‘It’s raining’ I conclude ‘The ground is wet’” can be perfectly good, but “From ‘It’s raining’ and ‘My socks are purple’ I conclude ‘The ground is wet’” may fail, because the socks have nothing to do with wet ground.
This doesn’t mean you can never add premises. It means that adding a premise to a relevant argument is like adding an ingredient to a recipe: you need to make sure the new ingredient actually contributes. The logic of the conditional, “if… then…,” changes too. In classical logic, you can always prove “If q, then p” from p alone: if p is true, then any q makes the conditional true. But in relevant logic, that step is not allowed because q may be utterly irrelevant to p. For relevant logicians, a conditional like “If my socks are purple, then it’s raining” is not automatically true just because it happens to be raining; you’d want the color of your socks to actually connect to the weather. By rejecting weakening, relevant logic makes the conditional care about relevance.
Can You Reuse Your Best Point?
Some premises behave like ingredients in a recipe. If a recipe calls for one egg, you can’t take the same egg out of the bowl and crack it again; it’s already used. Contraction is the structural rule that lets you reuse a premise as many times as you want. In classical logic, if you need a premise p to reach a conclusion, you can appeal to p twice, three times, or more, and then treat those many uses as just one. Contraction allows you to step from “p, p can prove q” to “p can prove q.”
In 1987 the logician Jean-Yves Girard introduced linear logic, which treats premises as resources that are consumed. In linear logic, contraction is not allowed. If a conclusion depends on using p twice, you need two separate copies of p. You can’t collapse them into one. This logic was designed to model real processes—computer programs, chemical reactions, economic transactions—where resources are limited. A simple example: suppose “If I have an egg, I can make a cake” is a conditional, and I have one egg. I can use that egg to make the cake. But if I try to use the same egg to also make an omelette, I can’t; after I crack it for the cake, it’s gone. Linear logic captures that “use once” feeling.
Without contraction, certain arguments that look obvious in classical logic break down. In classical logic you can prove “If p, then q” from “If p, then (if p, then q)”—that is, reusing p isn’t needed. But in linear logic, the extra copy of p matters, so that step is invalid unless you actually have two copies of p. Contraction is also at the heart of some famous paradoxes, like the liar paradox (“This sentence is false”) and Curry’s paradox. Because contradiction can hide a sleight of hand, logicians working on paradoxes—following ideas from Jan Łukasiewicz (1878–1956) and others—have explored contraction-free logics as a way to tame those puzzles.
Does the Order of Your Reasons Matter?
You probably think that “A and B” means the same as “B and A.” But in natural language, order sometimes matters a great deal. “John loves Mary” does not mean the same as “Mary loves John.” In the late 1950s, the mathematician Joachim Lambek (1922–2014) designed a logic—now called the Lambek calculus—to model how strings of words combine. In this logic, premise combination is not commutative: you cannot freely swap premises. And it is not necessarily associative either: grouping matters. The sequence (p, q), r might behave differently from p, (q, r), just as word groups in a sentence have structure.
Because order matters, the meaning of a conditional splits into two. The usual “if p then q” (written p → q) behaves like “whenever p appears to the left, q appears to the right.” But in a non-commutative setting you can also define a backward conditional ← such that q ← p says something like “q gets you p”—the order is reversed. In the Lambek calculus, both arrows naturally arise, and each has its own rules. This is why linguists have used these ideas to build grammatical categories: “John” is a noun phrase, “loves” is a transitive verb, and the whole sentence is well-formed only if the pieces combine in the right order.
So, if you’re ever in a debate where the sequence of your reasons makes a difference—like “first you need a passport, then you can buy a ticket, then you can board the plane”—you are already thinking in a substructural way.
When Logic Gets Flexible: Strict and Tolerant Truths
The substructural logics we’ve met so far keep the Cut rule and the Identity rule (every statement implies itself). But the most recent wave of substructural thinking questions even those. What if you keep all the usual combination rules but drop Cut? Cut says that if you can prove A from some premises, and from A (with others) you can prove B, then you can chain them and prove B. Without Cut, your reasoning can be non-transitive: you might have a path from premises to conclusion, but you can’t automatically glue paths together.
In the 2000s, philosophers like David Ripley and Pablo Cobreros developed a logic called ST (for Strict/Tolerant). It uses a neat trick with truth values. Picture a three-way evaluation: a sentence can be strictly true, strictly false, or somewhere in between (tolerantly true). The rule for a valid argument in ST is: if all the premises are strictly true, then the conclusion must be at least tolerantly true. Because the standard for the conclusion is lower, the Cut rule can fail, even though each individual step follows ordinary classical truth tables.
Why would anyone want a logic like that? The liar paradox provides a powerful reason. Take the sentence “This sentence is false.” In ST, we can show that it is tolerantly true and also tolerantly false, but not strictly true. That lets us accept both “The liar sentence is true” and “The liar sentence is false” without the whole system collapsing into nonsense. Cut fails for the very sentences that cause trouble, but ordinary reasoning stays safe. A cousin logic, TS (Tolerant/Strict), flips the standards and rejects Identity instead: nothing follows from itself, yet many meta-inferences still hold. These logics show that even the most basic rules—Cut and Identity—are choices, not sacred commands.
So, Which Set of Rules Is the Right One?
The point of substructural logics is not to announce that one system has finally gotten things right and the others are wrong. Rather, each logic illuminates a different aspect of how we actually reason—or how we might want to reason in a certain domain. If you care about relevance, a relevant logic is a sharp tool. If your argument uses up resources, linear logic models that scarcity. If you’re parsing a sentence, the Lambek calculus gives you a grammar of order. And if you’re facing a paradox that threatens to explode your whole system, ST and TS offer ways to keep your footing.
Next time you build an argument—whether you’re trying to convince a friend to share dessert, plan a project with limited supplies, or figure out what a messy sentence really means—pay attention to the invisible rules you’re using. Can you toss in an extra fact freely? Can you use the same reason twice? Does the order of your points change anything? The answers aren’t always “yes,” and noticing that is the first step into a rich landscape of logical options. That flexibility doesn’t make logic weak; it makes it more precise, more honest, and a lot more interesting.
Think about it
- If a friend says, “You should help me with chores because you’re my friend,” does adding “and the sky is blue” make their argument stronger, weaker, or just sillier? Why might a logician say it shouldn’t count at all?
- Imagine you need one spoonful of honey for a recipe. Could you use the same spoonful twice to double the sweetness? What would it mean if an argument tried to “re-use” a premise that had already done its work?
- Can you think of a situation where changing the order of “A and B” changes the message entirely—like giving directions or telling a story? Does that mean logic should care about order?
