What Follows From What? Medieval Puzzles About Logical Consequence

Imagine you’re in an argument with a friend. You say, “Socrates is a man, therefore Socrates is an animal.” Your friend says, “That doesn’t follow.” Who’s right? Most of us would say you are—if something is a man, it has to be an animal. But now imagine your friend says, “God doesn’t exist, therefore you are a donkey.” That seems ridiculous. But if “God doesn’t exist” can’t possibly be true (many medieval philosophers believed this), then doesn’t it follow that anything follows from it? If a premise is impossible, can it really lead to any conclusion at all?

These are the kinds of questions that medieval philosophers spent centuries arguing about. They were trying to understand what it means for one thing to “follow from” another—what we call consequence. And their answers are surprisingly sophisticated, and still debated today.

The Basic Puzzle

When you say “if this, then that,” what makes that claim true or false? Medieval philosophers noticed that there are at least two different things we might mean.

One idea is simple: a consequence is valid if it’s impossible for the premise to be true and the conclusion false. This is called necessary truth-preservation. If you say “it’s raining, therefore the ground is wet,” that’s a good consequence if it’s literally impossible for it to be raining and the ground dry at the same time. This seems obvious.

But here’s the problem. Consider: “Snow is white, therefore 2+2=4.” Is this a good consequence? Well, it’s impossible for the premise to be true and the conclusion false, because the conclusion can’t be false at all. So it satisfies necessary truth-preservation. But something feels wrong about it. The premise and conclusion have nothing to do with each other. The conclusion doesn’t follow from the premise in any meaningful way.

So medieval philosophers started asking: is necessary truth-preservation enough, or do we need something more? Do we need a real connection between the premise and conclusion? And if so, what kind of connection?

Two Big Ideas: Form and Containment

Medieval philosophers developed two main ways of thinking about what makes a consequence valid.

Valid in Virtue of Form

The first idea draws on a distinction that goes back to Aristotle: the difference between the form and the matter of an argument. Consider these two arguments:

Every man is mortal. Socrates is a man. Therefore, Socrates is mortal.
Every dog is an animal. Fido is a dog. Therefore, Fido is an animal.

These are different in their matter (they’re about different things), but they have the same form: “Every A is B. C is A. Therefore, C is B.” This form is what makes them valid. If you changed the subject matter, the argument would still work. “Every rock is heavy. This pebble is a rock. Therefore, this pebble is heavy.” Still valid.

This idea—that a consequence is valid in virtue of its form—became very important. A formal consequence is one that holds no matter what terms you substitute in, as long as you keep the same structure. “If it’s A, it’s B; it’s A; therefore it’s B.” Fill in anything: “If it’s a cat, it’s an animal; it’s a cat; therefore it’s an animal.” Still works.

But here’s where it gets tricky. What about “Socrates is a man, therefore Socrates is an animal”? According to the form criterion, this is NOT a formal consequence, because the structure alone doesn’t guarantee it. “Socrates is a man, therefore Socrates is a philosopher” has the same structure but is false. So something else must be making it valid—something about the meaning of the words.

Valid in Virtue of Containment

This leads to the second big idea: containment. The thought is that in a valid consequence, the conclusion is somehow already contained in, or understood from, the premise. When you understand that Socrates is a man, you already understand that he’s an animal, because “man” contains “animal” in its meaning. The conclusion doesn’t add anything new—it just makes explicit what was already there.

One medieval philosopher, a brilliant and controversial figure named Peter Abelard (who had a famously dramatic life—he fell in love with his student Heloise, they had a secret child, and her uncle had him castrated), argued that this containment was the heart of valid inference. He said that in a valid consequence, the sense of the antecedent requires the sense of the consequent. The conclusion is demanded by the premise.

This sounds nice, but it leads to problems too. If containment is required, then “God doesn’t exist, therefore you are a donkey” wouldn’t count as valid, because there’s no containment there. But many medieval philosophers thought this should count as valid—if the premise is impossible, it’s just true that anything follows.

Famous Rules (and the Trouble They Cause)

Medieval logicians worked out a bunch of rules for valid consequences. Some of them seem obvious. For example:

Transitivity: If A implies B, and B implies C, then A implies C. (If running implies moving, and moving implies being in space, then running implies being in space.)
Contraposition: If A implies B, then not-B implies not-A. (If being a dog implies being an animal, then not being an animal implies not being a dog.)

But two rules caused enormous debate:

From the impossible, anything follows (ex impossibili quodlibet sequitur). If your premise is impossible, any conclusion is valid. So “A square circle exists, therefore you are a dolphin” counts as a valid consequence.
The necessary follows from anything (ad necessarium ex quolibet sequitur). If your conclusion is necessarily true (it can’t be false), then it follows from any premise. So “You are reading this, therefore 2+2=4” counts as a valid consequence.

Some medieval philosophers accepted these rules. Others thought they were absurd. John Buridan, a 14th-century philosopher who wrote one of the most sophisticated treatises on consequence, accepted them. He argued that if you define consequence as necessary truth-preservation, then these rules simply follow—you can’t avoid them.

But others, especially a group of British philosophers later in the 14th century, disagreed. They thought that a real consequence required a real connection. Ralph Strode, for example, said that a formal consequence is one where understanding the antecedent already makes you understand the consequent. If I understand that you are a man, I already understand that you are an animal. But if I understand that God doesn’t exist, I do NOT already understand that you are a donkey. So the first is valid, the second isn’t.

The Big Split: Two Schools of Thought

By the mid-14th century, two main traditions had emerged.

The Parisian tradition, led by Buridan and his followers, defined formal consequences as those that hold for all substitutions of terms. For them, material consequences (like “Socrates is a man, therefore Socrates is an animal”) are still valid—they just aren’t formal. They’re valid because of the meanings of the words, not just the structure. But they can be reduced to formal consequences by adding a missing premise: “Socrates is a man; all men are animals; therefore Socrates is an animal.” Now it’s formal.

The British tradition, led by Strode, Richard Billingham, and others, defined formal consequences as those where the consequent is understood in the antecedent. For them, “Socrates is a man, therefore Socrates is an animal” is a formal consequence, because understanding “man” already gives you “animal.” But “God doesn’t exist, therefore you are a donkey” is only a material consequence—it’s valid by truth-preservation, but it lacks real connection.

This disagreement mattered. It meant that the two traditions had different views on what logic itself was about. Was logic about the structure of arguments, independent of what the words mean? Or was logic about the meaningful connections between concepts?

A Tricky Example: The Liar Paradox

Here’s where things get really strange. One medieval philosopher, writing under the name Pseudo-Scotus (we don’t know his real name), came up with a puzzle that shows how tricky the idea of consequence can be.

Consider this consequence: “God exists, therefore this argument is invalid.”

First, “God exists” is necessarily true (most medieval philosophers believed this). So if the consequence is valid, the conclusion must be true. But the conclusion says the argument is invalid. So if it’s valid, it’s invalid. That’s a contradiction.

So the argument must be invalid. But wait—if it’s invalid, then it’s not a good consequence. But then the conclusion “this argument is invalid” is true. And a true conclusion follows from a necessary premise, so it satisfies truth-preservation. So maybe it IS valid after all?

This is an early version of what philosophers now call Curry’s Paradox, and it shows that even the simple idea of “if this, then that” can lead to mind-bending puzzles. Medieval philosophers took these puzzles seriously. They didn’t dismiss them as silly word games.

Why This Still Matters

You might think this is all ancient history. But the questions medieval philosophers were asking about consequence are still alive today. Modern logicians still debate: Is logical consequence just about truth-preservation, or does it require something more? Do “from the impossible, anything follows” and “the necessary follows from anything” actually capture something true about logic, or are they mistakes?

Some modern philosophers have developed relevance logics that try to capture the idea that a valid consequence requires a real connection between premise and conclusion—just like the British tradition tried to do. Others stick with the truth-preservation view, like Buridan. Nobody has fully settled the debate.

The medieval philosophers also left us with a framework for thinking about these questions. Their distinctions between formal and material consequence, between simple and as-of-now consequence, and their careful analysis of what it means for something to “follow from” something else, shaped how logic developed for centuries afterward. When we talk about “logical form” today, we’re still using concepts that medieval thinkers helped invent.

What Nobody Really Knows

Here’s the honest truth: philosophers still don’t agree on what consequence really is. They don’t agree on whether “from the impossible, anything follows” is true. They don’t agree on whether formal consequence is about structure or about containment. They don’t even fully agree on what counts as a good definition of consequence.

What the medieval philosophers showed is that these questions are deep. They look simple at first—“of course if A then B, that just means you can’t have A without B”—but they lead to paradoxes, disagreements, and different views about what logic itself is. That’s part of what makes them fascinating. A smart 12-year-old can see the problem. Grown-up philosophers have been arguing about it for 800 years.

Key Terms

Term	What it does in this debate
Consequence	The relation between a premise and conclusion when the conclusion “follows from” the premise
Necessary truth-preservation	The idea that a consequence is valid if it’s impossible for the premise to be true and the conclusion false
Formal consequence	A consequence that is valid because of its structure (form), not because of the particular words used
Material consequence	A consequence that is valid because of the meanings of the words, but not just because of its structure
Containment	The idea that in a valid consequence, the conclusion is already “contained in” or “understood from” the premise
Substitution criterion	The test for formal validity: if you replace the key terms with others and the consequence still holds, it’s formal
Ex impossibili	The principle that from an impossible premise, any conclusion follows
Ad necessarium	The principle that a necessary conclusion follows from any premise

Key People

Peter Abelard (1079–1142) – A brilliant and controversial French philosopher who had a famously tragic love affair with his student Heloise. He developed one of the earliest theories of consequence, arguing that valid inference requires the conclusion to be “required by” the sense of the premise.
John Buridan (c. 1300–1360) – A French philosopher who wrote the most sophisticated medieval treatise on consequence. He defined formal consequences by the substitution criterion and accepted that “from the impossible, anything follows.”
Pseudo-Scotus (early 14th century) – An unknown philosopher writing under the name of John Duns Scotus. He discovered a paradox about consequence that resembles what modern philosophers call Curry’s Paradox.
Walter Burley (c. 1275–1344) – An English philosopher who wrote one of the earliest treatises on consequence and argued with Ockham about how to define formal and material consequence.
Ralph Strode (14th century) – An English philosopher (and possibly Chaucer’s friend) who defined formal consequences in terms of containment: the conclusion must be understood from the premise.

Things to Think About

Suppose someone says: “If you’re my friend, then you’re a human.” Does “you’re my friend, therefore you’re a human” count as a valid consequence? It’s hard to imagine a friend who isn’t human. But is that because of the meaning of the words, or because of the structure of the argument? How would you decide?
The principle “from the impossible, anything follows” means that if someone says “I can fly by flapping my arms” (which is false, but not impossible), that doesn’t give you any conclusion. But if someone says “I am a married bachelor” (which is impossible), then literally anything follows—including “therefore, the moon is made of cheese.” Does that seem right to you? Why or why not?
Think about times when you’ve been in an argument where someone said “that doesn’t follow.” What did they mean? Were they saying the conclusion wasn’t guaranteed by the premise? Or that the premise and conclusion weren’t connected in the right way? Can you think of an example where both conditions are met—the conclusion is guaranteed by the premise, but there’s no real connection?
The medieval distinction between formal and material consequence depends on what counts as “form” and what counts as “matter.” But is it always clear where to draw that line? Consider: “If something is red, it is colored.” Is this formal or material? What about “If something is scarlet, it is red”? Where does the form stop and the meaning begin?

Where This Shows Up

Math class: When you learn that if a=b and b=c, then a=c, you’re using the principle of transitivity that medieval logicians studied. When a teacher says “that doesn’t follow,” they’re talking about consequence.
Computer programming: “If-then” statements in code are based on logical consequence. Programmers constantly think about what follows from what when they’re writing conditions.
Law and arguments: Lawyers argue about what “follows from” the evidence. Judges decide whether a conclusion is “logically connected” to the premises. These are the same ideas medieval philosophers were exploring.
Everyday arguments: When you say “that doesn’t follow” to a friend, you’re doing philosophy of logic. The medieval philosophers just went deeper—they asked why it doesn’t follow, and what “following” even means.
Modern logic: The two traditions (Parisian and British) still echo today. Some logicians think logic is about formal structure; others think it needs to capture real connections of meaning. The debate is still alive.