How to Think About Propositions: Bernard Bolzano's Strange New Logic

The Puzzle That Started It

Imagine you say to a friend: “Socrates is mortal.” Then you say: “All men are mortal.” Then: “If all men are mortal, and Socrates is a man, then Socrates is mortal.”

Here’s a weird question that bothered a 19th-century philosopher named Bernard Bolzano: Where do these sentences even exist?

Not the written marks on paper—those exist on a page. Not the sounds in the air—those exist as vibrations. Not the thoughts in your head—those exist in your brain for a while and then disappear. But what about the meaning of these sentences? The thing that’s true or false? The thing that’s the same whether you say it in English or German or think it silently?

Bolzano pointed out that meanings don’t seem to exist anywhere. They don’t take up space. They don’t get born or die. They don’t cause anything to happen. And yet—here’s the strange part—they seem to be there somehow. There are infinitely many of them (there are infinitely many true mathematical statements, for instance). They have parts that fit together in specific ways. Some of them are true, some false, and nothing you do can change that.

Bolzano called these things propositions in themselves. He wasn’t sure they existed in the normal sense, but he was sure that there are such things. This article is about what Bolzano discovered when he took these strange entities seriously and built a whole new kind of logic around them.

What Bolzano Was Trying to Do

Bolzano was a mathematician and a priest living in Prague. He got kicked out of his teaching job by the Emperor for being too liberal with his ideas, and spent the next years writing a massive book called the Theory of Science (1837). His big project was figuring out how to organize all human knowledge into a perfect system.

But before you can organize knowledge, you need to understand what knowledge is made of. Bolzano thought the basic ingredients were:

Ideas (like [mortality], [Socrates], [triangle])
Propositions (like [Socrates is mortal]), which are combinations of ideas

These aren’t the thoughts in your head. They’re the contents of those thoughts—the things your thoughts are about. You can think about [Socrates is mortal] right now, forget it tomorrow, and think about it again next year. The proposition itself stays the same the whole time.

This might seem like a small distinction, but it turned out to matter a lot for logic.

How Bolzano Redefined Logic

Before Bolzano, most philosophers thought logic was about how people think. It was the “art of thinking”—a guide to making good judgments and avoiding mistakes. Bolzano thought this was wrong. Logic, he said, isn’t about your mental processes at all. It’s about the objective relations between propositions themselves, whether anyone happens to be thinking about them or not.

Think about the difference between:

Psychology: “When people see that all men are mortal and Socrates is a man, they tend to conclude that Socrates is mortal.”
Logic: “[All men are mortal] and [Socrates is a man] together force [Socrates is mortal] to be true, regardless of what anyone thinks.”

The first is about human minds. The second is about the propositions themselves—a relationship that would hold even if no humans existed. Bolzano thought logic should study the second kind of thing.

This seems obvious to us today, but it was a radical move in 1837. It’s one reason Bolzano is sometimes called “the grandfather of modern logic.”

The Key Move: Variation

Here’s where Bolzano’s most creative idea comes in. He noticed that if you take a proposition and start swapping out parts of it, interesting things happen.

Take the proposition: [Socrates is mortal]

Now imagine you keep the structure the same but swap out the name “Socrates” for other names. You get:

[Plato is mortal] — true
[Aristotle is mortal] — true
[Your dog is mortal] — true
[A rock is mortal] — also true (rocks wear down eventually)

In fact, every substitution you can make seems to give you a true proposition (as long as you replace “Socrates” with something that makes sense). Bolzano called this universal validity with respect to the variable idea [Socrates].

Now try the same thing with a different proposition: [Socrates is wealthy]

Swap out names:

[Plato is wealthy] — who knows?
[Aristotle is wealthy] — who knows?
[Your dog is wealthy] — that doesn’t really make sense

Some substitutions give true propositions, some give false ones. Bolzano called this neutral—it depends on what you substitute.

Now try: [Socrates is a round square]

Every substitution that makes sense gives you a false proposition. Bolzano called this universal invalidity.

What Makes a Proposition Analytic?

Bolzano realized this gave him a new way to understand what philosophers call analytic propositions. An analytic proposition is one whose truth (or falsity) depends only on its structure, not on what the specific ideas mean.

Here’s his definition: A proposition is analytic if it’s either universally valid or universally invalid with respect to some of its variable parts.

Consider: [A right-angled triangle is a triangle]

If you swap out [triangle] for other things:

[A right-angled square is a square] — true
[A right-angled dog is a dog] — true (though weird)
[A right-angled number is a number] — true

Pretty much anything works. The truth doesn’t depend on what “triangle” or “right-angled” specifically mean. It’s just built into the structure: an A that is B is an A.

Bolzano thought this was much more interesting than the old idea that analytic propositions were ones where the predicate was somehow “contained” in the subject. The real point, he said, is that certain parts of the proposition don’t matter for the truth-value. They’re vacuous—you could swap them out for anything and the truth stays the same.

The Big System: Relations Between Propositions

Bolzano didn’t stop at classifying single propositions. He built a whole system of relations between propositions with variable parts.

Here’s how it works. Imagine you have two propositions with some ideas marked as variable:

[Napoleon predeceased Wellington] (let [Napoleon] and [Wellington] be variable)
[Wellington predeceased Napoleon] (same variables)

Now ask: is there any collection of ideas you could substitute for [Napoleon] and [Wellington] that would make both propositions true?

Well, if you substitute [Socrates] for [Napoleon] and [Plato] for [Wellington], the first proposition says [Socrates predeceased Plato] and the second says [Plato predeceased Socrates]. Both can’t be true at the same time.

But what about different variables? If you make [predeceased] variable instead:

[Napoleon predeceased Wellington]
[Wellington predeceased Napoleon]

Now substitute [fought] for [predeceased]:

[Napoleon fought Wellington] — true
[Wellington fought Napoleon] — also true (they did fight)

So the propositions are incompatible with respect to one set of variables but compatible with respect to another.

This might sound like a game, but it’s actually a deep idea. Bolzano was essentially inventing what we now call semantic consequence—the idea that one proposition follows from others if every way of making the premises true also makes the conclusion true. Modern logicians use something very similar today.

Probability and Logic

Here’s another surprising move Bolzano made. He realized that his method of variation could also give him a definition of probability.

Take the premises: [The number is between 1 and 10] and [The number is odd].
And the conclusion: [The number is prime].

If you treat [the number] as variable and consider numbers 1 through 10 as possible substitutions:

Numbers that make both premises true: 1, 3, 5, 7, 9 (five numbers)
Among those, ones that also make the conclusion true: 3, 5, 7 (three numbers)

So the probability of the conclusion given the premises is 3/5, or 0.6.

Bolzano’s genius move was to see that deduction (certain inference) and induction (probable inference) are the same kind of thing, just at different ends of a spectrum. When the probability is 1, you have deduction. When it’s less than 1, you have probability. Same machinery, just different numbers.

This was the first time anyone had defined probability in terms of logic.

The Hard Part: Grounding

Bolzano wasn’t satisfied with just understanding when one proposition follows from others. He wanted to know which propositions explain which others—which ones are the reasons for the rest.

He called this relation grounding (the German word is Abfolge). It’s a relation between true propositions where one (or a group) is the real reason the other is true.

Here’s an example. Consider:

Proposition A: [Socrates was an Athenian]
Proposition B: [Socrates was a philosopher]
Proposition C: [Socrates was an Athenian and a philosopher]

C follows from A and B (if both A and B are true, C must be true). And in this case, A and B also ground C—they’re the reason C is true. The conjunction is true because each part is true.

But not all cases of deduction are cases of grounding. Consider:

Premise: [All ravens are black]
Conclusion: [This raven is black]

The conclusion follows from the premise, but does the premise ground the conclusion? Bolzano thought no. The general fact that all ravens are black doesn’t explain why this particular raven is black. If anything, it’s the other way around—the general fact is grounded in all the particular facts. (Though even that is complicated.)

Grounding is tricky. Bolzano admitted he couldn’t define it properly and wasn’t even sure it could be defined. But he thought it was real and important, especially for organizing knowledge. The goal of science, he thought, is to arrange truths in the order of grounding—starting with the most basic truths and working up to the ones that are grounded in them.

This idea of grounding has become a hot topic in philosophy again in recent years. Philosophers are still arguing about exactly what it is, whether it exists, and how it relates to other logical concepts.

Why This Still Matters

Bolzano’s logic is strange and sometimes frustrating. His definitions are messy. He leaves important questions unanswered. Some of his claims seem obviously wrong (like that every proposition has a subject-predicate form). And his work was largely ignored during his lifetime—most people didn’t understand what he was trying to do.

But later philosophers came to see him as a visionary. He basically invented the idea that logic is about objective relations between propositions, not about psychological processes. He came up with the variation method that anticipates modern model theory. He gave the first logical definition of probability. He introduced the concept of grounding that philosophers are still exploring today.

His work also showed something important: that logic isn’t just a boring set of rules for avoiding mistakes. It’s a way of discovering the hidden structure of reality. When you start asking “what is a proposition?” and “what does it mean for one truth to follow from another?” you end up in some very strange places—places that are still being explored, more than 180 years later.

Appendix

Key Terms

Term	What It Does in This Debate
Proposition in itself	The meaning of a sentence, considered as something that can be true or false, regardless of whether anyone thinks it or says it
Idea in itself	A part of a proposition (like [Socrates] or [mortality]), also existing independently of anyone’s thinking
Variation	The method of treating some parts of a proposition as replaceable, to see how the truth-value changes
Universal validity	When every possible substitution (under certain rules) gives a true proposition
Analytic	A proposition whose truth or falsity doesn’t depend on the specific ideas in certain positions—it stays the same no matter what you substitute
Deducibility	When every way of making the premises true (by substituting for the variable ideas) also makes the conclusion true
Grounding	The relation between true propositions where one is the real reason the other is true (not just a logical consequence)

Key People

Bernard Bolzano (1781–1848) — A mathematician, priest, and philosopher from Prague who lost his teaching job for his liberal views and spent years writing a massive work on logic that almost nobody read at the time.
Edmund Husserl (1859–1938) — A later philosopher who rediscovered Bolzano’s work and called him “one of the greatest logicians of all time,” helping to bring his ideas to a wider audience.

Things to Think About

Bolzano said propositions in themselves don’t exist in space or time, but that “there are” such things. Do we have any good reason to believe in entities that don’t exist in any normal sense? What would count as evidence for or against them?
Bolzano’s method of variation lets you decide which parts of a proposition to treat as variable. This means the same argument can be valid or invalid depending on what you choose. Does that mean validity is somehow “relative”? Or does every argument have a “correct” set of variable parts?
Bolzano’s definition of analytic truth includes propositions like [Truman, who was a US president, was male]—if you treat [Truman] as variable, every substitution gives a true proposition. But this doesn’t seem “analytic” in the normal sense (it depends on the fact that all US presidents so far have been men). Is Bolzano’s definition too broad? Is there a better way to draw the line?
Grounding seems like it should be important for science—we want to know not just what follows from what, but what explains what. But Bolzano couldn’t define it properly, and philosophers still disagree about it. Is there a real phenomenon here, or are we just confused by language?

Where This Shows Up

Computer science: The idea of treating some parts of a statement as variable and others as fixed is basically what programming languages do with parameters and functions. Bolzano’s variation method is an early version of this.
Artificial intelligence: Modern AI systems that do logical reasoning often use methods that look a lot like Bolzano’s variation approach—checking whether conclusions hold under all possible substitutions.
Mathematics education: When math teachers ask “what happens if you change this number?” they’re using Bolzano’s method without knowing it. The question “what stays the same when you vary this?” is one of the most powerful tools in mathematics.
Your own thinking: Every time you ask “what if?”—substituting different possibilities into a situation to see what follows—you’re doing something like Bolzano’s logic.