Do Truths Exist Before Anyone Discovers Them?
A Truth That Nobody Thinks: Bolzano’s Daring Question
Imagine a fact that no one has ever thought about. Maybe it’s a theorem about numbers that hasn’t been discovered yet, or a statement about a dinosaur that died millions of years before humans. Does that truth already exist? Or does it need a mind to make it real?
A shy priest in Prague named Bernard Bolzano (1781–1848) answered with a bold “yes, it already exists.” He wasn’t just speculating. He spent two decades writing a gigantic work called the Theory of Science (1837) to show how truths can be real even when no one is thinking them. In the process, he built a new kind of logic that still influences how we reason today.
At the heart of his project is a simple idea with a complicated name: propositions in themselves. A proposition is the meaning of a sentence — what it says about the world, something that can be true or false. Bolzano claimed that propositions aren’t just thoughts in someone’s head. They are objective, timeless items, like numbers, that don’t exist in space or time but are still there for us to discover.
Ideas in Themselves: The Invisible Atoms of Thought
If propositions are the meanings of sentences, what are they made of? Bolzano analyzed them into smaller parts he called ideas in themselves. An idea is any piece of a proposition that isn’t itself a full proposition. For example, in the proposition [Fido is a dog], the ideas would include [Fido], [dog], and even [is]. These ideas aren’t pictures in anyone’s imagination; they are the objective content that words point to.
Bolzano believed that all complex ideas are built out of simple ones, just as a molecule is built from atoms. A simple idea, like [red] or [and], can’t be defined by breaking it further — you just have to grasp it. This is called semantic atomism. It’s a bit like saying that every truth, no matter how complicated, is ultimately a construction from a set of primitive building blocks.
Some ideas represent objects, like [Earth] or [cat]; Bolzano called these objectual. Others, like [round square], have no object at all — they are objectless. He also distinguished between concepts (ideas with no intuition in them) and intuitions, which are simple singular ideas that point to a particular thing, like the idea expressed by pointing and saying “this.” Those intuitions are what make a proposition about the physical world, not just about pure concepts.
This might sound abstract, but it sets the stage for something remarkable: a way to test truths without relying on what anyone believes.
The Variation Machine: Testing Truths by Swapping Pieces
Here’s where Bolzano’s logic becomes almost like a game. Take a proposition like [All men are mortal]. Now pretend that [men] and [mortal] are variable ideas — slots that can be filled with other ideas. You might replace [men] with [birds] and [mortal] with [feathered]. The new proposition is [All birds are feathered]. Is it still true? In this case, yes.
Bolzano built an entire method out of this swapping. He called a proposition universally valid with respect to certain variables if, no matter what ideas you plug into those slots (as long as you keep the subject idea objectual), the resulting proposition is always true. If it’s always false, it’s universally invalid. If sometimes true and sometimes false, it’s neutral.
This technique allowed him to define analytic propositions in a new and much wider way. According to Bolzano, a proposition is analytic if it’s either universally valid or universally invalid relative to some variable parts. Unlike Kant, who thought analytic truths were just statements where the predicate was already contained in the subject, Bolzano focused on the fact that in an analytic truth, some parts don’t affect the truth-value at all — they occur, as the logician Quine later said, vacuously.
This means some very surprising statements count as analytic. For example, the proposition [Harry Truman, a 20th-century U.S. president, was male] is universally valid with respect to the variable idea [Harry Truman]. Why? Because the idea [Harry Truman] has exactly one object, and any admissible substitution that preserves objectuality will be another idea that also represents Truman. So the resulting proposition remains true. An everyday historical fact turns out to be analytic in Bolzano’s system — a strange and powerful consequence.
There’s also a narrower kind of analyticity: logical analyticity, where the only invariable parts are logical concepts like and, not, all, and is. These are truths that hold simply because of their logical form, such as [If it’s raining, then it’s raining].
When Truths Force Other Truths: Deducibility
Bolzano didn’t just want to inspect single truths. He wanted to understand how truths relate to each other objectively. He introduced the relation of deducibility (Ableitbarkeit). This isn’t about what we can personally prove, but about what follows from what in the realm of truths themselves.
The definition is breathtakingly precise: a set of propositions M is deducible from a set A with respect to certain variable ideas if, and only if, (1) there is at least one way to substitute ideas for the variables so that all propositions in A and M become true (that’s compatibility), and (2) every substitution that makes all of A true also makes all of M true.
Think of it like a truth-preserving machine. If you know that [All dogs are mammals] and [Fido is a dog], you can treat [dog], [mammal], and [Fido] as variable. Any idea you put in for [dog] and [mammal] that keeps the premises true will force the conclusion [Fido is a mammal] to be true as well. The conclusion is deducible from the premises.
Notice that deducibility doesn’t care about what seems obvious to us. Even if no human ever noticed the connection, the relation would still hold in the timeless world of propositions. It’s an objective network. And it comes in degrees: if the premises aren’t compatible (no substitution makes them all true), nothing is deducible from them. That’s why Bolzano built compatibility right into the definition.
Bolzano also defined a stricter kind of deduction called exact deducibility, where no premise can be left out and no part of any premise is idle — every piece does real work. This led him toward a notion of relevance that later logicians would pursue independently.
The Hidden Architecture of Reality: Grounding and Probability
Deducibility tells you that one truth follows from others, but Bolzano also wanted to know why. He called this deeper relation grounding (Abfolge), the objective connection of ground to consequence. A ground is the truth or truths that explain another truth, not just cause us to believe it, but actually make it true.
Grounding is different from deducibility: a falsehood can be deduced from a falsehood, but grounding only holds among true propositions. And while deducibility can go in circles, grounding is anti-symmetric — if A grounds B, B cannot ground A. Bolzano believed that eventually all truths trace back to basic truths, which have no grounds themselves, like the truth [There is something]. From these, a vast tree of supporting truths branches out.
In many cases, especially in mathematics, grounding and deducibility overlap. Bolzano called this formal grounding. He proposed tentative rules for when a deduction is also an explanation: the premises must be the simplest equivalents, and none can be more complex than the conclusion. The idea was that the best explanation is also the most efficient deduction.
Remarkably, Bolzano’s logic of variation also gave birth to the first logical definition of probability. He asked: out of all the substitutions that make the premises true, what fraction also make the conclusion true? That ratio is the conditional probability of the conclusion given the premises. If the premises are [The number of eggs in the nest is between 1 and 10] and [The number is odd], and the conclusion is [The number is prime], then among the admissible numbers (1–10) that are odd, 3 out of 5 are prime: probability 0.6. Deductive certainty is just the extreme case where the ratio equals 1.
This meant that for the first time, deductive logic and inductive reasoning were two sides of the same coin. It was a huge step toward modern probability theory.
Why Bolzano Still Matters to You
You might think all this talk of timeless propositions is just a dusty 19th-century curiosity. But Bolzano’s ideas are everywhere. His definition of logical consequence influenced Alfred Tarski, whose work underpins the semantics of formal languages. His substitutional approach prefigures the way computer scientists test program correctness. The notion that truths can be analytic without being obvious still fuels debates in philosophy.
His probability theory showed that the same pattern of thinking that gives us ironclad proofs also lets us weigh uncertain evidence — exactly what you do when you decide whether to trust a weather forecast or a friend’s story.
Bolzano even changed what it means to do logic. Instead of studying how people actually think, he studied the objective structure of truth itself. He showed that you can treat a whole branch of knowledge, like geometry or biology, as a self-contained web of propositions that hold together in a discoverable order. That spirit of mapping reality through pure reasoning is at the heart of modern science and philosophy.
Next time you solve a math puzzle or check your phone’s weather app, remember the quiet priest in Prague who believed that truths wait for us, like stars in a night sky, whether or not we ever look up.
Think about it
- If every truth already exists in a timeless realm, does that mean future discoveries are just “uncovering” what was always there, or do we create new truths when we invent things?
- Could a computer, using Bolzano’s idea of deducibility, ever discover every possible truth from a small set of basic truths? Why or why not?
- Suppose two people disagree about whether a statement is true, but the statement is analytic according to Bolzano’s definition. Does that settle the disagreement, or can reasonable people still argue?
