Can a Lie Be More Informative Than the Truth?
The Guessing Game
You and a friend are playing “Guess the Number.” Your friend thinks of a number from 1 to 100. You ask questions: “Is it even?” “Is it less than 50?” Each answer shrinks the list of possible numbers. The more the answer rules out, the more you learn. If your friend simply says, “It’s a number,” you learn nothing — that rules out zero possibilities. But if she says, “It’s 42,” you instantly eliminate every other number. That one short sentence is packed with information.
In 1952, two philosophers — Rudolf Carnap (1891–1970) and Yehoshua Bar‑Hillel (1915–1975) — had the same insight about sentences in general. They wanted a precise way to measure the semantic information carried by a sentence: how much it tells you about the world, regardless of who hears it or what they already know. Their key idea, now called the inverse range principle, was simple: the amount of information a sentence gives you is inversely proportional to how likely that sentence is to be true. The rarer the truth, the bigger the payoff in information.
Possible Worlds and Probability
To turn that gut feeling into numbers, Carnap and Bar‑Hillel imagined all the different ways the world could be — for them, every complete combination of basic facts. We can think of these as possible worlds. A sentence’s content, written Cont(s), is simply the set of possible worlds where that sentence is false. The larger that set, the more the sentence rules out.
If we assign each possible world an equal chance of being the real one, we can attach a probability p(s) to the sentence — the fraction of worlds in which it is true. Then the content measure of information becomes:
cont(s) = 1 − p(s)
A sentence that is true in almost all worlds, like “The sun will rise tomorrow,” has a probability near 1, so its cont is tiny. A very specific sentence, like “It will rain between 3:02 and 3:03 p.m. on this exact spot,” has a very low probability, so it carries a lot of information by this measure. Carnap and Bar‑Hillel also defined a second measure, inf, using logarithms, to handle cases where two sentences are inductively independent — meaning each one gives you no extra clue about the other. The details are technical, but the point is that both measures capture the deep link between surprise and information.
Their framework required a language with only finitely many basic predicates, so the number of possible worlds stayed finite. Later philosophers extended it to richer languages, but the core idea held.
Two Puzzles That Won’t Go Away
As soon as Carnap and Bar‑Hillel published their theory, two big philosophical puzzles appeared. The first is now called the Bar‑Hillel–Carnap Semantic Paradox (BCP). A self-contradiction, like “It is raining and it is not raining,” is false in every possible world. According to their definitions, its content is the set of all worlds — maximal content. Its cont is 1, and its inf is infinite. That makes a contradiction the most informative sentence possible. But that seems absurd: if someone tells you a blatant contradiction, you haven’t learned anything about the real world. You’ve just heard a jumble.
Carnap and Bar‑Hillel shrugged it off by saying that their notion of semantic information does not require truth. A false sentence that “says much” is still highly informative in their sense — it’s just “too informative to be true.” Many philosophers found that reply unsatisfying. For them, information should help you get closer to reality, not just scream the loudest.
The second puzzle is the Scandal of Deduction. Logical truths, like “Either it is raining or it is not raining,” are true in every possible world. Their content is empty — they rule out nothing. So their cont is 0 and inf is 0. But we often feel we learn something when we work through a long logical proof or a math deduction. Carnap and Bar‑Hillel answered that logical truths might carry psychological information, which can vary from person to person, but they carry zero semantic information. Still, the question lingers: if logic gives you no new information about the world, why does it sometimes feel so enlightening?
Floridi and the Veridicality Thesis
The contemporary philosopher Luciano Floridi looked at the paradoxes and decided something had gone wrong. He proposed that real semantic information must be factive — it has to be true. This requirement is known as the veridicality thesis. If a sentence is false, no matter how many possibilities it rules out, it simply doesn’t count as semantic information at all. It might be highly informative (it could still shape your thinking), but it’s not the kind of objective information that gets you closer to how things actually are.
Floridi measures the informativeness of a true sentence by how close it comes to describing a specific situation perfectly — he calls the situation a target situation z. Instead of probability, he uses distance: how far the sentence’s claim is from z, within the agreed level of detail. A true sentence that nails the situation exactly (like “There will be exactly 200 guests”) is maximally informative. A true sentence that remains hazy (like “There will be between 100 and 200 guests”) is less informative because it fits many more possible situations. The formula he gives is:
ι(σ) = 1 − Θ(σ)²
where Θ(σ) is the distance from the true sentence σ to z. Values range from 0 (vague) to 1 (precise). Crucially, a false sentence is simply not on this scale — it has zero semantic information, even if we find it useful.
Consider a wedding planner. She is told there will be 201 guests (false, actually 200), or “between 100 and 200” (true). The false figure is far more informative by any everyday standard, yet Floridi’s theory says it carries no semantic information. That sharp division — informative versus semantically informative — lets him avoid the paradox of contradictions. A self-contradiction is false everywhere, so it falls entirely outside the realm of strongly semantic information. It has no content worth measuring.
How Meaning Rides on Information
The search for a theory of what sentences mean has often circled back to information. Why does the word “cat” mean cat? One tempting answer: because our encounters with cats — seeing one, hearing one, thinking about one — reliably cause us to use the word “cat.” The meaning just is the information the term carries about its source. But there’s a snag. Sometimes a small dog or a ball of yarn makes us say “cat” by mistake. So why doesn’t “cat” mean cat‑or‑dog‑or‑ball‑of‑yarn? This is known as the disjunction problem.
Philosophers like Jerry Fodor (born 1935) and Fred Dretske (1932–2013) argued that meaning and information must partially separate. The true meaning of “cat” is what all correct uses have in common — a kind of mental label, or what Frege called a mode of presentation. Mistaken uses are possible only because correct uses exist first; they are asymmetrically dependent on the world-to-mind link that real cats provide. So the information a particular token carries may vary, but the meaning stays anchored.
Gareth Evans (1946–1980) added a twist. He thought that for a thought to be about a particular thing in the world, two things must line up: the object at the far end of the causal chain, and the object that fits the thinker’s mental “file” about it. Think of a detective building a file on a suspect. If the file’s description matches the person who is actually supplying the clues, the thought is well‑grounded. Later philosophers, like François Recanati, have developed the idea of mental files that store all the information we gather about one object, whether true or false. The file’s reference is fixed not by the info inside it, but by the information channel — the causal connection that makes the flow of information possible.
These debates show that long before a sentence hits the page, the very building blocks of meaning are shaped by the flow of information through the world.
Why Spotting Real Information Matters Now
Every day you navigate a flood of statements — in group chats, news feeds, school, and family conversations. Some are true, some false, some wildly precise, some uselessly vague. The puzzles uncovered by Carnap, Bar‑Hillel, and Floridi are not just ivory‑tower curiosities. They ask: when you learn something, does it have to be true to count as real information? And can a statement that is utterly false — a rumor, a clever lie — still shape your picture of the world more effectively than a safe, true one?
Think back to the guessing game. If your friend lies and says “It’s 42” when it’s really 37, you will act on precise, useful guidance that turns out to be wrong. You learned nothing accurate, yet your mental map narrowed dramatically. That tension lives inside every fact‑check, every social media post you evaluate, and every decision you make on incomplete information. Philosophers don’t have a final answer to whether information must be true, but their arguments give you sharper tools for asking the question yourself.
Think about it
- If a perfectly false statement can be more useful than a vague truth, is it ever okay to lie to someone in order to help them learn something important?
- When you work through a long math proof and finally see why the theorem holds, do you gain new information about the world, or only about your own thoughts?
- Imagine a chatbot that randomly guesses an answer and happens to be right. Did it give you information? What if it gave the same answer using a reliable calculation — does the information change?
