The Puzzle That Taught Philosophers How Knowledge Changes
The Muddy Children: How Can You Know Without Looking?
Imagine three children playing outside. They get mud on their clothes, and some on their foreheads too, though none can see their own face. Their father calls them inside and says: “At least one of you has mud on your forehead.” The children look at each other. Each can clearly see the foreheads of the other two.
Then father adds a rule: “If you know whether your forehead is dirty, step forward now.” Nobody moves. He says it again. Now a couple of children step forward. He says it a third time, and the rest step forward. How many children had muddy foreheads?
The answer is two. But why? To solve it, you have to think about what the children know—and what they know about what each other knows. At first, none of them steps forward. If only one child had been muddy, that child would have seen two clean foreheads and immediately known it must be them, so they’d have stepped forward. Because nobody moved, every child can rule that out: they all now know there are at least two muddy foreheads. On the second request, if a child saw one muddy forehead and one clean one, they’d realize their own must be muddy too, and step forward. Two do exactly that. The remaining child, seeing both others step forward, now knows they themselves are clean. On the third request, that child steps forward. The puzzle resolves entirely through changes in knowledge triggered by just a few public words and the observation that others didn’t move.
This kind of reasoning fascinated philosophers and logicians. They asked: can we build a precise logic that explains how our knowledge shifts whenever we hear something new—especially when what we hear is completely trustworthy? That is the question at the heart of Dynamic Epistemic Logic (DEL).
When Everyone Hears the Same Thing: Public Announcements
The simplest form of DEL is Public Announcement Logic (PAL). It studies what happens when a truthful announcement is made so that everyone hears it, everyone knows everyone heard it, and no one doubts it. PAL was developed by researchers including Jan Plaza in 1989.
Think of your own knowledge as a set of possible situations. Before you know a friend’s birthday, you imagine many dates. When the friend tells you, “It’s in July,” you immediately delete all the non-July dates from your mental list. That’s exactly how PAL works—but with formal precision.
Philosophers use a diagram called a Kripke model (named after Saul Kripke, born 1940) to picture possibilities and who considers which possibilities possible. Each world is a different way things could be. Arrows between worlds show an agent’s uncertainty: if an arrow points from world A to world B, then the agent in world A thinks world B might be the real one. When a public announcement happens, any world where the announced statement is false gets deleted—along with all its incoming and outgoing arrows. The remaining worlds now represent the new, updated knowledge.
A famous puzzle that works on exactly this principle is Cheryl’s Birthday. Cheryl gives Albert and Bernard a list of ten possible dates and secretly tells Albert the month, Bernard the day. Through a short public conversation—Albert saying he doesn’t know the date and knows Bernard doesn’t either, Bernard saying he now does know, and Albert finally saying he also knows—they eliminate possibilities until only one date remains. PAL shows that every line of dialogue acts as a public announcement that updates the model, trimming away impossible worlds.
The key insight is: knowledge doesn’t just sit still. It changes, and a public announcement provides a clean, well-defined rule for how it changes—cut out all the worlds where the announcement would be false.
Secrets and Mixed Signals: Action Models
Public announcements are just the beginning. In real life, people often share information privately or misleadingly. DEL handles this with action models, introduced by Alexandru Baltag, Larry Moss, and Slawomir Solecki in 1998.
An action model is like a tiny Kripke model, but instead of possible worlds, it contains possible events that might occur—each labeled with the formula being announced. For example, imagine you’re told a secret in private. The actual event is the announcement of the secret to you. But someone else, who doesn’t hear it, thinks the event is simply a boring “nothing new is announced” message. The model captures both: the real event, plus the alternative event others mistakenly believe is happening. Arrows show who thinks which event might be the actual one.
A private announcement of a fact p to agent a uses an action model with two events: one where p is announced, and one where a trivial truth (⊤) is announced. Agent a has an arrow only to the p event, so she knows it’s happening. All other agents have arrows only to the trivial event, so they think nothing interesting was said. The model gets “multiplied” with the current Kripke model (a process called product update) to yield a new situation where a knows p while others remain clueless.
Even more mischievous is the misleading private announcement: a learns p, but everyone else is tricked into believing that a ¬p private announcement occurred. The action model contains three events—one real (p) and two fake (¬p and a decoy). Non-a agents see only the ¬p version as possible, so they come away with entirely wrong beliefs about what a knows.
These examples show that DEL can model not just straightforward announcements, but also the intricate patterns of who knows what, who is mistaken, and how misdirection works. By drawing simple diagrams of events and arrows, philosophers can make precise predictions about the knowledge and beliefs that arise after complex communication.
When What You Hear Doesn’t Fit What You Believe
All the logics we’ve seen assume announcements are always trustworthy and never conflict with what agents already believe. But real life is messier. You might hear something that contradicts your existing beliefs—and you have to decide how much to trust it. This is the problem of belief revision.
A classic example is a Moore sentence (named after philosopher G. E. Moore): “It is raining, but you don’t know it.” If you are told this truthfully, you immediately learn it’s raining—which falsifies the second part (“you don’t know it”). The announcement makes itself false! In DEL terms, the formula is unsuccessful because after you hear it, the world where it was true no longer exists.
To handle such cases, DEL uses plausibility models instead of plain Kripke models. In addition to worlds and arrows, these models rank worlds by how plausible an agent considers them. The most plausible worlds form the agent’s current beliefs, while less plausible ones represent alternative possibilities the agent keeps in the background, in case new evidence arrives.
When new information contradicts an agent’s beliefs, there are two ways to revise. In static belief change, you update your beliefs about a fixed moment in the past—like learning from an old photograph what you used to think. Your present beliefs need not change. In dynamic belief change, the situation itself evolves, and you adjust your beliefs for the present. For example, if a trusted friend tells you the party is cancelled, you now believe it’s cancelled, and the world changes accordingly.
DEL formalizes such changes through action-priority update. An action model (with plausibility rankings on its events) combines with a plausibility model, and the new plausibility ordering gives priority to the action events. A gentle announcement like a lexicographic upgrade (“I heard q, but maybe it’s false”) can be captured by an action with two events: one where q is announced (more plausible) and one where ¬q is announced (less plausible). Agents end up believing q but still consider the possibility they’re wrong.
This approach, developed in detail by Alexandru Baltag and Sonja Smets in the 2000s, lets us model how we trust rumors, change our minds after a news alert, or recalibrate when we discover a friend has been misled.
Why It Matters: How We Update Our Minds Every Day
You’re constantly updating your knowledge. A text from a friend about where to meet, a weather alert on your phone, a teacher correcting a fact in class—each piece of information changes what you think the world is like. Understanding that process isn’t just a puzzle for logicians; it helps us see why gossip spreads the way it does, why misunderstandings happen, and how we come to trust (or distrust) sources.
The logics of DEL have been used to analyze everything from famous puzzles to real-world miscommunication scenarios. They also underpin some computer systems that need to track what users know, like secure chat protocols or AI programs that update their own “beliefs” about a changing situation.
At its core, DEL tells us that knowledge is not a fixed bucket. It’s a living structure that shifts every time we talk, listen, or observe. And by studying that shift with rigorous diagrams and rules, we learn something surprising: the way we think can be captured—and clarified—by logic.
Think about it
- In the muddy children puzzle, what would happen if the father had said “At least two of you have mud” instead? Would the children figure things out sooner, or would they get stuck? Try reasoning through it with two friends.
- Think of a time when you were certain about something and then found out you were wrong. When you received the new information, did you immediately trust it completely, or did you stay a little skeptical? How might that process be drawn as a plausibility model?
- If a secret message can be modeled with action models, could a secret lie be modeled as well? What would have to change in the diagrams for an agent to be intentionally misled by a false announcement?
