science

Mutual knowledge (logic)

Mutual knowledge (logic) is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Mutual knowledge (logic) rather than just read about it. In short: Mutual knowledge is a fundamental concept about information in game theory, (epistemic) logic, and epistemology. An event is mutual knowledge if all agents know that the event occurred.

Key takeaways

Mutual knowledge (logic) belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Mutual knowledge (logic) to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Mutual knowledge (logic) from memory before moving on to harder problems.

Reference excerpt

Mutual knowledge is a fundamental concept about information in game theory, (epistemic) logic, and epistemology. An event is mutual knowledge if all agents know that the event occurred. However, mutual knowledge by itself implies nothing about what agents know about other agents' knowledge: i.e. it is possible that an event is mutual knowledge but that each agent is unaware that the other agents know it has occurred. Common knowledge is a related but stronger notion; any event that is common knowledge is also mutual knowledge. The philosopher Stephen Schiffer, in his book Meaning, developed a notion he called "mutual knowledge" which functions quite similarly to David K. Lewis's "common knowledge". Communications (verbal or non-verbal) can turn mutual knowledge into common knowledge. For example, in the Muddy Children Puzzle with two children (Alice and Bob, G = { a , b } {\displaystyle G=\{a,b\}} ), if they both have muddy face (viz. M a ∧ M b {\displaystyle M_{a}\land M_{b}} ), both of them know that there is at least one muddy face. Written formally, let p = [ ∃ x ∈ G ( M x ) ] {\displaystyle p=[\exists x\!\in \!G(M_{x})]} , and then we have K a p ∧ K b p {\displaystyle K_{a}p\land K_{b}p} . However, neither of them know that the other child knows ( ( ¬ K a K b p ) ∧ ( ¬ K b K a p ) {\displaystyle (\neg K_{a}K_{b}p)\land (\neg K_{b}K_{a}p)} ), which makes p = [ ∃ x ∈ G ( M x ) ] {\displaystyle p=[\exists x\!\in \!G(M_{x})]} mutual knowledge. Now suppose if Alice tells Bob that she knows p {\displaystyle p} (so that K a p {\displaystyle K_{a}p} becomes common knowledge, i.e. C G K a p {\displaystyle C_{G}K_{a}p} ), and then Bob tells Alice that he knows p {\displaystyle p} as well (so that K b p {\displaystyle K_{b}p} becomes common knowledge, i.e. C G K b p {\displaystyle C_{G}K_{b}p} ), this will turn p {\displaystyle p} into common knowledge ( C G E G p ⇒ C G p {\displaystyle C_{G}E_{G}p\Rightarrow C_{G}p} ), which is equivalent to the effect of a public announcement "there is at least one muddy face".

See also Elephant in the room The Emperor's New Clothes Distributed knowledge

External links Steven Pinker. "The Stuff of Thought: Language as a window into human nature". RSA.

References

Worked examples

Example 1 — a first encounter with Mutual knowledge (logic)

Start with the simplest possible case. Write down what Mutual knowledge (logic) claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Mutual knowledge (logic) before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Mutual knowledge (logic) ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Mutual knowledge (logic)

In research: Mutual knowledge (logic) appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Mutual knowledge (logic) in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Mutual knowledge (logic) is common in secondary-school and first-year university syllabi. It links to neighbouring topics Concepts in epistemology, Epistemic logic, Game theory, so understanding it makes those chapters shorter.
In everyday life: Look for Mutual knowledge (logic) outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Mutual knowledge (logic) in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Mutual knowledge (logic) means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Mutual knowledge (logic) out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Mutual knowledge (logic) in simple terms?

Mutual knowledge is a fundamental concept about information in game theory, (epistemic) logic, and epistemology. An event is mutual knowledge if all agents know that the event occurred.

Why does Mutual knowledge (logic) matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Mutual knowledge (logic)?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Mutual knowledge (logic).