In possible worlds semantics, what does it mean for Kₐp ('agent a knows that p') to be true at world w?
Ap is true at world w, and agent a believes that p
Bp is true at every world that agent a considers possible from w
Cp is true at some world accessible to agent a from w
DAgent a has a justified true belief in p according to traditional epistemology
Kₐp is true at w just when p holds at every world in a's accessibility set from w — every world a cannot rule out. This is why knowledge requires p to be settled across all epistemic possibilities: if there is even one accessible world where p is false, the agent doesn't know p. Option A confuses the modal-logic framework with JTB epistemology. Option C is the condition for possibility (◇p), not knowledge. The universal quantifier over accessible worlds is what makes K behave like the necessity operator □.
Question 2 Multiple Choice
An agent knows p but does NOT know that she knows p. Which axiom of S5 epistemic logic does this situation violate?
AThe T axiom (Kₐp → p), because knowledge must be factive
BThe 4 axiom (Kₐp → KₐKₐp), the positive introspection axiom
CThe 5 axiom (¬Kₐp → Kₐ¬Kₐp), the negative introspection axiom
DThe K axiom (Kₐ(p→q) → (Kₐp → Kₐq)), the distribution axiom
The 4 axiom (Kₐp → KₐKₐp) states that if you know p, you know that you know p — positive introspection. If an agent knows p but doesn't know she knows p, this axiom is violated. Students often confuse the 4 and 5 axioms: the 5 axiom concerns *negative* introspection (not knowing → knowing you don't know). Dropping the 5 axiom while keeping T and 4 gives S4, often used for belief, where you can fail to recognize your own ignorance.
Question 3 True / False
In epistemic logic, if a proposition p is actually true in the world, then nearly every agent in the system knows p.
TTrue
FFalse
Answer: False
The T axiom runs in only one direction: knowledge implies truth (Kₐp → p). Truth does not imply knowledge. An agent may be in a world where p is true but still have accessible worlds where p is false — those worlds are ones the agent cannot rule out. In the card example from the explainer: the card being red is true, but before you see it, you don't know it because red-false worlds remain epistemically accessible to you.
Question 4 True / False
Common knowledge of p requires more than all agents individually knowing p — it additionally requires each agent to know that every other agent knows p, and this iteration continues infinitely.
TTrue
FFalse
Answer: True
Common knowledge CK(p) is an infinite conjunction: everyone knows p, AND everyone knows that everyone knows p, AND everyone knows that everyone knows that everyone knows p, and so on. This is strictly stronger than 'everyone knows p.' The muddy children puzzle illustrates why: even when all children can see which children are muddy (so each knows the relevant facts), no child can act on this until a public announcement creates the infinite chain of mutual knowledge.
Question 5 Short Answer
Explain what the accessibility relation represents in epistemic logic and how it encodes what an agent knows.
Think about your answer, then reveal below.
Model answer: The accessibility relation for agent a at world w is the set of worlds that a cannot distinguish from w — worlds that, from a's perspective, could be the actual world. An agent knows p at w if and only if p is true at every world in this set. A larger accessibility set means the agent knows less (more possibilities remain open); a smaller set means more is settled.
This is the central mechanism of the semantics. Gaining knowledge corresponds to shrinking the accessibility set: new information eliminates worlds that are now inconsistent with what the agent has learned. The axioms T, 4, and 5 correspond to structural properties of the accessibility relation: T requires reflexivity (the actual world is always accessible — you can't rule out reality), 4 requires transitivity, and 5 requires symmetry (Euclidean property), making the relation an equivalence relation in S5.