Possible worlds

The whole field rests on one move. To say what an agent knows, do not describe the agent's head — describe the situations it cannot rule out.

The move

Suppose a coin has been flipped and covered. It landed heads. Alice did not see it.

We could try to model Alice's ignorance by giving her a "knows-heads" flag set to false. That works until you ask a second question — does she know it is tails? Also false. Does she know it is heads-or-tails? True. Three flags now, and they have to be kept consistent with each other by hand.

Instead, list the situations Alice considers possible:

    w1: heads          w0: tails

She cannot tell them apart, so both are live for her. Now every question answers itself:

• Does Alice know it is heads? Only if heads is true in every world she considers possible. It is false in w0, so no.
• Does she know it is heads-or-tails? That holds in both, so yes — for free, with no extra bookkeeping.
• Is she ignorant of which? Yes: neither heads nor ¬heads holds throughout.

The consistency comes from the structure rather than from discipline. That is the payoff.

The three parts

A model is three things:

• Worlds — the situations in play. One of them is the actual one; the rest are there because somebody cannot rule them out.
• A valuation — which propositions are true in each world.
• A relation, per agent — which worlds that agent connects to which. Alice's relation links w1 and w0 because she cannot distinguish them.

Each agent gets its own relation. That is what lets Bob know the coin while Alice does not: Bob's relation links each world only to itself, so from w1 he sees only w1.

state {
  *w1 <- { heads }      // `*` marks the actual world
   w0 <- { }
  alice: w0 ~ w1        // alice cannot tell these apart
}                       // bob relates each world only to itself

That is real delhi syntax — delhi show prints models in exactly this form. Bob needs no line at all: with nothing declared, he distinguishes everything.

Reading the picture

Two habits to build, because they are what make the diagrams say anything.

Knowledge is what survives every arrow. To evaluate "Alice knows φ", stand in the actual world, follow every one of Alice's arrows, and check φ in each place you land. One counterexample is enough to defeat the claim. This is why knowledge is expensive: it takes all the possibilities agreeing.

More arrows mean less knowledge. An agent who connects everything to everything knows nothing beyond what is true everywhere. An agent whose arrows go only from each world to itself knows the actual world exactly. Learning something means deleting arrows.

That second habit is worth sitting with, because it inverts the intuition. Information does not add to the model. It cuts it down.

Where this is going

So far every arrow has meant "cannot tell these apart" — a symmetric, all-or-nothing sort of uncertainty. That gives you knowledge, and only knowledge.

Belief needs more, because a believer is not merely uncertain: they lean. Alice may consider tails possible while finding heads far more plausible, and if you tell her the coin was tails she should be able to change her mind without ever having been logically wrong. Plain arrows cannot express leaning.

The next chapter adds the missing structure — an ordering on the arrows — and shows why knowledge and belief end up obeying genuinely different laws.