Reading the diagrams

A diagram shows factors as boxes. Each box carries a letter and, beneath it, the name of the factor. A factor is present or absent in a case.

An arrow runs from a factor to the effect. The arrow from S to ST stands for a cluster of a single condition: in every case in which Suzie throws, her stone hits.

Here two arrows lead to the effect, from the short circuit and from oxygen. A golden arc ties them into one cluster: only both conditions together are sufficient for the fire.

A cluster can have more than two conditions. Here flammable liquid, spark and oxygen stand under one arc: only all three together are sufficient. The arc spans all arrows of the cluster and only these. An arrow without an arc is, as in the first chapter, a cluster of a single condition.

Both clusters now stand in one diagram. There is no arc between them: different clusters are alternatives. Each is sufficient for the fire on its own, and the fire can occur through the one or the other.

This or-combination is the minimal theory of the fire: in every case with a fire at least one cluster holds, and neither cluster can be struck out. The diagram shows it completely: one arc per cluster, no arc in between.

A condition of a cluster can also be the absence of a factor. Billy's stone hits the bottle only if Billy throws and Suzie's stone has not already hit it. The box with ¬ on the arrow from ST shows that the condition here is not the presence of ST but its absence.

The box belongs to exactly one arrow. The other conditions of the cluster, here B, are ordinary arrows under the same arc.

All heads are hollow. A hollow head stands for a regularity over kinds of events: if Suzie throws, her stone hits; if either stone hits, the bottle shatters. This holds on every day, not on a particular one.

Now a single day: Suzie and Billy both throw. The shaded boxes are present on this day, the dashed box is absent: Billy's stone does not hit.

Filled heads show the case. They mark the clusters whose conditions are all met on this day: Suzie's throw, her hit and, through it, the shattering. Billy's cluster does not belong to them, because the condition "Suzie's stone has not hit" is missing. Hollow means: kinds of events. Filled means: this case.

Every diagram can be translated into a formula and every formula into a diagram. Each arc is a cluster, the arrows beneath it are its conditions, joined by stars. The clusters stand side by side, joined by plus signs, and the arrow ⇒ leads to the effect.

The diagram also shows which factors are causally relevant for the recorded set of factors: each one whose presence or absence is a condition of a cluster of the minimal theory. In the picture that is every factor from which an arrow starts. The short circuit, oxygen, flammable liquid and spark are such factors.

What the picture does not show: it draws the minimal theory for the recorded factors. Whether a factor is causally relevant requires more, namely that for every set of factors that contains it and the effect there is such a theory. That is what the definition on the home page says, not the single diagram.