The warehouse fire

A warehouse burns down. The expert names a short circuit as the cause. The statement sounds simple. It becomes difficult as soon as one asks what it claims.

The short circuit is not sufficient for the fire. The table on the right records nine test situations. In situation 5 there is a short circuit but no oxygen, and there is no fire. The short circuit would be sufficient only if a fire followed it in every situation (Vol. III, §7.1).

Nor is it necessary. In situation 3 there is a fire without a short circuit: there a combustible liquid, a spark and oxygen come together. The short circuit would be necessary only if there were never a fire without it (Vol. III, §7.1).

Neither sufficient nor necessary, and yet we rightly say that it caused the fire. The following chapters show what it is instead.

The short circuit is sufficient together with oxygen. In every situation in which both are present (1 and 2), there is a fire (Vol. III, §7.1).

Neither of the two parts may be missing. Without oxygen (situation 5) there is no fire despite the short circuit, without a short circuit (situation 6) none despite the oxygen. The combination is therefore minimally sufficient: sufficient, and none of its parts can be removed without its ceasing to be sufficient (Vol. III, §7.1).

Such a minimally sufficient condition is here called a cluster. In formulas it is written SC * OX; the star reads “and”. The short circuit is an indispensable part of this cluster. That is the first half of what makes up its causal relevance.

Situation 3 has shown that there can be a fire without a short circuit, too. A combustible liquid, a spark and oxygen are sufficient together: where all three are present (2 and 3), there is a fire (Vol. III, §7.1).

Here too every part is indispensable. Situation 7 has liquid and oxygen without a spark, situation 8 a spark and oxygen without liquid, situation 9 liquid and a spark without oxygen. In none of them is there a fire. FL * SP * OX is a second cluster, an alternative to the first. Alternatives are shown in the diagram by separate arcs, in the formula by the plus, which reads “or” (Vol. III, §7.1).

Oxygen belongs to both clusters. It is a partner in every way in which a fire comes about here.

Together the two clusters cover every fire in the table: situation 1 through the first, situation 3 through the second, situation 2 through both. The disjunction of the clusters is therefore necessary for the fire (Vol. III, §7.1).

And it is minimally necessary: neither of the two clusters can be removed. Without the first, the fire in situation 1 would remain uncovered; without the second, the one in situation 3. A disjunction of minimally sufficient conditions that is minimally necessary in this sense is called a minimal theory of the effect (Graßhoff and May 2001) (Vol. III, §7.1).

Now the expert's question can be answered. A factor is causally relevant to an effect if and only if, for every set of factors that contains it and the effect, there is a minimal theory of the effect formed from that set in which its presence or its absence is a condition of a cluster. For the factor set of this table the requirement is met: the short circuit is a condition of the cluster SC * OX. Whether it is met for every other set of factors no single table decides; every further survey with additional factors tests it.

The definition needs neither counterfactual assumptions nor interventions nor probabilities. It requires regularities between kinds of events and a double minimisation: of the clusters and of their disjunction. The case of the Manchester hooters shows how important the second half is.