 Solution

A philosophical account of scientific explanation must accommodate causal explanations, but there is no good reason to insist that every explanation be causal or that every one explicitly contain an exceptionless universal law. The issue is what explanatory task an account performs, rather than whether it fits one syntactic template.

The deductive-nomological model explains by deriving a phenomenon from scientific laws and specified circumstances. It illuminates a real feature of physical reasoning: a result is intelligible because it follows from a more general structure, not merely because it has been listed. Yet valid derivability does not automatically supply the direction or relevance of explanation.^[ref]

For a vertical pole of height $h$ in a simple geometrical model, a solar elevation $\alpha$ gives a shadow length $s=h\cot \alpha$. Algebra permits inference from $h,\alpha$ to $s$, or from $s,\alpha$ to $h$. The equation alone does not decide which quantities are causal inputs. A causal explanation must supplement the relation with how the pole, illumination and surface generate the shadow, and with the changes meant by an intervention. Mere reversal of a calculation is not reversal of a physical mechanism.

Conversely, some explanations appeal to constraints rather than a mechanism producing an event. Twenty-three objects cannot be divided into three equal groups of whole objects because $3$ does not divide $23$. The relevant divisibility fact explains an impossibility without describing a cause that prevents each attempted arrangement. Physical applicability still requires that the objects count as indivisible for the task; the arithmetic alone is not a complete theory of the material.

Statistical explanation supplies another limit to a universal deductive requirement. A radioactive decay model gives a hazard rate and the probability $1-e^{-\lambda t}$ of decay within time $t$. It can explain rates and the distribution of waiting times without deductively fixing a particular atom's decay instant. A very unlikely event can still have a recognized mechanism. One must distinguish explaining why an event was possible or had a certain chance from explaining why this event occurred at precisely this time.

A demand for explicitly stated fundamental laws can also be too restrictive in a local mechanistic explanation. Identifying a broken circuit connection and the resulting interruption of current may answer a practical question without deriving the fault from microscopic dynamics. The explanation depends on physical regularities and background assumptions, but dependence is weaker than requiring every law to be displayed in the explanatory argument.

A useful plural account therefore asks whether the explanation identifies relevant dependence, states its domain and answers the actual contrastive question. Causal explanation, mathematical constraints, statistical accounts and explanatory unification can each contribute. This is not permission for arbitrary stories: relevance, independent support and the distinction between prediction and explanation remain constraints on all of them.