Past exam of the philosophy of science supplementary subject of the University of Oxford / 2015 / 7

The deductive-nomological model represents an explanation as a valid deduction of an event description from scientific laws and particular circumstances. In the schemathe premises are the explanans and $E$ the explanandum. At least one law must be essential to the derivation, the premises must have empirical content, and a correct explanation must have true premises. Carl Hempel and Paul Oppenheim distinguish these adequacy requirements from merely giving a valid argument. Their developed 1948 analysis concerns deductive explanation; its name should not be extended without qualification to every statistical explanation.^[ref]

The model captures something valuable. For an ideal simple harmonic oscillator, the equation of motion $m\ddot{x}=-kx$, a positive mass $m$, a positive spring constant $k$ and non-equilibrium initial conditions determine a periodThe derivation explains why the period changes with those parameters within the model. It is not just the report that an oscillation occurred. Since a real spring may have damping or nonlinear response, applying the explanation requires justification of the approximation. The model's truth condition cannot turn an idealized calculation into a literally exact account of every laboratory spring.

The account is not sufficient to settle explanatory direction. For a vertical pole of height $h$ on level ground, a solar elevation angle $0<\theta <\pi /2$ and parallel light rays give a shadow length $\ell =h\cot \theta$. Given $h$ and $\theta$, one can deduce $\ell$; given $\ell$ and $\theta$, one can equally deduce $h$. Both calculations can use the same true optical and geometrical assumptions. Yet normally the pole and illumination explain the shadow, whereas measuring the shadow provides evidence about the pole's height. Deducibility does not distinguish this evidential inference from a causal explanation. This is the problem of explanatory asymmetry.

Nor is strict deduction necessary for every good physical explanation. A radioactive decay model can explain a sample's decay rate and the chance of a nucleus decaying in a specified interval, while not entailing which particular nucleus decays first. The exponential distribution supplies a controlled statistical account. Adding the observed result as a premise would permit a deduction but would not supply the missing explanation. An inductive-statistical model is a different proposal, and even that needs an account of which statistical information is relevant.

These objections do not show that every explanation must instead recount a sequence of causes. Explanations based on conservation laws or symmetry can constrain a phenomenon without specifying a detailed mechanism. The important point is that the role of the premises, the question being asked and the relevant dependence matter, in addition to logical consequence. The deductive-nomological model is a useful pattern for some physical explanations, but it is not an adequate necessary-and-sufficient account of explanation throughout the physical sciences.