Past exam of the philosophy of science supplementary subject of the University of Oxford / 2021 / 2

Old evidence challenges a simple account of confirmation as actual conditional updating, but does not refute every probabilistic treatment of the raven paradox. The strongest response distinguishes support from newly acquired information and makes the sampling procedure explicit.

Let $H$ say that every raven is black. Its contrapositive says that every nonblack thing is a nonraven. Since these statements are logically equivalent, evidence should support them equally. Combining this equivalence requirement with the unrestricted rule that every positive instance confirms a universal generalization appears to make discovering a white shoe support $H$. The tension is between that result and the judgement that such a discovery is irrelevant.^[ref]

Bayesian confirmation replaces an automatic instance rule with a comparison of likelihoods. For background $K$, positive prior probability for both $H$ and its alternative, and evidence $E$ of positive probability,The equivalence follows directly from Bayes' theorem. A statement's grammatical appearance as a positive instance does not determine either likelihood.

Consider an intentionally simple finite model. There are $r$ ravens and $M$ nonblack objects, with these counts fixed under both hypotheses. Under $H$ none of the nonblack objects is a raven; under the sole alternative $H_1$, exactly one is. Selecting uniformly among the nonblack objects and discovering a nonraven has likelihoods $1$ and $1-1/M$. Selecting uniformly among the ravens and discovering a black one has likelihoods $1$ and $1-1/r$. With equal prior probabilities, the respective posterior probabilities of $H$ areIf $M\gg r$, the first discovery supplies far less support. For $M=10^6$ and $r=100$, the increases over $1/2$ are approximately $2.5\times 10^{-7}$ and $0.00251$. This illustrates a possible resolution of the intuitive disparity, not a theorem that every nonblack nonraven always confirms $H$. More complex alternatives can reverse the comparison, and examining something already known to be a shoe need supply no relevant new information.^[ref]

Now suppose $E$ is already in $K$. Then $P(E\mid K)=1$, and ordinary conditioning givesThere is no further increase from learning the same fact again. Yet an already established observation can become an important reason to accept a subsequently developed theory. Clark Glymour's problem of old evidence exposes this mismatch between actual updating and scientific assessments of evidential support.^[ref]

One response evaluates the observation relative to an earlier or counterfactual background that does not contain it. That can represent how strongly a theory accounts for the observation compared with alternatives. It must specify that background rather than pretend to remove knowledge uniquely from a person's present beliefs. Another response concerns learning that a theory successfully accounts for the observation. That discovery can be new even when the observation is old. Modelling it requires attention to limited reasoning and changing hypotheses; an ideal agent already knowing every logical consequence cannot simply learn an entailment it previously overlooked.

Neither manoeuvre licenses counting the same evidence twice, or ignoring that parameters were selected to fit it. Neither establishes a unique complete theory of confirmation. Nevertheless, the old-evidence objection concerns the interpretation and completeness of a probabilistic framework; it does not invalidate the likelihood calculation for genuinely new sampling information.

The preferable solution combines a background-sensitive account of support with rejection of unrestricted instance confirmation. It preserves logical equivalence while allowing a white shoe to provide tiny support, no support or contrary support under different justified models. This is more explanatory than assigning confirmation solely from logical form, although assessing old evidence requires additional work beyond routine conditioning.

Explicit reference to laws is not necessary for every good scientific explanation. Laws remain indispensable in many explanations, but their occurrence alone is not sufficient either. The issue turns partly on whether a law means a contingent regularity of nature, any supporting generalization, or even a mathematical necessity.

The deductive-nomological model offers a strong positive case. Laws and circumstances entail the phenomenon, and at least one law is essential to the derivation. For example, Newton's second law and an inverse-square gravitational force, with appropriate initial conditions, yield predictions for an idealized two-body orbit. A universal relation unifies cases that otherwise look like unrelated observations. Carl Hempel and Paul Oppenheim explicitly develop adequacy conditions for such deductive explanations; their 1948 paper should not be read as a finished treatment of every statistical explanation.^[ref]

However, derivability does not always give the right explanatory direction. In a simplified geometrical optics model, a vertical pole of height $h$ on level ground casts a shadow of length $\ell =h/\tan \theta$ at solar elevation $\theta$. Given $h$ and $\theta$, one derives $\ell$; given $\ell$ and $\theta$, one derives $h$. The same geometrical relations enter both derivations. Yet the shadow does not normally explain why the pole was built to that height. The reverse calculation is a way to measure its height. The difference concerns which features produce or constrain which others, rather than deductive validity alone.

Mechanistic explanation is another challenge to a compulsory law-based format. A clock stopped because debris blocked a particular wheel, thereby preventing the mechanism from advancing. Inspecting the obstruction, removing it and observing restored operation can answer the causal question with considerable precision. The explanation describes organization and a relevant difference-making event without stating a universal law. It is more informative about this failure than the observation that every component obeyed Newtonian mechanics.

A defender of the covering-law model can reply that the causal account relies implicitly on regularities: debris that offers no resistance would not stop the wheel. This is a fair warning against treating a bare chronology as an explanation. But it does not establish that explanatory understanding requires identifying exceptionless fundamental laws, or that replacing the account with a complete microscopic derivation would improve the answer. Stable, tested relations in the actual operating regime may be sufficient.

An explanation can also turn on a mathematical constraint. Suppose an engineer asks why a connected network with four odd-degree junctions cannot be inspected by a route using every segment exactly once. Represent junctions as vertices and segments as edges of an undirected graph. Every intermediate visit pairs an incoming and an outgoing edge, so only the two endpoints of an open trail can have odd vertex degree. Four such vertices therefore exclude an Eulerian trail. The impossibility persists if segment lengths, construction materials or walking speeds change while the connections remain fixed. This is the obstruction in the Seven Bridges of Königsberg.^[ref]

The physical application requires that the graph represent the allowed routes correctly. Once that is granted, the explanatory necessity comes from parity, rather than a contingent physical law. Calling every mathematical constraint a law would preserve the original claim only by greatly widening its meaning. A critic may restrict “scientific” to causal explanations, but that restriction needs justification when applied mathematics answers a concrete engineering question.

Thus laws are powerful explanatory resources, and good explanations normally support relevant what-if judgements. Those judgements can be grounded in laws, mechanisms or mathematical constraints. Requiring explicit reference to laws imposes an unnecessarily narrow format; requiring some disciplined account of dependence is much more defensible.