Past exam of the philosophy of science supplementary subject of the University of Oxford / 2015 / 2

Nelson Goodman's new riddle of induction asks which descriptions of an observed regularity are fit for projection. David Hume's problem of induction asks how a projection beyond experience can be justified at all. The problems are related, but accepting that experience supports some expectations does not settle which predicates an acceptable inference may project.

Fix a future time $t$ and let $E_t(x)$ mean that $x$ was examined before $t$. Define grue bySuppose every emerald examined so far is green. Each is then also grue. The same positive instances fit both a green generalization and a grue generalization, although the latter assigns blue to an emerald first examined after the cutoff. This does not require an already examined green emerald to change colour at $t$: its classification uses whether it was examined before $t$, not whether the present time is before $t$. The problem concerns incompatible projections to unexamined objects.^[ref]

It does not resolve the difficulty to object that grue is artificially disjunctive. Introduce the converse predicate, bleen, for examined-before-$t$ blue objects and other green objects. Green can then be defined using grue, bleen and $E_t$. Syntactic simplicity depends on which predicates are primitive. Nor does calling green a genuine natural kind settle matters unless one explains how the relevant natural structure is identified without simply presupposing the preferred projection. Evidence about stable material composition can provide a scientific reason to project a colour response, but that reason goes beyond counting instances.

David Hume's argument exposes a different gap. The denial of an unobserved continuation is not a contradiction. An appeal to the past success of extrapolation already extrapolates from past cases, so cannot supply a noncircular demonstration of its general reliability. His discussion of necessary connection similarly distinguishes experienced successions from an independently perceived necessity binding events together. These claims concern the basis of causal and predictive inference, not merely the spelling of a predicate.^[ref]

Predicate entrenchment is Nelson Goodman's proposed way of distinguishing established projective practices. Green has a history of actual use in projections which its newly constructed rival lacks. This gives a practice-based account of projectibility, rather than an unconditional guarantee that familiar classifications are true. Its weakness is that an account must still permit scientifically successful innovations and correction of entrenched mistakes.^[ref]

Thus Nelson Goodman sharpens the problem facing scientific confirmation: a purely formal positive-instance rule is too permissive. A defensible response combines explicit background assumptions with revisable standards for selecting predicates. That explains why two syntactically parallel generalizations need not receive equal support; it does not provide the noncircular guarantee demanded by the strongest reading of David Hume's challenge.

The raven paradox combines a plausible equivalence principle with an overly unrestricted account of positive-instance scientific confirmation. Let $R(x)$ and $B(x)$ mean that $x$ is a raven and black respectively. ThenA white shoe is a positive instance of the right-hand generalization. If every positive instance confirms its universal generalization, and logically equivalent statements receive the same evidential support, inspecting the shoe seems to confirm the original raven claim. There is no contradiction in the equivalence itself. The tension concerns the evidential rule and the intuition that an arbitrary shoe teaches us nothing about birds.^[ref]

The right response is to make background knowledge and sampling explicit. Under Bayesian confirmation, evidence increases support for $H$ precisely when its likelihood is greater under $H$ than under the alternative, assuming nondegenerate prior probabilities. A fact can therefore be compatible with $H$ yet provide no new support. The mere existence of a white shoe does not determine the needed likelihood ratio.^[ref]

Consider a deliberately simple finite model. There are $N$ objects, exactly $r$ ravens, and $M=N-r$ nonravens, with $M>r>1$. All nonravens are white. Under $H_0$ all ravens are black; under $H_1$ exactly one raven is white. Draw uniformly from the entire population. The event of drawing a white nonraven has probability $M/N$ under either hypothesis. Its Bayes factor is one and it leaves the prior odds unchanged. A black raven instead has likelihoods $r/N$ and $(r-1)/N$, so gives Bayes factor $r/(r-1)$ for $H_0$.

Now change the protocol: draw uniformly from the white objects, then discover that the selected object is a nonraven. This result has likelihoods $1$ and $M/(M+1)$. It gives the small positive Bayes factorFor $N=1000$ and $r=10$, this is $991/990$, whereas sampling a black raven from the full population gives $10/9$. The white nonraven can now help because the search could have revealed a white raven. The two experiments have different selection information, so their differing evidential values do not violate logical equivalence.

This model establishes possibilities, not a universal numerical law about shoes. Different alternatives, population information or selection procedures change the result. If the shoe's being a nonraven was already known, discovering that it is white need not add anything. Conversely, ruling out a potential counterexample can supply a small amount of information. Retain logical equivalence, but reject the unconditional instance-confirmation rule. Once the complete evidential situation is specified, neither zero support nor weak positive support is paradoxical, and neither amounts to proof of a universal law.