Scientific confirmation

Evidence supports a scientific hypothesis without necessarily proving it. Distinguish increasing its credibility from showing it highly credible in absolute terms; a low prior probability can rise substantially while remaining low.

Inference beyond what the premises deductively entail, often from observed cases to unobserved cases. This is distinct from mathematical induction.

The challenge of justifying inductive reasoning without either assuming the reliability being justified or demanding a deduction from premises that do not entail the conclusion.^[ref]

Nelson Goodman's challenge to explain why some predicates, rather than all predicates fitting the examined cases, are legitimate candidates for projection. It concerns projectibility, not a demonstration that every conceivable continuation is equally credible.^[ref]

The eligibility of a predicate or generalization for extending an inference beyond examined cases. Agreement with positive instances alone does not establish projectibility: grue and green can fit the same past examples while supporting incompatible predictions.

In Nelson Goodman's account, the standing a predicate acquires through a history of actual projections. It concerns the use of the predicate in inference, rather than merely the number of past objects satisfying it. Entrenchment is revisable and is not a proof of future reliability.^[ref]

For a fixed cutoff $t$, let $E_t(x)$ mean examined before $t$. Nelson Goodman's predicate is $G_t(x)⟺(E_t(x)\wedge \mathrm{green}(x))\vee (\neg E_t(x)\wedge \mathrm{blue}(x))$. An already examined green object remains grue after $t$ without a colour change. The conflicting projection concerns objects not examined before $t$.

From $T\wedge A\Rightarrow E$ and $\neg E$, infer only $\neg T\vee \neg A$. A failed prediction therefore does not logically isolate failure of the target $T$ from failure of auxiliary hypotheses $A$. Independent calibration and varied experiments can discriminate packages; the logical ambiguity does not imply that every package is equally supported.^[ref]

An assumption used with a target scientific hypothesis to obtain a prediction, for example an apparatus calibration or an initial condition. Independent evidence for an auxiliary hypothesis helps distinguish a failure of the target from a failure of the test setup.

Evidence $E$ confirms $H$ relative to background $K$ when it increases its conditional probability. With $0<P(H\mid K)<1$ and positive evidence probability, Bayes' theorem givesIndeed, writing $p=P(H\mid K)$, $a=P(E\mid H,K)$ and $b=P(E\mid \neg H,K)$ gives the posterior probability $pa/(pa+(1-p)b)$; comparison with $p$ reduces to $a>b$. Thus scientific confirmation depends on the alternatives and background, not just on a statement's logical form.

Fix $r>1$ ravens and $M>1$ nonblack objects in a finite population. Under $H_0$, all ravens are black; under $H_1$, exactly one is nonblack. The total numbers $r,M$ are held fixed under both hypotheses, so the classification of one nonraven also changes between hypotheses. This is a deliberately specified model for Bayesian confirmation, not a universal rule about how bird observations supply evidence.

A uniform draw among nonblack objects returns a nonraven with probabilities $1$ under $H_0$ and $1-1/M$ under $H_1$. A uniform draw among ravens returns a black object with probabilities $1$ and $1-1/r$. The respective likelihood ratios for $H_0$ over $H_1$ are therefore $(1-1/M{)}^{-1}$ and $(1-1/r{)}^{-1}$. When $M>r$, the former is closer to one, so supplies less support under these assumptions.

With equal prior probabilities, Bayes' theorem gives posterior probabilities $1/(2-1/M)$ and $1/(2-1/r)$. More generally, with prior probability $p$ for $H_0$, replace the two expressions by $p/[p+(1-p)(1-1/M)]$ and $p/[p+(1-p)(1-1/r)]$. Previously known information or a different sampling protocol requires different likelihoods; selecting an object already known to be a nonraven need not supply new evidence.

Already accepted evidence can support a newly considered theory even though conditioning again on a fact of probability one produces no increase in its posterior probability. Distinguish new learning, retrospective support and discovering a previously unrecognized theory–evidence connection.^[ref]

“All ravens are black” is logically equivalent to “all nonblack things are nonravens”. An unrestricted rule that positive instances confirm universal generalizations would make a white shoe confirm the first statement. A Bayesian confirmation account instead conditions on background knowledge and the sampling procedure.^[ref]