= Solution
<Scientific underdetermination> occurs when the relevant evidence does not select a unique theoretical account. Its force depends on the scope of “the evidence”. Several models can agree with a finite data set while disagreeing about new measurements; a suitable <experiment> can then discriminate them. The stronger claim of <empirical equivalence> says that rivals agree on every possible observation within a stated domain, so further evidence of that same kind cannot separate them.
The <Duhem problem> gives a related but distinct source of ambiguity. If a target theory $T$ and <auxiliary hypotheses> $A$ jointly imply an observation $E$, then observing $\neg E$ shows $\neg(T\land A)$, not uniquely $\neg T$. Independent evidence about the apparatus or background assumptions can redistribute support. It does not make the original conjunction logically true; it provides additional information relevant to which revision is warranted.
A finite-data example makes the limits clear. If $p_0$ interpolates observations at distinct $x_1,\ldots,x_n$, then every <polynomial> $p_c(x)=p_0(x)+c\prod_{i=1}^n(x-x_i)$ has the same sampled values. Thus <polynomial interpolation> does not determine a unique prediction outside the sampled points when the degree is unrestricted. A degree bound of $n-1$ would instead give the familiar uniqueness result, so the allowed model class matters. That does not show that every interpolant is equally credible for extrapolation. Smoothness assumptions, independently motivated mechanisms and further measurements can matter. But those extra commitments must be stated rather than smuggled into the claim that the original data uniquely determined a model.
In a probabilistic comparison, <Bayes' theorem> gives
$$
\frac{P(T_1\mid E)}{P(T_2\mid E)}=\frac{P(E\mid T_1)}{P(E\mid T_2)}\frac{P(T_1)}{P(T_2)},
$$
when the quantities are defined. Equal <likelihoods> leave the <prior odds> unchanged by this evidence. Neither equality nor unequal priors follows merely from both theories being logically compatible with $E$. Thus compatibility, equal fit in a particular measure and full <empirical equivalence> should not be conflated.
The philosophical significance is a constraint on inference, not a universal proof of irrationality. An <empiricist> may use it to motivate restraint about hidden structure; a <scientific realist> may defend further explanatory or intervention-based reasons for belief. A convincing sceptical argument must identify genuine rivals and the relevant equivalence, while a convincing realist reply must explain why its additional virtues track the claims at issue. Scientific knowledge can be selective and fallible even when evidence does not deductively choose a unique complete theory.
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