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<Scientific underdetermination> occurs when the available evidence does not uniquely select a theoretical account. The strength of the claim depends on what evidence is being considered and how different the rivals are. Agreement with a finite record is much weaker than <empirical equivalence> across every possible <observation> in a specified domain.
A straightforward finite-data example is <interpolation>. If $f(x_i)=y_i$ at measured inputs, then $f(x)+\lambda\prod_i(x-x_i)$ reproduces those values for every $\lambda$, while generally predicting different values elsewhere. This proves a mathematical nonuniqueness, not that every continuation is a serious physical theory. Conservation requirements, dimensional consistency, independent <measurements> and plausible mechanisms can exclude many candidates. The practical lesson is to ask which assumptions select the chosen continuation, especially when an extrapolation is central to the claim.
Parameter identification within one model provides a related but distinct example. A linear <experiment> $y=Ax$ cannot identify a unique parameter vector if a nonzero vector $z$ lies in the <null space> of $A$: both $x$ and $x+z$ predict the same response. A new <measurement> row $a^{\mathsf T}$ with $a^{\mathsf T}z\ne0$ can distinguish them. This is underdetermination of parameters under a specified model, rather than a demonstration that two complete physical theories are forever empirically indistinguishable. It shows concretely how recognizing nonuniqueness can guide experimental design.
The <Duhem problem> concerns another source of ambiguity. Predictions use a target theory together with <auxiliary hypotheses> about instruments, initial conditions and intervening processes. A failed prediction rejects their conjunction. It does not logically specify which member should change. Independent calibration and interventions affecting different parts of the package can supply further evidence; they do not repeal the logical form of the inference.<The Aim and Structure of Physical Theory by Pierre Duhem>[Part II, chapter VI, section 2: a test concerns a theoretical group rather than an isolated hypothesis]{ref}
A stronger philosophical claim posits rival accounts with the same observable consequences for every relevant <experiment>. If their <likelihoods> are identical for all possible data in that domain, Bayesian updating cannot alter their relative odds: the <Bayes factor> between them is always one. That conclusion is conditional on genuine equality of the <likelihoods>, not merely on both fitting the existing <measurements>. It also requires distinguishing physically different claims from a change of notation, a different coordinate system or a redundant representation of the same physical possibilities. Not every mathematical rewriting generates a new rival ontology.
There are several practical implications. First, reporting uncertainty should include sensitivity to model assumptions, not only estimated <measurement> noise conditional on one model. Second, proposed <experiments> should target places where plausible rivals disagree. Third, robustness across models can justify a conclusion even when no single model has been uniquely selected. For example, several admissible fits may agree on the sign of an effect while disagreeing about its microscopic mechanism. The evidence for those two claims need not have the same strength.
Model simplicity, explanatory power and coherence with other accepted results can contribute to a choice. Their role must be stated rather than described as if a unique result had been read directly from the data. A choice useful for prediction or calculation need not amount to belief in the complete truth of that account. Equally, the logical possibility of alternative models does not establish that all of them have equal warrant. Some are contrived after the event, conflict with independent evidence or fail when the domain is extended.
The implications for <scientific realism> are therefore qualified. Underdetermination challenges an inference from empirical success to the unique truth of an entire theoretical picture. It can motivate selective commitments or <constructive empiricism>. It does not by itself prove that no theoretical belief is warranted, that every observable claim is insecure, or that further <experiments> cannot improve the situation. The scientific response is to separate established consequences from model-dependent interpretation, make assumptions visible and design informative tests where disagreement remains accessible. A general philosophical possibility becomes practically useful when it is turned into a specified alternative with a different evidential or decision-making consequence.
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