= Solution
<Falsificationism> captures an important asymmetry between universal claims and counterexamples, but it is an incomplete account of scientific appraisal. It works best as a demand for serious testing within fallible theoretical packages, rather than as a mechanical rule for discarding a theory after any anomalous observation.
For an unrestricted universal claim, finitely many successful cases do not deductively establish every unobserved instance. One genuine contrary case can contradict the universal claim, provided the predicates and circumstances are correctly identified. <Karl Popper>'s emphasis on attempts at refutation consequently highlights something that mere accumulation of agreeable examples can miss: a test is informative partly because plausible alternative outcomes would have counted against the proposal.<Science: Conjectures and Refutations by Karl Popper>[chapter 1 excerpt: the contrast between confirmation-seeking and critical tests]{ref}
Real predictions nevertheless involve a conjunction. If $T\land A\Rightarrow E$ and $\neg E$, then
$$
\neg(T\land A)\equiv\neg T\lor\neg A.
$$
Here $A$ includes an apparatus model and initial conditions. The conclusion does not identify $T$ as the failed component. Independent calibration, varied experimental arrangements and agreement across instruments help distinguish the possibilities; they do not eliminate the logical structure of the <Duhem problem>. Insisting that every auxiliary assumption remain unrevisable would obstruct ordinary experimental repair.
Probabilistic theories create another difficulty for simple deductive falsification. If an ideal <Bernoulli distribution> model assigns a probability $0<q<1$ to success, every finite binary sequence has positive probability. An unusual sequence does not literally contradict the model. A <statistical hypothesis test> can instead use a predeclared rejection region with a controlled <type I error>. Its possible mistaken rejection is part of the method, not a contradiction to be ignored. Comparing alternatives also requires likelihoods or other explicit criteria, rather than a binary division into refuted and unrefuted proposals.
Finally, survival alone is not enough. A vague conjecture and a quantitatively successful theory might both be unrefuted; they need not be equally supported. Scientific decisions also concern approximation error, independent explanatory connections and whether a proposed modification predicts anything new. These considerations need not restore conclusive verification, but they require more than deductive refutation.
The plausible core is therefore methodological: specify what could count against a claim, expose it to demanding tests and resist changes made only to accommodate failure. The stronger thesis that this principle supplies a complete logic of discovery, theory choice and confirmation is not established. Critical testing remains valuable precisely when its assumptions and statistical limitations are stated honestly.
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