For Popper, a theory is falsifiable if there exist possible observations that would disprove it; falsifiability is the mark of science. This criterion elegantly separates science from pseudoscience and unfalsifiable metaphysics. However, it faces counterexamples where unfalsifiable statements seem scientific, and auxiliary hypotheses complicate the picture.
From your prerequisite on Popper, you know that he proposed falsificationism as an alternative to inductivism: rather than confirming theories with positive evidence, science advances by subjecting theories to severe tests and surviving attempted refutations. The demarcation criterion — falsifiability as the *line* between science and non-science — is both the sharpest application of that logic and its most controversial extension.
The criterion is elegant in its simplicity. A theory is falsifiable if there exist possible observations that would contradict it. Einstein's general relativity predicted that light bends around massive objects by a specific amount; that was a risky, testable prediction that could have been refuted when Eddington measured starlight deflection in 1919. Contrast this with the claim that "everything happens for a reason" — no possible observation could contradict it. It says nothing about what we will observe, so it tells us nothing about the world. For Popper, the asymmetry between confirmation and falsification is decisive: a million white swans can't prove all swans are white, but one black swan disproves it. So the scientific attitude is not "how can I confirm this?" but "what would prove this wrong, and does it?"
This criterion does powerful work against what Popper called pseudoscience — theories that accommodate any evidence. Freudian psychoanalysis and Adlerian psychology both struck Popper as examples: whatever a patient did, the analyst could explain it as confirmation of the theory. A theory that predicts everything predicts nothing. By contrast, Marx's historical materialism made specific predictions about the development of capitalism that failed — but Marxists kept revising auxiliary hypotheses to protect the core. Here the complication arises: the Duhem-Quine problem shows that no single hypothesis is ever tested in isolation. When an observation fails to match a prediction, you can always blame an auxiliary hypothesis rather than the core theory. Neptune was predicted before it was observed precisely by modifying an auxiliary assumption (that Newtonian mechanics had been applied to all the relevant bodies) rather than abandoning Newton's laws.
This creates a puzzle for the demarcation criterion. If scientists can always protect a core theory by adjusting auxiliaries, what makes science different from pseudoscience after all? Popper's answer invokes scientific method and attitude: science requires specifying in advance which observations would count as refutations, and it is intellectually dishonest to make that specification after the fact. Imre Lakatos later refined this into the idea of research programmes with a hard core and a protective belt of auxiliaries; a programme is progressive if it generates novel successful predictions, degenerative if it only accommodates old ones. The demarcation criterion, on this view, is less a sharp line than a spectrum of methodological integrity. Falsifiability remains the dominant intuition in scientific practice, even if its philosophical foundations require these qualifications.
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