I read recently a comment by a philosopher that Karl Popper's "falsifiability" theory is considered obsolete. Is this so? I always found it to be quite useful. If it's obsolete, what rendered it so, and by what was it replaced?

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There are several considerations that count strongly against Popper's "falsifiability" criterion, but I'll mention just two.

Remember that Popper's criterion is intended to distinguish science from non-science (or pseudoscience) on the grounds that a theory is scientific if and only if it is 'falsifiable', i.e., there are possible observations that would logically contradict it. Now for two quick arguments against this view:

1) Consider a statistical hypothesis, such as "This coin has a 50% chance of landing heads and a 50% chance of landing tails on any toss." Statistical hypotheses play a very important part in many important scientific theories (such quantum mechanics, evolutionary biology, statistical mechanics). But no possible sequence of coin toss outcomes logically contradicts this '50% chance' hypothesis. Some outcomes support the hypothesis; some disconfirm it. None falsifies it.

2) Many significant scientific hypothesis make no observable predictions all by themselves, but can be coaxed into making observable predictions only with the aid of additional hypotheses. (Consider Newton's second law of motion, which makes predictions only when it has been combined with hypotheses about forces, or the principle of natural selection, which makes predictions only when it has been combined with hypotheses about selection pressures, or the atomic theory, which makes predictions only when combined with hypotheses about the behavior of various measuring devices and the nature of temperature and pressure, etc.) If Popper's criterion is underestood as requiring that a scientific theory be falsifiable when the theory is taken in isolation, then none of these famous theories qualifies as a scientific theory. On the other hand, if Popper's criterion is understood as requiring that a scientific theory be falsifiable when the theory is combined with some additional hypotheses or other, then every hypothesis will count as scientific, since empirical predictions can be coaxed out of any theory if it is combined with crafty enough "auxiliary hypotheses". This nest of ideas is often referred to as "the Duhem-Quine thesis".

I'll add a third problem to Popper's views... it classifies obvious psuedo-sciences as sciences such as astrology, so long as they make potentially falsifiable predictions. Furthermore, it does nothing to distinguish something radically implausible like astrology from something more plausible, but not falsifiable such as ad-hoc psychological analysis. Popper's views and others similar to it (verificationism and logical positivism) belong to an era of philosophy when it was believed philosophy could be made 'scientific'. It has not really been replaced because few philosophers still hold to that belief.

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