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A verificationist is someone who adheres to the verification principle, a criterion for meaningfulness most associated with the logical positivist movement which had its roots in inter-war Vienna. This criterion requires that a (non-analytic) meaningful sentence be either verifiable, though it was disputed whether this must be possible in practice rather than merely in principle.
Note that verificationists need not be logical positivists; Willard Van Orman Quine is a famous example of a verificationist who does not accept logical positivism on grounds of semantic holism. He suggests that, for theoretical sentences as opposed to observation sentences, meaning is "infected by theory". That theoretical sentences are reducible to observation sentences is one of the ‘dogmas of empiricism’ he rejects as incompatible with semantic holism.
Similar ideas were expressed by the American pragmatists, in particular Charles Peirce and William James. James coined the famous verificationist motto: "A difference that makes no difference is no difference".
It is commonly believed that Karl Popper rejected the requirement that meaningful sentences be verifiable, demanding instead that they be falsifiable. However, Popper later claimed that his demand for falsifiability was not meant as a theory of meaning, but rather as a methodological norm, i.e. demarcation for the sciences.