The criterion of verifiability that was most strongly propounded by the neo-positivists, though implicit in the teachings of earlier empirically minded philosopers, amounts to the two theses that there must be given some clear criterion to distinguish scientific statements from non-scientific statements, such as metaphysical or theological statements, and that the criterion that does this is that scientific statements are such as to be verifiable by their logical consequences.

One may well be somewhat skeptical about the first claim, for even if there is a fundamental distinction between science and non-science it is not necessary that this corresponds to some clear criterion that is applicable to all statements and unmistakenly selects just those statements that are properly scientific.

And one may also hold that there is no simple criterion that makes such a distinction for all of the sciences, and that this is not necessary either, for what characterizes any real science is usually clear enough: Empirical investigation of aspects of the real world, by methods that are objective and capable of being employed by anybody qualified, and directed by empirical findings and logical argumentations, all done in an objective, honest and controllable way.

It seemed to the neo-positivists that it is quite important to distinguish science from non-science, and especially from metaphysics and theology, and it seemed to them that they had found a simple criterion to make the distinction: A statement is scientific, the neo-positivists claimed, if and only if the statement is verifiable, and then by an appropriate scientific method, included observation and measurement.

There are quite a few possible rational objections to this, and one important one was strongly argued by Popper: The problem for a scientific theory that gets verified by some of its logical consequences, is that it does not get proved true by such confirmations, and to believe that it does is to assume the fallacy of affirming the consequent. See: Criterion of falsifiability.

It may be said with some justification that this criticism is not just, since the criterion was proposed to distinguish between scientific and non-scientific statements, and the criterion did not say anything about their proof, nor does it need to.

A related objection, that is much stronger, is that because theories must go beyond the evidence - namely: to predict anything at all - the very ideas of verifiability and falsifiability of a theorem or a statement, in brief, a hypothesis,  seem to require at least two additional hypotheses of a principal kind: that there is some invariance (to test the hypothesis), and that there is some irrelevance (that may be abstracted from).

And finally, Bayesian Reasoning, that involves probability, gives much more, explains much better, and is much more subtle than any mere verification or falsification of consequences plus bivalent logic, which indeed seems - with the stress on the need for verifiability - committed also to the fallacy of affirming the consequence.