Here is my understanding of Karl Popper and Nelson Goodman. Both talk about whether and when observations may corroborate a given hypothesis. Popper concludes that observations may falsify, but never affirmatively prove, a statement. Goodman’s New Riddle says nothing about falsification directly, but creates a hypothetical where corroboration and falsification can never be known.
First, Popper. From Popper, "Demarcation", in the Stanford Encyclopedia of Philosophy. http://plato.stanford.edu/entries/popper/#ProbDema
Popper … repudiates induction…, substituting falsifiability in its
place. It is easy… to obtain evidence in favour of virtually any
theory, and… such ‘corroboration’, as he terms it, should count
scientifically only if it is the positive result of a genuinely
‘risky’ prediction, which might conceivably have been false…. In a
critical sense, Popper’s theory of demarcation is based upon his
perception of the logical asymmetry which holds between verification
and falsification: it is logically impossible to conclusively verify a
universal proposition by reference to experience…, but a single
counter-instance conclusively falsifies the corresponding universal
Every genuine scientific theory then, in Popper’s view, is
prohibitive, in the sense that it forbids, by implication, particular
events or occurrences. As such it can be tested and falsified, but
never logically verified. (emphasis in original)
By destroying the corroborative value of an observation, Goodman seems to call even falsification into question. Assume first that, before a future time t, each emerald examined is green. But then introduce a new category, grue. Grue is the color of things examined before time t just in case they are green, but also to other things, not examined before time t, just in case they are blue. If, before time t, a set of emeralds is examined and each is green, then this series confirms the hypothesis that all emeralds are green. However, the same observations, made before time t, also confirm that all emeralds are grue. Thus, before time t, the two hypotheses, "All emeralds are green" and "All emeralds are grue" are both confirmed by the same set of observations. (“The New Riddle of Induction”, in Fact, Fiction, and Forecast. 1983, Harvard University Press, p. 73-75).
Although the truth of one hypothesis implies the falsehood of the other, both are confirmed by observations before time t. Popper's argument does not resolve the riddle.