Two concepts of conditionality

Birnbaum's proof that C and M imply L, would lose its force if it is shown that in some situations M is not acceptable. Godambe (1979) has shown that Birnbaum's M is not as obvious or intuitive as the concept that a ‘mere relabelling’ of sample points should make no difference to the interference that can appropriately be drawn from a particular outcome of a given experiment. Akaike (1982) has shown that in certain situations M amounts to the assertion that a relabelling of sample points involving a false reporting of the outcome of an experiment should make no difference to the inference drawn from a particular outcome of a given experiment. It is shown in this paper that in the situation discussed by Akaike, even if M were to be considered acceptable, it is only a modified conditionality principle C? and M which can formally imply L; Birnbaum's conditionality principle C and M do not imply L.