it is easy to see that on the Neyman-Pearson theory of testing, a test of "All P's are Q" is not necessarily a test of "All non-Q's are non-P" or vice versa. A test of "All P's are Q" requires reference to some alternative statistical hypothesis of the form of all P's are Q, , whereas a test of "All non-Q's are non-P" requires reference to some statistical alternative of the form of all non-Q's are non-P, . But these two sets of possible alternatives are different ... Thus one could have a test of without having a test of its contrapositive. Rejecting material implication The following propositions all imply one another: "Every object is either black or not a raven", "Every raven is black", and "Every non-black object is a non-raven." They are therefore, by definition, logically equivalent. However, the three propositions have different domains: the first proposition says something about "every object", while the second says something about "every raven". The first proposition is the only one whose domain of quantification is unrestricted ("all objects"), so this is the only one that can be expressed in first-order logic. It is logically equivalent to: and also to where indicates the material conditional, according to which "If then can be understood to mean It has been argued by several authors that material implication does not fully capture the meaning of "If then (see the paradoxes of material implication). "For every object, is either black or not a raven" is true when there are no ravens. It is because of this that "All ravens are black" is regarded as true when there are no ravens. Furthermore, the arguments that Good and Maher used to criticize Nicod's criterion (see , above) relied on this fact – that "All ravens are black" is highly probable when it is highly probable that there are no ravens. To say that all ravens are black in the absence of any ravens is an empty statement. It refers to nothing. "All ravens are white" is equally relevant and true, if this statement is considered to have any truth or relevance. Some approaches to the paradox have sought to find other ways of interpreting "If then and "All are which would eliminate the perceived equivalence between "All ravens are black" and "All non-black things are non-ravens." One such approach involves introducing a many-valued logic according to which "If