Abstract

A - "All ravens are black." B - "All non-black things are non-ravens." C - "All the elements of the set of ravens are elements of the set of black ravens." D - "All the elements of the set of the things that are not black ravens are elements of the set of the things that are not ravens." The propositions A, B, C, D are logically equivalent: if one proposition is true, the other three propositions are also true; if one proposition is false, the other three propositions are also false. The generalizations A and C have the same instances: a black raven is an instance of A and is an instance of C. The generalizations B and D have some instances in common: a non-black object that is a non-raven is an instance of B and is an instance of D; in addition, the generalization D also has as its instances black objects that are non-ravens (for example, a black shoe is an instance of D but is not an instance of B). E - "All ravens are non-black." F - "All black things are non-ravens." G - "All the elements of the set of ravens are elements of the set of the things that are not black ravens". H - "All the elements of the set of black ravens are elements of the set of things that are not ravens." The generalizations E and G have the same instances: a non-black raven is an instance of E and is an instance of G. By contrast, the generalizations F and H do not have the same instances: a black thing that is a non-raven is an instance of F; by contrast, H is a generalization that cannot have instances because a black raven that is a non-raven is a contradiction in terms. To summarize: A and C have the same instances; B and D have some instances in common; E and G have the same instances; by contrast, F and H do not have instances in common because H has no instances, H cannot have instances. In other words: let's call "W" the set of the things that are not black ravens I - "All the elements of the set of ravens are elements of the set W." L - "All the things that are not elements of the set W are elements of the set of the things that are not ravens." A raven that is not a black raven is an instance of I. By contrast, L has no instances because a black raven that is not a raven is a self-contradictory concept.