In our discussion of existential import it will not be necessary that we should make any attempt to determine the ultimate nature of reality. The questions at issue are, however, not exactly easy of solution, and various sources of misunderstanding are apt to arise.
There is one sense in which the existence of something corresponding to the terms employed must be postulated in all predication. For in order to make use of any term in an intelligible sense we must mentally attach some meaning to it. Hence there must be something in the mind corresponding to every term we use. Even in cases where there cannot be said to be any corresponding mental product, there must at any rate 212 be some corresponding mental process. This applies even to such terms as round square or non-human man or root of minus one. We are not indeed able to form an image of a round square or an idea of a non-human man, nor can we evaluate the root of minus one. But we attach a meaning to these terms, and they must therefore have a mental equivalent of some sort. In the case of “round square” or “non-human man” this is not the actual combination in imagination or idea of “round” with “square” or “non-human” with “man,” for such combinations are impossible. But it is the idea of the combination, regarded as a problem presented for solution, and perhaps involving an unsuccessful effort to effect the combination in thought. It is apparently of existence of this kind that some writers are thinking when they maintain that of necessity every proposition implies logically the existence of its subject. But our meaning is something quite different when we speak of existence in the universe of discourse. The nature of the distinction may be made more clear by the following considerations.
It will be admitted that whatever else is included in the full implication of a universal proposition, it at least denies the existence of a certain class of objects. No S is P denies the existence of objects that are both S and P ; All S is P denies the existence of objects that are S without also being P. In these propositions, however, we do not intend to deny the existence of SP (or SPʹ) as objects of thought. For example, in the proposition No roses are blue it is not our intention to deny that we can form an idea of blue roses ; nor in the proposition All ruminant animals are cloven-hoofed is it our intention to deny that ruminant animals without cloven hoofs can exist as objects of thought. These illustrations may help us to understand more clearly what is meant by existence in the universe of discourse. The universe of discourse in the case of the proposition No S is P is the universe (whatever it may be) in which the existence of SP is denied. The universe of discourse in the case of a universal affirmative proposition may be defined similarly. As regards particulars it may be best to seek an interpretation through the universals by which the particulars 213 are contradicted. Thus, the universe of discourse in the case of the proposition Some S is P may be defined as the universe (whatever it may be) in which the existence of SP would be understood to be denied in the corresponding universal negative. The proposition Some S is not P may be dealt with similarly.
The question whether a categorical proposition is to be interpreted as formally implying that its terms are the names of existing things may then be interpreted as follows: Given a categorical proposition with S and P as subject and predicate, is the existence of S or of P formally implied in that sphere (whatever it may be) in which the existence of SP (or SPʹ) is denied by the proposition (or by its contradictory)?
The question may be somewhat differently expressed as follows. Such a proposition as No S is P denies the existence of a certain complex of attributes, namely, SP. But with rare exceptions, S itself signifies a certain complex of attributes; and so does P. Does the proposition affirm the existence of these latter complexes in the same sense as that in which it denies the existence of the former complex?
No general criterion can be laid down for determining what is actually the universe of discourse in any particular case. It may, however, be said that knowledge as to what is the universe referred to is involved in understanding the meaning of any given proposition; and cases in which there can be any practical doubt are exceptional.[215] Thus, in the propositions No roses are blue, All men are mortal, All ruminant animals are cloven-hoofed, the reference clearly is to the actual physical universe; in The wrath of the Olympian gods is very terrible to the universe of the Greek mythology;[216] in Fairies are able to assume different forms to the universe of folk-lore;[217] in Two straight lines cannot enclose a space to the universe of spatial intuitions.
[215] It must at the same time be admitted that controversies sometimes turn upon an unrecognised want of agreement between the controversialists as to the universe of discourse to which reference is made.
[216] The universe of the Greek mythology does not consist of gods, heroes, centaurs, &c., but of accounts of such beings currently accepted in ancient Greece, and handed down to us by Homer and other authors. As regards the reference to reality, therefore, such a proposition as The wrath of the Olympian gods is very terrible is elliptical in a sense already explained.
[217] Here again there is an ellipsis. The universe of folk-lore does not consist of fairies, elves, &c., but of descriptions of them, based on popular beliefs, and conventionally accepted when such beings are referred to. Of course for anyone who really believed in the existence of fairies there would be no ellipsis, and the universe of discourse would be different.
214 With respect to the existential import of propositions the following questions offer themselves for consideration:
(1) Is the problem one with which logic, and more particularly formal logic, is properly concerned?
(2) How should the propositions belonging to the traditional schedule be interpreted as regards their existential implications?
(3) Can we formulate a schedule of propositions which directly affirm or deny existence, and how will such a schedule be related to the traditional schedule?
(4) How are ordinary logical doctrines affected by the answer given to the second of these questions?