Part 2 — CHAPTER VIII.

THE EXISTENTIAL IMPORT OF CATEGORICAL PROPOSITIONS.212

153. Existence and the Universe of Discourse.—It has been shew in section 49 that every judgment involves an objective reference, or—as it may otherwise be expressed—a reference to some system of reality distinct from the act of judgment itself. The reference may be to the total system of reality without limitation, or it may be to some particular aspect or portion of that system. Whatever it may be, we may speak of it as the universe of discourse.213 The universe of discourse may be limited in various ways; for example, to physical objects, or to psychical events, or again with reference to time or space. But in all cases it is a universe of reality in the sense in which that term has been used in section 49. The nature of the reference in propositions relating to fictitious objects, for example, to the characters and occurrences in a play or a novel, may be specially considered. We may say that in a case of this kind the universe of discourse consists of a series of statements about persons and events made by a certain author; and it is clear that such statements have objective reality, although the persons and events themselves are fictitious. It follows that, as regards the reference to reality, such a proposition as “Hamlet killed Polonius” must be considered elliptical. For the reference is not to real persons or to the actual course of events in the past history of the world, as it is when we say “Mary Stuart was beheaded,” but to a series of descriptions given by Shakespeare in a particular play. These descriptions have, however, a reality of their own, and (the different nature of the reference being clearly understood) I am no more free to say that Hamlet did not kill Polonius (that is, that Shakespeare did not describe Hamlet as killing Polonius) than I am to say that Mary Stuart was not beheaded.

The substance of the above has been expressed by saying that reality is the ultimate subject of every proposition. Every proposition makes an affirmation about a certain universe of discourse, and the universe of discourse (whatever it may be) has some real content. In this sense then every proposition has an existent subject.214 A further question may, however, be raised, namely, whether—using the word “subject” in its ordinary logical signification—all or any propositions should be interpreted as implying the existence (or occurrence) of their subjects within the universe of discourse (or particular portion of reality) to which reference is made. It is mainly with this problem, and the ways in which ordinary logical doctrines are affected by its solution, that we shall be concerned in the present chapter.

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 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 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.

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?

It is clear that the first and fourth of these questions are connected, since if the fourth admits of any positive answer at all, the first is thereby answered in the affirmative. Since, however, the first question blocks the way and seems to demand an answer before we carry the discussion further, it will be well to deal with it briefly at the outset.

The second and third questions are also closely connected together.

Between the second and fourth questions an important distinction must be drawn. The second question is one of interpretation, and within certain limits the answer to it is a matter of convention. Hence a given solution may be preferred on grounds that would not justify the rejection of other solutions as altogether erroneous, although they may be considered inconvenient or unsuitable. But the answer to the fourth question is not similarly a matter of convention. On the basis of any given interpretation of propositional forms, the manner in which logical doctrines are affected can admit of only one correct solution.

It is to be observed further that the fourth question can be dealt with hypothetically, that is to say, we can work out the consequences of interpretations which we have no intention of adopting; and it is desirable that we should work out such consequences before deciding upon the adoption of any given interpretation. Hence we propose to deal with the fourth question before discussing the second. The third question may conveniently be taken after the first.

154. Formal Logic and the Existential Import of Propositions.—We have then, in the first place, briefly to consider the question whether the problem of existential import is one with which logic has any proper concern. It may be urged that formal logic, at any rate, cannot from its very nature be concerned with questions relating to existence in any other sphere than that of thought. The function of the formal logician, it may be said, is to distinguish between that which is self-consistent and that which is self-contradictory; it is his business to distinguish between what can and what cannot exist in the world of thought. But beyond this he cannot go. Any considerations relating to objective existence are beyond the scope of formal logic.

We may meet the above argument by clearly defining our position. It is of course no function of logic to determine whether or not certain classes actually exist in any given universe of discourse, any more than it is the function of logic to determine whether given propositions are true or false. But it does not follow that logic has, therefore, no concern with any questions relating to objective existence. For, just as, certain propositions being given true, logic determines what other propositions will as a consequence also be true, so given an assertion or a set of assertions to the effect that certain combinations do or do not exist in a given universe of discourse, it can determine what other assertions about existence in the same universe of discourse follow therefrom.218 As a matter of fact, the premisses in any argument necessarily contain certain implications in regard to existence in the particular universe of discourse to which reference is made, and the same is true of the conclusion; it is accordingly essential that the logician should make sure that the latter implications are clearly warranted by the former.

Without at present going into any detail we may very briefly indicate one or two existential questions that cannot be altogether excluded from consideration in formal logic. Universal propositions, as we have seen, assert non-existence in some sphere of reality; and it is not possible to bring out their full import without calling attention to this fact. Again, the proposition All S is P at least involves that if there are any S’s in the universe of discourse, there must also be some P’s, while it does not seem necessarily to involve that if there are any P’s there must be some S’s. But now convert the proposition. The result is Some P is S, and this does involve that if there are any P’s there must be some S’s.219 How then can the process of conversion be shewn to be valid without some assumption which will serve to justify this latter implication? Similarly, in passing from All S is P to Some not-S is not-P, it must at least be assumed that if S does not constitute the entire universe of discourse, neither does P do so. It is indeed quite impossible to justify the process of inversion in any case without having some regard to the existential interpretation of the propositions concerned.220

155. The Existential Formulation of Propositions.—We may define an existential proposition as one that directly affirms or denies existence (or occurrence) in the universe of discourse (or portion of reality) to which reference is made. Such propositions are of course met with in ordinary forms of speech: for example, God exists, It rains, There are white hares, It does not rain, Unicorns are non-existent. There is no rose without a thorn. Sometimes the affirmation or denial of existence takes a less simple form, but is none the less direct: for example, The assassination of Caesar is an historical event, D’Artagnan is not an imaginary person, The centaur is a fiction of the poets, The large copper butterfly is extinct.

In the formal expression of existential propositions it will be convenient to make use of certain symbols described in the preceding chapter. Thus, the affirmation of the existence of S may be written in the form S > 0, and the denial of the existence of S in the form S = 0. We shall then have an existential schedule of propositions if we reduce our statements to one or other of these forms or to a conjunctive or disjunctive combination of them. The relation between the traditional schedule and an existential schedule of this kind will be discussed in the next section but one.

It may here be pointed out that since the universe of discourse is itself assumed to be real and hence cannot be entirely emptied of content, any denial of existence involves also an affirmation of existence. For if we deny the existence of S, we thereby implicitly affirm the existence of not-S, since by the law of excluded middle everything in the universe of discourse must be either S or not-S. It follows that every proposition contains directly or indirectly an affirmation of existence.221

156. Various Suppositions concerning the Existential Import of Categorical Propositions.—Several different views may be taken as to what implication with regard to existence, if any, is involved in categorical propositions of the traditional type. The following may be formulated for special discussion:—222

(1) It may be held that every categorical proposition should be interpreted as implying the existence both of objects denoted by the terms directly involved and also of objects denoted by their contradictories; that, for example, All S is P should be regarded as implying the existence of S, not-S, P, not-P. This view is implied in Jevons’s Criterion of Consistency mentioned in the note on page 217. It is also practically adopted by De Morgan.223

(2) It may be held that every proposition should be interpreted as implying simply the existence of its subject. This is Mill’s view (as regards real propositions); for he holds that we cannot give information about a non-existent subject.224 This is no doubt the view that, at any rate on a first consideration of the subject, appears to be at once the most reasonable and the most simple.

(3) It may be held that we should not regard propositions as necessarily implying the existence either of their subjects or of their predicates. On this view, the full implication of All S is P may be expressed by saying that it denies the existence of anything that is at the same time S and not-P. Similarly No S is P implies the existence neither of S nor of P, but merely denies the existence of anything that is both S and P. Some S is P (or is not P) may be read Some S, if there is any S, is P (or is not P). Here we neither affirm nor deny the existence of any class absolutely;225 the sum total of what we affirm is that if any S exists, then something which is both S and P (or S and not-P) also exists. On this interpretation, therefore, particular propositions have a hypothetical and not a purely categorical character.

(4) It may be held that universal propositions should not be interpreted as implying the existence of their subjects, but that particular propositions should be interpreted as doing so.226 On this view All S is P merely denies the existence of anything that is both S and not-P; No S is P denies the existence of anything that is both S and P ; Some S is P affirms the existence of something that is both S and P ; Some S is not P affirms the existence of something that is both S and not-P. Thus, universals are interpreted as having existentially a negative force, while particulars have an affirmative force. This hypothesis will be found to lead to certain paradoxical results, but it will also be shewn to lead to a more satisfactory and symmetrical treatment of logical problems than is otherwise possible.227

157. Reduction of the traditional forms of proposition to the form of Existential Propositions.—Without at present attempting to decide between the different possible suppositions as to the existential import of the traditional forms of proposition, we may enquire how on the different suppositions they may be reduced to existential form. It will be assumed throughout that both the traditional forms and the existential forms are interpreted assertorically. In the case of each of the traditional forms it will suffice to deal with the two fundamental suppositions, namely, that it does and that it does not imply the existence of its subject.

The universal affirmative. (1) If SaP is interpreted as not carrying with it any existential implication in regard to its separate terms, it is equivalent to the existential proposition SPʹ = 0. Dr Wolf denies this on the ground that SaP contains further the implication “If there are any S’s, they must all be P’s”; and hence that, while on the supposition in question SPʹ = 0 is an inference from SaP, it is not equivalent to it. It is of course a very elementary truth that inferences are not always the exact equivalents of their premisses. But in the above argument Dr Wolf has apparently overlooked the fact that SPʹ = 0, equally with SaP, contains the implication “If there are any S’s they are all P’s.”228 By the law of excluded middle, every S (if there are any S’s) must be P or not P, and since SPʹ = 0, the above inference clearly follows. SPʹ = 0 carries with it in fact the two implications If S > 0 then P > 0, If P > 0 then Sʹ > 0. These may also be written in the forms Either S = 0 or P > 0, Either Pʹ = 0 or Sʹ > 0.

(2) If SaP is interpreted as implying the existence of S, then it may be expressed existentially S > 0 and SPʹ = 0. These existential forms carry with them the implications P > 0, Either Pʹ = 0 or Sʹ > 0.

The universal negative. Taking the same two suppositions the corresponding existentials will be:— (1) SP = 0 (carrying with it the implications Either S = 0 or Pʹ > 0, Either P = 0 or Sʹ > 0); (2) S > 0 and SP = 0 (with the implications Pʹ > 0, Either P = 0 or Sʹ > 0).

These results need no separate discussion.

The particular affirmative. (1) On the supposition that SiP does not carry with it any implication as to the separate existence of its terms, it can be expressed existentially Either S = 0 or SP > 0. It might also be written in the form If S > 0 then SP > 0. Complications resulting from the introduction of considerations of modality will, however, be more easily avoided if the hypothetical form is not made use of.

(2) On the supposition that the existence of S is implied, SiP is reducible to the form SP > 0.

The particular negative. Here the corresponding results are (1) Either S = 0 or SPʹ > 0; (2) SPʹ > 0.

We may sum up our results with reference to the third and fourth of the suppositions formulated in the preceding section.

Let no proposition be interpreted as implying the existence of its separate terms. Then corresponding to the traditional schedule we have the following existential schedule:—

A,—SPʹ = 0;

E,—SP = 0;

I,—Either S = 0 or SP > 0;

O,—Either S = 0 or SPʹ > 0.

This represents what may be regarded as the minimum existential import of each of the traditional propositions (interpreted assertorically).

It must be remembered that SPʹ = 0 carries with it the implications Either S = 0 or P > 0, Either Pʹ = 0 or Sʹ > 0.

Let particulars be interpreted as implying, while universals are not interpreted as implying, the existence of their subjects. We then have:—

A,—SPʹ = 0;

E,—SP = 0;

I,—SP > 0;

O,—SPʹ > 0.

158. Immediate Inferences and the Existential Import of Propositions.—It has been already suggested that before coming to any decision in regard to the existential import of propositions, it will be well to enquire how certain logical doctrines are affected by the different existential assumptions upon which we may proceed. This discussion will as far as possible be kept distinct from the enquiry as to which of the assumptions ought normally to be adopted. The latter question is of a highly controversial nature, but the logical consequences of the various suppositions ought to be capable of demonstration, so as to leave no room for differences of opinion.

We shall in the present section enquire how far different hypotheses regarding the existential import of propositions affect the validity of obversion and conversion and the other immediate inferences based upon these. In the next section we shall consider inferences connected with the square of opposition.

We may take in order the suppositions formulated in section 156.

(1) Let every proposition he understood to imply the existence of both its subject and its predicate and also of their contradictories. It is clear that on this hypothesis the validity of conversion, obversion, contraposition, and inversion will not be affected by existential considerations. The terms of the original proposition together with their contradictories being in each case identical with the terms of the inferred proposition together with their contradictories, the latter cannot possibly contain any existential implication that is not already contained in the original proposition.229

(2) Let every proposition he understood to imply simply the existence of its subject. (a) The validity of obversion is not affected. (b) The conversion of A is valid, and also that of I. If All S is P and Some S is P imply directly the existence of S, then they clearly imply indirectly the existence of P ; and this is all that is required in order that their conversion may be legitimate. The conversion of E is not valid; for No S is P implies neither directly nor indirectly the existence of P, whilst its converse does imply this. (c) The contraposition of E is valid, and also that of O. No S is P and Some S is not P both imply on our present supposition the existence of S, and since by the law of excluded middle every S is either P or not-P, it follows that they imply indirectly the existence of not-P. The contraposition of A is not valid; for it involves the conversion of E, which we have already seen not to be valid.230 (d) The process of inversion is not valid; for it involves in the case of both A and E the conversion of an E proposition.231 If along with an E proposition we are specially given the information that P exists, or if this is implied in some other proposition given us at the same time, then the E proposition may of course be converted. In corresponding circumstances the contraposition and inversion of A and the inversion of E may be valid.232 Or again, given simply No S is P, we may infer Either P is non-existent or no P is S ; and similarly in other cases.

(3) Let no proposition he understood to imply the existence either of its subject or of its predicate. Having now got rid of the implication of the existence either of subject or predicate in the case of all propositions, we might naturally suppose that in no case in which we make an immediate inference need we trouble ourselves with any question of existence at all. As already indicated, however, this conclusion would be erroneous. (а) The process of obversion is still valid. Take, for example, the obversion of No S is P. The obverse All S is not-P implies that if there is any S there is also some not-P. But this is necessarily implied in the proposition No S is P itself. If there is any S it is by the law of excluded middle either P or not-P; therefore, given that No S is P, it follows immediately that if there is any S there is some not-P. (b) The conversion of E is valid. Since No S is P denies the existence of anything that is both S and P, it implies that if there is any S there is some not-P and that if there is any P there is some not-S ; and these are the only implications with regard to existence involved in its converse. The conversion of A, however, is not valid; nor is that of I. For Some P is S implies that if there is any P there is also some S ; but this is not implied either in All S is P or in Some S is P. (c) That the contraposition of A is valid follows from the fact that the obversion of A and the conversion of E are both valid.233 That the contraposition of E and that of O are invalid follows from the fact that the conversion of A and that of I are both invalid. (d) That inversion is invalid follows similarly. On our present supposition then the following are valid: the obversion and contraposition of A, the obversion of I, the obversion and conversion of E, the obversion of O; the following are invalid: the conversion and inversion of A, the conversion of I, the contraposition and inversion of E, the contraposition of O.234

(4) Let particulars be understood to imply, while universals are not understood to imply, the existence of their subjects. (a) The validity of obversion is again obviously unaffected.235 (b) The conversion of E is valid, and also that of I, but not that of A.236 (c) The contraposition of A is valid, and also that of O, but not that of E. (d) The process of inversion is not valid. These results are obvious; and the final outcome is—as might have been anticipated—that we may infer a universal from a universal, or a particular from a particular, but not a particular from a universal.237 An important point to notice is that in the immediate inferences which remain valid on this supposition (namely, obversion, simple conversion, and simple contraposition) there is no loss of logical force; while at the best the reverse would be the case in those that are no longer valid (namely, conversion per accidens, contraposition per accidens, and inversion).

159. The Doctrine of Opposition and the Existential Import of Propositions.—The ordinary doctrine of opposition, in its application to the traditional schedule of propositions, is as follows: (a) The truth of Some S is P follows from that of All S is P, and the truth of Some S is not P from that of No S is P (doctrine of subalternation); (b) All S is P and Some S is not P cannot both be true and they cannot both be false, similarly for Some S is P and No S is P (doctrine of contradiction); (c) All S is P and No S is P cannot both be true but they may both be false (doctrine of contrariety); (d) Some S is P and Some S is not P may both be true but they cannot both be false (doctrine of sub-contrariety). We will now examine how far these several doctrines hold good under various suppositions respecting the existential import of propositions.238

It should be added that, throughout the discussion, the propositions are supposed to be interpreted assertorically, as has always been the custom with the traditional schedule. The necessity for this proviso will from time to time be pointed out.

(1) Let every proposition be interpreted as implying the existence both of its subject and of its predicate and also of their contradictories.239

On this supposition, if either the subject or the predicate of a proposition is the name of a class which is unrepresented in the universe of discourse or which exhausts that universe, then that proposition is false; for it implies what is inconsistent with fact. It follows that a pair of contradictories as usually stated, and also a pair of sub-contraries, may both be false. For example, All S is P and Some S is not P both imply the existence of S in the universe of discourse. In the case then in which S does not exist in that universe, these propositions would both be false.

If a concrete illustration is desired, we may take the propositions, None of the answers to the question shewed originality, Some of the answers to the question shewed originality, and assume that each of these propositions includes as part of its implication the actual occurrence of its subject in the universe of discourse. Then our position is that if there were no answers to the question at all, the truth of both the propositions must be denied. The fact of there having been no answers does not render the propositions meaningless; but it renders them false, their full import being assumed to be, respectively, There were answers to the question but none of them shewed originality, There were answers to the question and some of them shewed originality.

We must not of course say that under our present supposition true contradictories cannot be found; for this is always possible. The true contradictory of All S is P is Either some S is not P, or else either S or not-S or P or not-P is non-existent. Similarly in other cases. The ordinary doctrines of subalternation and contrariety remain unaffected.

(2) Let every proposition be interpreted as implying the existence of its subject. For reasons similar to those stated above, the ordinary doctrines of contradiction and sub-contrariety again fail to hold good. The true contradictory of All S is P now becomes Either some S is not P, or S is non-existent. The ordinary doctrines of subalternation and contrariety again remain unaffected.

(3) Let no proposition be interpreted as implying the existence either of its subject or of its predicate. (a) The ordinary doctrine of subalternation holds good. (b) The ordinary doctrine of contradiction does not hold good. All S is P, for example, merely denies the existence of any S’s that are not P’s; Some S is not P merely asserts that if there are any S’s some of them are not P’s. In the case in which S does not exist in the universe of discourse we cannot affirm the falsity of either of these propositions.240 (c) The ordinary doctrine of contrariety does not hold good. For if there is no implication of the existence of the subject in universal propositions we are not actually precluded from asserting together two propositions that are ordinarily given as contraries. All S is P merely denies that there are any S not-P’s, No S is P that there are any SP’s. We may, therefore, without inconsistency affirm both All S is P and No S is P ; but this is virtually to deny the existence of S.241 (d) The ordinary doctrine of sub-contrariety remains unaffected.

(4) Let particulars be interpreted as implying, while universals are not interpreted as implying, the existence of their subjects. (a) The ordinary doctrine of subalternation does not hold good. Some S is P, for example, implies the existence of S, while this is not implied by All S is P. (b) The ordinary doctrine of contradiction holds good. All S is P denies that there is any S that is not-P; Some S is not P affirms that there is some S that is not-P. It is clear that these propositions cannot both be true; it is also clear that they cannot both be false. Similarly for No S is P and Some S is P. (c) The ordinary doctrine of contrariety does not hold good. All S is P and No S is P are not inconsistent with one another, but the force of asserting both of them is to deny that there are any S’s.242 This follows just as in the case of our third supposition.243 (d) The ordinary doctrine of sub-contrariety does not hold good.244 Some S is P and Some S is not P are both false in the case in which S does not exist in the universe of discourse.

The relation between contradictories is by far the most important relation with which we are concerned in dealing with the opposition of propositions, and it will be observed that the last of the above suppositions is the only one under which the ordinary doctrine of contradiction holds good.

160. The Opposition of Modal Propositions considered in connexion with their Existential Import.—The propositions discussed in the preceding sections have been the propositions belonging to the traditional schedule interpreted assertorically. Turning now to the corresponding modal schedule, we may briefly consider how the doctrine of opposition is affected, if at all, on the supposition that the propositions included in the schedule are not interpreted as implying the existence of their subjects. We find that on this supposition S as such is P and S need not be P are true contradictories.

S as such is P (interpreted as not necessarily implying the existence of S) does more than deny the actual occurrence of the conjunction S not-P, it denies the possibility of such a conjunction; and all that is necessary in order to contradict this is to affirm the possibility of the conjunction. This is done by the proposition S need not be P (also interpreted as not necessarily implying the existence of S). On the same supposition, S as such is P, S as such is other than P, are true contraries.

Here, however, another problem suggests itself. Leaving on one side the question as to any implication of actuality, are modal propositions to be interpreted as containing any implication in regard to the possibility of their antecedents? And, further, how does our answer to this question affect the opposition of modals? The consideration of this problem may be deferred until we come to deal with the opposition of conditional propositions (see section 176).

161. Jevons’s Criterion of Consistency.—In passing to the explicit discussion of the existential import of categorical propositions, we may consider first the Criterion of Consistency, which is laid down by Jevons (following De Morgan):—Any two or more propositions are contradictory when, and only when, after all possible substitutions are made, they occasion the total disappearance of any term, positive or negative, from the Logical Alphabet. The criterion amounts to this, that every proposition must be understood to imply the existence of things denoted by every simple term contained in it, and also of things denoted by the contradictories of such terms. If, for example, we have the proposition All S is P, this implies that among the members of the universe of discourse are to be found S’s and P’s, not-S’s, and not-P’s. In defence of this doctrine Jevons appears to rely mainly upon the psychological law of relativity, namely, that we cannot think at all without separating what we think about from other things. Hence if either a term or its contradictory represents nonentity, that term cannot be either subject or predicate in a significant proposition.245 It is clear, however, that this psychological argument falls away as soon as it is allowed that we may be confining ourselves to a limited universe of discourse, or indeed if we confine ourselves to any universe less extensive than that which covers the whole realm of the conceivable. Of course the more limited the universe to which our proposition is supposed to relate the more easily may S or P either exhaust it or be absent from it; but with very complex subjects and predicates the contradictory of one or both of our terms may easily exhaust even an extended universe. Take, for example, the proposition, No satisfactory solution of the problem of squaring the circle has ever been published by Mr A. Here the subject is non-existent; and it may happen also that Mr A. has never published anything at all.246 Further, if I am not allowed to negative X, why should I be allowed to negative AB? There is nothing to prevent X from representing a class formed by taking the part common to two other classes. In certain combinations indeed it may be convenient to substitute X for AB, or vice versâ. It would appear then that what is contradictory when we use a certain set of symbols may not be contradictory when we use another set of symbols. This argument has a special bearing on the complex propositions which are usually relegated to symbolic logic, but to which Jevons’s criterion is intended particularly to apply.

No doubt Jevons’s criterion is sometimes a convenient assumption to make; provisionally, for example, in working out the doctrine of immediate inferences on the traditional lines. But it is an assumption that should always be explicitly referred to when made; and it ought not to be regarded as having an axiomatic and binding force, so as to make it necessary to base the whole of logic upon it.

162. The Existential Import of the Propositions included in the Traditional Schedule.—We may now turn to the consideration of the question whether the propositions SaP, SeP, SiP, SoP should or should not be interpreted as implying the existence of their subjects in the universe of discourse to which reference is made. In this section it will be assumed that the import of all the propositions under discussion is assertoric, not modal.

A brief reference may be made to two sources of misunderstanding to which attention has already been called. (а) All propositions contain affirmations relating to some system of reality; and by analysis every proposition may be made to yield an “ultimate subject” which is real, namely, the system of reality to which the proposition relates. This system of reality is what we mean by the universe of discourse; and, as we have seen, the universe of discourse can never be entirely emptied of content. It must then be understood that if we decide that certain propositional forms are not to be interpreted as containing as part of their import the affirmation of the existence of their subjects, it is far from being thereby intended that propositions falling into these forms contain no affirmation relating to reality.247 (b) We must put on one side a very summary solution of our problem, which, if it were correct, would render any further discussion needless. How, it may be asked, can we possibly speak about anything and at the same time exclude it from the universe of discourse? This question suggests a certain ambiguity which may attach to the phrase universe of discourse, but which can hardly remain an ambiguity after the explanations already given. The answer is that we can certainly think and speak about a thing with reference to a given universe of discourse without implying, or even believing in, its existence in that universe. Suppose, for example, that I say there are no such things as unicorns. If this statement is to be accepted, it must be interpreted literally (not elliptically); and it is clear that the universe of discourse referred to is the material universe.248 I speak then of unicorns with reference to the material universe, but deny that such creatures are to be found (or exist) in it.

The question we have to discuss is one of the interpretation of propositional forms,249 and the solution will therefore be to some extent a matter of convention. We shall be guided in our solution partly by the ordinary usage of language, and partly by considerations of logical convenience and suitability.

As regards the ordinary usage of language there can be no doubt that we seldom do as a matter of fact make predications about non-existent subjects. For such predications would in general have little utility or interest for us. “The practical exigencies of life,” as Dr Venn remarks, “confine most of our discussions to what does exist, rather than to what might exist” (Symbolic Logic, p. 131). We must, however, consider whether there are not exceptional cases; and if we can find any in which it is clear that the speaker would not necessarily intend to imply the existence of the subject, we may draw the conclusion that the propositional form of which he makes use is not in popular usage uniformly intended to convey such an implication.

Universal Affirmatives. If a universal affirmative proposition is obtained by a process of exhaustive enumeration (e.g., All the Apostles were Jews, All the books on that shelf are bound in morocco), or if it is obtained by empirical generalisation based on the examination of individual instances (e.g., All ruminant animals are cloven-hoofed), then it is clear that the existence of the subject is a presupposition of the affirmation. We may, however, note certain other classes of cases in which such a presupposition is not necessary.

(a) We may affirm an abstract connexion of attributes, based on considerations of a deductive character or at any rate not obtained by direct generalisation from observed instances of the subject, and the existence of the subject is then not essential. For example, The impact of two perfectly elastic bodies leads to no diminution of kinetic energy ; Every body, not compelled by impressed forces to change its state, continues in a state of rest or of uniform motion in a straight line.

It may perhaps be said that all propositions falling within this category will be really apodeictic, and that our present discussion has been limited to assertoric propositions. There is some force in this criticism. It is, however, to be remembered that the assertoric SaP can be inferred from the apodeictic SaP, so that if we can have the latter without any implication as to the existence of S we may have the former also, unless indeed we decide to differentiate between them in regard to their existential implication. The examples that we have given are moreover expressed in ordinary assertoric form, and not in any distinctive apodeictic form, such as S as such is P, It is inherent in the nature of S to be P.

(b) The proposition SaP may express a rule laid down, and remaining in force, without any actual instance of its application having arisen. For example, All candidates arriving five minutes late are fined one shilling, All candidates who stammer are excused reading aloud, All trespassers are prosecuted.

If it is argued that, in such cases as these,250 the propositions ought properly to be written in the conditional and not in the categorical form (e.g., If any candidate arrives five minutes late, that candidate is fined one shilling), the reply is that this is to misunderstand the point just now at issue, which is whether we meet with propositions in ordinary discourse which are categorical in form and yet are hypothetical so far as the existence of their subjects is concerned. It is of course open to us to decide that for logical purposes we will so interpret categorical propositions that in such cases as the above the categorical form can no longer be used. But for the present we are merely discussing popular usage.

(c) Assertions in regard to possible future events are sometimes thrown into the form SaP. For example, Who steals my purse steals trash, Those who pass this examination an lucky men. The first of these propositions would not be invalidated supposing my purse never to be stolen, and the latter, as Dr Venn remarks,251 would be tacitly supplemented by the clause “if any such there be.”

(d) There are cases in which the intended implication of a proposition of the form All S is P is to deny that there are any S’s; for example, An honest miller has a golden thumb, All the carts that come to Crowland are shod with silver.252

Universal Negatives. It is still easier to find instances from common speech in which universal negative propositions, that is, propositions of the form No S is P, are not to be regarded as necessarily implying the existence of their subjects.

(a) There are again cases in which the proposition is reached by a process of abstract reasoning about a subject the actual existence or occurrence of which is not presupposed; for example, A planet moving in a hyperbolic orbit can never return to any position it once occupied.253

(b) The import of the proposition may be distinctly to imply, if not definitely to affirm, the non-existence of the subject; for example, No ghosts have troubled me, No unicorns have ever been seen.254

(c) A denial of the conjunction ABC may be expressed in the form No AB is C without any intention of thereby affirming the conjunction AB ; for example, No satisfactory solution of the problem of squaring the circle has been published, No woman candidate for the Theological Tripos has been educated at Newnham College, No Advanced Student in Law is on the boards of Trinity College.255

Particulars. In the case of particular propositions, it is far less easy to give examples, such as might be met with in ordinary discourse, in which there is no implication of the existence of the subjects of the propositions. There may be exceptions, but at any rate the cases are exceedingly rare in which in ordinary speech we predicate anything of a non-existent subject without doing so universally. The main reason for this is, as Dr Venn points out, that “an assertion confined to ‘some’ of a class generally rests upon observation or testimony rather than on reasoning or imagination, and therefore almost necessarily postulates existent data, though the nature of this observation and consequent existence is, as already remarked, a perfectly open question. ‘Some twining plants turn from left to right,’ ‘Some griffins have long claws,’ both imply that we have looked in the right quarters to assure ourselves of the fact. In one case I may have observed in my own garden, and in the other on crests or in the works of the poets, but according to the appropriate tests of verification, we are in each case talking of what is.”256 If we look at the question from the other side, we find that when our primary object is to affirm the existence of a class of objects, our assertion very naturally takes the form of a particular proposition. If, for example, we desire to affirm the existence of black swans, we say Some swans are black. The existential implication of a proposition of this kind in ordinary discourse is one of its most fundamental characteristics.

On the whole it cannot be said that the usages of ordinary speech afford a decisive solution of the problem under discussion. It has, however, been shewn (1) that we seldom or never make statements about non-existent subjects in the form Some S is P or the form Some S is not P ; (2) that, although it is also true that we do not as a rule do so in the form All S is P or the form No S is P, still there are several classes of cases in which the use of these latter forms is not to be understood as necessarily carrying with it the implication that S is existent. Hence we should be departing very little from ordinary usage if we were to decide to interpret particulars as implying the existence of their subjects, but universals as not doing so (that is, as not doing so by their bare form).

I do not, however, regard this solution as necessitated by popular usage. It is, for instance, still open to anyone to adopt the convention that, for logical purposes, the categorical form shall only be used when the implication of the existence of the subject is intended. On this interpretation, the conditional or hypothetical form must be adopted whenever the existence of the subject is left an open question. Thus, if we are doubtful about the existence of S (or, at any rate, do not wish to affirm its existence), we must be careful to say, If there is any S, then all S is P, instead of simply All S is P ; in other words, the hypothetical character of the proposition so far as the existence of its subject is concerned must be made explicit.

The problem then not being decided by considerations of popular usage alone, we must go on to enquire how the question is affected by considerations of logical convenience and suitability. Here again there is no one solution that is inevitable. Reasons can, however, be urged in favour of interpreting particulars as implying, but universals as not implying, the existence of their subjects;257 and this, as we have seen, is a solution that derives some sanction from popular usage.

(1) A consideration of the manner in which the validity of immediate inferences is affected by the existential import of propositions affords reasons for the adoption of this interpretation.258 The most important immediate inferences are simple conversion (i.e., the conversion of E and of I) and simple contraposition (i.e., the contraposition of A and of O). If, however, universals are regarded as implying the existence of their subjects, then, as shewn in section 158, neither the conversion of E nor the contraposition of A is valid, irrespective of some farther assumption; whereas, if universals are not regarded as implying the existence of their subjects, then both these operations are legitimate without qualification. On the other hand, the conversion of I and the contraposition of O are valid only if particulars do imply the existence of their subjects.259

Turning to immediate inferences of another kind, it is clear that if universal propositions formally imply the existence of their subjects, we cannot legitimately pass from All X is Y to All AX is Y.260 For it is possible that there may be X’s and yet no AX’s, and in this case the former proposition may be true, while the latter will certainly be false. Again, given that A is X, B is Y, C is Z, we cannot infer that ABC is XYZ. Such restrictions as these would constitute an almost insurmountable bar to progress in inference as soon as we have to do with complex propositions.261

(2) We may next consider the existential import of propositions with reference to the doctrine of opposition. It has been shewn in section 159 that if particulars are interpreted as implying the existence of their subjects, while universals are not so interpreted, then A and O, E and I, are true contradictories; but that this is not the case under any of the other suppositions discussed in the same section.262 There can, however, be no doubt that one of the most important functions of particular propositions is to contradict the universal propositions of opposite quality; and hence we have a strong argument in favour of a view of the existential import of propositions which will leave the ordinary doctrine of contradiction unaffected.

As regards the doctrines of subalternation, contrariety, and subcontrariety, our results (namely, that I does not follow from A, or O from E, that A and E may both be true, and that I and O may both be false) are no doubt paradoxical. But this objection is far more than counterbalanced by the fact that the doctrine of contradiction is saved. For as compared with the relation between contradictories, these other relations are of little importance. We may specially consider the relation between A and I. Some S is P cannot now without qualification be inferred from All S is P, since the former of these propositions implies the existence of S, while the latter does not. But as a matter of fact this is an inference which we never have occasion to make. If their existential import is the same why should we ever lay down a particular proposition when the corresponding universal is at our service? On the other hand, the view that we are advocating gives Some S is P a status relatively to All S is P as well as relatively to No S is P which it could not otherwise possess; and similarly for Some S is not P. Our result as regards the relation between SaP and SiP has been described as equivalent to saying “that a statement of partial knowledge carries more real information than a statement of full knowledge; since if we only possess limited information, and so can only assert SiP, we thereby affirm the existence of S ; but if we have sufficient knowledge to speak of all S (S remaining the same) the statement of that full knowledge immediately casts a doubt upon that existence.” This way of putting it is, however, misleading if not positively erroneous. On the view in question it is incorrect to say simply that SiP and SaP give “partial” and “full” knowledge respectively, for SiP while giving less knowledge than SaP in one direction gives more in another. In other words, the knowledge which is “full” relatively to SiP is not expressed by SaP by itself, but by SaP together with the statement that there are such things as S.263

(3) There is one further point of importance to be noted, and that is, that the interpretation of A, E, I, O propositions under consideration is the only interpretation according to which each one of these propositions is resolved into a single categorical statement. For if A and E imply the existence of their subjects they express double, not single, judgments, being equivalent respectively to the statements: There are S’s, but there are no SPʹ’s ; There are S’s, but there are no SP’s ; whereas on the interpretation here proposed they simply express respectively the single judgments: There are no SPʹ’s ; There are no SP’s. On the other hand, if I and O do not imply the existence of their subjects, instead of expressing categorical judgments, they express somewhat complex hypothetical ones, being equivalent respectively to the statement: If there are any S’s then there are some SP’s ; If there are any S’s then there are some SPʹ’s ; whereas on our interpretation they express respectively the categorical judgments: There are SP’s ; There are SPʹ’s.264

On the whole, there is a strong cumulative argument in favour of interpreting particulars, but not universals, as implying formally the existence of their subjects.265 This solution is to be regarded as partly of the nature of a convention. We arrive, however, at the conclusion that no other solution can equally well suffice as the basis of a scientific treatment of the traditional schedule of propositions, so long, at any rate, as the propositions included in the schedule are regarded as assertoric and not modal.

163. The Existential Import of Modal Propositions.—Of apodeictic propositions it may be said still more emphatically than of assertoric universals that they do not necessarily imply the existence of their subjects. For they assert a necessary relation between attributes, the ground of which is frequently to be sought in abstract reasoning rather than in concrete experiences. And the same is true of the denial of apodeictic propositions. We may on abstract grounds assert the possibility of a certain concomitance (or non-concomitance) of attributes without having had actual experience of that concomitance (or non-concomitance), and without intending to imply its actuality. Hence we should not interpret the proposition S may be P, any more than the proposition S must be P, as by its bare form affirming the existence of S.

It has been shewn that in order that the propositions All S is P and Some S is not P may be true contradictories, one or other of them must be interpreted as implying the existence of S. It follows, however, from what has been said above that the same condition need not be fulfilled in order that S must be P and S need not be P may be true contradictories.266

But to this it has to be added that, in order that these two propositions may be true contradictories, one or other of them must be interpreted as implying the possible existence of S. This line of thought has been suggested in section 160, and it will be pursued farther in sections 176 and 179.

EXERCISES.

164. The particular judgment has, from different stand-points, been identified (a) with the existential judgment, (b) with the problematic judgment, (c) with the narrative judgment. Comment on each of these views. [C.]

The student may find that to write a detailed answer to this question will help to clear up his views respecting the particular proposition. No detailed answer will here be given; but attention may be called to one or two points. (a) Two kinds of existential judgments may be distinguished. (i) Those which affirm existence indefinitely, that is, somewhere in the universe of discourse; for example, There are white hares, There is a devil. (ii) Those which affirm existence with reference to some definite time and place; for example, It rains, I am hungry. The particular may perhaps be identified with (i), hardly with (ii). (b) We may be justified in affirming the problematical S may be P, when we cannot affirm the particular Some S is P. There are reasons for interpreting the latter judgment existentially as regards its subject, which do not apply to the former judgment. (c) The narrative judgment need not have the indefinite character of the particular. We may, however, hold that the two kinds of judgment have this in common that there are grounds for interpreting both existentially as regards their subjects.

165. Discuss the relation between the propositions All S is P and All not-S is P.

This is an interesting case to notice in connexion with the discussion raised in sections 158 and 159.

We have

SaP = SePʹ = PʹeS ; SʹaP = SʹePʹ = PʹeSʹ = PʹaS.

The given propositions come out, therefore, as contraries. On the view that we ought not to enter into any discussion concerning existence in connexion with immediate inference, we must, I suppose, rest content with this statement of the case. It seems, however, sufficiently curious to demand further investigation and explanation. We may as before take different suppositions with regard to the existential import of propositions. (1) If every proposition implies the existence of both subject and predicate and their contradictories, then it is at once clear that the two propositions cannot both be true together; for between them they deny the existence of not-P. (2) On the view that propositions imply simply the existence of their subjects, it has been shewn in section 158 that we are not justified in passing from All not-S is P to All not-P is S unless we are given independently the existence of not-P. But it will be observed that in the case before us the given propositions make this impossible. Since all S is P and all not-S is P, and everything is either S or not-S by the law of excluded middle, it follows that nothing is not-P. In order, therefore, to reduce the given propositions to such a form that they appear as contraries (and consequently267 as inconsistent with each other) we have to assume the very thing that taken together they really deny. (3) and (4). On the view that at any rate universal propositions do not imply the existence of their subjects, we have found in section 159 that the propositions No not-P is S, All not-P is S, are not necessarily inconsistent, for they may express the fact that P constitutes the entire universe of discourse. But this fact is just what is given us by the propositions in their original form. Under each hypothesis, then, the result obtained is satisfactorily accounted for and explained.

166. “The boy is in the garden.” “The centaur is a creation of the poets.” “A square circle is a contradiction.” Discuss the above propositions as illustrating different functions of the verb “to be”; or as bearing upon the logical conception of different universes of discourse or of different kinds of existence. [C.]

167. Discuss the existential import of singular propositions. “The King of Utopia did not die on Tuesday last.” Examine carefully the meaning to be attached to the denial of this proposition. [K.]

168. Some logicians hold that from All S is P we may infer Some not-S is not-P. Take as an illustration, All human actions are foreseen by the Deity. [C.]

169. Discuss the validity of the following inference:—All trespassers will be prosecuted, No trespassers have been prosecuted, therefore, There have been no trespassers. [C.]

170. On the assumption that particulars are interpreted as implying while universals are not interpreted as implying the existence of their subjects in the universe of discourse, examine (stating your reasons) the validity of the following inferences; All S is P and Some R is not S therefore, Some not-S is not P ; All S is P and Some R is not P, therefore, Some not-S is not P ; All S is P and Some R is S, it is, therefore, false that No P is S ; All S is P and Some R is P, it is, therefore, false that No P is S. [K.]

171. Discuss the formal validity of the following arguments, (i) on the supposition that all categorical propositions are to be interpreted as implying the existence of their subjects in the universe of discourse, (ii) on the supposition that no categorical propositions are to be so interpreted: (a) All P is Q, therefore, All AP is AQ ; (b) All AP is AQ, therefore, Some P is Q. [K.]

172. Work out the doctrine of Opposition and the doctrine of Immediate Inferences on the hypothesis that universals are to be interpreted as implying, while particulars are not to be interpreted as implying, the existence of their subjects in the universe of discourse. [K.]