Studies and Exercises in Formal Logic

²²⁸ Dr Wolf perhaps draws a distinction between the proposition “If there are any S’s they must all be P’s” and the proposition “If there are any S’s they are all P’s,” giving to the former an apodeictic, and to the latter a merely assertoric, force. But if so, then the former is implied by All S is P, only if this proposition is apodeictic, not if it is merely assertoric. The argument is in this case irrelevant so far as the position which I take is concerned, since it is only the assertoric SaP that I regard as equivalent to SPʹ = 0. Dr Wolf can hardly maintain that all propositions of the form All S is P are apodeictic. His whole treatment of the subject with which we are now dealing appears, however, to be valid only if it relates to a modal schedule of propositions. At the same time he nowhere clearly indicates a limitation of this kind, and many of the doctrines which he criticises are intended by those who adopt them to apply only to an assertoric schedule.

222 (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. 223 We then have:—

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

²²⁹ The reader may be reminded that in our first working out of these immediate inferences we provisionally assumed, apart from any implication contained in the propositions themselves, that the terms involved and also their contradictories represented existing classes.

224 (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.

²³⁰ Or we might argue directly that the contraposition of A is not valid, since All S is P does not imply the existence of not-P, whilst its contrapositive does imply this.

²³¹ Or again we might argue directly from the fact that neither All S is P nor No S is P implies the existence of not-S.

²³² For example, given (α) No S is P, (β) All R is P, we may under our present supposition convert (α), since (β) implies indirectly the existence of P ; and we may contraposit (β), since (α) implies indirectly the existence of not-P. It will also he found that, given these two propositions together, they both admit of inversion.

(3) Let no proposition he understood to imply the existence either of its subject or of its predicate.
225 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

²³³ Or we might argue directly as follows; since the proposition All S is P denies the existence of anything that is both S and not-P, it implies that if there is any S there is some P and that if there is any not-P there is some not-S ; and these are the only implications with regard to existence involved in its contrapositive.

²³⁴ Dr Wolf holds in opposition to the view here expressed that on the supposition in question all the ordinary immediate inferences remain valid. This conclusion is based on the doctrine that Some S is P does not imply that if there is any S there is also some P. “All S is P and Some S is P, it is true, do not imply that ‘if there is any P there is also some S.’ But then Some P is S does not necessarily imply that either. There can, therefore, be no objection, on that score, against inferring, by conversion, Some P is S from All S is P or Some S is P. With the vindication of conversion all the remaining supposed illegitimate inferences connected with it are also vindicated. We may, therefore, conclude that to let no propositional form as such necessarily imply the existence of either its subject or its predicate in no way affects the validity of any of the traditional inferences of logic” (Studies in Logic, p. 147). I have dealt with Dr Wolf’s position in the note on page 216; and it is unnecessary to repeat the argument here. If importance is attached to concrete examples, I may suggest, as an example for conversion, All blue roses are blue (a formal proposition which must be regarded as valid on the existential supposition under discussion); and, as an example for inversion, All human actions are foreseen by the Deity. There are, moreover, certain difficulties connected with syllogistic and more complex reasonings that need a brief separate discussion, even when the case of conversion has been disposed of.

226 (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
227 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).

²³⁵ Obversion thus remains valid on all the suppositions which have been specially discussed above. If, however, affirmatives are interpreted as implying the existence of their subjects while negatives are not so interpreted, then of course we cannot pass by obversion from E to A, or from O to I.

²³⁶ But from the two propositions, All S is P, Some R is S, we can infer Some P is S ; and similarly in other cases.

²³⁷ On the assumption, however, that the universe of discourse can never be entirely emptied of content, Something is P may be inferred from Everything is P, and Something is not P may be inferred from Nothing is P. Again, as is shewn by Dr Venn (Symbolic Logic, pp. 142–9), the three universals All S is P, No not-S is P, All not-S is P, together establish the particular Some S is P. Any universe of discourse contains à priori four classes—(1) SP, (2) S not-P, (3) not-S P, (4) not-S not-P. All S is P negatives (2); No not-S is P negatives (3); All not-S is P negatives (4). Given these three propositions, therefore, we are able to infer that there is some SP, for this is all that we have left in the universe of discourse. As already pointed out, the assumption that the universe of discourse can never be entirely emptied of content is a necessary assumption, since it is an essential condition of a significant judgment that it relate to reality. If the universe of discourse is entirely emptied of content we must either fail to satisfy this condition, or else unconsciously transcend the assumed universe of discourse and refer to some other and wider one in which the former is affirmed not to exist.

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

²³⁸ Of course the doctrine of contradiction always holds good in the sense that a pair of real contradictories cannot both be true or both false; and similarly with the other doctrines. The doctrines that we have to consider are not these, but whether SaP and SoP are really contradictories irrespective of the existential interpretation of the propositions, whether SaP and SeP are really contraries, and so on.

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 228 existence both of its subject and of its predicate and also of their contradictories.239

²³⁹It would be quite a different problem if we were to assume the existence of S and P independently of the affirmation of the given proposition. A failure to distinguish between these problems is probably responsible for a good deal of the confusion and misunderstanding that has arisen in connexion with the present discussion. But it is clearly one thing to say (a) “All S is P and S is assumed to exist,” and another to say (b) “all S is P,” meaning thereby “S exists and is always P.” In case (a) it is futile to go on to make the supposition that S is non-existent; in case (b), on the other hand, there is nothing to prevent our making the supposition, and we find that, if it holds good, the given proposition is false.

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.

229 (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
230 (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.

²⁴⁰ Dr Wolf (Studies in Logic, p. 132) denies the validity of this reasoning. He admits apparently that the existential propositions SPʹ = 0 and Either S = 0 or SPʹ > 0 are not contradictories; but he denies that on the supposition under discussion SaP and SPʹ = 0 are equivalent. His main ground for taking this view is that SaP carries with it the implication If there are any S’s they are all P’s, while SPʹ = 0 does not carry with it any such implication. This position has been already criticized in section 157. Dr Wolf relies partly upon concrete examples, but in so doing he complicates the discussion by introducing modal forms of expression. Thus for the proposition “Some successful candidates do not receive scholarships,” we find substituted in the course of his argument “If there are any successful candidates then some of them do not (or need not) receive scholarships,” and the insertion of the words in brackets yields a proposition which, although an inference from the original proposition, is not really equivalent to it, unless the original proposition is itself interpreted modally. Later on Dr Wolf explicitly alters the whole problem by assuming that what is under consideration is a modal schedule of propositions. Thus he goes on to say, “What SaP and SeP really express severally is the necessity and the impossibility of S being P”; and for the purpose of contradicting SaP and SeP, “SiP and SoP need mean no more than S may be P and S need not be P.” The question how far SaP and SeP should be interpreted modally is discussed elsewhere. All I would point out here is that it is a distinct question from that raised in the text, which is a question relating to the traditional schedule of propositions interpreted assertorically. The whole question of existential import is indeed one that cannot be discussed to any purpose until the character of the schedule of propositions under consideration has been defined. From the mixing up of schedules and interpretations nothing but confusion can result. In the following section the opposition of modals will be briefly considered in connexion with their existential import.

²⁴¹ Of course on the view under consideration we ought not to continue to speak of these two propositions as contraries.

(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
231 (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.

²⁴² If, however, we are given No S is P and also Some S is P, then we are able to infer that All S is P is false. The second of these propositions affirms the existence of S, and therefore destroys the hypothesis on which alone the first and third can be treated as compatible.

²⁴³ The above doctrine has been criticized on the ground that it practically amounts to saying that neither of the given propositions has any meaning whatever, but that each is a mere sham and pretence of predication; and a request is made for concrete examples. The following example may perhaps suffice to illustrate the particular point now at issue: “An honest miller has a golden thumb”; “Well, I am sure that no miller, honest or otherwise, has a golden thumb.” These two propositions are in the form of what would ordinarily be called contraries; but taken together they may quite naturally be interpreted as meaning that no such person can be found as an honest miller. The former proposition would indeed probably be intended to be supplemented by the latter or by some proposition involving the latter, and so to carry inferentially the denial of the existence of its subject.

Another example is contained in the following quotation from Mrs Ladd Franklin: “All x is y, No x is y, assert together that x is neither y nor not-y, and hence that there is no x. It is common among logicians to say that two such propositions are incompatible; but that is not true, they are simply together incompatible with the existence of x. When the schoolboy has proved that the meeting point of two lines is not on the right of a certain transversal and that it is not on the left of it, we do not tell him that his propositions are incompatible and that one or other of them must be false, but we allow him to draw the natural conclusion that there is no meeting point, or that the lines are parallel” (Mind, 1890, p. 77 n.).

Dr Wolf (Studies in Logic, p. 140), criticizing Mrs Ladd Franklin’s concrete example, maintains that the two propositions given by her are sub-contraries (I and O), not contraries (A and E). A moment’s consideration will, however, shew that this is not the case since neither of the propositions is particular. At the same time it is true that a little manipulation is required to bring them to the forms A and E. There is also the assumption that “on the right” and “on the left” exhaust the possibilities and are therefore contradictory terms. Granting this assumption, the two propositions may be expressed symbolically in the forms No S is P, No S is not P, and it then needs only the obversion of one of them to bring them to the forms A and E.

²⁴⁴ It may be worth observing that, given (b), (d) might be deduced from (c) or vice versâ.

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 232 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 233 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.

²⁴⁵ This point is put somewhat tentatively in a passage in Jevons’s Principles of Science (chapter 6, § 5) where he remarks: “If A were identical with ‘B or not-B,’ its negative not-A would be non-existent. This result would generally be an absurd one, and I see much reason to think that in a strictly logical point of view it would always be absurd. In all probability we ought to assume as a fundamental logical axiom that every term has its negative in thought. We cannot think at all without separating what we think about from other things, and these things necessarily form the negative notion. If so, it follows that any term of the form ‘B or not-B’ is just as self-contradictory as one of the form ‘B and not-B’.”

²⁴⁶ Other examples will be given in the following section.

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 234 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 235 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.

²⁴⁷ Compare Sigwart, Logic, i. p. 97 n.

²⁴⁸ It is hardly necessary to point out that ideas of unicorns exist in imagination, and that statements about unicorns are to be met with in fairy tales.

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.

²⁴⁹ See section 48.

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 236 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.