CHAPTER VIII.
THE EXISTENTIAL IMPORT OF CATEGORICAL PROPOSITIONS.[212]
[212] It will be advisable for students, on a first reading, to omit this chapter.
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 211 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.
[213] “The universe of discourse is sometimes limited to a small portion of the actual universe of things, and is sometimes co-extensive with that universe” (Boole, Laws of Thought, p. 166). On the conception of a limited universe of discourse, compare also De Morgan, Syllabus of a Proposed System of Logic, §§ 122, 3, and Formal Logic, p. 55; Venn, Symbolic Logic, pp. 127, 8; and Jevons, Principles of Science, chapter 3, § 4.
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.
[214] Compare Bradley, Principles of Logic, p. 41.
In our discussion of existential import it will not be necessary that we should make any attempt to determine the ultimate nature of reality. The questions at issue are, however, not exactly easy of solution, and various sources of misunderstanding are apt to arise.
There is one sense in which the existence of something corresponding to the terms employed must be postulated in all predication. For in order to make use of any term in an intelligible sense we must mentally attach some meaning to it. Hence there must be something in the mind corresponding to every term we use. Even in cases where there cannot be said to be any corresponding mental product, there must at any rate 212 be some corresponding mental process. This applies even to such terms as round square or non-human man or root of minus one. We are not indeed able to form an image of a round square or an idea of a non-human man, nor can we evaluate the root of minus one. But we attach a meaning to these terms, and they must therefore have a mental equivalent of some sort. In the case of “round square” or “non-human man” this is not the actual combination in imagination or idea of “round” with “square” or “non-human” with “man,” for such combinations are impossible. But it is the idea of the combination, regarded as a problem presented for solution, and perhaps involving an unsuccessful effort to effect the combination in thought. It is apparently of existence of this kind that some writers are thinking when they maintain that of necessity every proposition implies logically the existence of its subject. But our meaning is something quite different when we speak of existence in the universe of discourse. The nature of the distinction may be made more clear by the following considerations.
It will be admitted that whatever else is included in the full implication of a universal proposition, it at least denies the existence of a certain class of objects. No S is P denies the existence of objects that are both S and P ; All S is P denies the existence of objects that are S without also being P. In these propositions, however, we do not intend to deny the existence of SP (or SPʹ) as objects of thought. For example, in the proposition No roses are blue it is not our intention to deny that we can form an idea of blue roses ; nor in the proposition All ruminant animals are cloven-hoofed is it our intention to deny that ruminant animals without cloven hoofs can exist as objects of thought. These illustrations may help us to understand more clearly what is meant by existence in the universe of discourse. The universe of discourse in the case of the proposition No S is P is the universe (whatever it may be) in which the existence of SP is denied. The universe of discourse in the case of a universal affirmative proposition may be defined similarly. As regards particulars it may be best to seek an interpretation through the universals by which the particulars 213 are contradicted. Thus, the universe of discourse in the case of the proposition Some S is P may be defined as the universe (whatever it may be) in which the existence of SP would be understood to be denied in the corresponding universal negative. The proposition Some S is not P may be dealt with similarly.
The question whether a categorical proposition is to be interpreted as formally implying that its terms are the names of existing things may then be interpreted as follows: Given a categorical proposition with S and P as subject and predicate, is the existence of S or of P formally implied in that sphere (whatever it may be) in which the existence of SP (or SPʹ) is denied by the proposition (or by its contradictory)?
The question may be somewhat differently expressed as follows. Such a proposition as No S is P denies the existence of a certain complex of attributes, namely, SP. But with rare exceptions, S itself signifies a certain complex of attributes; and so does P. Does the proposition affirm the existence of these latter complexes in the same sense as that in which it denies the existence of the former complex?
No general criterion can be laid down for determining what is actually the universe of discourse in any particular case. It may, however, be said that knowledge as to what is the universe referred to is involved in understanding the meaning of any given proposition; and cases in which there can be any practical doubt are exceptional.[215] Thus, in the propositions No roses are blue, All men are mortal, All ruminant animals are cloven-hoofed, the reference clearly is to the actual physical universe; in The wrath of the Olympian gods is very terrible to the universe of the Greek mythology;[216] in Fairies are able to assume different forms to the universe of folk-lore;[217] in Two straight lines cannot enclose a space to the universe of spatial intuitions.
[215] It must at the same time be admitted that controversies sometimes turn upon an unrecognised want of agreement between the controversialists as to the universe of discourse to which reference is made.
[216] The universe of the Greek mythology does not consist of gods, heroes, centaurs, &c., but of accounts of such beings currently accepted in ancient Greece, and handed down to us by Homer and other authors. As regards the reference to reality, therefore, such a proposition as The wrath of the Olympian gods is very terrible is elliptical in a sense already explained.
[217] Here again there is an ellipsis. The universe of folk-lore does not consist of fairies, elves, &c., but of descriptions of them, based on popular beliefs, and conventionally accepted when such beings are referred to. Of course for anyone who really believed in the existence of fairies there would be no ellipsis, and the universe of discourse would be different.
214 With respect to the existential import of propositions the following questions offer themselves for consideration:
(1) Is the problem one with which logic, and more particularly formal logic, is properly concerned?
(2) How should the propositions belonging to the traditional schedule be interpreted as regards their existential implications?
(3) Can we formulate a schedule of propositions which directly affirm or deny existence, and how will such a schedule be related to the traditional schedule?
(4) How are ordinary logical doctrines affected by the answer given to the second of these questions?
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 215 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 216 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.
[218] The latter part of this statement is indeed nothing more than a repetition of the former part from a rather different point of view. The doctrine that the conclusions reached by the aid of formal logic can never do more than relate to what is merely conceivable is a very mischievous error. The material truth of the conclusion of a formal reasoning is only limited by the material truth of the premisses.
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 217 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]
[219] Dr Wolf denies this. His argument is, however, based mainly on the misinterpretation of a single concrete example. “Let us,” he says, “take a concrete example. Some things that children fear are ghosts. Does this proposition imply that if there is anything that children fear then there are also ghosts? Surely one may legitimately make such an assertion while believing that there are things that children fear, and yet absolutely disbelieving in the existence of ghosts. In fact the above proposition might very well be used in conjunction with an express denial of the existence of ghosts in order to prove that, while some things that children fear are real, they are also afraid of things that do not exist, but are merely imaginary” (Studies in Logic, p. 144). Any speciousness that this argument may possess arises from the ambiguity of the words “thing” and “real.” It is clear that in order to make the proposition in question intelligible the word “things” must be interpreted to mean “things, real or imaginary.” Moreover “imaginary things” have a reality of their own, though it is not a physical, material reality. Ghosts, therefore, do exist in the universe of discourse to which reference is made. The objects denoted by the predicate of the proposition have in fact just the same kind of existence as certain of the objects denoted by the subject. Looking at the matter from a slightly different point of view, it is clear that if by “things” in the subject we mean things having material existence, then unless ghosts have a similar existence the proposition is not true.
Bearing in mind the constant ambiguity of language, and the ways in which verbal forms may fail to represent adequately the judgments they are intended to express, it would in any case be unsatisfactory to allow a question of the kind we are here discussing to be decided by a single concrete example. Dr Wolf’s view is that Some S is P does not imply that if there are any S’s there are also some P’s. Suppose then that there are some S’s and that there are no P’s. It follows that there are S’s but not a single one of them is P. What in these circumstances the proposition Some S is P can mean it is difficult to understand.
So far as Dr Wolf’s argument is independent of the above concrete example, it appears to depend upon an identification of the proposition Some S is P with the proposition S may be P. The latter is a modal form, and is undoubtedly consistent with the existence of S and the non-existence of P. But I venture to think that the identification of the two forms runs entirely counter to the current use of language. I am quite prepared to admit that if All S is P is interpreted as an unconditional universal, meaning S as such is P, its true contradictory is S may be P, not Some S is P. But this is just because I do not think that Some S is P would be understood to express merely the abstract compatibility of S and P. Certainly Dr Wolf’s own concrete example, referred to above, cannot bear this interpretation. For some further observations on modals in connexion with existential import, see sections [160] and [163].
[220] Jevons remarks that he does not see how there can be in deductive logic any question about existence, and observes, with reference to the opposite view taken by De Morgan, that “this is one of the few points in which it is possible to suspect him of unsoundness “ (Studies in Deductive Logic, p. 141). It is, however, impossible to attach any meaning to Jevons’s own “Criterion of Consistency,” unless it has some reference to “existence.” “It is assumed as a necessary law that every term must have its negative. Thence arises what I propose to call the Criterion of Consistency, stated as follows:—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” (p. 181). What can this mean but that although we may deny the existence of the combination AB, we cannot without contradiction deny the existence of A itself, or not-A, or B, or not-B? This assumption regarding the existential implication of propositions runs through the whole of Jevons’s equational logic. The following passage, for example, is taken almost at random: “There remain four combinations, ABC, aBC, abC, abc. But these do not stand on the same logical footing, because if we were to remove ABC, there would be no such thing as A left; and if we were to remove abc there would be no such thing as c left. Now it is the criterion or condition of logical consistency that every separate term and its negative shall remain. Hence there must exist some things which are described by ABC, and other things described by abc” (p. 216).
218 155. 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]
[221] In an article in Baldwin’s Dictionary of Philosophy and Psychology, Mrs Ladd Franklin points out that the proposition All S is P is equivalent to the proposition Everything is P or not-S, and hence necessarily implies the existence of either P or not-S. Write x for not-S and y for P, so that the original proposition becomes All but x is y ; it then implies, as its minimum existential import, the existence of either x or y.
156. Various Suppositions concerning the Existential Import of Categorical Propositions.—Several different views may be 219 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]
[222] The suppositions that follow are not intended to be exhaustive. We might, for instance, regard propositions as implying the existence both of their subjects and their predicates, but not of the contradictories of these; or we might regard universals as always implying the existence of their subjects, but particulars as not necessarily implying the existence of theirs (see note [3] on p. 241); or affirmatives as always implying the existence of their subjects, but negatives as not necessarily implying the existence of theirs. This last supposition represents the view of Ueberweg. Still another view is taken by Lewis Carroll, who regards all categorical propositions, except universal negatives, as implying the existence of their subjects. “In every proposition beginning with some or all, the actual existence of the subject is asserted. If, for instance, I say ‘all misers are selfish,’ I mean that misers actually exist. If I wished to avoid making this assertion, and merely to state the law that miserliness necessarily involves selfishness, I should say ‘no misers are unselfish,’ which does not assert that any misers exist at all, but merely that, if any did exist, they would be selfish” (Game of Logic, p. 19). It would take too much space, however, to give a separate discussion to suppositions other than those mentioned in the text.
(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]
[223] “By the universe (of a proposition) is meant the collection of all objects which are contemplated as objects about which assertion or denial may take place. Let every name which belongs to the whole universe be excluded as needless: this must be particularly remembered. Let every object which has not the name X (of which there are always some) be conceived as therefore marked with the name x meaning not-X” (Syllabus, pp. 12, 13). Compare, also, De Morgan’s Formal Logic, p. 55.
(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 220 consideration of the subject, appears to be at once the most reasonable and the most simple.
[224] “An accidental or non-essential affirmation does imply the real existence of the subject, because in the case of a non-existent subject there is nothing for the proposition to assert” (Logic, I. 6, § 2).
(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.
[225] Jevons lays down the dictum that “we cannot make any statement except a truism without implying that certain combinations of terms are contradictory and excluded from thought” (Principles of Science, 2nd edition, p. 32). This is true of universals (though somewhat loosely expressed), but it does not seem to be true of particular propositions, whatever view may be taken of them.
(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]
[226] Dr Venn advocates this doctrine with special reference to the operations of symbolic logic; but there is no reason why it should not be extended to ordinary formal logic.
[227] The hypothesis in question has been already provisionally adopted in the scheme of logical equivalences given in section [108], and also in the symbolic scheme of propositions given on page [193].
221 157. 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.
[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:—
| 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]
[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]
[239]It 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.
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.
[250] This argument might be used with, reference to cases coming under (a) or (c) as well as with reference to those coming under (b).
(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 237 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.”
[251] Symbolic Logic, p. 132.
(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]
[252] Both these propositions are naturally to be interpreted as containing an indirect denial of the existence of their subjects. “Crowland is situated in such moorish rotten ground in the Fens, that scarce a horse, much less a cart, can come to it” (Bohn’s Handbook of Proverbs, p. 211). It would appear, however, that this proverb has now lost its force, inasmuch as “since the draining, in summer time, carts may go thither.”
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]
[253] This example is taken from Dixon, Essay on Reasoning, p. 62.
(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]
[254] The universe of discourse must here be taken to be the material universe. With reference to this example, however, a critic writes, “But surely the universe of imagination is the only one applicable; for unicorns have long been known not to belong to the actual material universe.” The universe of imagination may be required in order to sustain the position that the subject of the proposition exists in the universe of discourse; but any person making the statement would certainly not be referring to the world of imagination or the universe of heraldry, for the simple reason that in either of these cases the proposition (which must then be interpreted elliptically) would obviously not be true. On the other hand, we can quite well suppose the statement made with reference to the material universe: “Whether unicorns exist or not, at any rate they have never been seen.” Again, to take another example of a similar kind where the reference is also to the phenomenal universe, we can quite well suppose the statement made: “Whether there are ghosts or not, at any rate none have ever troubled me.” In order to avoid misapprehension, it is important to distinguish the above examples from such (elliptical) propositions as the following: “The wrath of the Homeric gods is very terrible,” “Fairies are able to assume different forms.” In each of these cases, the subject of the proposition (properly interpreted) exists in the particular universe to which reference is made. See notes [2] and [3] on page 213.
238 (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]
[255] “As an instance of a possibly non-existent subject of a negative proposition, take the following: ‘No person condemned for witchcraft in the reign of Queen Anne was executed.’” (Venn, Symbolic Logic, p. 132.)
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 239 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.
[256] Symbolic Logic, p. 131. Again, in such a proposition as “Some sea-serpents are not half a mile long” (meaning your so-called sea-serpents), the subject of the proposition exists in the universe to which reference is made, namely, the universe which may be described as the universe of travellers’ tales. We are here regarding the proposition as elliptical in a sense that has been already explained.
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 240 subjects;[257] and this, as we have seen, is a solution that derives some sanction from popular usage.
[257] On this view whenever it is desired specially to affirm the existence in the universe of discourse of the subject of a universal proposition, a separate statement to this effect must be made. For example, There are S’s, and all of them are P’s. If, on the other hand, it is ever desired to affirm a particular proposition without implying the existence of the subject, then recourse must be had to the hypothetical or conditional form of statement. Thus, if we do not intend to imply the existence of S, instead of writing Some S’s are P’s, we must write, If there are any S’s, then in some such cases they are also P’s.
(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]
[258] It has been objected that to base our view of the existential import of propositions upon the validity or invalidity of immediate inferences is to argue in a circle. “Whether,” it is said, “the immediate inferences are valid or not must be a consequence of the view taken of the existential import of the proposition and should not, therefore, be made a portion of the ground on which that view is based.” This objection involves a confusion between different points of view from which the problem of the relation between the existential import of propositions and the validity of logical operations may be regarded. In section [158] the logical consequences of various assumptions were worked out without any attempt being made to decide between these assumptions. Our point of view is now different; we are investigating the grounds on which one of the assumptions may be preferred to the others, and there is no reason why the consequences previously deduced should not form part of our data for deciding this question. The argument contains nothing that is of the nature of a circulus in probando.
[259] Thus, the table of equivalences given in section [106] is valid on the interpretation with which we are now dealing. The dependence of the table given in section [108] upon the same supposition is still more obvious. It has been already pointed out that the remaining immediate inferences based on conversion and obversion are of much less importance; see page [227].
241 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]
[260] It will be observed further that upon the same assumption we cannot even affirm the formal validity of the proposition All X is X. For X might be non-existent, and the proposition would then be false.
[261] Hence Mrs Ladd Franklin is led to the conclusion that “no consistent logic of universal propositions is possible except with the convention that they do not imply the existence of their terms” (Mind, 1890, p. 88).
(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.
[262] A and O, E and I, will also be true contradictories if universals are interpreted as implying the existence of their subjects, while particulars are not so interpreted. It would be interesting, if space permitted, to work out the results of this supposition in detail. If the student does this for himself, he will find that this is the only supposition, under which the ordinary doctrine of opposition holds good throughout. All other considerations, however, are opposed to its adoption. It altogether conflicts with popular usage; it renders the processes of simple conversion and simple contraposition illegitimate; and whilst making universals double judgments, it destroys the categorical character of particulars altogether. In regard to this last point see page [220].
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 242 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]
[263] The position taken above in regard to subalternation is very well expressed by Mrs Ladd Franklin. “Nothing of course is now illogical that was ever logical before. It is merely a question of what convention in regard to the existence of terms we adopt before we admit the warm-blooded sentences of real life into the iron moulds of logical manipulation. With the old convention (which was never explicitly stated) subalternation ran thus: No x’s are y’s (and we hereby mean to imply that there are x’s, whatever x may be), therefore, Some x’s are non-y’s. With the new convention the requirement is simply that if it is known that there are x’s (as it is known, of course, in by far the greater number of sentences that it interests us to form) that fact must be expressly stated. The argument then is: No x’s are y’s, There are x’s, therefore, There are x’s which are non-y’s.”
243 (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]
[264] Compare sections [156], [157].
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 244 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.
[265] We may briefly discuss in a note one or two objections to this view which have not yet been explicitly considered.
(а) Mill argues that a synthetical proposition necessarily implies “the real existence of the subject, because in the case of a non-existent subject there is nothing for the proposition to assert” (Logic, i. 6, § 2). In answer to this it is sufficient to point out that a non-existent thing will be described as possessing attributes which are separately attributes of existing things, although that particular combination of them may not anywhere be found, and if we know (as we may do) that certain of these attributes are always accompanied by other attributes we may predicate the latter of the non-existent thing, thereby obtaining a real proposition which does not involve the actual existence of its subject. As an argument ad hominem it may further be pointed out that Mill inclines to deny the existence of perfect straight lines or perfect circles. Would he therefore affirm that we can make no real assertions about such things?
(b) Mr Welton repeats several times that a proposition which relates to a non-existent subject must be a mere jumble of words, a predication in appearance only. “That the meaning of a universal proposition can be expressed as a denial is true, but this is not its primary import. And this denial itself must rest upon what the proposition affirms. Unless SaP implies the existence of S, and asserts that it possesses P, we have no data for denying the existence of SPʹ. For if S is non-existent the denial that it is non-P can have no intelligible meaning” (Logic, p. 241). The examples which we have already given are sufficient to dispose of this objection; but it may be worth while to add a further argument. According to Mr Welton, an E proposition implies the existence of its subject but not of its predicate. We cannot then infer PeS from SeP because we have no assurance of the existence of P. But in accordance with the position taken by Mr Welton, we ought to go further and say that PeS must be a mere jumble of words unless we are assured of the existence of P. It is impossible, however, to regard PeS as a mere unmeaning jumble of words, a predication in appearance only, when SeP is a significant and true proposition. PeS may be false, or it may be an unnatural form of statement, but it cannot be meaningless if SeP has a meaning. Take, for example, the propositions—No woman is now hanged for theft in England, No person now hanged for theft in England is a woman. The second of these propositions is false if it is taken to imply that there are at the present time persons who are hanged for theft in England, but how it can possibly be regarded as meaningless I cannot understand.
(c) Miss Jones argues that if some carries with it an implication of existence, when used with a subject-term, it must do so equally when used with a predicate-term; but the predicate of an A proposition being undistributed is practically qualified by some ; hence, if Some S is P implies the existence of S and therefore of P, All S is P must imply the existence of P and therefore of S. In reply to this argument it may be pointed out, first, that a distinction may fairly be drawn without any risk of confusion between a term explicitly quantified by the word some and a term which we can shew to be undistributed but which is not explicitly quantified at all; and, secondly, that the position which we have taken is based upon a consideration of the import of propositions as a whole, not upon the force of signs of quantity considered in the abstract. The irrelevancy of the argument will be apparent if it is taken in connexion with the reasons which we have urged for holding that particulars should be interpreted as implying the existence of their subjects.
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 245 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]
[266] It is because Dr Wolf identifies the ordinary particular proposition with the problematic proposition that he is led to the conclusion that SaP and SoP are true contradictories although neither of them is interpreted as implying the existence of S.
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, 246 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 247 nothing is not-P. In order, therefore, to reduce the given propositions to such a form that they appear as contraries (and consequently[267] 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.
[267] It will be remembered that under suppositions (1) and (2) the ordinary doctrine of contrariety holds good.
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 248 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.]