Studies and Exercises in Formal Logic

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).

²⁵¹ Symbolic Logic, p. 132.

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.”

²⁵³ This example is taken from Dixon, Essay on Reasoning, p. 62.

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.

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.)

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.

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.

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.

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).

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.

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.”

²⁶⁴ Compare sections 156, 157.

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.

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.

²⁶⁸ For the distinction indicated in the present section I was in the first instance indebted to an essay, written in 1884, by Mr W. E. Johnson. This essay has not been published in its original form; but the substance of it has been included in some papers on The Logical Calculus by Mr Johnson which appeared in Mind in 1892.

²⁶⁹ The above distinction has been adopted in some recent treatises on Logic, but it must be borne in mind that most logicians use the terms conditional and hypothetical as synonymous or else draw a distinction between them different from the above.

²⁷⁰ Conditionals can generally be reduced to the last of these three forms without much difficulty, and such reduction is sometimes useful. A consideration of the concrete examples already given will, however, shew that a certain amount of manipulation may be required in order to effect the reduction. The following are examples: If any child is spoilt, then that child will have suffering parents ; If any two straight lines are such that another straight line falling upon them makes the alternate angles equal to one another, then those two straight lines are parallel to one another.