CHAPTER VIII ORDER OF TERMS, EULER'S DIAGRAMS, LOGICAL EQUATIONS, EXISTENTIAL IMPORT OF PROPOSITIONS Still, if the question is, how we may best cast a literary sentence into logical form, good grounds for a definite answer may perhaps be found. We must not try to stand upon the naturalness of expression, for Dark is the fate of man is quite as natural as Man is mortal. When the purpose is not merely to state a fact, but also to express our feelings about it, to place the grammatical predicate first may be perfectly natural and most effective. But the grounds of a logical order of statement must be found in its adaptation to the purposes of proof and inference. Now general propositions are those from which most inferences can be drawn, which, therefore, it is most important to establish, if true; and they are also the easiest to disprove, if false; since a single negative instance suffices to establish the contradictory. It follows that, in re-casting a literary or colloquial sentence for logical purposes, we should try to obtain a form in which the subject is distributed—is either a singular term or a general term predesignate as 'All' or 'No.' Seeing, then, that most adjectives connote a single attribute, whilst most substantives connote more than one attribute; and that therefore the denotation of adjectives is usually wider than that of substantives; in any proposition, one term of which is an adjective and the other a substantive, if either can be distributed in relation to the other, it is nearly sure to be the substantive; so that to take the substantive term for subject is our best chance of obtaining an universal proposition. These considerations seem to justify the practice of Logicians in selecting their examples. For similar reasons, if both terms of a proposition are substantive, the one with the lesser denotation is (at least If, however, a controvertist has no other object in view than to refute some general proposition laid down by an opponent, a particular proposition is all that he need disentangle from any statement that serves his purpose. These circles represent the denotation of the terms. Suppose the proposition to be All hollow-horned animals ruminate: then, if we could collect all ruminants upon a prairie, and enclose them with a circular palisade; and segregate from amongst them all the hollow-horned beasts, and enclose them with another ring-fence inside the other; one way of interpreting the proposition (namely, in denotation) would be figured to us thus: Fig. 1. An Universal Affirmative may also state a relation between two terms whose denotation is co-extensive. A definition always does this, as Man is a rational animal; and this, of course, we cannot represent by two distinct Fig. 2. The Particular Affirmative Proposition may be represented in several ways. In the first place, bearing in mind that 'Some' means 'some at least, it may be all,' an I. proposition may be represented by Figs. 1 and 2; for it is true that Some horned animals ruminate, and that Some men are rational. Secondly, there is the case in which the 'Some things' of which a predication is made are, in fact, not all; whilst the predicate, though not given as distributed, yet might be so given if we wished to state the whole truth; as if we say Some men are Chinese. This case is also represented by Fig. 1, the outside circle representing 'Men,' and the inside one 'Chinese.' Thirdly, the predicate may appertain to some only of the subject, but to a great many other things, as in Some horned beasts are domestic; for it is true that some are not, and that certain other kinds of animals are, domestic. This case, therefore, must be illustrated by overlapping circles, thus: Fig. 3. The Universal Negative is sufficiently represented by a single Fig. (4): two circles mutually exclusive, thus: Fig. 4. That is, No horned beasts are carnivorous. Lastly, the Particular Negative may be represented by any of the Figs. 1, 3, and 4; for it is true that Some ruminants are not hollow-horned, that Some horned animals are not domestic, and that Some horned beasts are not carnivorous. Besides their use in illustrating the denotative force of propositions, these circles may be employed to verify the results of Obversion, Conversion, and the secondary modes of Immediate Inference. Thus the Obverse of A. is clear enough on glancing at Figs. 1 and 2; for if we agree that whatever term's denotation is represented by a given circle, the denotation of the contradictory term shall be represented by the space outside that circle; then if it is true that All hollow horned animals are ruminants, it is at the same time true that No hollow-horned animals are not-ruminants; since none of the hollow-horned are found outside the palisade that encloses the ruminants. The Obverse of I., E. or O. may be verified in a similar manner. As to the Converse, a Definition is of course susceptible of Simple Conversion, and this is shown by Fig. 2: 'Men are rational animals' and 'Rational animals are men.' But any other A. proposition is presumably convertible only by limitation, and this is shown by Fig. 1; where All hollow-horned animals are ruminants, but we can only say that Some ruminants are hollow-horned. That I. may be simply converted may be seen in Fig. 3, which represents the least that an I. proposition can mean; and that E. may be simply converted is manifest in Fig. 4. As for O., we know that it cannot be converted, and this is made plain enough by glancing at Fig. 1; for that represents the O., Some ruminants are not hollow-horned, but also shows this to be compatible with All hollow-horned animals are ruminants (A.). Now in conversion there is (by definition) no change of quality. The Converse, then, of Some ruminants are not hollow-horned must be a negative proposition, having 'hollow-horned' for its subject, either in E. or O.; but these would be respectively the contrary and contradictory of All hollow-horned animals are ruminants; and, therefore, if this be true, they must both be false. But (referring still to Fig. 1) the legitimacy of contrapositing O. is equally clear; for if Some ruminants are not hollow-horned, Some animals that are not hollow-horned are ruminants, namely, all the animals between the two ring-fences. Similar inferences may be illustrated from Figs. 3 and 4. And the Contraposition of A. may be verified by Figs. 1 and 2, and the Contraposition of E. by Fig. 4. Lastly, the Inverse of A. is plain from Fig. 1—Some things that are not hollow-horned are not ruminants, namely, things that lie outside the outer circle and are neither 'ruminants' nor 'hollow-horned.' And the Inverse of E may be studied in Fig. 4—Some things that are not-horned beasts are carnivorous. Notwithstanding the facility and clearness of the demonstrations thus obtained, it may be said that a diagrammatic method, representing denotations, is not properly logical. Fundamentally, the relation asserted (or denied) to exist between the terms of a proposition, is a relation between the terms as determined by their attributes or connotation; whether we take Mill's view, that a proposition asserts that the connotation of the subject is a mark of the connota But what is that sense? Clearly not the same as that in which mathematical terms are equated, namely, in respect of some mode of quantity. For if we may say Some X is some Y, these Xs that are also Ys are not merely the same in number, or mass, or figure; they are the same in every When we are told that logical propositions are to be considered as equations, we naturally expect to be shown some interesting developments of method in analogy with the equations of Mathematics; but from Hamilton's innovations no such thing results. This cannot be said, however, of the equations of Symbolic Logic; which are the starting-point of very remarkable processes of ratiocination. As the subject of Symbolic Logic, as a whole, lies beyond the compass of this work, it will be enough to give Dr. Venn's equations corresponding with the four propositional forms of common Logic. According to this system, universal propositions are to be regarded as not necessarily implying the existence of their terms; and therefore, instead of giving them a positive form, they are translated into symbols that express what they deny. For example, the proposition All devils are ugly need not imply that any such things as 'devils' really exist; but it certainly does imply that Devils that are not ugly do A. xȳ = 0. That is, x that is not y is nothing; or, Devils that are not-ugly do not exist. And, similarly, writing x for 'angels' and y for 'ugly,' we may express E., the universal negative, thus: E. xy = 0. That is, x that is y is nothing; or, Angels that are ugly do not exist. On the other hand, particular propositions are regarded as implying the existence of their terms, and the corresponding equations are so framed as to express existence. With this end in view, the symbol v is adopted to represent 'something,' or indeterminate reality, or more than nothing. Then, taking any particular affirmative, such as Some metaphysicians are obscure, and writing x for 'metaphysicians,' and y for 'obscure,' we may express it thus: I. xy = v. That is, x that is y is something; or, Metaphysicians that are obscure do occur in experience (however few they may be, or whether they all be obscure). And, similarly, taking any particular negative, such as Some giants are not cruel, and writing x for 'giants' and y for 'not-cruel,' we may express it thus: O. xȳ = v. That is, x that is not y is something; or, giants that are not-cruel do occur—in romances, if nowhere else. Clearly, these equations are, like Hamilton's, concerned with denotation. A. and E. affirm that the compound terms xȳ and xy have no denotation; and I. and O. declare that xȳ and xy have denotation, or stand for something. Here, however, the resemblance to Hamilton's system In the first place, Logic treats primarily of the relations implied in propositions. This follows from its being the science of proof for all sorts of (qualitative) propositions; since all sorts of propositions have nothing in common except the relations they express. But, secondly, relations without terms of some sort are not to be thought of; and, hence, even the most formal illustrations of logical doctrines comprise such terms as S and P, X and Y, or x and y, in a symbolic or representative character. Terms, therefore, of some sort are assumed to exist (together with their negatives or contradictories) for the purposes of logical manipulation. Thirdly, however, that Formal Logic cannot as such directly involve the existence of any particular concrete terms, such as 'man' or 'mountain,' used by way of illustration, is implied in the word 'formal,' that is, 'confined to what is common or abstract'; since the only thing common to all terms is to be related in some way to other terms. The actual existence of any concrete thing can only be known by experience, as with 'man' or 'mountain'; or by methodically justifiable inference from experience, as with 'atom' or 'ether.' If 'man' or 'mountain,' or Nevertheless, fourthly, the existence or non-existence of particular terms may come to be implied: namely, wherever the very fact of existence, or of some condition of existence, is an hypothesis or datum. Thus, given the proposition All S is P, to be P is made a condition of the existence of S: whence it follows that an S that is not P does not exist (xȳ = 0). On the further hypothesis that S exists, it follows that P exists. On the hypothesis that S does not exist, the existence of P is problematic; but, then, if P does exist we cannot convert the proposition; since Some P is S (P existing) would involve the existence of S; which is contrary to the hypothesis. Assuming that Universals do not, whilst Particulars do, imply the existence of their subjects, we cannot infer the subalternate (I. or O.) from the subalternans (A. or E.), for that is to ground the actual on the problematic; and for the same reason we cannot convert A. per accidens. Assuming, again, a certain suppositio or universe, to which in a given discussion every argument shall refer, then, any propositions whose terms lie outside that suppositio are irrelevant, and for the purposes of that discussion are sometimes called "false"; though it seems better to call them irrelevant or meaningless, seeing that to call them false implies that they might in the same case be true. Thus propositions which, according to the doctrine of Opposition, appear to be Contradictories, may then cease to be so; for of Contradictories one is true and the other false; but, in the case supposed, both are meaningless. If the subject of discussion be Zoology, all propositions about centaurs or unicorns are absurd; and such specious In Formal Logic, in short, we may make at discretion any assumption whatever as to the existence, or as to any condition of the existence of any particular term or terms; and then certain implications and conclusions follow in consistency with that hypothesis or datum. Still, our conclusions will themselves be only hypothetical, depending on the truth of the datum; and, of course, until this is empirically ascertained, we are as far as ever from empirical reality. (Venn: Symbolic Logic, c. 6; Keynes: Formal Logic, Part II. c. 7: cf. Wolf: Studies in Logic.) |