Logic: Deductive and Inductive Part 9

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 ill.u.s.trating 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 s.p.a.ce 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 ill.u.s.trated 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 a.s.serted (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 a.s.serts that the connotation of the subject is a mark of the connotation of the predicate; or Dr. Venn's view, that things denoted by the subject (as having its connotation) have (or have not) the attribute connoted by the predicate; or, the Conceptualist view, that a judgment is a relation of concepts (that is, of connotations). With a few exceptions artificially framed (such as 'kings now reigning in Europe'), the denotation of a term is never directly and exhaustively known, but consists merely in 'all things that have the connotation.' If the value of logical training depends very much upon our habituating ourselves to construe propositions, and to realise the force of inferences from them, according to the connotation of their terms, we shall do well not to turn too hastily to the circles, but rather to regard them as means of verifying in denotation the conclusions that we have already learnt to recognise as necessary in connotation.

-- 3. The equational treatment of propositions is closely connected with the diagrammatic. Hamilton thought it a great merit of his plan of quantifying the predicate, that thereby every proposition is reduced to its true form--an equation. According to this doctrine, the proposition _All X is all Y_ (U.) equates X and Y; the proposition _All X is some Y_ (A.) equates X with some part of Y; and similarly with the other affirmatives (Y. and I.). And so far it is easy to follow his meaning: the Xs are identical with some or all the Ys. But, coming to the negatives, the equational interpretation is certainly less obvious. The proposition _No X is Y_ (E.) cannot be said in any sense to equate X and Y; though, if we obvert it into _All X is some not-Y_, we have (in the same sense, of course, as in the above affirmative forms) X equated with part at least of 'not-Y.'

But what is that sense? Clearly not the same as that in which mathematical terms are equated, namely, in respect of some mode of quant.i.ty. For if we may say _Some X is some Y_, these Xs that are also Ys are not merely the same in number, or ma.s.s, or figure; they are the same in every respect, both quant.i.tative and qualitative, have the same positions in time and place, are in fact identical. The proposition 2+2=4 means that any two things added to any other two are, _in respect of number_, equal to any three things added to one other thing; and this is true of all things that can be counted, however much they may differ in other ways. But _All X is all Y_ means that Xs and Ys are the same things, although they have different names when viewed in different aspects or relations. Thus all equilateral triangles are equiangular triangles; but in one case they are named from the equality of their angles, and in the other from the equality of their sides. Similarly, 'British subjects' and 'subjects of King George V' are the same people, named in one case from the person of the Crown, and in the other from the Imperial Government. These logical equations, then, are in truth ident.i.ties of denotation; and they are fully ill.u.s.trated by the relations of circles described in the previous section.

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 a.n.a.logy 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 compa.s.s 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 not exist_. Similarly, the proposition _No angels are ugly_ implies that _Angels that are ugly do not exist_. Therefore, writing _x_ for 'devils,' _y_ for 'ugly,' and _y_ for 'not-ugly,' we may express A., the universal affirmative, thus:

A. _xy_ = 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. _xy_ = 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 xy and xy have no denotation; and I. and O. declare that xy and xy have denotation, or stand for something. Here, however, the resemblance to Hamilton's system ceases; for the Symbolic Logic, by operating upon more than two terms simultaneously, by adopting the algebraic signs of operations, +,-, , (with a special signification), and manipulating the symbols by quasi-algebraic processes, obtains results which the common Logic reaches (if at all) with much greater difficulty. If, indeed, the value of logical systems were to be judged of by the results obtainable, formal deductive Logic would probably be superseded. And, as a mental discipline, there is much to be said in favour of the symbolic method.

But, as an introduction to philosophy, the common Logic must hold its ground. (Venn: _Symbolic Logic_, c. 7.)

-- 4. Does Formal Logic involve any general a.s.sumption as to the real existence of the terms of propositions?

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 ill.u.s.trations 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 a.s.sumed 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 ill.u.s.tration, 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 'Cuzco' be used to ill.u.s.trate logical forms, they bring with them an existential import derived from experience; but this is the import of language, not of the logical forms. 'Centaur' and 'El Dorado' signify to us the non-existent; but they serve as well as 'man' and 'London' to ill.u.s.trate Formal Logic.

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 (_xy_ = 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.

a.s.suming 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_.

a.s.suming, 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 Contradictories as _No centaurs play the lyre--Some centaurs do play the lyre_; or _All unicorns fight with lions--Some unicorns do not fight with lions_, are both meaningless, because in Zoology there are no centaurs nor unicorns; and, therefore, in this reference, the propositions are not really contradictory. But if the subject of discussion or _suppositio_ be Mythology or Heraldry, such propositions as the above are to the purpose, and form legitimate pairs of Contradictories.

In Formal Logic, in short, we may make at discretion any a.s.sumption 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_.)

CHAPTER IX

FORMAL CONDITIONS OF MEDIATE INFERENCE

-- 1. A Mediate Inference is a proposition that depends for proof upon two or more other propositions, so connected together by one or more terms (which the evidentiary propositions, or each pair of them, have in common) as to justify a certain conclusion, namely, the proposition in question. The type or (more properly) the unit of all such modes of proof, when of a strictly logical kind, is the Syllogism, to which we shall see that all other modes are reducible. It may be exhibited symbolically thus:

M is P; S is M: ? S is P.

Syllogisms may be cla.s.sified, as to quant.i.ty, into Universal or Particular, according to the quant.i.ty of the conclusion; as to quality, into Affirmative or Negative, according to the quality of the conclusion; and, as to relation, into Categorical, Hypothetical and Disjunctive, according as all their propositions are categorical, or one (at least) of their evidentiary propositions is a hypothetical or a disjunctive.

To begin with Categorical Syllogisms, of which the following is an example:

All authors are vain; Cicero is an author: ? Cicero is vain.

Here we may suppose that there are no direct means of knowing that Cicero is vain; but we happen to know that all authors are vain and that he is an author; and these two propositions, put together, unmistakably imply that he is vain. In other words, we do not at first know any relation between 'Cicero' and 'vanity'; but we know that these two terms are severally related to a third term, 'author,' hence called a Middle Term; and thus we perceive, by mediate evidence, that they are related to one another. This sort of proof bears an obvious resemblance (though the relations involved are not the same) to the mathematical proof of equality between two quant.i.ties, that cannot be directly compared, by showing the equality of each of them to some third quant.i.ty: A = B = C ? A = C. Here B is a middle term.

We have to inquire, then, what conditions must be satisfied in order that a Syllogism may be formally conclusive or valid. A specious Syllogism that is not really valid is called a Parasyllogism.

-- 2. General Canons of the Syllogism.

(1) A Syllogism contains three, and no more, distinct propositions.

(2) A Syllogism contains three, and no more, distinct univocal terms.

These two Canons imply one another. Three propositions with less than three terms can only be connected in some of the modes of Immediate Inference. Three propositions with more than three terms do not show that connection of two terms by means of a third, which is requisite for proving a Mediate Inference. If we write--

All authors are vain; Cicero is a statesman--

there are four terms and no middle term, and therefore there is no proof. Or if we write--

All authors are vain; Cicero is an author: ? Cicero is a statesman--