Logic: Deductive and Inductive Part 11

Some M is P; No S is M: ? ------

Here the minor and the middle terms are both distributed, but not the major (P); and, therefore, a negative conclusion is impossible.

-- 3. First Principle or Axiom of the Syllogism.--Hitherto in this chapter we have been a.n.a.lysing the conditions of valid mediate inference. We have seen that a single step of such inference, a Syllogism, contains, when fully expressed in language, three propositions and three terms, and that these terms must stand to one another in the relations required by the fourth, fifth, and sixth Canons. We now come to a principle which conveniently sums up these conditions; it is called the _Dictum de omni et nullo_, and may be stated thus:

Whatever is predicated (affirmatively or negatively) of a term distributed,

With which term another term can be (partly or wholly) identified,

May be predicated in like manner (affirmatively or negatively) of the latter term (or part of it).

Thus stated (nearly as by Whately in the introduction to his _Logic_) the _Dictum_ follows line by line the course of a Syllogism in the First Figure (see chap. X. -- 2). To return to our former example: _All authors are vain_ is the same as--Vanity is predicated of all authors; _Cicero is an author_ is the same as--Cicero is identified as an author; therefore _Cicero is vain_, or--Vanity may be predicated of Cicero. The _Dictum_ then requires: (1) three propositions; (2) three terms; (3) that the middle term be distributed; (4) that one premise be affirmative, since only by an affirmative proposition can one term be identified with another; (5) that if one premise be negative the conclusion shall be so too, since whatever is predicated of the middle term is predicated _in like manner_ of the minor.

Thus far, then, the _Dictum_ is wholly a.n.a.lytic or verbal, expressing no more than is implied in the definitions of 'Syllogism' and 'Middle Term'; since (as we have seen) all the General Canons (except the third, which is a still more general condition of formal proof) are derivable from those definitions. However, the _Dictum_ makes a further statement of a synthetic or real character, namely, that _when these conditions are fulfilled an inference is justified_; that then the major and minor terms are brought into comparison through the middle, and that the major term may be predicated affirmatively or negatively of all or part of the minor. It is this real a.s.sertion that justifies us in calling the _Dictum_ an Axiom.

-- 4. Whether the Laws of Thought may not fully explain the Syllogism without the need of any synthetic principle has, however, been made a question. Take such a syllogism as the following:

All domestic animals are useful; All pugs are domestic animals: ? All pugs are useful.

Here (an ingenious man might urge), having once identified pugs with domestic animals, that they are useful follows from the Law of Ident.i.ty.

If we attend to the meaning, and remember that what is true in one form of words is true in any other form, then, all domestic animals being useful, of course pugs are. It is merely a case of subalternation: we may put it in this way:

All domestic animals are useful: ? Some domestic animals (e.g., pugs) are useful.

The derivation of negative syllogisms from the Law of Contradiction (he might add) may be shown in a similar manner.

But the force of this ingenious argument depends on the participial clause--'having once identified pugs with domestic animals.' If this is a distinct step of the reasoning, the above syllogism cannot be reduced to one step, cannot be exhibited as mere subalternation, nor be brought directly under the law of Ident.i.ty. If 'pug,' 'domestic,' and 'useful'

are distinct terms; and if 'pug' and 'useful' are only known to be connected because of their relations to 'domestic': this is something more than the Laws of Thought provide for: it is not Immediate Inference, but Mediate; and to justify it, scientific method requires that its conditions be generalised. The _Dictum_, then, as we have seen, does generalise these conditions, and declares that when such conditions are satisfied a Mediate Inference is valid.

But, after all (to go back a little), consider again that proposition _All pugs are domestic animals_: is it a distinct step of the reasoning; that is to say, is it a Real Proposition? If, indeed, 'domestic' is no part of the definition of 'pug,' the proposition is real, and is a distinct part of the argument. But take such a case as this:

All dogs are useful; All pugs are dogs.

Here we clearly have, in the minor premise, only a verbal proposition; to be a dog is certainly part of the definition of 'pug.' But, if so, the inference 'All pugs are useful' involves no real mediation, and the argument is no more than this:

All dogs are useful; ? Some dogs (e.g., pugs) are useful.

Similarly, if the major premise be verbal, thus:

All men are rational; Socrates is a man--

to conclude that 'Socrates is rational' is no Mediate Inference; for so much was implied in the minor premise, 'Socrates is a man,' and the major premise adds nothing to this.

Hence we may conclude (as antic.i.p.ated in chap. vii. -- 3) that 'any apparent syllogism, having one premise a verbal proposition, is really an Immediate Inference'; but that, if both premises are real propositions, the Inference is Mediate, and demands for its explanation something more than the Laws of Thought.

The fact is that to prove the minor to be a case of the middle term may be an exceedingly difficult operation (chap. xiii. -- 7). The difficulty is disguised by ordinary examples, used for the sake of convenience.

-- 5. Other kinds of Mediate Inference exist, yielding valid conclusions, without being truly syllogistic. Such are mathematical inferences of Equality, as--

A = B = C ? A = C.

Here, according to the usual logical a.n.a.lysis, there are strictly four terms--(1) A, (2) equal to B, (3) B, (4) equal to C.

Similarly with the argument _a fortiori_,

A > B > C ? (much more) A > C.

This also is said to contain four terms: (1) A, (2) greater than B, (3) B, (4) greater than C. Such inferences are nevertheless intuitively sound, may be verified by trial (within the limits of sense-perception), and are generalised in appropriate axioms of their own, corresponding to the _Dictum_ of the syllogism; as 'Things equal to the same thing are equal to one another,' etc.

Now, surely, this is an erroneous application of the usual logical a.n.a.lysis of propositions. Both Logic and Mathematics treat of the _relations_ of terms; but whilst Mathematics employs the sign = for only one kind of relation, and for that relation exclusive of the terms; Logic employs the same signs (_is_ or _is not_) for all relations, recognising only a difference of quality in predication, and treating every other difference of relation as belonging to one of the terms related. Thus Logicians read _A--is--equal to B_: as if _equal to B_ could possibly be a term co-relative with A. Whence it follows that the argument _A = B = C ? A = C_ contains four terms; though everybody sees that there are only three.

In fact (as observed in chap. ii. -- 2) the sign of logical relation (_is_ or _is not_), whilst usually adequate for cla.s.s-reasoning (coinherence) and sometimes extensible to causation (because a cause implies a cla.s.s of events), should never be stretched to include other relations in such a way as to sacrifice intelligence to formalism. And, besides mathematical or quant.i.tative relations, there are others (usually considered qualitative because indefinite) which cannot be justly expressed by the logical copula. We ought to read propositions expressing time-relations (and inferences drawn accordingly) thus:

B--is before--C; A--is before--B: ? A--is before--C.

And in like manner _A--is simultaneous with--B; etc._ Such arguments (as well as the mathematical) are intuitively sound and verifiable, and might be generalised in axioms if it were worth while: but it is not, because no method could be founded on such axioms.

The customary use of relative terms justifies some Mediate Inferences, as, _The father of a father is a grand-father_.

Some cases, however, that at first seem obvious, are really delusive unless further data be supplied. Thus _A co-exists with B, B with C; ? A with C_--is not sound unless _B_ is an instantaneous event; for where B is perdurable, _A_ may co-exist with it at one time and _C_ at another.

Again: _A is to the left of B, B of C; ? A of C_. This may pa.s.s; but it is not a parallel argument that if _A is north of B and B west of C_, then _A is north-west of C_: for suppose that A is a mile to the north of B, and B a yard to the west of C, then A is practically north of C; at least, its westward position cannot be expressed in terms of the mariner's compa.s.s. In such a case we require to know not only the directions but the distances of A and C from B; and then the exact direction of A from C is an affair of mathematical calculation.

Qualitative reasoning concerning position is only applicable to things in one dimension of s.p.a.ce, or in time considered as having one dimension. Under these conditions we may frame the following generalisation concerning all Mediate Inferences: Two terms definitely related to a third, and one of them positively, are related to one another as the other term is related to the third (that is, positively or negatively); provided that the relations given are of the same kind (that is, of Time, or Coinherence, or Likeness, or Equality).

Thus, to ill.u.s.trate by relations of Time--

B is simultaneous with C; A is not simultaneous with B: ? A is not simultaneous with C.

Here the relations are of the same kind but of different logical quality, and (as in the syllogism) a negative copula in the premises leads to a negative conclusion.

An examination in detail of particular cases would show that the above generalisation concerning all Mediate Inferences is subject to too many qualifications to be called an Axiom; it stands to the real Axioms (the _Dictum_, etc.) as the notion of the Uniformity of Nature does to the definite principles of natural order (_cf._ chap. xiii. -- 9).

CHAPTER X

CATEGORICAL SYLLOGISMS

-- 1. The type of logical, deductive, mediate, categorical Inference is a Syllogism directly conformable with the _Dictum_: as--

All carnivores (M) are excitable (P); Cats (S) are carnivores (M): ? Cats (S) are excitable (P).

In this example P is predicated of M, a term distributed; in which term, M, S is given as included; so that P may be predicated of S.

Many arguments, however, are of a type superficially different from the above: as--