Logic Part 6

If it be asked whether it is true that every term, itself significant, has a significant contradictory, and not merely a formal contradictory, generated by force of the word 'not,' it is difficult to give any better answer than was indicated in ---- 3-5, without venturing further into Metaphysics. I shall merely say, therefore, that, granting that some such term as 'Universe' or 'Being' may have no significant contradictory, if it stand for 'whatever can be perceived or thought of'; yet every term that stands for less than 'Universe' or 'Being' has, of course, a contradictory which denotes the rest of the universe. And since every argument or train of thought is carried on within a special 'universe of discourse,' or under a certain _suppositio_, we may say that _within the given suppositio every term has a contradictory_, and that every predication concerning a term implies some predication concerning its contradictory. But the name of the _suppositio_ itself has no contradictory, except with reference to a wider and inclusive _suppositio_.

The difficulty of actual reasoning, not with symbols, but about matters of fact, does not arise from the principles of Logic, but sometimes from the obscurity or complexity of the facts, sometimes from the ambiguity or clumsiness of language, sometimes from the deficiency of our own minds in penetration, tenacity and lucidity. One must do one's best to study the facts, and not be too easily discouraged.

CHAPTER VII

IMMEDIATE INFERENCES

-- 1. Under the general t.i.tle of Immediate Inference Logicians discuss three subjects, namely, Opposition, Conversion, and Obversion; to which some writers add other forms, such as Whole and Part in Connotation, Contraposition, Inversion, etc. Of Opposition, again, all recognise four modes: Subalternation, Contradiction, Contrariety and Sub-contrariety. The only peculiarities of the exposition upon which we are now entering are, that it follows the lead of the three Laws of Thought, taking first those modes of Immediate Inference in which Ident.i.ty is most important, then those which plainly involve Contradiction and Excluded Middle; and that this method results in separating the modes of Opposition, connecting Subalternation with Conversion, and the other modes with Obversion. To make up for this departure from usage, the four modes of Opposition will be brought together again in -- 9.

-- 2. Subalternation.--Opposition being the relation of propositions that have the same matter and differ only in form (as A., E., I., O.), propositions of the forms A. and I. are said to be Subalterns in relation to one another, and so are E. and O.; the universal of each quality being distinguished as 'subalternans,' and the particular as 'subalternate.'

It follows from the principle of Ident.i.ty that, the matter of the propositions being the same, if A. is true I. is true, and that if E. is true O. is true; for A. and E. predicate something of _All S_ or _All men_; and since I. and O. make the same predication of _Some S_ or _Some men_, the sense of these particular propositions has already been predicated in A. or E. If _All S is P, Some S is P_; if _No S is P, Some S is not P_; or, if _All men are fond of laughing, Some men are_; if _No men are exempt from ridicule, Some men are not_.

Similarly, if I. is false A. is false; if O. is false E. is false. If we deny any predication about _Some S_, we must deny it of _All S_; since in denying it of _Some_, we have denied it of at least part of _All_; and whatever is false in one form of words is false in any other.

On the other hand, if I. is true, we do not know that A. is; nor if O.

is true, that E. is; for to infer from _Some_ to _All_ would be going beyond the evidence. We shall see in discussing Induction that the great problem of that part of Logic is, to determine the conditions under which we may in reality transcend this rule and infer from _Some_ to _All_; though even there it will appear that, formally, the rule is observed. For the present it is enough that I. is an immediate inference from A., and O. from E.; but that A. is not an immediate inference from I., nor E. from O.

-- 3. Connotative Subalternation.--We have seen (chap. iv. -- 6) that if the connotation of one term is only part of another's its denotation is greater and includes that other's. Hence genus and species stand in subaltern relation, and whatever is true of the genus is true of the species: If _All animal life is dependent on vegetation, All human life is dependent on vegetation_. On the other hand, whatever is not true of the species or narrower term, cannot be true of the whole genus: If it is false that '_All human life is happy_,' it is false that '_All animal life is happy_.'

Similar inferences may be drawn from the subaltern relation of predicates; affirming the species we affirm the genus. To take Mill's example, if _Socrates is a man, Socrates is a living creature_. On the other hand, denying the genus we deny the species: if _Socrates is not vicious, Socrates is not drunken_.

Such cases as these are recognised by Mill and Bain as immediate inferences under the principle of Ident.i.ty. But some Logicians might treat them as imperfect syllogisms, requiring another premise to legitimate the conclusion, thus:

_All animal life is dependent on vegetation; All human life is animal life; ? All human life is dependent on vegetation._

Or again:

_All men are living creatures; Socrates is a man; ? Socrates is a living creature._

The decision of this issue turns upon the question (_cf._ chap. vi. -- 3) how far a Logician is ent.i.tled to a.s.sume that the terms he uses are understood, and that the ident.i.ties involved in their meanings will be recognised. And to this question, for the sake of consistency, one of two answers is required; failing which, there remains the rule of thumb.

First, it may be held that no terms are understood except those that are defined in expounding the science, such as 'genus' and 'species,'

'connotation' and 'denotation.' But very few Logicians observe this limitation; few would hesitate to subst.i.tute 'not wise' for 'foolish.'

Yet by what right? Malvolio being foolish, to prove that he is not-wise, we may construct the following syllogism:

_Foolish is not-wise; Malvolio is foolish; ? Malvolio is not-wise._

Is this necessary? Why not?

Secondly, it may be held that all terms may be a.s.sumed as understood unless a definition is challenged. This principle will justify the subst.i.tution of 'not-wise' for 'foolish'; but it will also legitimate the above cases (concerning 'human life' and 'Socrates') as immediate inferences, with innumerable others that might be based upon the doctrine of relative terms: for example, _The hunter missed his aim_: therefore, _The prey escaped_. And from this principle it will further follow that all apparent syllogisms, having one premise a verbal proposition, are immediate inferences (_cf._ chap. ix. -- 4).

Closely connected with such cases as the above are those mentioned by Archbishop Thomson as "Immediate Inferences by added Determinants"

(_Laws of Thought_, -- 87). He takes the case: '_A negro is a fellow-creature_: therefore, _A negro in suffering is a fellow-creature in suffering_.' This rests upon the principle that to increase the connotations of two terms by the same attribute or determinant does not affect the relationship of their denotations, since it must equally diminish (if at all) the denotations of both cla.s.ses, by excluding the same individuals, if any want the given attribute. But this principle is true only when the added attribute is not merely the same verbally, but has the same significance in qualifying both terms. We cannot argue _A mouse is an animal_; therefore, _A large mouse is a large animal_; for 'large' is an attribute relative to the normal magnitude of the thing described.

-- 4. Conversion is Immediate Inference by transposing the terms of a given proposition without altering its quality. If the quant.i.ty is also unaltered, the inference is called 'Simple Conversion'; but if the quant.i.ty is changed from universal to particular, it is called 'Conversion by limitation' or '_per accidens._' The given proposition is called the 'convertend'; that which is derived from it, the 'converse.'

Departing from the usual order of exposition, I have taken up Conversion next to Subalternation, because it is generally thought to rest upon the principle of Ident.i.ty, and because it seems to be a good method to exhaust the forms that come only under Ident.i.ty before going on to those that involve Contradiction and Excluded Middle. Some, indeed, dispute the claims of Conversion to ill.u.s.trate the principle of Ident.i.ty; and if the sufficient statement of that principle be 'A is A,' it may be a question how Conversion or any other mode of inference can be referred to it. But if we state it as above (chap. vi. -- 3), that whatever is true in one form of words is true in any other, there is no difficulty in applying it to Conversion.

Thus, to take the simple conversion of I.,

_Some S is P; ? Some P is S._ _Some poets are business-like; ? Some business-like men are poets._

Here the convertend and the converse say the same thing, and this is true if that is.

We have, then, two cases of simple conversion: of I. (as above) and of E. For E.:

_No S is P; ? No P is S._ _No ruminants are carnivores; ? No carnivores are ruminants._

In converting I., the predicate (P) when taken as the new subject, being preindesignate, is treated as particular; and in converting E., the predicate (P), when taken as the new subject, is treated as universal, according to the rule in chap. v. -- 1.

A. is the one case of conversion by limitation:

All S is P; ? Some P is S.

All cats are grey in the dark; ? Some things grey in the dark are cats.

The predicate is treated as particular, when taking it for the new subject, according to the rule not to go beyond the evidence. To infer that _All things grey in the dark are cats_ would be palpably absurd; yet no error of reasoning is commoner than the simple conversion of A.

The validity of conversion by limitation may be shown thus: if, _All S is P_, then, by subalternation, _Some S is P_, and therefore, by simple conversion, _Some P is S_.

O. cannot be truly converted. If we take the proposition: _Some S is not P_, to convert this into _No P is S_, or _Some P is not S_, would break the rule in chap. vi. -- 6; since _S,_ undistributed in the convertend, would be distributed in the converse. If we are told that _Some men are not cooks_, we cannot infer that _Some cooks are not men_.

This would be to a.s.sume that '_Some men_' are identical with '_All men_.'

By quantifying the predicate, indeed, we may convert O. simply, thus:

_Some men are not cooks_ ? _No cooks are some men._

And the same plan has some advantage in converting A.; for by the usual method _per accidens_, the converse of A. being I., if we convert this again it is still I., and therefore means less than our original convertend. Thus:

_All S is P ? Some P is S ? Some S is P._

Such knowledge, as that _All S_ (the whole of it) _is P_, is too precious a thing to be squandered in pure Logic; and it may be preserved by quantifying the predicate; for if we convert A. to Y., thus--

_All S is P ? Some P is all S--_

we may reconvert Y. to A. without any loss of meaning. It is the chief use of quantifying the predicate that, thereby, every proposition is capable of simple conversion.

The conversion of propositions in which the relation of terms is inadequately expressed (see chap. ii., -- 2) by the ordinary copula (_is_ or _is not_) needs a special rule. To argue thus--

_A is followed by B_ ? _Something followed by B is A_--

would be clumsy formalism. We usually say, and we ought to say--

_A is followed by B_ ? _B follows A_ (or _is preceded by A_).

Now, any relation between two terms may be viewed from either side--_A: B_ or _B: A_. It is in both cases the same fact; but, with the altered point of view, it may present a different character. For example, in the Immediate Inference--_A > B_ ? _B < a_--a="" diminishing="" turns="" into="" an="" increasing="" ratio,="" whilst="" the="" fact="" predicated="" remains="" the="" same.="" given,="" then,="" a="" relation="" between="" two="" terms="" as="" viewed="" from="" one="" to="" the="" other,="" the="" same="" relation="" viewed="" from="" the="" other="" to="" the="" one="" may="" be="" called="" the="" reciprocal.="" in="" the="" cases="" of="" equality,="" co-existence="" and="" simultaneity,="" the="" given="" relation="" and="" its="" reciprocal="" are="" not="" only="" the="" same="" fact,="" but="" they="" also="" have="" the="" same="" character:="" in="" the="" cases="" of="" greater="" and="" less="" and="" sequence,="" the="" character="">