CHAPTER VII
IMMEDIATE INFERENCES

§ 1. Under the general title 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 Identity 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 Identity 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 Identity. 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 entitled to assume that the terms he uses are understood, and that the identities 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 substitute '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 assumed as understood unless a definition is challenged. This principle will justify the substitution 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 classes, 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 quantity is also unaltered, the inference is called 'Simple Conversion'; but if the quantity 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 Identity, and because it seems to be a good method to exhaust the forms that come only under Identity before going on to those that involve Contradiction and Excluded Middle. Some, indeed, dispute the claims of Conversion to illustrate the principle of Identity; 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 assume 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 alters.

We may, then, state the following rule for the conversion of propositions in which the whole relation explicitly stated is taken as the copula: Transpose the terms, and for the given relation substitute its reciprocal. Thus—

A is the cause of B ∴ B is the effect of A.

The rule assumes that the reciprocal of a given relation is definitely known; and so far as this is true it may be extended to more concrete relations—

A is a genus of B ∴ B is a species of A

A is the father of B ∴ B is a child of A.

But not every relational expression has only one definite reciprocal. If we are told that A is the brother of B, we can only infer that B is either the brother or the sister of A. A list of all reciprocal relations is a desideratum of Logic.

§ 5. Obversion (otherwise called Permutation or Æquipollence) is Immediate Inference by changing the quality of the given proposition and substituting for its predicate the contradictory term. The given proposition is called the 'obvertend,' and the inference from it the 'obverse.' Thus the obvertend being—Some philosophers are consistent reasoners, the obverse will be—Some philosophers are not inconsistent reasoners.

The legitimacy of this mode of reasoning follows, in the case of affirmative propositions, from the principle of Contradiction, that if any term be affirmed of a subject, the contradictory term may be denied (chap. vi. § 3). To obvert affirmative propositions, then, the rule is—Insert the negative sign, and for the predicate substitute its contradictory term.