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