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.
In agreement with this mode of inference, we have the rule of modern English grammar, that 'two negatives make an affirmative.'
Again, by the principle of Excluded Middle, if any term be denied of a subject, its contradictory may be affirmed: to obvert negative propositions, then, the rule is—Remove the negative sign, and for the predicate substitute its contradictory term.
Thus, by obversion, each of the four propositions retains its quantity but changes its quality: A. to E., I. to O., E. to A., O. to I. And all the obverses are infinite propositions, the affirmative infinites having the sense of negatives, and the negative infinites
