Studies and Exercises in Formal Logic

by John Neville Keynes

from which may be inferred

All X is a₁ or a₂ … or a_n or P₁P₂ … or Q₁Q₂ … or &c.

Any alternant in the predicate of this proposition which contains x may clearly be omitted.

If all the alternants contain x, then the information afforded with regard to X is that it is non-existent.

If some alternants are left, then the proposition will afford information concerning X unless, when the predicate has been simplified to the fullest possible extent,[502] one of the alternants is itself X uncombined with any other term, in which case it is clear that we are left with a merely formal proposition.

[⁵⁰²] All superfluous terms being omitted, but the predicate still consisting of a series of alternants which themselves contain only simple determinants.

Now one of these alternants will be X in the following cases, and only in these cases:—
First, If one of the alternants in the predicate of the original proposition, when reduced to the affirmative form, is X.
Secondly, If any set of alternants in the predicate of the original proposition, when reduced to the affirmative form, constitutes a development of X, since any development (for example, AX or aX, ABX or AbX or aBX or abX) is equivalent to X simply.[503]
Thirdly, If one of the alternants in the predicate of the original proposition, when reduced to the affirmative form, contains X in combination solely with some determinant that is also a determinant of the subject or the contradictory of some other alternant of the predicate; since in either of these cases such alternant is equivalent to X simply.[504]
Fourthly, If one of the determinants of the subject is x ; since in that case we shall after contraposition have X as one of the alternants of the predicate.

[⁵⁰³] See section [430].

[⁵⁰⁴] By section [445], rule (2), All AB is AX or D is equivalent to All AB is X or D ; and by the law of exclusion (section [432]) A or aX is equivalent to A or X.

The above may be summed up in the following proposition:—Any non-formal universal proposition will afford information with regard to any term X, unless, after it has been brought to the affirmative form, (1) one of the alternants of the predicate is X, or (2) any set of alternants in the predicate constitutes a development of X, or (3) any alternant of the predicate contains X in combination 506 solely with some determinant that is also a determinant of the subject or the contradictory of some other alternant of the predicate, or (4) x is a determinant of the subject.