If, after the proposition has been reduced to the affirmative form, all superfluous terms are omitted in accordance with the rules given in chapters [1] and [2], then the criterion becomes more simple:—Any non-formal universal proposition will afford information with regard to any term X, unless (after it has been brought to the affirmative form and its predicate has been so simplified that it contains no superfluous terms) X is itself an alternant of the predicate or x is a determinant of the subject.[505]
[505] It may be added that every universal proposition, unless it be purely formal, will afford information either with regard to X or with regard to x. For if both X and x appear as alternants of the predicate, or as determinants of the subject of a universal affirmative proposition, then the proposition will necessarily be formal.
If instead of X we have a complex term XYZ, then no determinant of this term must appear by itself as an alternant of the predicate, and there must be at least one alternant in the subject which does not contain as a determinant the contradictory of any determinant of this complex term; i.e., no alternant in the predicate must be X, Y, or Z, or any combination of these, and some alternant of the subject must contain neither x, y, nor z.
The above criterion is of simple application.
488. Information jointly afforded by a series of universal propositions with regard to any given term.—The great majority of direct problems[506] involving complex propositions may be brought under the general form, Given any number of universal propositions involving any number of terms, to determine what is all the information that they jointly afford with regard to any given term or combination of terms. If the student turns to Boole, Jevons, or Venn, he will find that this problem is treated by them as the central problem of symbolic logic.[507]
[506] Inverse problems will be discussed in the following [chapter].
[507] “Boole,” says Jevons, “first put forth the problem of Logical Science in its complete generality:—Given certain logical premisses or conditions, to determine the description of any class of objects under those conditions. Such was the general problem of which the ancient logic had solved but a few isolated cases—the nineteen moods of the syllogism, the sorites, the dilemma, the disjunctive syllogism, and a few other forms. Boole shewed incontestably that it was possible, by the aid of a system of mathematical signs, to deduce the conclusions of all these ancient modes of reasoning, and an indefinite number of other conclusions. Any conclusion, in short, that it was possible to deduce from any set of premisses or conditions, however numerous and complicated, could be calculated by his method” (Philosophical Transactions, 1870). Compare also Principles of Science, 6, § 5.
507 A general method of solution is as follows:—
Let X be the term concerning which information is desired. Find what information each proposition gives separately with regard to X, thus obtaining a new set of propositions of the form All X is P1 or P2 … or Pn.
This is always possible by the aid of the rules for obversion and contraposition given in [chapter 3]. By the aid of the rule given in the preceding [section] those propositions which do not afford any information at all with regard to X may at once be left out of account.
Next let the propositions thus obtained be combined in the manner indicated in section [475]. This will give the desired solution.
If information is desired with regard to several terms, it will be convenient to bring all the propositions to the form
Everything is P1 or P2 … or Pn ;
and to combine them at once, thus summing up in a single proposition all the information given by the separate propositions taken together. From this proposition all that is known concerning X may immediately be deduced by omitting every alternant that contains x, all that is known concerning Y by omitting every alternant that contains y, and so on.