to find a set of propositions not involving any alternative combination of terms, which shall together be equivalent to it.[520]
[520] The problem may also be stated as follows:—Given a universal affirmative complex proposition containing alternative terms to find an equivalent compound conjunctive proposition all the determinants of which are affirmative and free from alternative terms.
It may be observed that Jevons does not definitely exclude alternative terms in his solutions of inverse problems, though he generally seeks to avoid them. The problem cannot, however, be defined with accuracy unless such terms are explicitly excluded.
The inverse problem is in a sense indeterminate, for we may find a number of sets of propositions, not involving any alternative combination of terms, which are precisely equivalent in logical force, and hence any inverse problem may admit of a number of solutions. But it is not necessary to have recourse to a series of guesses in order to solve any inverse problem, nor need the method of solution be described as wholly tentative. Several systematic methods of solution applicable to any inverse problem are formulated in the following sections. Since, however, more solutions than one are possible, some of which are simpler than others, the process may be regarded as more or less tentative in so far as we seek to obtain the most satisfactory solution.
The following may be taken as our criterion of simplicity. Comparing two equivalent sets of propositions, not involving any 527 alternative combination of terms, that set may be regarded as the simpler which contains the smaller number of propositions. If each set contains the same number of propositions, then we may count the number of terms involved in their subjects and predicates taken together, and regard that one as the simpler which involves the fewer terms.
535. A General Solution of the Inverse Problem.—Let us suppose, then, that we are given a complex proposition involving alternative combination, and that we are to find a set of propositions, not involving alternative combination, which shall together be equivalent to it.
The data may be written in the form
Everything is P or Q or S or T or &c.,
where P, Q, &c., are themselves complex terms involving conjunctive, but not alternative, combination.[521]
[521] The proposition in its original form may admit of simplification in accordance with the rules laid down in [chapter 1]. It will generally speaking be found advantageous to have recourse to such simplification before proceeding further with the solution.