The notation thus explained enables us to solve any problems in a simple manner. The expression in its final form may be read equally well in columns or in rows, i.e., as a determinative or as an alternative synthesis. Of course, a precisely similar process may be used, if we started with determinatively given or mixed data” (Mind, 1892, p. 351).
539. The Inverse Problem and Schröder’s Law of Reciprocal Equivalences.—The inverse problem may also be solved, though somewhat laboriously, by the aid of the reciprocal relation between the laws of distribution given in section [428], this reciprocal relation depending upon the law that to every equivalence there corresponds another equivalence in which conjunctive combination is throughout substituted for alternative combination and vice versâ. Thus, by the first law of distribution, (A or B) and (C or D) = AC or AD or BC or BD, and hence follows the corresponding equivalence AB or CD = (A or C) and (A or D) and (B or C) and (B or D). In this way any inverse problem may be practically resolved into the more 535 familiar problem of conjunctively combining a series of alternative terms.[530]
[530] It will be observed that the inverse problem involves the transformation of a logical expression consisting of a series of alternants into an equivalent expression consisting of a series of determinants. Schröder’s Law of Reciprocity shews that the process required for this transformation is practically the same as that by which an expression consisting of a series of determinants is transformed into an equivalent expression consisting of a series of alternants.
Taking as an example the first problem given in section [535], we may proceed as follows: (A or B or C) and (A or b or c) and (a or B or C) and (a or b or C) = (A or Bc or bC) and (a or C) = AC or aBc or bC. Therefore, we have the corresponding equivalence ABC or Abc or aBC or abC = (A or C) and (a or B or c) and (b or C). Hence the proposition Everything is ABC or Abc or aBC or abC may be resolved into the three propositions, Everything is A or C, Everything is a or B or c, Everything is b or C ; and we have for our solution of the inverse problem: All c is A, All bC is a, All c is b ; or, combining the first and last of these propositions, All c is Ab, All bC is a.
Similarly, the second problem in section [535] may be solved as follows:—(A or C or e) (a or B or C or e) (a or B or c or d or E) (a or b or C or e) (a or b or c or E) = aC or bCd or CE or ce. Hence the corresponding equivalence ACe or aBCe or aBcdE or abCe or abcE = (a or C) (b or C or d) (C or E) (c or e); and we have for our solution of the inverse problem, All A is C, All BD is C, All c is E, All C is e ; or, combining the first and third of these propositions, All c is aE, All BD is C, All C is e.
EXERCISES.
540. Find propositions that leave only the following combinations, ABCD, ABcD, AbCd, aBCd, abcd. [Jevons, Studies, p. 254.]
Jevons gives this as the most difficult of his series of inverse problems involving four terms. It may be solved as follows:—
Everything is ABCD or ABcD or AbCd or aBCd or abcd ; therefore, by contraposition and the reduction of dual terms, Whatever is not either AbCd or aBCd is ABD or abcd.
536 Therefore, Whatever is AB or ab or c or D is ABD or abcd ; and this is resolvable into the four following propositions:
| ⎧ ⎨ ⎩ | All AB is D, (1) |
| All ab is cd, (2) | |
| All c is ABD or abd, (3) | |
| All D is AB. (4) |
Since by (4) All D is AB, and by (2) All ab is d, (3) may be reduced to All c is D or ab, and therefore to All cd is ab. Also, by (4) All ab is d, and hence (2) may be reduced to All ab is c.