But whatever degree of complexity a term may reach, it will consist of a series of conjunctive and alternative combinations; and it may be successively resolved into the combination of pairs of relatively simple terms till it is at last shewn to result from the combination of absolutely simple terms. For example,—ABC or DE or FG results from the alternative combination of ABC or DE with FG ; ABC or DE results from the alternative combination of ABC with DE ; FG results from the conjunctive combination of F with G ; and ABC, DE may be resolved similarly.
Hence the successive application of the above rule, for finding the contradictory of a complex term where we are dealing with a single pair of determinants or alternants, will result in our ultimately substituting for each simple term involved its contradictory, and reversing the nature of their combination throughout.[470] We may, therefore, lay down the following rule for obtaining the contradictory of any complex term: Replace each constituent simple term by its contradictory and throughout substitute conjunctive combination for alternative combination and vice versâ.[471] This rule is of simple application, and it is of fundamental importance in the treatment of complex propositions adopted in the following pages.
[470] Thus, taking the term ABC or DE or FG, and in the first instance denoting the contradictory of a complex term by a bar drawn across it, we have successively,—
| ABC or DE or FG | |
| = | ABC (DE or FG) |
| = | (AB or c) DE . FG |
| = | (a or b or c) (d or e) (f or g). |
[471] Compare Schröder, Der Operationskreis des Logikkalkuls, p. 18.
Thus, the contradictory of A or BC
is a and (b or c),
i.e., ab or ac ;
and the contradictory of ABC or ABD
is (a or b or c) and (a or b or d),
which, by the aid of rules presently to be given, is reducible to the form