a or b or cd.
It is possible for two complex terms to be formally inconsistent or repugnant without being true contradictories. This will be the case if they contain contradictory determinants without between them exhausting the universe of discourse. The terms AB and bC afford an example: nothing can be both AB and bC (for, if this 472 were so, something would be both B and not-B), but we cannot say à priori that everything is either AB or bC (since something may be Abc, which is neither AB nor bC).
427. Duality of Formal Equivalences in the case of Complex Terms.—It will be shewn in the following sections that certain complex terms are formally equivalent to other complex terms or to simple terms (for example, A or aB = A or B, A or AB = A); and it is important to notice at the outset that such formal equivalences always go in pairs. For if two terms are equivalent, their contradictories must also be equivalent; and hence, applying the rule for obtaining contradictories given in the preceding section, we are enabled to formulate the simple law that to every formal equivalence there corresponds another formal equivalence in which conjunctive combination is throughout substituted for alternative combination and vice versâ.[472] This law may be more precisely established as follows:—A formal equivalence that holds good for any given set of terms must equally hold good for any other set of terms; and, therefore, whatever holds good for the terms A, B, &c. must hold good for their contradictories a, b, &c. Hence, given any equivalence, we may first replace each simple term by its contradictory, and then take the contradictory of each side of the equivalence. The result of this double transformation will be that we shall obtain another equivalence in which every conjunctive combination has been replaced by an alternative combination, and conversely, while the term-symbols involved have remained unchanged. This proves what was required.
[472] This is pointed out by Schröder, Der Operationskreis des Logikkalkuls, p. 3. The two equivalences which are thus mutually deducible the one from the other may be said to be reciprocal.
The application of the above law will be fully illustrated in the sections that immediately follow.
428. Laws of Distribution.—In order to combine a simple term conjunctively with an alternative term, we must conjunctively combine it with every alternant of the alternative.[473] A and (B or C)[474] denotes whatever is A and at the same time either B or C, and hence is equivalent to AB or AC. It follows that in order to combine two alternative terms conjunctively, we must conjunctively combine every alternant of the one with every alternant of the other. Thus, 473 (A or B)(C or D) denotes whatever is either A or B and at the same time either C or D, and is equivalent to AC or AD or BC or BD.[475]
[473] Compare Jevons, Principles of Science, 5, § 7.
[474] In such a case as this the use of brackets is necessary in order to avoid ambiguity. Thus, A and B or C might mean AB or C, or as above AB or AC.
[475] Whether or not we introduce algebraic symbols into logic, there is here a very close analogy with algebraic multiplication which cannot be disguised.
We have then