A(B or C) = AB or AC,
and applying the law of duality of formal equivalences given in the preceding section, we have at once another equivalence, namely,
A or BC = (A or B)(A or C).[476]
[476] This equivalence might also be established independently by the aid of certain of the equivalences given in the following sections.
These two equivalences are called by Schröder the Laws of Distribution.[477] They are of the greatest importance in the manipulation and simplification of complex terms.
[477] Der Operationskreis des Logikkalkuls, pp. 9, 10.
429. Laws of Tautology.—The following rules may be laid down for the omission of superfluous terms from a complex term:
(a) The repetition of any given determinant is superfluous.
Out of the class A to select the A’s is a process that leaves us just where we began. In other words, what is both A and A is identical with what is A. Thus, such terms as AA, ABB, are tautologous; the former merely denotes the class A, and the latter the class AB. Hence the above rule, which is called by Jevons the Law of Simplicity.[478]
(b) The repetition of any given alternant is superfluous.
To say that anything is A or A is equivalent to saying simply that it is A. Hence such terms as A or A, A or BC or BC, are tautologous; and we have the above rule, which is called by Jevons the Law of Unity.[479]
[478] See Pure Logic, § 42; and Principles of Science, 2, § 8. The corresponding equation x2 = x is in Boole’s system fundamental; see Laws of Thought, p. 31.
[479] See Pure Logic, § 69; and Principles of Science, 5, § 4.
It will be seen by reference to the rule given in section [427] that the Law of Simplicity (AA = A) and the Law of Unity (A or A = A) are reciprocal; that is, the former is deducible from the latter and vice versâ. For the only difference between them is that conjunctive combination in the one is replaced by alternative combination in the other.[480]