Conversely, the process of passing from (A or B) (A or b) to A, or from AB or Ab to A, may be called the reduction of a term by means 475 of the law of contradiction or the law of excluded middle, as the case may be.
Following Jevons, we may speak of an alternative term of the type AB or Ab as a dual term, and of the substitution of A for AB or Ab as the reduction of a dual term.[481]
[481] Pure Logic, § 103. The conjunctive term (A or B) (A or b) may also be spoken of as a dual term, and its reduction to A as the reduction of a dual term.
431. Laws of Absorption.—It may be shewn that any alternant which is merely a subdivision of another alternant may be indifferently introduced or omitted from a complex term. Thus, AB being a subdivision of A, the terms A or AB and A are equivalent. This rule (which is called by Schröder the Law of Absorption[482]) may be established as follows: By the development of A with reference to B, A or AB becomes AB or Ab or AB ; but, by the law of unity, this is equivalent to AB or Ab ; and by reduction this is equivalent to A.
[482] Der Operationskreis des Logikkalkuls, p. 12. This Law of Absorption is equivalent to one of Boole’s “Methods of Abbreviation” (Laws of Thought, p. 130). Compare, also, Jevons, Pure Logic, § 70.
Applying the rule given in section [427] we obtain a second law of absorption, namely, A (A or B) = A, which is the reciprocal of the first law of absorption, A or AB = A.
432. Laws of Exclusion and Inclusion.—The contradictory of any alternant in a complex term may be indifferently introduced or omitted as a determinant of any other alternant; that is to say, the terms A or aB and A or B are equivalent. This may be established as follows: By the law of absorption A or aB is equivalent to A or AB or aB, and by reduction this yields A or B. The above equivalence may be called the Law of Exclusion on the ground that by passing from A or B to A or aB we make the alternants mutually exclusive.
The reciprocal equivalence A (a or B) = AB may be expressed as follows: The contradictory of any determinant in a complex term may be indifferently introduced or omitted as an alternant of any other determinant. This equivalence may be called the Law of Inclusion on the ground that by passing from AB to A (a or B) we make the determinants collectively inclusive of the entire universe of discourse.
433. Summary of Formal Equivalences of Complex Terms.—The following is a summary of the formal equivalences contained in the five preceding sections (those that are bracketed together being 476 in each case related to one another reciprocally in the manner indicated in section [427]):—
| (1) | A (B or C) = AB or AC, | ⎱ | Laws of Distribution ; |
| (2) | A or BC = (A or B) (A orC), | ⎰ | |
| (3) | AA = A, | ⎱ | Laws of Tautology (Law ofSimplicity and Law of Unity) ; |
| (4) | A or A = A, | ⎰ | |
| (5) | A = A or Bb = (A or B) (A or b), | ⎱ | Laws of Development and Reduction ; |
| (6) | A = A (B or b) =AB or Ab, | ⎰ | |
| (7) | A or AB = A, | ⎱ | Laws of Absorption ; |
| (8) | A (A or B) = A, | ⎰ | |
| (9) | A or B = A or aB, | ⎱ | Law of Exclusion and Law of Inclusion. |
| (10) | AB = A (a or B), | ⎰ |