[480] It may assist the reader in following the reasoning in section [427] if we work through this particular case independently. If AA = A, then aa = a, for whatever is formally valid in the case of A must also be formally valid in the case of any other term. But if two terms are equivalent their contradictories must be equivalent. Hence from aa = a, it follows that A or A = A. And it is clear that we might pass similarly from A or A = A to AA = A.
474 430. Laws of Development and Reduction.—Important formal equivalences are yielded by the laws of contradiction and excluded middle.
By the law of contradiction a term containing contradictory determinants (for example, Bb) cannot represent any existing class. Hence A or Bb is equivalent to A simply; in other words, the conjunctive combination of contradictories may be indifferently introduced or omitted as an alternant.
Again, by the law of excluded middle a term containing contradictory alternants (for example, B or b) represents the entire universe of discourse. Hence A (B or b) is equivalent to A simply; in other words, the alternative combination of contradictories may be indifferently introduced or omitted as a determinant.
It will be observed that the above equivalences, namely,
A or Bb = A,
A (B or b) = A,
are reciprocal.
Applying further the Laws of Distribution given in section [428] we have the following:
A = A or Bb = (A or B) (A or b),
A = A (B or b) = AB or Ab.
These may be taken as formulae for the development and the reduction of terms. Thus, the substitution of (A or B) (A or b) for A may be called the development of a term by means of the law of contradiction; and the substitution of AB or Ab for A the development of a term by means of the law of excluded middle. In both the above cases the term A is developed with reference to the term B. Similarly by developing A with reference to B and C, we should have (A or B or C) (A or B or c) (A or b or C) (A or B or c) if we make use of the law of contradiction, or ABC or ABc or AbC or Abc if we make use of the law of excluded middle. Development by means of the law of excluded middle is the more useful of the two processes in the manipulation of complex terms, and it may be understood that this is meant when the development of a term is spoken of without further qualification.