The question, as we shall afterwards see more fully, is one of the greatest theoretical importance, because it concerns the true distinction between the sciences of Logic and Mathematics. It is the foundation of number that every unit shall be distinct from every other unit; but Boole imported the conditions of number into the science of Logic, and produced a system which, though wonderful in its results, was not a system of logic at all.
Laws of the Disjunctive Relation.
In considering the combination or synthesis of terms (p. [30]), we found that certain laws, those of Simplicity and Commutativeness, must be observed. In uniting terms by the disjunctive symbol we shall find that the same or closely similar laws hold true. The alternatives of either member of a disjunctive proposition are certainly commutative. Just as we cannot properly distinguish between rich and rare gems and rare and rich gems, so we must consider as identical the expression rich or rare gems, and rare or rich gems. In our symbolic language we may say
A ꖌ B = B ꖌ A.
The order of statement, in short, has no effect upon the meaning of an aggregate of alternatives, so that the Law of Commutativeness holds true of the disjunctive symbol.
As we have admitted the possibility of joining as alternatives terms which are not really different, the question arises, How shall we treat two or more alternatives when they are clearly shown to be the same? If we have it asserted that P is Q or R, and it is afterwards proved that Q is but another name for R, the result is that P is either R or R. How shall we interpret such a statement? What would be the meaning, for instance, of “wreath or anadem” if, on referring to a dictionary, we found anadem described as a wreath? I take it to be self-evident that the meaning would then become simply “wreath.” Accordingly we may affirm the general law
A ꖌ A = A.
Any number of identical alternatives may always be reduced to, and are logically equivalent to, any one of those alternatives. This is a law which distinguishes mathematical terms from logical terms, because it obviously does not apply to the former. I propose to call it the Law of Unity, because it must really be involved in any definition of a mathematical unit. This law is closely analogous to the Law of Simplicity, AA = A; and the nature of the connection is worthy of attention.
Few or no logicians except De Morgan have adequately noticed the close relation between combined and disjunctive terms, namely, that every disjunctive term is the negative of a corresponding combined term, and vice versâ. Consider the term
Malleable dense metal.