# Defining Negation in Mathematics

0.15 Definition of NegationThe introduction of the terms 0 and 1 makes it possible for us to define _negation_. This is a “uni-nary” operation which transforms a single term into another term called its _negative_.25 The negative of $a$ is called not-$a$ and is written $a^{\prime}$.26 Its formal definition implies the following postulate of existence27:[In French] the same word _negation_ denotes both the operation and its result, which becomes equivocal. The result ought to be denoted by another word, like [the English] “negative”. Some authors say, “supplementary” or “supplement”, [_e.g._ Boole and Huntington on ], Classical logic makes use of the term “contradictory” especially for propositions.We adopt here the notation of MacColl; Schröder indicates not-$a$ by $a_{1}$ which prevents the use of indices and obliges us to express them as exponents. The notation $a^{\prime}$ has the advantage of excluding neither indices nor exponents. The notation $\bar{a}$ employed by many authors is inconvenient for typographical reasons. When the negative affects a proposition written in an explicit form (with a copula ) it is applied to the copula $_contradictories_ ; the principle of excluded middle must be added which equally deserves the name of principle of contradiction. This is why Mrs. Ladd-Franklin proposes to call them respectively the _principle of exclusion_ and the _principle of exhaustion_, inasmuch as, according to the first, two contradictory terms are _exclusive_ (the one of the other); and, according to the second, they are _exhaustive_ (of the universe of discourse).C. I.: 1. The classes \(a$ and $a^{\prime}$ have nothing in common; in other words, no element can be at the same time both $a$ and not-$a$.2. The classes $a$ and $a^{\prime}$ combined form the whole; in other words, every element is either $a$ or not-$a$.P. I.: 1. The simultaneous affirmation of the propositions $a$ and not-$a$ is false; in other words, these two propositions cannot both be true at the same time.2. The alternative affirmation of the propositions $a$ and not-$a$ is true; in other words, one of these two propositions must be true