Defining Negation in Mathematics

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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


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