Definition of Negation and its Properties

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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 \(


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