Law of Double Negation: Reciprocity and Transposition of Negatives

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0.17 Law of Double NegationMoreover this reciprocity is general: if a term \(b\) is the negative of the term \(a\), then the term \(a\) is the negative of the term \(b\). These two statements are expressed by the same formulas\[ab=0,\quad a+b=1,\]and, while they unequivocally determine \(b\) in terms of \(a\), they likewise determine \(a\) in terms of \(b\). This is due to the symmetry of these relations, that is to say, to the commutativity of multiplication and addition. This reciprocity is expressed by the _law of double negation_\[(a^{\prime})^{\prime}=a,\]which may be formally proved as follows: \(a^{\prime}\) being by hypothesis the negative of \(a\), we have\[aa^{\prime}=0,\quad a+a^{\prime}=1.\]On the other hand, let \(a^{\prime\prime}\) be the negative of \(a^{\prime}\); we have, in the same way,\[a^{\prime}a^{\prime\prime}=0,\quad a^{\prime}+a^{\prime\prime}=1.\]But, by the preceding lemma, these four equalities involve the equality\[a=a^{\prime\prime}.\]Q. E. D.This law may be expressed in the following manner:If \(b=a^{\prime}\), we have \(a=b^{\prime}\), and conversely, by symmetry.This proposition makes it possible, in calculations, to transpose the negative from one member of an equality to the other.The law of double negation makes it possible to conclude the equality of two terms from the equality of their negatives (if \(a^{\prime}=b^{\prime}\) then \(a=b\)), and therefore to cancel the negation of both members of an equality.From the characteristic formulas of negation together with the fundamental properties of \(0\) and \(1\), it results that every product which contains two contradictory factors is null, and that every sum which contains two contradictory summands is equal to \(1\).In particular, we have the following formulas:\[a=ab+ab^{\prime},\quad a=(a+b)(a+b^{\prime}),\]which may be demonstrated as follows by means of the distributive law:\[a=a\times 1=a(b+b^{\prime})=ab+ab^{\prime},\] \[a=a+0=a+bb^{\prime}=(a+b)(a+b^{\prime}).\]These formulas indicate the principle of the method of development which we shall explain in detail later (SSSS0.21 sqq.)


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