The Law of Duality: Symmetry between Addition and Multiplication

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0.14 The Law of DualityWe have proved that a perfect symmetry exists between the formulas relating to multiplication and those relating to addition. We can pass from one class to the other by interchanging the signs of addition and multiplication, on condition that we also interchange the terms 0 and 1 and reverse the meaning of the sign negation which have not yet been stated. We shall see that these laws possess the same property and consequently preserve the duality, but they do not originate it; and duality would exist even if the idea of negation were not introduced. For instance, the equality (SS0.12)Boole thus derives it (_Laws of Thought_, London 1854, Chap. III, Prop. IV).\[ab+ac+bc=(a+b)(a+c)(b+c)\] is its own reciprocal by duality, for its two members are transformed into each other by duality.It is worth remarking that the law of duality is only applicable to primary propositions. We call [after Boole ] those propositions _primary_ which contain but one copula (\(


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