Definition of 0 and 1: Introduction to Logical Calculus

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0.13 Definition of 0 and 1We shall now define and introduce into the logical calculus two special terms which we shall designate by 0 and by 1, because of some formal analogies that they present with the zero and unity of arithmetic. These two terms are formally defined by the two following principles which affirm or postulate their existence.Ax. 6 There is a term 0 such that whatever value may be given to the term \(x\), we have\[0 null ” or “void” class which contains no element (Nothing or Naught), 1 denotes the class which contains all classes; hence it is the totality of the elements which are contained within it. It is called, after Boole, the “universe of discourse” or simply the “whole”.P. I.: 0 denotes the proposition which implies every proposition; it is the “false” or the “absurd”, for it implies notably all pairs of contradictory propositions, 1 denotes the proposition which is implied in every proposition; it is the “true”, for the false may imply the true whereas the true can imply only the true.By definition we have the following inclusions\[0 null class is contained in the _whole_.23The rendering “Nothing is everything” must be avoided.P. I.: The false implies the true.By the definitions of 0 and 1 we have the equivalences\[(a null ” [zero] is to say that each of the summands is null ; to say that a product is equal to 1 is to say that each of its factors is equal to 1.Thus we have\[(a+b=0) null ; it by no means follows that either one or the other of these classes is null . The second denotes that these two classes combined form the whole; it by no means follows that either one or the other is equal to the whole.The following formulas comprising the rules for the calculus of 0 and 1, can be demonstrated:\[a\times 0=0,\quad a+1=1,\] \[a+0=a,\quad a\times 1=a.\]For\[(0 null class is the null class; the sum of any class whatever and of the whole is the whole. The sum of the null class and of any class whatever is equal to the latter; the part common to the whole and any class whatever is equal to the latter.P. I.: The simultaneous affirmation of any proposition whatever and of a false proposition is equivalent to the latter (_i.e._, it is false); while their alternative affirmation is equal to the former. The simultaneous affirmation of any proposition whatever and of a true proposition is equivalent to the former; while their alternative affirmation is equivalent to the latter (_i.e._, it is true)._Remark.–_If we accept the four preceding formulas as axioms, because of the proof afforded by the double interpretation, we may deduce from them the paradoxical formulas\[0

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