Formulas for Propositions: Principle of Assertion and Logical Equivalences

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All the formulas which we have hitherto noted are valid alike for propositions and for concepts. We shall now establish a series of formulas which are valid only for propositions, because all of them are derived from an axiom peculiar to the calculus of propositions, which may be called the _principle of assertion_.This axiom is as follows:Ax. 10\[(a=1)=a.\]P. I.: To say that a proposition a is true is to state the proposition itself. In other words, to state a proposition is to affirm the truth of that proposition.58We can see at once that this formula is not susceptible of a conceptual interpretation (C. I.); for, if \(a\) is a concept, (\(a=1\)) is a proposition, and we would then have a logical equality [identity] between a concept and a proposition, which is absurd._Corollary_:\[a^{\prime}=(a^{\prime}=1)=(a=0).\]P. I.: The negative of a proposition \(a\) is equivalent to the affirmation that this proposition is false.By Ax. 9 (SS0.20), we already have\[(a=1)(a=0)=0,\]”A proposition cannot be both true and false at the same time”, for(Syll.)\[(a=1)(a=0)


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