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There is one circumstance of particular interest, namely, that the algebra in question, like logic, is susceptible of two distinct interpretations, the parallelism between them being almost perfect, according as the letters represent concepts or propositions. Doubtless we can, with Boole, reduce the two interpretations to one, by considering the concepts on the one hand and the propositions on the other as corresponding to ''assemblages'' or ''classes''; since a concept determines the class of objects to which it is applied (and which in logic is called its ''extension''), and a proposition determines the class of the instances or moments of time in which it is true (and which by analogy can also be called its extension). Accordingly [[predicate calculus]] and [[propositional calculus]] become reduced to but one, the calculus of classes or, as Leibniz called it, the theory of the whole and part, of that which contains and that which is contained. But as a matter of fact, predicate calculus and propositional calculus present certain differences which prevent their complete identification from the formal point of view and consequently their reduction to a single calculus of classes.
In particular, the principle of ''assertion'' <math>(a = 1) = a</math> is peculiar to propositional calculus, and is interpreted as follows: To state a proposition is to affirm the truth of that proposition. Clearly, this formula is not susceptible of a conceptual interpretation, for, if <math>a</math> is a concept, <math>(a = 1)</math> is a proposition, and we would then have a logical equality between a concept and a proposition, which is absurd. From this formula combined with the [[Law of non contradiction]] we deduce the law of [[Bivalence and related laws|Bivalence]]. In fact, propositional calculus is equivalent to the calculus of classes when the classes can possess only the two values 0 and 1.
The equivalence of an implication and a disjunction <math>(a \Rightarrow b) \Leftrightarrow (\bar{a} \vee b)</math> is no less fundamental to propositional calculus, as it makes possible to reduce secundary, tertiary, etc. propositions to [[primary proposition]]s, or even to sums of elementary propositions.
Accordingly we have in reality three distinct calculi, or, in the part common to all, three different interpretations of the same calculus. In any case one must not forget that the logical value and the deductive sequence of the formulas does not in the least depend upon the interpretations which may be given them. These interpretations shall serve only to render the formulas intelligible, to give them clearness and to make their meaning at once obvious, but never to justify them. They may be omitted without destroying the formal rigidity of the system.
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