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[[Mathematical logic]] is the use of formal logic to study mathematical reasoning. At the beginning of the twentieth century, philosophical logicians including ([[Gottlob Frege|Frege]], [[Bertrand Russell|Russell]]) attempted to prove that mathematics could be entirely reduced to logic. They held that in discovering the "logical form" of a sentence, you were somehow revealing the "right" way to say it, or uncovering some previously hidden essence. The reduction failed, but in the process, logic took on much of the notation and methodology of mathematics, and nowadays logic is accepted as an accurate way to describe mathematical reasoning.
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.
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.
In order not to favor either interpretation one might say that the letters represent ''terms''; these terms may be either concepts or propositions according to the case in hand.
==Philosophical logic==
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