THE LOGIC OF RATIONALISM Spinoza.—A fundamental contrast to the school of Bacon and Locke is afforded by the great systems of reason, owning Cartesian inspiration, which are identified with the names of Spinoza and Leibniz. Spinoza's philosophy is expounded ordine geometrico and with Euclidean cogency from a relatively small number of definitions, axioms and postulates. But how we reach our as surance of the necessity of these principles is not made specifi cally clear. The invaluable tractate De Intellectus emendatione, in which the agreement with and divergence from Descartes on the question of method could have been fully elucidated, is un happily not finished. We know that we need to pass from what Spinoza terms vague experience, where imagination with its f rag mentary apprehension is liable to error and neither necessity nor impossibility can be predicated, right up to intellection. And what Spinoza has to say of the requisites of definition and the marks of intellection makes it clear that insight comes with co herence, and that the work of method on the "inductive" side is by means of the unravelling of all that makes for artificial limitation to lay bare what can then be seen to exhibit nexus in the one great system. When all is said, however, the geometric method as universalized in philosophy is rather used by Spinoza than expounded.
The extreme case of course is the human subject. "The indi vidual notion of each person includes once for all what is to befall it, world without end," and "it would not have been our Adam but another, if he had had other events." Existent subjects con taining eternally all their successive predicates in the time-series are substances, which when the problems connected with their activity or, dynamically speaking, their force, have been resolved, demand—and supply—the, metaphysic of the Monadology.
Complex truths of reason or essence raise the problem of definition, which consists in their analysis into simpler truths and ultimately into simple—i.e., indefinable ideas, with primary prin ciples of another kind—axioms, and postulates that neither need nor admit of proof. These are identical in the sense that the opposite contains an express contradiction. In the case of non identical truths, too, there is a priori proof drawn from the notion of the terms, "though it is not always in our power to arrive at this analysis" so that the question arises, specially in connection with the possibility of a calculus, whether the con tingent is reducible to the necessary or identical at the ideal limit.
Leibniz's remaining legacy to later logicians is the conception of Characteristica Universalis and Ars Combinatoria, a universal denoting by symbols and a calculus working by substitutions and the like. The two positions that a subject contains all its predi cates, and that all non-contingent propositions, i.e., all proposi tions not concerned with the existence of individual facts ulti mately analyse out into identities—obviously lend themselves to the design of this algebra of thought, though the mathematician in Leibniz should have been aware that a significant equation is never an identity. Leibniz, fresh from the battle of the calculus in the mathematical field, and with his conception of logic, at least in some of its aspects, as a generalized mathematic, found a fruitful inspiration, harmonizing well with his own metaphysic, in Bacon's alphabet of nature. He, too, was prepared to offer a new instrument. That the most important section, the list of forms of combination, was never achieved—this, too, was after the Baconian example while the mode of symbolization was crude with a=ab and the like—matters little. A new technique of manipulation—it is, of course, no more—had been evolved.