Gottfried Leibniz, like Descartes, contributed well to mathematics as he did to philosophy. Albeit he was a rationalist, he thought that rationality wasn’t enough to justify pure reason and that knowledge must be obtained from the external world too. He identified two kinds of truths, those that could be established through reason alone, and those that required empirical evidence. These are known today as analytical and synthetic statements. Analytical statements, like “All bachelors are unmarried” are true because there is nothing contradictory about the sentence itself. Synthetic statements, such as “Water boils at 100 degrees Celsius”, require empirical evidence to verify.
The basis for Leibniz’s philosophy is pure logical analysis. Every proposition, he believed, can be expressed in subject-predicate form. What is more, every true proposition is a statement of identity whose predicate is wholly contained in its subject, like “2 + 3 = 5.” In this sense, all propositions are analytic for Leibniz. But since the required analysis may be difficult, he distinguished two kinds of true propositions:
Truths of Reason are explicit statements of identity, or reducible to explicit identities by substitution of the definitions of their terms. Since a finite analysis always reveals the identity structure of such truths, they cannot be denied without contradiction and are perfectly necessary.
Truths of Fact, on the other hand, are implicit statements of identity, the grounds for whose truth may not be evident to us. These truths are merely contingent and may be subject to dispute since only an infinite analysis could show them to be identities.
Anything that human beings can believe or know, Leibniz held, must be expressed in one or the other of these two basic forms. The central insight of Leibniz’s system is that all existential propositions are truths of fact, not truths of reason. For the philosophical world, this simple doctrine had many significant consequences.
