Two great principles of reasoning

Gottfried Wilhelm Leibniz (1646–1716) — mathematician, diplomat, co-inventor of the calculus — wanted a philosophy resting on principles so secure that reasoning could, in principle, become calculation. He isolates two great principles on which “our reasonings are founded” (Monadology §31–32). Everything else in his system — the monads, the harmony, the best world — is squeezed out of these two.

The first is the Principle of Non-Contradiction. A proposition cannot be both true and false; whatever contains or implies a contradiction is false, and its opposite true. This principle governs the truths of reason (vérités de raison) — truths that are necessary, holding in every possible world. Their denial is not merely false but impossible: it collapses into contradiction.

So necessary truths are known by analysis in a finite number of steps: you break the subject down until the predicate visibly sits inside it, terminating in a bare identity (A is A). Because the analysis finishes, a finite mind can, in principle, prove them with certainty. Geometry, arithmetic, logic — all truths of reason.

The second is the Principle of Sufficient Reason (principium rationis sufficientis): nothing is the case, no fact is real, no proposition true, “without a sufficient reason why it is so and not otherwise” (Monadology §32). Nothing is simply brute. This principle governs the truths of fact (vérités de fait) — truths that are contingent: their opposite is perfectly possible, implies no contradiction, yet they are true anyway. For those, there is always a reason — even if we cannot survey it.

Underneath both principles lies Leibniz’s boldest thesis: the predicate-in-notion theory of truth. In every true affirmative proposition, “the concept of the predicate is in some way contained in the concept of the subject” (Discourse on Metaphysics §8). To say something true of a thing is to unfold what was already packed into the complete concept of that thing.

This has a startling consequence: every truth is, in principle, analytic — provable by unpacking the subject. For God, who grasps complete concepts, there is no difference in kind between 2+2=4 and Caesar crossed the Rubicon; both are contained in their subjects. The difference is in the analysis.

And here is the resolution. For a necessary truth, the analysis of the subject into the predicate terminates — finite steps, ending in an identity. For a contingent truth, the analysis is infinite: you could unfold the concept forever and never reach a bare identity; the predicate is contained, but only an infinite mind can see that it is. This “infinite analysis” is Leibniz’s technical mark of contingency.