Volume 24, December 2014
Solving the Lucky and Guaranteed Proof Problems
Leibniz’s infinite-analysis theory of contingency says a truth is contingent if and only if it cannot be proved via analysis in finitely many steps. Some have argued that this theory faces the Problem of Lucky Proof—we might, by luck, complete our proof early in the analysis, and thus have a finite proof of a contingent truth—and the related Problem of Guaranteed Proof—even if we do not complete our proof early in the analysis, we are guaranteed to complete it in finitely many steps. I aim to solve both problems. For Leibniz, analysis is constrained by three rules: an analysis begins with the conclusion; subsequent steps replace a term by (part of) its real definition; and the analysis is finished only when an identity is reached. Furthermore, real definitions of complete concepts are infinitely complex, and Leibniz thinks infinities lack parts. From these observations, a solution to our problems follows: an analysis of a truth containing a complete concept cannot be completed in a finite number of steps—indeed, the first step of the analysis cannot be completed. I conclude by defusing some alleged counterexamples to my account.