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The Leibniz Review

Volume 29, December 2019

Dedicated to Daniel Garber

Wolfgang Lenzen
Pages 59-81

“Ex nihilo nihil fit”
On Leibniz’s “Principia Calculi rationalis”

In the essay “Principia Calculi rationalis” Leibniz attempts to prove the theory of the syllogism within his own logic of concepts. This task would be quite easy if one made unrestricted use of the fundamental laws discovered by Leibniz, e.g., in the “General Inquiries” of 1686. In the essays of August 1690, Leibniz had developed some similar proofs which, however, he considered as unsatisfactory because they presupposed the unproven law of contraposition: “If concept A contains concept B, then conversely Non-B contains Non-A”. The proof in “Principia Calculi rationalis” appears to reach its goal without resorting to this law. However, it contains a subtle flaw which results from failing to postulate that the ingredient concepts have to be “possible”, i.e. self-consistent. Once this flaw is corrected, it turns out that the proof – though formally valid – would not have been approved by Leibniz because, again, it rests on an unproven principle even stronger than the law of contraposition.

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