Leibniz Logical Calculus

Origin. Gottfried Wilhelm Leibniz, in working drafts spanning 1679–1690, none published in his lifetime: the characteristic-number essays of April 1679 (Elementa calculi and its companions), the Generales Inquisitiones de Analysi Notionum et Veritatum (1686), and the calculus of real addition of about 1690 (Gerhardt, Philosophische Schriften VII, 236–247; translated by Parkinson as "A Study in the Calculus of Real Addition"). Nineteenth-century editions by Erdmann and Gerhardt printed fragments; Louis Couturat's La Logique de Leibniz (1901) and Opuscules et fragments inédits de Leibniz (1903) recovered the program whole — the Generales Inquisitiones appeared there for the first time — after Peirce and Frege had already built modern logic without it.

Models. Term logic read intensionally. "A est B" relates concepts, not classes: it is true when the concept of A contains the concept of B — praedicatum inest subjecto, Leibniz's containment theory of truth, asserted for every true affirmative proposition, necessary or contingent alike. Extension inverts intension — A's concept contains B's exactly when A's individuals all fall under B — and Leibniz states both readings in the Nouveaux Essais and works with comprehension. On Wolfgang Lenzen's reconstruction the resulting algebra of concepts is deductively equivalent to Boolean algebra: Boolean algebra avant la lettre, intensionally read, which is the standard modern assessment and a theorem about the fragments rather than a result Leibniz stated.

Formalism.

Algebra of concepts (Generales Inquisitiones, 1686): Terms compose: AB, commutative and idempotent (AA = A). A est B — containment of B's concept in A's; coincidence A = B is containment both ways. non-A — concept negation. A est ens — A is a possible, self-consistent concept; on Lenzen's reconstruction, the ens qualification carries existential import, with "some A is B" as the possibility of the composite AB. Propositions as terms: a proposition is itself a term and truth is containment again — the device at the center of Malink and Vasudevan's formalization. Principle 129 ties the calculus to the 1679 arithmetic: anything can be proved by numbers, provided aa and a are equivalent.

Characteristic numbers (April 1679): Elementa calculi with Elementa characteristicae universalis, Calculi universalis elementa, Calculi universalis investigationes, and Regulae de bonitate consequentiarum. A composite concept's number is the product of its components': man = rational · animal = 2 · 3 = 6. 'Every S is P' is true when S's number is divisible by P's (s = xp) — so 'every man is rational' because 3 divides 6. Particular and negative propositions defeated the single-number scheme; the Regulae refine it to pairs of coprime integers, and the truth conditions were rewritten from manuscript to manuscript without being settled. Sotirov ("Arithmetizations of Syllogistic à la Leibniz", Journal of Applied Non-Classical Logics 9, 1999): two arithmetical models built on Leibniz's idea — divisibility for the universal affirmative, a g.c.d. condition for the particular — are adequate for the assertoric syllogistic and extend to term negation and conjunction.

Calculus of real addition (about 1690): A ⊕ B — real addition: commutative, associative, and idempotent; A ⊕ A = A is the axiom separating it from arithmetical addition. Containment becomes being a summand: A is in B when B = A ⊕ C for some C. The plus-minus fragments (Non inelegans specimen demonstrandi in abstractis) add a partial subtraction inverse to ⊕. The calculus is stated abstractly, without fixing the interpretation — concepts and classes are both models.

The program: characteristica universalis — real characters, one per simple concept, their composition mirroring conceptual composition, over an alphabet of human thought. calculus ratiocinator — computation on the characters in place of dispute: "Let us calculate" (The Art of Discovery, 1685). What they were: a notation scheme and decision procedures for containment among composite concepts. What they were not: carried out — no alphabet of simple concepts was produced, and no calculus of relations was attempted.

Symbols.

Metatheory. Everything proved about these calculi is modern, and that is itself the historical finding. Lenzen ("Leibniz's Logic", Handbook of the History of Logic, vol. 3, 2004, pp. 1–83) reconstructed the algebra of concepts from the Generales Inquisitiones and the 1690 papers and showed it deductively equivalent to Boolean algebra, with existential import handled through the ens operator; Malink and Vasudevan ("The logic of Leibniz's Generales Inquisitiones de Analysi Notionum et Veritatum", Review of Symbolic Logic 9, 2016, pp. 686–751) formalized the GI calculus whole, propositional terms included; Sotirov (1999) built the two arithmetical semantics the 1679 essays were reaching for and proved both adequate for the assertoric syllogistic — the characteristic-number idea was sound, and what Leibniz lacked was a workable truth condition for particular propositions, not the arithmetic. Couturat's verdict of 1901, that Leibniz already possessed the principles of the algebra of logic, and Russell's, that he had reached a level regained only two hundred years later, are the standard assessments — and both postdate the field they compare him to, since Boole, Peirce, and Frege worked without these manuscripts. The calculi anticipated the algebra of logic and influenced none of it.

Applies to. History of logic. Intensional semantics of terms and concepts. Computational metaphysics — the automation of the concept algebra in Zalta's theory of concepts (Alama, Oppenheimer, and Zalta, CADE 2015). The theory of notation — Leibniz attributed his discoveries in mathematics to his notations. Comparative assessment of term logics against Boolean algebra.

Limitations. The calculi are term logics: composition, negation, and containment of unary concepts, with no logic of relations — Russell traced Leibniz's metaphysics to exactly this exclusion — and no quantification beyond the four categorical forms. The characteristic numbers presuppose what the characteristica universalis never supplied, a factorization of every concept into simples; without the alphabet of human thought the arithmetic has nothing to number, and Leibniz's own truth conditions for particular and negative propositions never worked — the adequate versions are Sotirov's. The corpus is working notes: the Generales Inquisitiones revises its own definitions as it proceeds, which fragments constitute "the calculus" is an editorial decision, and so the system proved equivalent to Boolean algebra is in part the reconstructor's — the same situation the Stoic material presents, with Couturat and Lenzen in the role of Mates and Bobzien.

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