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Gottfried Wilhelm Leibniz
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===Formal logic<!--'Algebra of concepts' and 'Leibniz's theory of concepts' redirect here-->=== {{Main|Algebraic logic}} Leibniz has been noted as one of the most important logicians between the times of Aristotle and [[Gottlob Frege]].<ref>Lenzen, W., 2004, "Leibniz's Logic," in ''Handbook of the History of Logic'' by D. M. Gabbay/J. Woods (eds.), volume 3: ''The Rise of Modern Logic: From Leibniz to Frege'', Amsterdam et al.: Elsevier-North-Holland, pp. 1–83.</ref> Leibniz enunciated the principal properties of what we now call [[logical conjunction|conjunction]], [[disjunction]], [[negation]], [[Identity (mathematics)|identity]], set [[subset|inclusion]], and the [[empty set]]. The principles of Leibniz's logic and, arguably, of his whole philosophy, reduce to two: # All our ideas are compounded from a very small number of simple ideas, which form the [[alphabet of human thought]]. # Complex ideas proceed from these simple ideas by a uniform and symmetrical combination, analogous to arithmetical multiplication. The formal logic that emerged early in the 20th century also requires, at minimum, [[unary function|unary]] negation and [[Quantification (logic)|quantified]] [[variable (mathematics)|variables]] ranging over some [[universe of discourse]]. Leibniz published nothing on formal logic in his lifetime; most of what he wrote on the subject consists of working drafts. In his ''[[A History of Western Philosophy|History of Western Philosophy]]'', [[Bertrand Russell]] went so far as to claim that Leibniz had developed logic in his unpublished writings to a level which was reached only 200 years later. Russell's principal work on Leibniz found that many of Leibniz's most startling philosophical ideas and claims (e.g., that each of the fundamental [[Monad (philosophy)|monads]] mirrors the whole universe) follow logically from Leibniz's conscious choice to reject ''relations'' between things as unreal. He regarded such relations as (real) ''qualities'' of things (Leibniz admitted [[unary function|unary]] [[Predicate (mathematical logic)|predicates]] only): For him, "Mary is the mother of John" describes separate qualities of Mary and of John. This view contrasts with the relational logic of [[Augustus De Morgan|De Morgan]], [[Charles S. Peirce|Peirce]], [[Ernst Schröder (mathematician)|Schröder]] and Russell himself, now standard in [[predicate logic]]. Notably, Leibniz also declared space and time to be inherently relational.<ref>{{Cite book|title=A Critical Exposition of the Philosophy of Leibniz|publisher=The University Press, Cambridge|date=1900|first=Bertrand|last=Russell}}</ref> Leibniz's 1690 discovery of his '''algebra of concepts'''<!--boldface per WP:R#PLA--><ref>''Leibniz: Die philosophischen Schriften'' VII, 1890, [https://archive.org/details/diephilosophisc00gerhgoog/page/n251/mode/2up pp. 236]–247; translated as [http://171.67.193.21/cm/leibniz/leibniz-1690.pdf "A Study in the Calculus of Real Addition" (1690)] {{Webarchive|url=https://web.archive.org/web/20210719231443/http://171.67.193.21/cm/leibniz/leibniz-1690.pdf |date=19 July 2021 }} by G. H. R. Parkinson, ''Leibniz: Logical Papers – A Selection'', Oxford 1966, pp. 131–144.</ref><ref>[[Edward N. Zalta]], [https://mally.stanford.edu/Papers/leibniz.pdf "A (Leibnizian) Theory of Concepts"], ''Philosophiegeschichte und logische Analyse / Logical Analysis and History of Philosophy'', 3 (2000): 137–183.</ref> (deductively equivalent to the [[Boolean algebra]])<ref>{{cite IEP |url-id=leib-log |title=Leibniz: Logic |last=Lenzen |first=Wolfgang}}</ref> and the associated metaphysics, are of interest in present-day [[computational metaphysics]].<ref>Jesse Alama, Paul E. Oppenheimer, [[Edward N. Zalta]], [https://mally.stanford.edu/Papers/cade.pdf "Automating Leibniz's Theory of Concepts"], in A. Felty and A. Middeldorp (eds.), ''Automated Deduction – CADE 25: Proceedings of the 25th International Conference on Automated Deduction'' (Lecture Notes in Artificial Intelligence: Volume 9195), Berlin: Springer, 2015, pp. 73–97.</ref>
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