Editing Gottfried Wilhelm Leibniz (section)

== Philosophy ==
Leibniz's philosophical thinking appears fragmented because his philosophical writings consist mainly of a multitude of short pieces: journal articles, manuscripts published long after his death, and letters to correspondents. He wrote two book-length philosophical treatises, of which only the ''Théodicée'' of 1710 was published in his lifetime.

Leibniz dated his beginning as a philosopher to his ''[[Discourse on Metaphysics]]'', which he composed in 1686 as a commentary on a running dispute between [[Nicolas Malebranche]] and [[Antoine Arnauld]]. This led to an extensive correspondence with Arnauld;<ref>Ariew & Garber, 69; Loemker, §§36, 38</ref> it and the ''Discourse'' were not published until the 19th century. In 1695, Leibniz made his public entrée into European philosophy with a journal article titled "New System of the Nature and Communication of Substances".<ref>Ariew & Garber, 138; Loemker, §47; Wiener, II.4</ref> Between 1695 and 1705, he composed his ''[[New Essays on Human Understanding]]'', a lengthy commentary on [[John Locke]]'s 1690 ''[[An Essay Concerning Human Understanding]]'', but upon learning of Locke's 1704 death, lost the desire to publish it, so that the ''New Essays'' were not published until 1765. The ''[[Monadology|Monadologie]]'', composed in 1714 and published posthumously, consists of 90 aphorisms.

Leibniz also wrote a short paper, "Primae veritates" ("First Truths"), first published by [[Louis Couturat]] in 1903 (pp.&nbsp;518–523)<ref>Later translated as Loemker 267 and Woolhouse and Francks 30</ref> summarizing his views on [[metaphysics]]. The paper is undated; that he wrote it while in Vienna in 1689 was determined only in 1999, when the ongoing critical edition finally published Leibniz's philosophical writings for the period 1677–1690.<ref>A VI, 4, n. 324, pp. 1643–1649 with the title: ''Principia Logico-Metaphysica''</ref> Couturat's reading of this paper influenced much 20th-century thinking about Leibniz, especially among [[analytic philosophy|analytic philosophers]]. After a meticulous study (informed by the 1999 additions to the critical edition) of all of Leibniz's philosophical writings up to 1688, Mercer (2001) disagreed with Couturat's reading.{{clarify|date=April 2024}}

Leibniz met [[Baruch Spinoza]] in 1676, read some of his unpublished writings, and had since been influenced by some of Spinoza's ideas.{{citation needed|date=December 2024}} While Leibniz befriended him and admired Spinoza's powerful intellect, he was also dismayed by Spinoza's conclusions,<ref>Ariew & Garber, 272–284; Loemker, §§14, 20, 21; Wiener, III.8</ref> especially when these were inconsistent with Christian orthodoxy.

Unlike Descartes and Spinoza, Leibniz had a university education in philosophy. He was influenced by his [[Leipzig]] professor [[Jakob Thomasius]], who also supervised his BA thesis in philosophy.<ref name="Arthur p. 13">Arthur 2014, p. 13.</ref> Leibniz also read [[Francisco Suárez]], a Spanish [[Society of Jesus|Jesuit]] respected even in [[Lutheranism|Lutheran]] universities. Leibniz was deeply interested in the new methods and conclusions of Descartes, Huygens, Newton, and [[Robert Boyle|Boyle]], but the established philosophical ideas in which he was educated influenced his view of their work.

=== Principles ===<!--Linked from "Difference (philosophy)"-->
Leibniz variously invoked one or another of seven fundamental philosophical Principles:<ref>Mates (1986), chpts. 7.3, 9</ref>
* [[Identity (mathematics)|Identity]]/[[contradiction]]. If a proposition is true, then its negation is false and vice versa.
* [[Identity of indiscernibles]]. Two distinct things cannot have all their properties in common. If every predicate possessed by ''x'' is also possessed by ''y'' and vice versa, then entities ''x'' and ''y'' are identical; to suppose two things indiscernible is to suppose the same thing under two names. The "identity of indiscernibles" is frequently invoked in modern logic and philosophy. It has attracted the most controversy and criticism, especially from corpuscular philosophy and quantum mechanics. The [[converse (logic)|converse]] of this is often called ''Leibniz's law'', or the ''indiscernibility of identicals'', which is mostly uncontroversial. 
* [[principle of sufficient reason|Sufficient reason]]. "There must be a sufficient reason for anything to exist, for any event to occur, for any truth to obtain."<ref>Loemker 717</ref>
* [[Pre-established harmony]].<ref>See Jolley (1995: 129–131), Woolhouse and Francks (1998), and Mercer (2001).</ref> "[T]he appropriate nature of each substance brings it about that what happens to one corresponds to what happens to all the others, without, however, their acting upon one another directly." (''Discourse on Metaphysics'', XIV) A dropped glass shatters because it "knows" it has hit the ground, and not because the impact with the ground "compels" the glass to split.
* [[Law of continuity]]. ''[[Natura non facit saltus]]''<ref name="Saltus">Gottfried Leibniz, [[New Essays on Human Understanding|''New Essays'']], IV, 16: "''la nature ne fait jamais des sauts''". ''Natura non-facit saltus'' is the Latin translation of the phrase (originally put forward by [[Carolus Linnaeus|Linnaeus]]' ''[[Philosophia Botanica]]'', 1st ed., 1751, Chapter III, § 77, p. 27. See also {{Cite SEP|url-id=continuity/|title=Continuity and Infinitesimals|date=Mar 16, 2022|edition=Spring 2022|last=Bell|first=John L.}} See also [[Alexander Baumgarten]], ''Metaphysics: A Critical Translation with Kant's Elucidations'', Translated and Edited by Courtney D. Fugate and John Hymers, Bloomsbury, 2013, "Preface of the Third Edition (1750)", [https://books.google.com/books?id=Jw-Q3hfXTqoC&q=%22must+also+have+in+mind+Leibniz%27s+%22natura+non+facit+saltus%22+%5Bnature+does+not%22&pg=PA79 p. 79 n.d.<!--footnote alphabet-number-->]: "[Baumgarten] must also have in mind Leibniz's "''natura non-facit saltus'' [nature does not make leaps]" ([[Nouveaux essais sur l'entendement humain|NE]]<!--the abbreviation is not italicized in the original--> IV, 16)."). A variant translation is "''natura non-saltum facit''" (literally, "Nature does not make a jump") ({{cite book|last1=Britton|first1=Andrew|url=https://books.google.com/books?id=goW6JsEUz4EC|title=Ökonomische Theorie und christlicher Glaube|last2=Sedgwick|first2=Peter H.|last3=Bock|first3=Burghard|publisher=LIT Verlag Münster|year=2008|isbn=978-3-8258-0162-5|page=289}} [https://books.google.com/books?id=goW6JsEUz4EC&pg=PA289 Extract of page 289].)</ref> (literally, "Nature does not make jumps").
* [[Philosophical optimism|Optimism]]. "God assuredly always chooses the best."<ref>Loemker 311</ref>
* [[Principle of plenitude|Plenitude]]. Leibniz believed that the best of all possible worlds would actualize every genuine possibility, and argued in ''Théodicée'' that this best of all possible worlds will contain all possibilities, with our finite experience of eternity giving no reason to dispute nature's perfection.<ref>[[Arthur Lovejoy]], ''The [[Great Chain of Being]]''. Harvard University Press, 1936, Chapter V "Plenitude and Sufficient Reason in Leibniz and Spinoza", pp.&nbsp;144–182.</ref>

Leibniz would on occasion give a rational defense of a specific principle, but more often took them for granted.<ref>For a precis of what Leibniz meant by these and other Principles, see Mercer (2001: 473–484). For a classic discussion of Sufficient Reason and Plenitude, see Lovejoy (1957).</ref>

===Monads<!--'Well-founded phenomenon' redirects here-->===
[[File:Leibniz Monadology 1.jpg|thumb|A page from Leibniz's manuscript of the ''[[Monadology]]'']]
Leibniz's best known contribution to [[metaphysics]] is his theory of [[Monad (philosophy)|monads]], as exposited in ''[[Monadology|Monadologie]]''. He proposes his theory that the universe is made of an infinite number of simple substances known as monads.<ref>{{cite book|last1=O'Leary-Hawthorne|first1=John|last2=Cover|first2=J. A.|title=Substance and Individuation in Leibniz|publisher=Cambridge University Press|isbn=978-0-521-07303-5|page=65|date=2008-09-04}}</ref> Monads can also be compared to the corpuscles of the [[mechanical philosophy]] of René Descartes and others. These simple substances or monads are the "ultimate units of existence in nature". Monads have no parts but still exist by the qualities that they have. These qualities are continuously changing over time, and each monad is unique. They are also not affected by time and are subject to only creation and annihilation.<ref>{{cite book|last1=Rescher|first1=Nicholas|title=G. W. Leibniz's Monadology: an edition for students|url=https://archive.org/details/gwleibnizsmonado00resc|url-access=limited|date=1991|publisher=University of Pittsburgh Press|location=Pittsburgh|isbn=978-0-8229-5449-1|page=[https://archive.org/details/gwleibnizsmonado00resc/page/n40 40]}}</ref> Monads are centers of [[force]]; substance is force, while [[space]], [[matter]], and [[Motion (physics)|motion]] are merely phenomenal. He argued, against Newton, that [[space]], [[time]], and motion are completely relative:<ref name=Ferraro>{{cite book |title =Einstein's Space-Time: An Introduction to Special and General Relativity|page= 1| url=https://books.google.com/books?id=wa3CskhHaIgC&q=time+%22absolute+space%22&pg=PA1|first=Rafael |last=Ferraro |isbn=978-0-387-69946-2 |publisher=Springer |year=2007}}</ref> "As for my own opinion, I have said more than once, that I hold space to be something merely relative, as time is, that I hold it to be an order of coexistences, as time is an order of successions."<ref name="See H. G pp. 25">See H. G. Alexander, ed., ''The [[Leibniz-Clarke Correspondence]]'', Manchester: Manchester University Press, pp. 25–26.</ref> Einstein, who called himself a "Leibnizian", wrote in the introduction to [[Max Jammer]]'s book ''Concepts of Space'' that Leibnizianism was superior to Newtonianism, and his ideas would have dominated over Newton's had it not been for the poor technological tools of the time; Joseph Agassi argues that Leibniz paved the way for Einstein's [[theory of relativity]].<ref>{{Cite journal|last=Agassi|first=Joseph|title=Leibniz's Place in the History of Physics|journal=Journal of the History of Ideas|url=https://www.jstor.org/stable/2708561|date=September 1969|volume=30|issue=3|pages=331–344|doi=10.2307/2708561|jstor=2708561}}</ref>

Leibniz's proof of God can be summarized in the ''[[Théodicée]]''.<ref name="Leibniz: A Guide for the Perplexed">{{cite book|last1=Perkins|first1=Franklin|title=Leibniz: A Guide for the Perplexed|publisher=Bloomsbury Academic|isbn=978-0-8264-8921-0|pages=22|date=2007-07-10}}</ref> Reason is governed by the [[principle of contradiction]] and the [[principle of sufficient reason]]. Using the principle of reasoning, Leibniz concluded that the first reason of all things is God.<ref name="Leibniz: A Guide for the Perplexed"/> All that we see and experience is subject to change, and the fact that this world is contingent can be explained by the possibility of the world being arranged differently in space and time. The contingent world must have some necessary reason for its existence. Leibniz uses a geometry book as an example to explain his reasoning. If this book was copied from an infinite chain of copies, there must be some reason for the content of the book.<ref>{{cite book|last1=Perkins|first1=Franklin|title=Leibniz: A Guide for the Perplexed|publisher=Bloomsbury Academic|isbn=978-0-8264-8921-0|page=23|date=2007-07-10}}</ref> Leibniz concluded that there must be the "''monas monadum''" or God.

The [[ontology|ontological]] essence of a monad is its irreducible simplicity. Unlike atoms, monads possess no material or spatial character. They also differ from atoms by their complete mutual independence, so that interactions among monads are only apparent. Instead, by virtue of the principle of [[pre-established harmony]], each monad follows a pre-programmed set of "instructions" peculiar to itself, so that a monad "knows" what to do at each moment. By virtue of these intrinsic instructions, each monad is like a little mirror of the universe. Monads need not be "small"; e.g., each human being constitutes a monad, in which case [[free will]] is problematic.

Monads are purported to have gotten rid of the problematic:
* interaction between [[mind]] and matter arising in the system of [[Descartes]];
* lack of [[Principle of individuation|individuation]] inherent to the system of [[Spinoza]], which represents individual creatures as merely accidental.

===Theodicy and optimism===
{{further|Best of all possible worlds|Philosophical optimism}}

The ''[[Théodicée|Theodicy]]''<ref>Rutherford (1998) is a detailed scholarly study of Leibniz's [[theodicy]].</ref> tries to justify the apparent imperfections of the world by claiming that it is [[Best of all possible worlds|optimal among all possible worlds]]. It must be the best possible and most balanced world, because it was created by an all powerful and all knowing God, who would not choose to create an imperfect world if a better world could be known to him or possible to exist. In effect, apparent flaws that can be identified in this world must exist in every possible world, because otherwise God would have chosen to create the world that excluded those flaws.<ref>{{cite journal |last1=Franklin |first1=James |date=2022 |title=The global/local distinction vindicates Leibniz's theodicy |journal=Theology and Science |volume=20 |issue=4 |pages= 445–462|doi=10.1080/14746700.2022.2124481|s2cid=252979403 |doi-access=free |hdl=1959.4/unsworks_80586 |hdl-access=free }}</ref>

Leibniz asserted that the truths of theology (religion) and philosophy cannot contradict each other, since reason and faith are both "gifts of God" so that their conflict would imply God contending against himself. The ''Theodicy'' is Leibniz's attempt to reconcile his personal philosophical system with his interpretation of the tenets of Christianity.<ref>Magill, Frank (ed.). ''Masterpieces of World Philosophy''. New York: Harper Collins (1990).</ref> This project was motivated in part by Leibniz's belief, shared by many philosophers and theologians during the [[Age of Enlightenment|Enlightenment]], in the rational and enlightened nature of the Christian religion. It was also shaped by Leibniz's belief in the perfectibility of human nature (if humanity relied on correct philosophy and religion as a guide), and by his belief that metaphysical necessity must have a rational or logical foundation, even if this metaphysical causality seemed inexplicable in terms of physical necessity (the natural laws identified by science).

In the view of Leibniz, because reason and faith must be entirely reconciled, any tenet of faith which could not be defended by reason must be rejected. Leibniz then approached one of the central criticisms of Christian theism:<ref>Magill, Frank (ed.) (1990)</ref> if God is [[Omnibenevolence|all good]], [[Omniscience|all wise]], and [[Omnipotence|all powerful]], then how did [[Problem of evil|evil come into the world]]? The answer (according to Leibniz) is that, while God is indeed unlimited in wisdom and power, his human creations, as creations, are limited both in their wisdom and in their will (power to act). This predisposes humans to false beliefs, wrong decisions, and ineffective actions in the exercise of their [[free will]]. God does not arbitrarily inflict pain and suffering on humans; rather he permits both ''moral evil'' (sin) and ''physical evil'' (pain and suffering) as the necessary consequences of ''metaphysical evil'' (imperfection), as a means by which humans can identify and correct their erroneous decisions, and as a contrast to true good.<ref>{{Cite book|title=The Golden Book About Leibniz|last=Anderson Csiszar|first=Sean|date=26 July 2015|isbn=978-1515243915|pages=20|publisher=CreateSpace Independent Publishing Platform }}</ref>

Further, although human actions flow from prior causes that ultimately arise in God and therefore are known to God as metaphysical certainties, an individual's free will is exercised within natural laws, where choices are merely contingently necessary and to be decided in the event by a "wonderful spontaneity" that provides individuals with an escape from rigorous predestination.

=== ''Discourse on Metaphysics'' ===
For Leibniz, "God is an absolutely perfect being". He describes this perfection later in section VI as the simplest form of something with the most substantial outcome (VI). Along these lines, he declares that every type of perfection "pertains to him (God) in the highest degree" (I). Even though his types of perfections are not specifically drawn out, Leibniz highlights the one thing that, to him, does certify imperfections and proves that God is perfect: "that one acts imperfectly if he acts with less perfection than he is capable of", and since God is a perfect being, he cannot act imperfectly (III). Because God cannot act imperfectly, the decisions he makes pertaining to the world must be perfect. Leibniz also comforts readers, stating that because he has done everything to the most perfect degree; those who love him cannot be injured. However, to love God is a subject of difficulty as Leibniz believes that we are "not disposed to wish for that which God desires" because we have the ability to alter our disposition (IV). In accordance with this, many act as rebels, but Leibniz says that the only way we can truly love God is by being content "with all that comes to us according to his will" (IV).

Because God is "an absolutely perfect being" (I), Leibniz argues that God would be acting imperfectly if he acted with any less perfection than what he is able of (III). His syllogism then ends with the statement that God has made the world perfectly in all ways. This also affects how we should view God and his will. Leibniz states that, in lieu of God's will, we have to understand that God "is the best of all masters" and he will know when his good succeeds, so we, therefore, must act in conformity to his good will—or as much of it as we understand (IV). In our view of God, Leibniz declares that we cannot admire the work solely because of the maker, lest we mar the glory and love God in doing so. Instead, we must admire the maker for the work he has done (II). Effectively, Leibniz states that if we say the earth is good because of the will of God, and not good according to some standards of goodness, then how can we praise God for what he has done if contrary actions are also praiseworthy by this definition (II). Leibniz then asserts that different principles and geometry cannot simply be from the will of God, but must follow from his understanding.<ref>Leibniz, Gottfried Wilhelm. ''Discourse on Metaphysics. The Rationalists: Rene Descartes – Discourse on Method, Meditations''. N.Y.: Dolphin., n.d., n.p.,</ref>

Leibniz wrote: "[[Problem of why there is anything at all|Why is there something rather than nothing?]] The sufficient reason ... is found in a substance which ... is a necessary being bearing the reason for its existence within itself."<ref>''[[Monadology|Monadologie]]'' (1714). [[Nicholas Rescher]], trans., 1991. ''The Monadology: An Edition for Students''. Uni. of Pittsburgh Press, p. 135.</ref> [[Martin Heidegger]] called this question "the fundamental question of metaphysics".<!--"Warum ist überhaupt Seiendes und nicht vielmehr Nichts?"--><ref>{{cite web|title=The Fundamental Question|url=https://www.hedweb.com/witherall/existence.htm|publisher=hedweb.com|access-date=26 April 2017}}</ref><ref>{{cite book|last1=Geier|first1=Manfred|title=Wittgenstein und Heidegger: Die letzten Philosophen|publisher=Rowohlt Verlag|isbn=978-3-644-04511-8|url=https://books.google.com/books?id=JUiFDQAAQBAJ&pg=PP166|access-date=26 April 2017|language=de|date=2017-02-17}}</ref>

===Symbolic thought and rational resolution of disputes===
Leibniz believed that much of human reasoning could be reduced to calculations of a sort, and that such calculations could resolve many differences of opinion:

{{blockquote|The only way to rectify our reasonings is to make them as tangible as those of the Mathematicians, so that we can find our error at a glance, and when there are disputes among persons, we can simply say: Let us calculate, without further ado, to see who is right.<ref>{{Cite SEP|url-id=leibniz-mind|title=Leibniz's Philosophy of Mind |date=June 29, 2020|edition=Winter 2020 |author-last1=Kulstad|author-first1= Mark |author-last2=Carlin |author-first2=Laurence}}</ref><ref>{{Cite web |last=Gray |first=Jonathan |title="Let us Calculate!": Leibniz, Llull, and the Computational Imagination |url=https://publicdomainreview.org/essay/let-us-calculate-leibniz-llull-and-the-computational-imagination/ |access-date=2023-06-22 |website=The Public Domain Review |language=en}}</ref><ref>''The Art of Discovery'' 1685, Wiener 51</ref>}}

Leibniz's [[calculus ratiocinator]], which resembles [[Mathematical logic|symbolic logic]], can be viewed as a way of making such calculations feasible. Leibniz wrote memoranda<ref>Many of his memoranda are translated in [[George Henry Radcliffe Parkinson|Parkinson]] 1966.</ref> that can now be read as groping attempts to get symbolic logic—and thus his ''calculus''—off the ground. These writings remained unpublished until the appearance of a selection edited by Carl Immanuel Gerhardt (1859). [[Louis Couturat]] published a selection in 1901; by this time the main developments of modern logic had been created by [[Charles Sanders Peirce]] and by [[Gottlob Frege]].

Leibniz thought [[symbol]]s were important for human understanding. He attached so much importance to the development of good notations that he attributed all his discoveries in mathematics to this. His notation for [[calculus]] is an example of his skill in this regard. Leibniz's passion for symbols and notation, as well as his belief that these are essential to a well-running logic and mathematics, made him a precursor of [[semiotics]].<ref>Marcelo Dascal, ''Leibniz. Language, Signs and Thought: A Collection of Essays'' (''Foundations of Semiotics'' series), John Benjamins Publishing Company, 1987, p. 42.</ref>

But Leibniz took his speculations much further. Defining a [[Grapheme|character]] as any written sign, he then defined a "real" character as one that represents an idea directly and not simply as the word embodying the idea. Some real characters, such as the notation of logic, serve only to facilitate reasoning. Many characters well known in his day, including [[Egyptian hieroglyphics]], [[Chinese character]]s, and the symbols of [[astronomy]] and [[chemistry]], he deemed not real.<ref>Loemker, however, who translated some of Leibniz's works into English, said that the symbols of chemistry were real characters, so there is disagreement among Leibniz scholars on this point.</ref><!--is this paragraph correct up to this point?--> Instead, he proposed the creation of a ''[[characteristica universalis]]'' or "universal characteristic", built on an [[alphabet of human thought]] in which each fundamental concept would be represented by a unique "real" character:

{{blockquote|It is obvious that if we could find characters or signs suited for expressing all our thoughts as clearly and as exactly as arithmetic expresses numbers or geometry expresses lines, we could do in all matters ''insofar as they are subject to reasoning'' all that we can do in arithmetic and geometry. For all investigations which depend on reasoning would be carried out by transposing these characters and by a species of calculus.<ref>''Preface to the General Science'', 1677. Revision of Rutherford's translation in Jolley 1995: 234. Also Wiener I.4</ref>}}

Complex thoughts would be represented by combining characters for simpler thoughts. Leibniz saw that the uniqueness of [[prime factorization]] suggests a central role for [[prime numbers]] in the universal characteristic, a striking anticipation of [[Gödel numbering]]. Granted, there is no intuitive or [[mnemonic]] way to number any set of elementary concepts using the prime numbers.

Because Leibniz was a mathematical novice when he first wrote about the ''characteristic'', at first he did not conceive it as an [[algebra]] but rather as a [[universal characteristic|universal language]] or script. Only in 1676 did he conceive of a kind of "algebra of thought", modeled on and including conventional algebra and its notation. The resulting ''characteristic'' included a logical calculus, some combinatorics, algebra, his ''analysis situs'' (geometry of situation), a universal concept language, and more. What Leibniz actually intended by his ''characteristica universalis'' and calculus ratiocinator, and the extent to which modern formal logic does justice to calculus, may never be established.<ref>A good introductory discussion of the "characteristic" is Jolley (1995: 226–240). An early, yet still classic, discussion of the "characteristic" and "calculus" is Couturat (1901: chpts. 3, 4).</ref> Leibniz's idea of reasoning through a universal language of symbols and calculations remarkably foreshadows great 20th-century developments in formal systems, such as [[Turing completeness]], where computation was used to define equivalent universal languages (see [[Turing degree]]).

===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>