Editing History of logic

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The '''history of logic''' deals with the study of the development of the science of valid [[inference]] ([[logic]]). Formal logics developed in ancient times in [[Indian logic|India]], [[Logic in China|China]], and [[Greek philosophy|Greece]]. Greek methods, particularly [[Aristotelian logic]] (or term logic) as found in the ''[[Organon]]'', found wide application and acceptance in Western science and mathematics for millennia.<ref name="Boehner p. xiv">Boehner p. xiv</ref> The [[Stoicism|Stoics]], especially [[Chrysippus]], began the development of [[predicate logic]].

[[Christian philosophy|Christian]] and [[Logic in Islamic philosophy|Islamic]] philosophers such as [[Boethius]] (died 524), [[Avicenna]] (died 1037), [[Thomas Aquinas]] (died 1274) and [[William of Ockham]] (died 1347) further developed Aristotle's logic in the [[Medieval philosophy#High Middle Ages|Middle Ages]], reaching a high point in the mid-fourteenth century, with [[Jean Buridan]]. The period between the fourteenth century and the beginning of the nineteenth century saw largely decline and neglect, and at least one historian of logic regards this time as barren.<ref name="ReferenceA">Oxford Companion p. 498; Bochenski, Part I Introduction, ''passim''</ref> [[Empirical methods]] ruled the day, as evidenced by Sir [[Francis Bacon]]'s ''[[Novum Organon]]'' of 1620.

Logic revived in the mid-nineteenth century, at the beginning of a revolutionary period when the subject developed into a rigorous and formal discipline which took as its exemplar the exact method of [[mathematical proof|proof]] used in [[mathematics]], a hearkening back to the Greek tradition.<ref>{{cite book |url=http://www.naturalthinker.net/trl/texts/Frege,Gottlob/Frege,%20Gottlob%20-%20The%20Foundations%20of%20Arithmetic%20%281953%29%202Ed_%207.0-2.5%20LotB.pdf |title=The Foundations of Arithmetic |author-first=Gottlob |author-last=Frege |page=1 |access-date=2016-02-03  |archive-date=2018-09-20  |archive-url=https://web.archive.org/web/20180920172024/http://www.naturalthinker.net/trl/texts/Frege,Gottlob/Frege,%20Gottlob%20-%20The%20Foundations%20of%20Arithmetic%20(1953)%202Ed_%207.0-2.5%20LotB.pdf |url-status=dead }}</ref> The development of the modern "symbolic" or "mathematical" logic during this period by the likes of [[George Boole|Boole]], [[Gottlob Frege|Frege]], [[Bertrand Russell|Russell]], and [[Giuseppe Peano|Peano]] is the most significant in the two-thousand-year history of logic, and is arguably one of the most important and remarkable events in human [[intellectual history]].<ref name="Oxford Companion p. 500">Oxford Companion p. 500</ref>

Progress in [[mathematical logic]] in the first few decades of the twentieth century, particularly arising from the work of [[Kurt Gödel|Gödel]] and [[Alfred Tarski|Tarski]], had a significant impact on [[analytic philosophy]] and [[philosophical logic]], particularly from the 1950s onwards, in subjects such as [[modal logic]], [[temporal logic]], [[deontic logic]], and [[relevance logic]].

==Logic in India==
{{Main|Indian logic}}

=== Hindu logic ===

==== Origin ====
The [[Nasadiya Sukta]] of the ''[[Rigveda]]'' ([[Mandala 10|RV 10]].129) contains [[ontological]] speculation in terms of various logical divisions that were later recast formally as the four circles of ''[[catuskoti]]'': "A", "not A", "A and 'not A{{'"}}, and "not A and not not A".
{{Rquote|right|"Who really knows? <br/>Who will here proclaim it? <br/>Whence was it produced? Whence is this creation? <br/>The gods came afterwards, with the creation of this universe. <br/>Who then knows whence it has arisen?"|[[Nasadiya Sukta]], concerns the [[origin of the universe]], [[Rig Veda]], ''10:129-6'' <ref name="Kramer1986">{{cite book |author-first=Kenneth |author-last=Kramer |title=World Scriptures: An Introduction to Comparative Religions |url=https://books.google.com/books?id=RzUAu-43W5oC&pg=PA34 |date=January 1986 |publisher=Paulist Press |isbn=978-0-8091-2781-8 |pages=34–}}</ref><ref name="Christian2011">{{cite book |author-first=David |author-last=Christian |title=Maps of Time: An Introduction to Big History |url=https://books.google.com/books?id=7RdVmDjwTtQC&pg=PA18|date=1 September 2011 |publisher=University of California Press |isbn=978-0-520-95067-2 |pages=18–}}</ref><ref name="Singh2008">{{cite book |author-first=Upinder |author-last=Singh |title=A History of Ancient and Early Medieval India: From the Stone Age to the 12th Century |url=https://books.google.com/books?id=H3lUIIYxWkEC&pg=PA206 |date=2008 |publisher=Pearson Education India |isbn=978-81-317-1120-0 |pages=206–}}</ref>}}
Logic began independently in [[ancient India]] and continued to develop to early modern times without any known influence from Greek logic.<ref>Bochenski p. 446</ref>

==== Before Gautama ====
Though the origins in India of public debate (''pariṣad''), one form of rational inquiry, are not clear, we know that public debates were common in preclassical India, for they are frequently alluded to in various ''[[Upanishads|Upaniṣads]]'' and in the early Buddhist literature. Public debate is not the only form of public deliberations in preclassical India. Assemblies (''pariṣad'' or ''[[sabhā]]'') of various sorts, comprising relevant experts, were regularly convened to deliberate on a variety of matters, including administrative, legal and religious matters.{{Citation needed|date=January 2025}}

==== Dattatreya ====
A philosopher named Dattatreya is stated in the [[Bhagavata Purana]] to have taught Anviksiki to Aiarka, Prahlada and others. It appears from the [[Markandeya Purana|Markandeya purana]] that the Anviksiki-vidya expounded by him consisted of a mere disquisition on soul in accordance with the yoga philosophy. Dattatreya expounded the philosophical side of Anviksiki and not its logical aspect.<ref>{{cite book |author-last=Vidyabhusana |author-first=S. C. |url=http://archive.org/details/in.ernet.dli.2015.213362 |title=History Of Indian Logic. |date=1921 |pages=11}}</ref><ref>{{cite book |author-first=Satis Chandra Vidya |author-last=Bhusana |url=http://archive.org/details/in.ernet.dli.2015.489008 |title=A History Of Indian Logic |date=1921}}</ref>

==== Medhatithi Gautama ====
While the teachers mentioned before dealt with some particular topics of Anviksiki, the credit of founding the Anviksiki in its special sense of a science is to be attributed to [[Indian logic|Medhatithi Gautama]] (c. 6th century BC). Guatama founded the ''[[anviksiki]]'' school of logic.<ref>S. C. Vidyabhusana (1971). ''A History of Indian Logic: Ancient, Mediaeval, and Modern Schools'', pp. 17–21.</ref> The ''[[Mahabharata]]'' (12.173.45), around the 5th century BC, refers to the ''anviksiki'' and ''tarka'' schools of logic.

==== Panini ====
{{IAST|[[Pāṇini]]}} (c. 5th century BC) developed a form of logic (to which [[Boolean logic]] has some similarities) for his formulation of [[Vyakarana|Sanskrit grammar]]. Logic is described by [[Chanakya]] (c. 350–283 BC) in his ''[[Arthashastra]]'' as an independent field of inquiry.<ref>R. P. Kangle (1986). ''The Kautiliya Arthashastra'' (1.2.11). Motilal Banarsidass.</ref>

==== Nyaya-Vaisheshika ====

Two of the six Indian schools of thought deal with logic: [[Nyaya]] and [[Vaisheshika]]. The [[Nyāya Sūtras]] of [[Aksapada Gautama]] (c. 2nd century AD) constitute the core texts of the Nyaya school, one of the six orthodox schools of [[Hindu]] philosophy. This [[Philosophical realism|realist]] school developed a rigid five-member schema of [[inference]] involving an initial premise, a reason, an example, an application, and a conclusion.<ref>Bochenski p. 417 and ''passim''</ref> The [[Idealism|idealist]] [[Buddhist philosophy]] became the chief opponent to the Naiyayikas.

=== Jain logic ===
[[File:उमास्वामी आचार्यजी.jpg|thumb|Umaswati (2nd century AD), author of first Jain work in Sanskrit, [[Tattvartha Sutra|Tattvārthasūtra]], expounding the [[Jain philosophy]] in a most systematized form acceptable to all sects of Jainism.]]
[[Jainism|Jains]] made its own unique contribution to this mainstream development of logic by also occupying itself with the basic epistemological issues, namely, with those concerning the nature of knowledge, how knowledge is derived, and in what way knowledge can be said to be reliable.

The Jains have doctrines of [[Relativism|relativity]] used for logic and reasoning:

* [[Anekantavada|Anekāntavāda]] – the theory of relative pluralism or manifoldness;
* [[Syadvada|Syādvāda]] – the theory of conditioned predication and;
* [[Anekantavada#Nayav%C4%81da|Nayavāda]] – The theory of partial standpoints.

These concepts in [[Jain philosophy]] made important contributions to the thought, especially in the areas of skepticism and relativity. [http://www.jainworld.com/jainbooks/firstep-2/indianjaina-1-2.htm]<ref>{{cite journal |author-last=Ganeri |author-first=Jonardon |title=Jaina Logic and the Philosophical Basis of Pluralism |url=https://www.academia.edu/2146233 |journal=History and Philosophy of Logic |date=2002 |language=en |volume=23 |issue=4 |pages=267–281 |doi=10.1080/0144534021000051505 |s2cid=170089234 |issn=0144-5340}}</ref>

=== Buddhist logic ===

==== Nagarjuna ====
[[Nagarjuna]] (c. 150–250 AD), the founder of the [[Madhyamaka]] ("Middle Way") developed an analysis known as the [[catuṣkoṭi]] (Sanskrit), a "four-cornered" system of argumentation that involves the systematic examination and rejection of each of the four possibilities of a proposition, ''P'':
# ''P''; that is, being.
# not&nbsp;''P''; that is, not being.
# [[File:Eight Patriarchs of the Shingon Sect of Buddhism Nagarjuna Cropped.jpg|thumb|Painting of Nāgārjuna from the ''Shingon Hassozō'', a series of scrolls authored by the [[Shingon]] school of Buddhism. Japan, [[Kamakura period]] (13th–14th century)]]''P'' and not&nbsp;''P''; that is, being and not being.
# not (''P'' or not&nbsp;''P''); that is, neither being nor not being.{{Paragraph break}}Under [[propositional logic]], [[De Morgan's laws]] would imply that the fourth case is equivalent to the third case, and would be therefore superfluous, with only 3 actual cases to consider.

=== Dignaga ===
However, [[Dignāga]] (c 480–540 AD) is sometimes said to have developed a formal syllogism,<ref>Bochenski pp. 431–437</ref> and it was through him and his successor, [[Dharmakirti]], that [[Buddhist logic]] reached its height; it is contested whether their analysis actually constitutes a formal syllogistic system. In particular, their analysis centered on the definition of an inference-warranting relation, "[[vyapti]]", also known as invariable concomitance or pervasion.<ref name="Matilal">{{cite book |author-last=Matilal |author-first=Bimal Krishna |title=The Character of Logic in India |date=1998 |publisher=State University of New York Press |location=Albany, New York, USA |isbn=9780791437407 |pages=12, 18}}</ref> To this end, a doctrine known as "apoha" or differentiation was developed.<ref>Bochenksi p. 441</ref> This involved what might be called inclusion and exclusion of defining properties.

Dignāga's famous "wheel of reason" (''[[Hetucakra]]'') is a method of indicating when one thing (such as smoke) can be taken as an invariable sign of another thing (like fire), but the inference is often inductive and based on past observation. Matilal remarks that Dignāga's analysis is much like John Stuart Mill's Joint Method of Agreement and Difference, which is inductive.<ref>Matilal, 17</ref>

==Logic in China==
{{Main|Logic in China}}
In China, a contemporary of [[Confucius]], [[Mozi]], "Master Mo", is credited with founding the [[Mohism|Mohist school]], whose canons dealt with issues relating to valid inference and the conditions of correct conclusions. In particular, one of the schools that grew out of Mohism, the [[School of Names|Logicians]], are credited by some scholars for their early investigation of [[formal logic]]. Due to the harsh rule of [[Legalism (Chinese philosophy)|Legalism]] in the subsequent [[Qin dynasty]], this line of investigation disappeared in China until the introduction of Indian philosophy by [[Buddhism|Buddhists]].

==Logic in the ancient Mediterranean==

===Prehistory of logic===
Valid reasoning has been employed in all periods of human history. However, logic studies the ''principles'' of valid reasoning, inference and demonstration. It is probable that the idea of demonstrating a conclusion first arose in connection with [[geometry]], which originally meant the same as "land measurement".<ref>Kneale, p. 2</ref> The [[ancient Egypt]]ians discovered [[Egyptian mathematics|geometry]], including the formula for the volume of a [[Frustum|truncated pyramid]].<ref name="Kneale3">Kneale p. 3</ref> [[Babylonian mathematics|Ancient Babylon]] was also skilled in mathematics. [[Esagil-kin-apli]]'s medical ''Diagnostic Handbook'' in the 11th century BC was based on a logical set of [[axiom]]s and assumptions,<ref name="Stol-99">H. F. J. Horstmanshoff, Marten Stol, Cornelis Tilburg (2004), ''Magic and Rationality in Ancient Near Eastern and Graeco-Roman Medicine'', p. 99, [[Brill Publishers]], {{ISBN|90-04-13666-5}}.</ref> while [[Babylonian astronomy|Babylonian astronomers]] in the 8th and 7th centuries BC employed an [[internal logic]] within their predictive planetary systems, an important contribution to the [[philosophy of science]].<ref name="Brown">D. Brown (2000), ''Mesopotamian Planetary Astronomy-Astrology '', Styx Publications, {{ISBN|90-5693-036-2}}.</ref>

===Ancient Greece before Aristotle===
While the ancient Egyptians empirically discovered some truths of geometry, the great achievement of the ancient Greeks was to replace empirical methods by demonstrative [[Mathematical proof|proof]]. Both [[Thales]] and [[Pythagoras]] of the [[Pre-Socratic philosophers]] seemed aware of geometric methods.

Fragments of early proofs are preserved in the works of Plato and Aristotle,<ref>Heath, ''Mathematics in Aristotle'', cited in Kneale, p. 5</ref> and the idea of a deductive system was probably known in the Pythagorean school and the [[Platonic Academy]].<ref name="Kneale3"/> The proofs of [[Euclid of Alexandria]] are a paradigm of Greek geometry. The three basic principles of geometry are as follows:

* Certain propositions must be accepted as true without demonstration; such a proposition is known as an [[axiom]] of geometry.
* Every proposition that is not an axiom of geometry must be demonstrated as following from the axioms of geometry; such a demonstration is known as a [[Mathematical proof|proof]] or a "derivation" of the proposition.
* The proof must be ''formal''; that is, the derivation of the proposition must be independent of the particular subject matter in question.<ref name="Kneale3"/>
Further evidence that early Greek thinkers were concerned with the principles of reasoning is found in the fragment called ''[[dissoi logoi]]'', probably written at the beginning of the fourth century BC. This is part of a protracted debate about truth and falsity.<ref>Kneale, p. 16</ref> In the case of the classical Greek city-states, interest in argumentation was also stimulated by the activities of the [[Rhetoric]]ians or Orators and the [[Sophists]], who used arguments to defend or attack a thesis, both in legal and political contexts.<ref>{{cite web |url=http://www.britannica.com/EBchecked/topic/346217/history-of-logic#toc65918 |title=History of logic |website=britannica.com |access-date=2 April 2018}}</ref>
[[File:Thales' Theorem.svg|thumb|130px|left|Thales Theorem]]

====Thales====
It is said Thales, most widely regarded as the first philosopher in the [[Greek philosophy|Greek tradition]],<ref>[[Aristotle]], Metaphysics Alpha, 983b18.</ref><ref name="CPM">{{cite book |author-last=Smith |author-first=William |title=Dictionary of Greek and Roman biography and mythology |date=1870 |url=https://archive.org/stream/dictionaryofgree03smituoft#page/1016 |page=1016 |publisher=Boston, Little}}</ref> measured the height of the [[pyramids]] by their shadows at the moment when his own shadow was equal to his height. Thales was said to have had a sacrifice in celebration of discovering [[Thales' theorem]] just as Pythagoras had the [[Pythagorean theorem]].<ref>T. Patronis & D. Patsopoulos {{cite book |url=http://journals.tc-library.org/index.php/hist_math_ed/article/viewFile/189/184 |title=The Theorem of Thales: A Study of the naming of theorems in school Geometry textbooks |publisher=[[Patras University]] |access-date=2012-02-12 |archive-url=https://web.archive.org/web/20160303171258/http://journals.tc-library.org/index.php/hist_math_ed/article/viewFile/189/184 |archive-date=2016-03-03 |url-status=usurped}}</ref>

Thales is the first known individual to use [[deductive reasoning]] applied to geometry, by deriving four corollaries to his theorem, and the first known individual to whom a mathematical discovery has been attributed.<ref>{{harv|Boyer|1991|loc="Ionia and the Pythagoreans" p. 43}}</ref> [[Indian mathematics|Indian]] and Babylonian mathematicians knew his theorem for special cases before he proved it.<ref>de Laet, Siegfried J. (1996). ''History of Humanity: Scientific and Cultural Development''. [[UNESCO]], Volume 3, p. 14. {{ISBN|92-3-102812-X}}</ref> It is believed that Thales learned that an angle inscribed in a [[semicircle]] is a right angle during his travels to [[Babylon]].<ref>Boyer, Carl B. and [[Uta Merzbach|Merzbach, Uta C.]] (2010). ''A History of Mathematics''. John Wiley and Sons, Chapter IV. {{ISBN|0-470-63056-6}}</ref>

====Pythagoras====
[[File:Illustration to Euclid's proof of the Pythagorean theorem.svg|thumb|180px|Proof of the Pythagorean Theorem in Euclid's ''Elements'']]
Before 520&nbsp;BC, on one of his visits to Egypt or Greece, Pythagoras might have met the c.&nbsp;54 years older Thales.<ref>C. B. Boyer (1968)</ref> The systematic study of proof seems to have begun with the school of Pythagoras (i. e. the Pythagoreans) in the late sixth century BC.<ref name="Kneale3"/> Indeed, the Pythagoreans, believing all was number, are the first philosophers to emphasize ''form'' rather than ''matter''.<ref>{{cite book |page=11 |author=Samuel Enoch Stumpf |title=Socrates to Sartre}}</ref>

====Heraclitus and Parmenides====
The writing of [[Heraclitus]] (c. 535 – c. 475 BC) was the first place where the word ''[[logos]]'' was given special attention in ancient Greek philosophy,<ref>F.E. Peters, ''Greek Philosophical Terms'', New York University Press, 1967.</ref> Heraclitus held that everything changes and all was fire and conflicting opposites, seemingly unified only by this ''Logos''. He is known for his obscure sayings.
{{blockquote|This ''logos'' holds always but humans always prove unable to understand it, both before hearing it and when they have first heard it. For though all things come to be in accordance with this ''logos'', humans are like the inexperienced when they experience such words and deeds as I set out, distinguishing each in accordance with its nature and saying how it is. But other people fail to notice what they do when awake, just as they forget what they do while asleep.|[[Diels-Kranz]]|22B1}}
[[File:Busto di Parmenide (cropped).jpg|thumb|160px|Parmenides has been called the discoverer of logic.]]

In contrast to Heraclitus, [[Parmenides]] held that all is one and nothing changes. He may have been a dissident Pythagorean, disagreeing that One (a number) produced the many.<ref>{{cite book |author-last=Cornford |author-first=Francis MacDonald |url=https://www.bard.edu/library/arendt/pdfs/Cornford-Parmenides.pdf |title=Plato and Parmenides: Parmenides' ''Way of Truth'' and Plato's ''Parmenides'' translated with an introduction and running commentary |publisher=Liberal Arts Press |date=1957 |orig-date=1939}}</ref> "X is not" must always be false or meaningless. What exists can in no way not exist. Our sense perceptions with its noticing of generation and destruction are in grievous error. Instead of sense perception, Parmenides advocated ''logos'' as the means to Truth. He has been called the discoverer of logic,<ref>{{cite book |title=Western Philosophy: an introduction |author=R. J. Hollingdale |date=1974 |page=73}}</ref><ref>{{cite book |author-last=Cornford |author-first=Francis MacDonald |url=https://www.wilbourhall.org/pdfs/From_religion_to_philosophy.pdf |title=From religion to philosophy: A study in the origins of western speculation |publisher=Longmans, Green and Co. |date=1912}}</ref>

{{blockquote|For this view, that That Which Is Not exists, can never predominate. You must debar your thought from this way of search, nor let ordinary experience in its variety force you along this way, (namely, that of allowing) the eye, sightless as it is, and the ear, full of sound, and the tongue, to rule; but (you must) judge by means of the Reason ([[Logos]]) the much-contested proof which is expounded by me.|B 7.1–8.2}}

[[Zeno of Elea]], a pupil of Parmenides, had the idea of a standard argument pattern found in the method of proof known as ''[[reductio ad absurdum]]''. This is the technique of drawing an obviously false (that is, "absurd") conclusion from an assumption, thus demonstrating that the assumption is false.<ref>Kneale p. 15</ref> Therefore, Zeno and his teacher are seen as the first to apply the art of logic.<ref>{{cite web |url=https://books.google.com/books?id=DPoqAAAAMAAJ&pg=PA170 |title=The Numismatic Circular |date=2 April 2018 |access-date=2 April 2018 |via=Google Books |last1=Son |first1=Spink }}</ref> Plato's dialogue [[Parmenides (dialogue)|Parmenides]] portrays Zeno as claiming to have written a book defending the [[monism]] of Parmenides by demonstrating the absurd consequence of assuming that there is plurality. Zeno famously used this method to develop his [[Zeno's Paradoxes|paradoxes]] in his arguments against motion. Such ''dialectic'' reasoning later became popular. The members of this school were called "dialecticians" (from a Greek word meaning "to discuss").

====Plato====
{{blockquote|Let no one ignorant of geometry enter here.|Inscribed over the entrance to Plato's Academy.}}
[[File:MANNapoli 124545 plato's academy mosaic.jpg|alt=Mosaic: seven men standing under a tree|thumb|200px|[[Plato's Academy mosaic]]]]
None of the surviving works of the great fourth-century philosopher [[Plato]] (428–347 BC) include any formal logic,<ref>Kneale p. 17</ref> but they include important contributions to the field of [[philosophical logic]]. Plato raises three questions:

* What is it that can properly be called true or false?
* What is the nature of the connection between the assumptions of a valid argument and its conclusion?
* What is the nature of definition?

The first question arises in the dialogue ''[[Theaetetus (dialogue)|Theaetetus]]'', where Plato identifies thought or opinion with talk or discourse (''logos'').<ref>"forming an opinion is talking, and opinion is speech that is held not with someone else or aloud but in silence with oneself" ''Theaetetus'' 189E–190A</ref> The second question is a result of Plato's [[theory of Forms]]. Forms are not things in the ordinary sense, nor strictly ideas in the mind, but they correspond to what philosophers later called [[universals]], namely an abstract entity common to each set of things that have the same name. In both the ''[[The Republic (Plato)|Republic]]'' and the ''[[Sophist (dialogue)|Sophist]]'', Plato suggests that the necessary connection between the assumptions of a valid argument and its conclusion corresponds to a necessary connection between "forms".<ref>Kneale p. 20.  For example, the proof given in the ''Meno'' that the square on the diagonal is double the area of the original square presumably involves the forms of the square and the triangle, and the necessary relation between them</ref> The third question is about [[definition]]. Many of Plato's dialogues concern the search for a definition of some important concept (justice, truth, the Good), and it is likely that Plato was impressed by the importance of definition in mathematics.<ref>Kneale p. 21</ref> What underlies every definition is a Platonic Form, the common nature present in different particular things. Thus, a definition reflects the ultimate object of understanding, and is the foundation of all valid inference. This had a great influence on Plato's student [[Aristotle]], in particular Aristotle's notion of the [[essence]] of a thing.<ref>Zalta, Edward N. "[http://plato.stanford.edu/entries/aristotle-logic/#Def Aristotle's Logic]". [[Stanford University]], 18 March 2000. Retrieved 13 March 2010.</ref>

===Aristotle===
{{Main article|Term logic}}
[[File:Aristotle Altemps Inv8575.jpg|thumb|160px|Aristotle]]
The logic of [[Aristotle]], and particularly his theory of the [[syllogism]], has had an enormous influence in [[Western thought]].<ref>See e.g. [http://plato.stanford.edu/entries/aristotle-logic/ Aristotle's logic], Stanford Encyclopedia of Philosophy</ref> Aristotle was the first logician to attempt a systematic analysis of [[logical syntax]], of noun (or ''[[terminology|term]]''), and of verb. He was the first ''formal logician'', in that he demonstrated the principles of reasoning by employing variables to show the underlying [[logical form]] of an argument.<ref>{{cite book |author-last=Sowa |author-first=John F. |title=Knowledge representation: logical, philosophical, and computational foundations |date=2000 |publisher=Brooks/Cole |isbn=0-534-94965-7 |location=Pacific Grove |pages=2 |oclc=38239202}}</ref> He sought relations of dependence which characterize necessary inference, and distinguished the [[Validity (logic)|validity]] of these relations, from the truth of the premises. He was the first to deal with the principles of [[Principle of contradiction|contradiction]] and [[Law of excluded middle|excluded middle]] in a systematic way.<ref name="Bochenski p. 63">Bochenski p. 63</ref> 
[[File:Aristoteles Logica 1570 Biblioteca Huelva.jpg|alt=Front cover of book, titled "Aristotelis Logica", with an illustration of eagle on a snake|240px|thumb|left|Aristotle's logic was still influential in the [[Renaissance]].]]

====The Organon====
His logical works, called the ''[[Organon]]'', are the earliest formal study of logic that have come down to modern times. Though it is difficult to determine the dates, the probable order of writing of Aristotle's logical works is:
* ''[[Categories (Aristotle)|The Categories]]'', a study of the ten kinds of primitive term.
* ''[[Topics (Aristotle)|The Topics]]'' (with an appendix called ''[[On Sophistical Refutations]]''), a discussion of dialectics.
* ''[[De Interpretatione|On Interpretation]]'', an analysis of simple [[categorical proposition]]s into simple terms, negation, and signs of quantity. 
* ''[[Prior Analytics|The Prior Analytics]]'', a formal analysis of what makes a [[syllogism]] (a valid argument, according to Aristotle).
* ''[[Posterior Analytics|The Posterior Analytics]]'', a study of scientific demonstration, containing Aristotle's mature views on logic.
[[File:Square of opposition, set diagrams.svg|thumb|180px|This diagram shows the contradictory relationships between [[categorical proposition]]s in the [[square of opposition]] of [[Term logic|Aristotelian logic]].]]
These works are of outstanding importance in the history of logic. In the ''Categories'', he attempts to discern all the possible things to which a term can refer; this idea underpins his philosophical work ''[[Metaphysics (Aristotle)|Metaphysics]]'', which itself had a profound influence on Western thought.

He also developed a theory of non-formal logic (''i.e.,'' the theory of [[logical fallacy|fallacies]]), which is presented in ''Topics'' and ''Sophistical Refutations''.<ref name="Bochenski p. 63"/>

''On Interpretation'' contains a comprehensive treatment of the notions of [[Square of opposition|opposition]] and conversion; chapter 7 is at the origin of the [[square of opposition]] (or logical square); chapter 9 contains the beginning of [[modal logic]].

The ''Prior Analytics'' contains his exposition of the "syllogism", where three important principles are applied for the first time in history: the use of variables, a purely formal treatment, and the use of an axiomatic system.

===Stoics===
{{Main|Stoic logic}}
The other great school of Greek logic is that of the [[Stoicism|Stoics]].<ref>"Throughout later antiquity two great schools of logic were distinguished, the Peripatetic which was derived from Aristotle, and the Stoic which was developed by Chrysippus from the teachings of the Megarians" – Kneale p. 113</ref> Stoic logic traces its roots back to the late 5th century BC philosopher [[Euclid of Megara]], a pupil of [[Socrates]] and slightly older contemporary of Plato, probably following in the tradition of Parmenides and Zeno. His pupils and successors were called "[[Megarian school|Megarians]]", or "Eristics", and later the "Dialecticians". The two most important dialecticians of the Megarian school were [[Diodorus Cronus]] and [[Philo the Dialectician|Philo]], who were active in the late 4th century BC. 
[[File:Chrysippos BM 1846.jpg|alt=Stone bust of a bearded, grave-looking man|thumb|160px|[[Chrysippus]] of Soli]]
The Stoics adopted the Megarian logic and systemized it. The most important member of the school was [[Chrysippus]] (c. 278 – c. 206 BC), who was its third head, and who formalized much of Stoic doctrine. He is supposed to have written over 700 works, including at least 300 on logic, almost none of which survive.<ref>''Oxford Companion'', article "Chrysippus", p. 134</ref><ref>[http://plato.stanford.edu/entries/logic-ancient/] Stanford Encyclopedia of Philosophy: [[Susanne Bobzien]], ''Ancient Logic''</ref> Unlike with Aristotle, we have no complete works by the Megarians or the early Stoics, and have to rely mostly on accounts (sometimes hostile) by later sources, including prominently [[Diogenes Laërtius]], [[Sextus Empiricus]], [[Galen]], [[Aulus Gellius]], [[Alexander of Aphrodisias]], and [[Cicero]].<ref>K. Hülser, Die Fragmente zur Dialektik der Stoiker, 4 vols, Stuttgart 1986–1987</ref>

Three significant contributions of the Stoic school were (i) their account of [[Modal logic|modality]], (ii) their theory of the [[Material conditional]], and (iii) their account of [[Meaning (philosophy of language)|meaning]] and [[truth]].<ref>Kneale 117–158</ref>

* ''Modality''. According to Aristotle, the Megarians of his day claimed there was no distinction between [[Potentiality and actuality (Aristotle)|potentiality and actuality]].<ref>''Metaphysics'' Eta 3, 1046b 29</ref> Diodorus Cronus defined the possible as that which either is or will be, the impossible as what will not be true, and the contingent as that which either is already, or will be false.<ref>[[Boethius]], ''Commentary on the Perihermenias'', Meiser p. 234</ref> Diodorus is also famous for what is known as his [[Master argument (Diodorus Cronus)|Master argument]], which states that each pair of the following 3 propositions contradicts the third proposition:
:* Everything that is past is true and necessary.
:* The impossible does not follow from the possible.
:* What neither is nor will be is possible.
: Diodorus used the plausibility of the first two to prove that nothing is possible if it neither is nor will be true.<ref>[[Epictetus]], ''Dissertationes'' ed. Schenkel ii. 19. I.</ref> Chrysippus, by contrast, denied the second premise and said that the impossible could follow from the possible.<ref>Alexander p. 177</ref>

* ''Conditional statements''. The first logicians to debate [[Material conditional|conditional statements]] were Diodorus and his pupil Philo of Megara. Sextus Empiricus refers three times to a debate between Diodorus and Philo. Philo regarded a conditional as true unless it has both a true [[antecedent (logic)|antecedent]] and a false [[consequent]]. Precisely, let ''T<sub>0</sub>'' and ''T<sub>1</sub>'' be true statements, and let ''F<sub>0</sub>'' and ''F<sub>1</sub>'' be false statements; then, according to Philo, each of the following conditionals is a true statement, because it is not the case that the consequent is false while the antecedent is true (it is not the case that a false statement is asserted to follow from a true statement):
:* If ''T<sub>0</sub>'', then ''T<sub>1</sub>''
:* If ''F<sub>0</sub>'', then ''T<sub>0</sub>''
:* If ''F<sub>0</sub>'', then ''F<sub>1</sub>''
: The following conditional does not meet this requirement, and is therefore a false statement according to Philo:
:* If ''T<sub>0</sub>'', then ''F<sub>0</sub>''
: Indeed, Sextus says "According to &#91;Philo&#93;, there are three ways in which a conditional may be true, and one in which it may be false."<ref name="sextus-adv-math">Sextus Empiricus, ''Adv. Math.'' viii, Section 113</ref> Philo's criterion of truth is what would now be called a [[truth-functional]] definition of "if ... then"; it is the definition used in [[predicate logic|modern logic]].

:In contrast, Diodorus allowed the validity of conditionals only when the antecedent clause could never lead to an untrue conclusion.<ref name="sextus-adv-math"/><ref>Sextus Empiricus, ''Hypotyp.'' ii. 110, comp.</ref><ref>Cicero, ''Academica'', ii. 47, ''de Fato'', 6.</ref>
* ''Meaning and truth''. The most important and striking difference between Megarian-Stoic logic and Aristotelian logic is that Megarian-Stoic logic concerns propositions, not terms, and is thus closer to modern [[propositional logic]].<ref>See e.g. Lukasiewicz p. 21</ref> The Stoics distinguished between utterance (''phone''), which may be noise, speech (''lexis''), which is articulate but which may be meaningless, and discourse (''logos''), which is meaningful utterance. The most original part of their theory is the idea that what is expressed by a sentence, called a ''lekton'', is something real; this corresponds to what is now called a ''proposition''. Sextus says that according to the Stoics, three things are linked together: that which signifies, that which is signified, and the object; for example, that which signifies is the word ''Dion'', and that which is signified is what Greeks understand but barbarians do not, and the object is Dion himself.<ref>Sextus Bk viii., Sections 11, 12</ref>

==Medieval logic==

===Logic in the Middle East===
{{Main|Logic in Islamic philosophy}}
{{See also|Avicennism#Avicennian logic|l1=Avicennian logic}}
[[File:Canon-Avicenna-small.jpg|alt=Arabic text in pink and blue|thumb|A text by [[Avicenna]], founder of [[Avicennism#Avicennian logic|Avicennian logic]] ]]

The works of [[Al-Kindi]], [[Al-Farabi]], [[Avicenna]], [[Al-Ghazali]], [[Averroes]] and other Muslim logicians were based on Aristotelian logic and were important in communicating the ideas of the ancient world to the medieval West.<ref>See e.g. [http://www.rep.routledge.com/article/H057 Routledge Encyclopedia of Philosophy Online Version 2.0]  {{webarchive |url=https://web.archive.org/web/20220606082214/https://www.rep.routledge.com/articles/islamic-philosophy;jsessionid=B31B033F077DD5E68E09CC9D35C02105 |date=2022-06-06}}, article 'Islamic philosophy'</ref> [[Al-Farabi]] (Alfarabi) (873–950) was an Aristotelian logician who discussed the topics of [[future contingent]]s, the number and relation of the categories, the relation between [[logic]] and [[grammar]], and non-Aristotelian forms of [[inference]].<ref name="Britannica"/> Al-Farabi also considered the theories of [[conditional syllogism]]s and [[Analogy|analogical inference]], which were part of the [[Stoicism|Stoic]] tradition of logic rather than the Aristotelian.<ref>{{cite journal |issn=0022-362X |volume=61 |issue=22 |pages=724–734 |author-last=Feldman |author-first=Seymour |title=Rescher on Arabic Logic |journal=The Journal of Philosophy |date=1964-11-26 |jstor=2023632 |doi=10.2307/2023632 |publisher=Journal of Philosophy, Inc.}} [726]. {{cite book |publisher=Cambridge University Press |isbn=0-521-27556-3 |author-last1=Long |author-first1=A. A. |author-first2=D. N. |author-last2=Sedley |title=The Hellenistic Philosophers. Vol 1: Translations of the principal sources with philosophical commentary |location=Cambridge |date=1987}}</ref>

[[Maimonides]] (1138-1204) wrote a ''Treatise on Logic'' (Arabic: ''Maqala Fi-Sinat Al-Mantiq''), referring to Al-Farabi as the "second master", the first being Aristotle.

[[Avicenna|Ibn Sina]] (Avicenna) (980–1037) was the founder of [[Avicennian logic]], which replaced Aristotelian logic as the dominant system of logic in the Islamic world,<ref name="Hasse">{{cite encyclopedia |author-first=Dag Nikolaus |author-last=Hasse |title=Influence of Arabic and Islamic Philosophy on the Latin West |encyclopedia=[[Stanford Encyclopedia of Philosophy]] |date=19 September 2008 |url=http://plato.stanford.edu/entries/arabic-islamic-influence/ |access-date=2009-10-13}}</ref> and also had an important influence on Western medieval writers such as [[Albertus Magnus]].<ref>Richard F. Washell (1973), "Logic, Language, and Albert the Great", ''Journal of the History of Ideas'' '''34''' (3), pp. 445–450 [445].</ref> Avicenna wrote on the [[hypothetical syllogism]]<ref name="Goodman"/> and on the [[propositional calculus]], which were both part of the Stoic logical tradition.<ref>Goodman, Lenn Evan (1992); ''Avicenna'', p. 188, [[Routledge]], {{ISBN|0-415-01929-X}}.</ref> He developed an original "temporally modalized" syllogistic theory, involving [[temporal logic]] and [[modal logic]].<ref name="Britannica">[http://www.britannica.com/ebc/article-65928 History of logic: Arabic logic], ''[[Encyclopædia Britannica]]''.</ref> He also made use of [[inductive reasoning|inductive logic]], such as the [[Mill's Methods|methods of agreement, difference, and concomitant variation]] which are critical to the [[scientific method]].<ref name="Goodman">Goodman, Lenn Evan (2003), ''Islamic Humanism'', p. 155, [[Oxford University Press]], {{ISBN|0-19-513580-6}}.</ref> One of Avicenna's ideas had a particularly important influence on Western logicians such as [[William of Ockham]]: Avicenna's word for a meaning or notion (''ma'na''), was translated by the scholastic logicians as the Latin ''intentio''; in medieval logic and [[epistemology]], this is a sign in the mind that naturally represents a thing.<ref>Kneale p. 229</ref> This was crucial to the development of Ockham's [[conceptualism]]: A universal term (''e.g.,'' "man") does not signify a thing existing in reality, but rather a sign in the mind (''intentio in intellectu'') which represents many things in reality; Ockham cites Avicenna's commentary on ''Metaphysics'' V in support of this view.<ref>Kneale: p. 266; Ockham: [[Summa Logicae]] i. 14; Avicenna: ''Avicennae Opera'' Venice 1508 f87rb</ref>

[[Fakhr al-Din al-Razi]] (b. 1149) criticised Aristotle's "[[Syllogism|first figure]]" and formulated an early system of inductive logic, foreshadowing the system of inductive logic developed by [[John Stuart Mill]] (1806–1873).<ref name="Iqbal">[[Muhammad Iqbal]], ''[[The Reconstruction of Religious Thought in Islam]]'', "The Spirit of Muslim Culture" ([[cf.]] [http://www.allamaiqbal.com/works/prose/english/reconstruction] and [http://www.witness-pioneer.org/vil/Books/MI_RRTI/chapter_05.htm])</ref> Al-Razi's work was seen by later Islamic scholars as marking a new direction for Islamic logic, towards a [[Logic in Islamic philosophy#Post-Avicennian logic|Post-Avicennian logic]]. This was further elaborated by his student Afdaladdîn al-Khûnajî (d. 1249), who developed a form of logic revolving around the subject matter of [[concept]]ions and [[Grammar of Assent|assents]]. In response to this tradition, [[Nasir al-Din al-Tusi]] (1201–1274) began a tradition of Neo-Avicennian logic which remained faithful to Avicenna's work and existed as an alternative to the more dominant Post-Avicennian school over the following centuries.<ref name="Stanford"/>

The [[Illuminationist philosophy|Illuminationist school]] was founded by [[Shahab al-Din Suhrawardi]] (1155–1191), who developed the idea of "decisive necessity", which refers to the reduction of all modalities (necessity, [[Logical possibility|possibility]], [[Contingency (philosophy)|contingency]] and [[Epistemic possibility|impossibility]]) to the single mode of necessity.<ref>Lotfollah Nabavi, [http://public.ut.ac.ir/html/fac/lit/articles.html Sohrevardi's Theory of Decisive Necessity and kripke's QSS System] {{webarchive|url=https://web.archive.org/web/20080126100838/http://public.ut.ac.ir/html/fac/lit/articles.html |date=2008-01-26 }}, ''Journal of Faculty of Literature and Human Sciences''.</ref> [[Ibn al-Nafis]] (1213–1288) wrote a book on Avicennian logic, which was a commentary of Avicenna's ''Al-Isharat'' (''The Signs'') and ''Al-Hidayah'' (''The Guidance'').<ref name="Roubi">Abu Shadi Al-Roubi (1982), "Ibn Al-Nafis as a philosopher", ''Symposium on Ibn al-Nafis'', Second International Conference on Islamic Medicine: Islamic Medical Organization, Kuwait ([[cf.]] [http://www.islamset.com/isc/nafis/drroubi.html Ibn al-Nafis As a Philosopher] {{webarchive |url=https://web.archive.org/web/20080206072116/http://www.islamset.com/isc/nafis/drroubi.html |date=2008-02-06}}, ''Encyclopedia of Islamic World'').</ref> [[Ibn Taymiyyah]] (1263–1328), wrote the ''Ar-Radd 'ala al-Mantiqiyyin'', where he argued against the usefulness, though not the validity, of the [[syllogism]]<ref>See pp.&nbsp;253–254 of {{cite book |publisher=Cambridge University Press |isbn=978-0-521-52069-0 |pages=247–265 |editor1=Peter Adamson |editor2=Richard C. Taylor |author-last=Street |author-first=Tony |title=The Cambridge Companion to Arabic Philosophy |chapter=Logic |date=2005}}</ref> and in favour of [[inductive reasoning]].<ref name="Iqbal"/> Ibn Taymiyyah also argued against the certainty of [[syllogism|syllogistic arguments]] and in favour of [[analogy]]; his argument is that concepts founded on [[inductive reasoning|induction]] are themselves not certain but only probable, and thus a syllogism based on such concepts is no more certain than an argument based on analogy. He further claimed that induction itself is founded on a process of analogy. His model of analogical reasoning was based on that of juridical arguments.<ref>{{cite journal |author=Ruth Mas |title=Qiyas: A Study in Islamic Logic |journal=Folia Orientalia |volume=34 |pages=113–128 |date=1998 |url=http://www.colorado.edu/ReligiousStudies/faculty/mas/LOGIC.pdf |issn=0015-5675}}</ref><ref name="Sowa">{{cite conference |author1=John F. Sowa |author2=Arun K. Majumdar |title=Analogical reasoning |book-title=Conceptual Structures for Knowledge Creation and Communication, Proceedings of ICCS 2003 |publisher=Springer-Verlag |date=2003 |location=Berlin |url=http://www.jfsowa.com/pubs/analog.htm |author-link1=John F. Sowa}}, pp. 16–36</ref> This model of analogy has been used in the recent work of [[John F. Sowa]].<ref name="Sowa"/>

The ''Sharh al-takmil fi'l-mantiq'' written by Muhammad ibn Fayd Allah ibn Muhammad Amin al-Sharwani in the 15th century is the last major Arabic work on logic that has been studied.<ref>[[Nicholas Rescher]] and Arnold vander Nat, "The Arabic Theory of Temporal Modal Syllogistic", in George Fadlo Hourani (1975), ''Essays on Islamic Philosophy and Science'', pp. 189–221, [[State University of New York Press]], {{ISBN|0-87395-224-3}}.</ref> However, "thousands upon thousands of pages" on logic were written between the 14th and 19th centuries, though only a fraction of the texts written during this period have been studied by historians, hence little is known about the original work on Islamic logic produced during this later period.<ref name="Stanford">{{cite encyclopedia |author=Tony Street |title=Arabic and Islamic Philosophy of Language and Logic |encyclopedia=[[Stanford Encyclopedia of Philosophy]] |date=23 July 2008 |url=http://plato.stanford.edu/entries/arabic-islamic-language |access-date=2008-12-05}}</ref>

===Logic in medieval Europe===
[[File:Britoquestionsonoldlogic.jpg|alt=Top left corner of early printed text, with an illuminated S, beginning "Sicut dicit philosophus"|thumb|[[Radulphus Brito|Brito's]] questions on the ''Old Logic'']]

"Medieval logic" (also known as "Scholastic logic") generally means the form of Aristotelian logic developed in [[Middle Ages|medieval Europe]] throughout roughly the period 1200–1600.<ref name="Boehner p. xiv"/> For centuries after Stoic logic had been formulated, it was the dominant system of logic in the classical world. When the study of logic resumed after the [[Dark Ages (historiography)|Dark Ages]], the main source was the work of the Christian philosopher [[Boethius]], who was familiar with some of Aristotle's logic, but almost none of the work of the Stoics.<ref name="Kneale198">Kneale p. 198</ref> Until the twelfth century, the only works of Aristotle available in the West were the ''Categories'', ''On Interpretation'', and Boethius's translation of the [[Isagoge]] of [[Porphyry (philosopher)|Porphyry]] (a commentary on the Categories). These works were known as the "Old Logic" (''Logica Vetus'' or ''Ars Vetus''). An important work in this tradition was the ''Logica Ingredientibus'' of [[Peter Abelard]] (1079–1142). His direct influence was small,<ref>Stephen Dumont, article "Peter Abelard" in Gracia and Noone p. 492</ref> but his influence through pupils such as [[John of Salisbury]] was great, and his method of applying rigorous logical analysis to theology shaped the way that theological criticism developed in the period that followed.<ref>Kneale, pp. 202–203</ref> The proof for the [[principle of explosion]], also known as the principle of Pseudo-Scotus, the law according to which any proposition can be proven from a contradiction (including its negation), was first given by the 12th century French logician [[William of Soissons]].

By the early thirteenth century, the remaining works of Aristotle's ''Organon'', including the ''[[Prior Analytics]]'', ''[[Posterior Analytics]]'', and the ''[[Sophistical Refutations]]'' (collectively known as the ''[[Logica Nova]]'' or "New Logic"), had been recovered in the West.<ref>See e.g. Kneale p. 225</ref> Logical work until then was mostly paraphrasis or commentary on the work of Aristotle.<ref>Boehner p. 1</ref> The period from the middle of the thirteenth to the middle of the fourteenth century was one of significant developments in logic, particularly in three areas which were original, with little foundation in the Aristotelian tradition that came before. These were:<ref>Boehner pp. 19–76</ref>

* The theory of [[Supposition theory|supposition]]. Supposition theory deals with the way that predicates (''e.g.,'' 'man') range over a domain of individuals (''e.g.,'' all men).<ref>Boehner p. 29</ref> In the proposition 'every man is an animal', does the term 'man' range over or 'supposit for' men existing just in the present, or does the range include past and future men? Can a term supposit for a non-existing individual? Some medievalists have argued that this idea is a precursor of modern [[first-order logic]].<ref>Boehner p. 30</ref> "The theory of supposition with the associated theories of ''copulatio'' (sign-capacity of adjectival terms), ''ampliatio'' (widening of referential domain), and ''distributio'' constitute one of the most original achievements of Western medieval logic".<ref>Ebbesen 1981</ref>
* The theory of [[Syncategorematic term|syncategoremata]]. Syncategoremata are terms which are necessary for logic, but which, unlike ''categorematic'' terms, do not signify on their own behalf, but 'co-signify' with other words. Examples of syncategoremata are 'and', 'not', 'every', 'if', and so on.
* The theory of [[Logical consequence|consequences]]. A consequence is a hypothetical, conditional proposition: two propositions joined by the terms 'if ... then'. For example, 'if a man runs, then God exists' (''Si homo currit, Deus est'').<ref>Boehner pp. 54–55</ref> A fully developed theory of consequences is given in Book III of [[William of Ockham]]'s work [[Summa Logicae]]. There, Ockham distinguishes between 'material' and 'formal' consequences, which are roughly equivalent to the modern [[Material conditional|material implication]] and [[logical implication]] respectively. Similar accounts are given by [[Jean Buridan]] and [[Albert of Saxony (philosopher)|Albert of Saxony]].

The last great works in this tradition are the ''Logic'' of John Poinsot (1589–1644, known as [[John of St Thomas]]), the ''Metaphysical Disputations'' of [[Francisco Suarez]] (1548–1617), and the ''Logica Demonstrativa'' of [[Giovanni Girolamo Saccheri]] (1667–1733).

==Traditional logic==

===The textbook tradition===
[[File:Fennerartoflogic-small.jpg|alt=Frontispiece, with title beginning "The Artes of Logike and Rethorike, plainlie set foorth in the English tounge, easie to be learned and practised".|thumb|[[Dudley Fenner]]'s ''Art of Logic'' (1584)]]
''Traditional logic'' generally means the textbook tradition that begins with [[Antoine Arnauld]]'s and [[Pierre Nicole]]'s ''Logic, or the Art of Thinking'', better known as the ''[[Port-Royal Logic]]''.<ref>''Oxford Companion'' p. 504, article "Traditional logic"</ref> Published in 1662, it was the most influential work on logic after Aristotle until the nineteenth century.<ref name="Buroker xxiii">Buroker xxiii</ref> The book presents a loosely Cartesian doctrine (that the proposition is a combining of ideas rather than terms, for example) within a framework that is broadly derived from Aristotelian and medieval [[term logic]]. Between 1664 and 1700, there were eight editions, and the book had considerable influence after that.<ref name="Buroker xxiii"/> The Port-Royal introduces the concepts of [[extension (semantics)|extension]] and [[intension]]. The account of [[proposition]]s that [[John Locke|Locke]] gives in the ''Essay'' is essentially that of the Port-Royal: "Verbal propositions, which are words, [are] the signs of our ideas, put together or separated in affirmative or negative sentences. So that proposition consists in the putting together or separating these signs, according as the things which they stand for agree or disagree."<ref>(Locke, ''An Essay Concerning Human Understanding'', IV. 5. 6)</ref>

[[Dudley Fenner]] helped popularize [[Ramist]] logic, a reaction against Aristotle. Another influential work was the ''[[Novum Organum]]'' by [[Francis Bacon]], published in 1620. The title translates as "new instrument". This is a reference to [[Aristotle]]'s work known as the ''[[Organon]]''. In this work, Bacon rejects the syllogistic method of Aristotle in favor of an alternative procedure "which by slow and faithful toil gathers information from things and brings it into understanding".<ref>Farrington, 1964, 89</ref> This method is known as [[inductive reasoning]], a method which starts from empirical observation and proceeds to lower axioms or propositions; from these lower axioms, more general ones can be induced. For example, in finding the cause of a ''phenomenal nature'' such as heat, three lists should be constructed:
* The presence list: a list of every situation where heat is found.
* The absence list: a list of every situation that is similar to at least one of those of the presence list, except for the lack of heat.
* The variability list: a list of every situation where heat can vary.
Then, the ''form nature'' (or cause) of heat may be defined as that which is common to every situation of the presence list, and which is lacking from every situation of the absence list, and which varies by degree in every situation of the variability list.

Other works in the textbook tradition include [[Isaac Watts]]'s ''Logick: Or, the Right Use of Reason'' (1725), [[Richard Whately]]'s ''Logic'' (1826), and [[John Stuart Mill]]'s ''A System of Logic'' (1843). Although the latter was one of the last great works in the tradition, Mill's view that the foundations of logic lie in introspection<ref>N. Abbagnano, "Psychologism" in P. Edwards (ed) ''The Encyclopaedia of Philosophy'', MacMillan, 1967</ref> influenced the view that logic is best understood as a branch of psychology, a view which dominated the next fifty years of its development, especially in Germany.<ref>Of the German literature in this period, Robert Adamson wrote "''Logics'' swarm as bees in springtime..."; Robert Adamson, ''A Short History of Logic'', Wm. Blackwood & Sons, 1911, page 242</ref>

===Logic in Hegel's philosophy===
[[File:G.W.F. Hegel (by Sichling, after Sebbers).jpg|thumb|Georg Wilhelm Friedrich Hegel]]
[[G.W.F. Hegel]] indicated the importance of logic to his philosophical system when he condensed his extensive ''[[Science of Logic]]'' into a shorter work published in 1817 as the first volume of his ''Encyclopaedia of the Philosophical Sciences.'' The "Shorter" or "Encyclopaedia" ''Logic'', as it is often known, lays out a series of transitions which leads from the most empty and abstract of categories—Hegel begins with "Pure Being" and "Pure Nothing"—to the "[[Absolute (philosophy)|Absolute]]", the category which contains and resolves all the categories which preceded it. Despite the title, Hegel's ''Logic'' is not really a contribution to the science of valid inference. Rather than deriving conclusions about concepts through valid inference from premises, Hegel seeks to show that thinking about one concept compels thinking about another concept (one cannot, he argues, possess the concept of "Quality" without the concept of "Quantity"); this compulsion is, supposedly, not a matter of individual psychology, because it arises almost organically from the content of the concepts themselves. His purpose is to show the rational structure of the "Absolute"—indeed of rationality itself. The method by which thought is driven from one concept to its contrary, and then to further concepts, is known as the Hegelian [[dialectic]].

Although Hegel's ''Logic'' has had little impact on mainstream logical studies, its influence can be seen elsewhere:
* [[Carl von Prantl]]'s ''Geschichte der Logik im Abendland'' (1855–1867).<ref>Carl von Prantl (1855–1867), ''Geschichte von Logik in Abendland'', Leipzig:  S. Hirzl, anastatically reprinted in 1997, Hildesheim:  Georg Olds.</ref>
* The work of the [[British Idealism|British Idealists]], such as F. H. Bradley's ''Principles of Logic'' (1883).
* The economic, political, and philosophical studies of [[Karl Marx]], and in the various schools of [[Marxism]].

===Logic and psychology===
Between the work of Mill and Frege stretched half a century during which logic was widely treated as a descriptive science, an empirical study of the structure of reasoning, and thus essentially as a branch of [[psychology]].<ref>See e.g. [http://plato.stanford.edu/entries/psychologism Psychologism], Stanford Encyclopedia of Philosophy</ref> The German psychologist [[Wilhelm Wundt]], for example, discussed deriving "the logical from the psychological laws of thought", emphasizing that "psychological thinking is always the more comprehensive form of thinking."<ref>Wilhelm Wundt, ''Logik'' (1880–1883); quoted in Edmund Husserl, ''Logical Investigations,'' translated J. N. Findlay, Routledge, 2008, Volume 1, pp. 115–116.</ref> This view was widespread among German philosophers of the period:
* [[Theodor Lipps]] described logic as "a specific discipline of psychology".<ref>Theodor Lipps, ''Grundzüge der Logik'' (1893); quoted in Edmund Husserl, ''Logical Investigations,'' translated J. N. Findlay, Routledge, 2008, Volume 1, p. 40</ref>
* [[Christoph von Sigwart]] understood logical necessity as grounded in the individual's compulsion to think in a certain way.<ref>Christoph von Sigwart, ''Logik'' (1873–1878); quoted in Edmund Husserl, ''Logical Investigations,'' translated J. N. Findlay, Routledge, 2008, Volume 1, p. 51</ref>
* [[Benno Erdmann]] argued that "logical laws only hold within the limits of our thinking".<ref>Benno Erdmann, ''Logik'' (1892); quoted in Edmund Husserl, ''Logical Investigations,'' translated J. N. Findlay, Routledge, 2008, Volume 1, p. 96</ref>
Such was the dominant view of logic in the years following Mill's work.<ref>Dermot Moran, "Introduction"; Edmund Husserl, ''Logical Investigations,'' translated J. N. Findlay, Routledge, 2008, Volume 1, p. xxi</ref> This psychological approach to logic was rejected by [[Gottlob Frege]]. It was also subjected to an extended and destructive critique by [[Edmund Husserl]] in the first volume of his ''Logical Investigations'' (1900), an assault which has been described as "overwhelming".<ref>Michael Dummett, "Preface"; Edmund Husserl, ''Logical Investigations,'' translated J. N. Findlay, Routledge, 2008, Volume 1, p. xvii</ref> Husserl argued forcefully that grounding logic in psychological observations implied that all logical truths remained unproven, and that [[skepticism]] and [[relativism]] were unavoidable consequences.

Such criticisms did not immediately extirpate what is called "[[psychologism]]". For example, the American philosopher [[Josiah Royce]], while acknowledging the force of Husserl's critique, remained "unable to doubt" that progress in psychology would be accompanied by progress in logic, and vice versa.<ref>Josiah Royce, "Recent Logical Enquiries and their Psychological Bearings" (1902) in John J. McDermott (ed) ''The Basic Writings of Josiah Royce'' Volume 2, Fordham University Press, 2005, p. 661</ref>

==Rise of modern logic==
The period between the fourteenth century and the beginning of the nineteenth century had been largely one of decline and neglect, and is generally regarded as barren by historians of logic.<ref name="ReferenceA"/> The revival of logic occurred in the mid-nineteenth century, at the beginning of a revolutionary period where the subject developed into a rigorous and formalistic discipline whose exemplar was the exact method of proof used in [[mathematics]]. The development of the modern "symbolic" or "mathematical" logic during this period is the most significant in the 2000-year history of logic, and is arguably one of the most important and remarkable events in human intellectual history.<ref name="Oxford Companion p. 500"/>

A number of features distinguish modern logic from the old Aristotelian or traditional logic, the most important of which are as follows:<ref>Bochenski, p. 266</ref> Modern logic is fundamentally a ''calculus'' whose rules of operation are determined only by the ''shape'' and not by the ''meaning'' of the symbols it employs, as in mathematics. Many logicians were impressed by the "success" of mathematics, in that there had been no prolonged dispute about any truly mathematical result. [[Charles Sanders Peirce|C. S. Peirce]] noted<ref>Peirce 1896</ref> that even though a mistake in the evaluation of a definite integral by [[Laplace]] led to an error concerning the moon's orbit that persisted for nearly 50 years, the mistake, once spotted, was corrected without any serious dispute. Peirce contrasted this with the disputation and uncertainty surrounding traditional logic, and especially reasoning in [[metaphysics]]. He argued that a truly "exact" logic would depend upon mathematical, i.e., "diagrammatic" or "iconic" thought. "Those who follow such methods will ... escape all error except such as will be speedily corrected after it is once suspected". Modern logic is also "constructive" rather than "abstractive"; i.e., rather than abstracting and formalising theorems derived from ordinary language (or from psychological intuitions about validity), it constructs theorems by formal methods, then looks for an interpretation in ordinary language. It is entirely symbolic, meaning that even the logical constants (which the medieval logicians called "[[Syncategorematic term|syncategoremata]]") and the categoric terms are expressed in symbols.

==Modern logic==
{{See also|History of mathematical logic}}

The development of modern logic falls into roughly five periods:<ref>See Bochenski p. 269</ref>
* The ''embryonic period'' from [[Gottfried Wilhelm Leibniz|Leibniz]] to 1847, when the notion of a logical calculus was discussed and developed, particularly by Leibniz, but no schools were formed, and isolated periodic attempts were abandoned or went unnoticed.
* The ''algebraic period'' from [[Boole]]'s Analysis to [[Ernst Schröder (mathematician)|Schröder]]'s ''Vorlesungen''. In this period, there were more practitioners, and a greater continuity of development.
* The ''[[logicist]] period'' from the [[Begriffsschrift]] of [[Frege]] to the ''[[Principia Mathematica]]'' of [[Bertrand Russell|Russell]] and [[A. N. Whitehead|Whitehead]]. The aim of the "logicist school" was to incorporate the logic of all mathematical and scientific discourse in a single unified system which, taking as a fundamental principle that all mathematical truths are logical, did not accept any non-logical terminology. The major logicists were [[Gottlob Frege|Frege]], [[Bertrand Russell|Russell]], and the early [[Ludwig Wittgenstein|Wittgenstein]].<ref>''Oxford Companion'' p. 499</ref> It culminates with the ''Principia'', an important work which includes a thorough examination and attempted solution of the [[antinomy|antinomies]] which had been an obstacle to earlier progress.
* The ''metamathematical period'' from 1910 to the 1930s, which saw the development of [[metalogic]], in the [[finitist]] system of [[David Hilbert|Hilbert]], and the non-finitist system of [[Leopold Löwenheim|Löwenheim]] and [[Skolem]], the combination of logic and metalogic in the work of [[Gödel]] and [[Alfred Tarski|Tarski]]. Gödel's [[incompleteness theorem]] of 1931 was one of the greatest achievements in the history of logic. Later in the 1930s, Gödel developed the notion of [[set-theoretic constructibility]].
* The ''period after World War II'', when [[mathematical logic]] branched into four inter-related but separate areas of research: [[model theory]], [[proof theory]], [[computability theory]], and [[set theory]], and its ideas and methods began to influence [[philosophy]].

===Embryonic period===
[[File:Gottfried Wilhelm Leibniz, Bernhard Christoph Francke.jpg|thumb|upright|Leibniz]]
The idea that inference could be represented by a purely mechanical process is found as early as [[Ramon Llull|Raymond Llull]], who proposed a (somewhat eccentric) method of drawing conclusions by a system of concentric rings. The work of logicians such as the [[Oxford Calculators]]<ref>Edith Sylla (1999), "Oxford Calculators", in ''The Cambridge Dictionary of Philosophy'', Cambridge, Cambridgeshire:  Cambridge.</ref> led to a method of using letters instead of writing out logical calculations (''calculationes'') in words, a method used, for instance, in the ''Logica magna'' by [[Paul of Venice]]. Three hundred years after Llull, the English philosopher and logician [[Thomas Hobbes]] suggested that all logic and reasoning could be reduced to the mathematical operations of addition and subtraction.<ref>El. philos. sect. I de corp 1.1.2.</ref> The same idea is found in the work of [[Gottfried Wilhelm Leibniz|Leibniz]], who had read both Llull and Hobbes, and who argued that logic can be represented through a combinatorial process or calculus. But, like Llull and Hobbes, he failed to develop a detailed or comprehensive system, and his work on this topic was not published until long after his death. Leibniz says that ordinary languages are subject to "countless ambiguities" and are unsuited for a calculus, whose task is to expose mistakes in inference arising from the forms and structures of words;<ref>Bochenski p. 274</ref> hence, he proposed to identify an [[alphabet of human thought]] comprising fundamental concepts which could be composed to express complex ideas,<ref>Rutherford, Donald, 1995, "Philosophy and language" in Jolley, N., ed., ''The Cambridge Companion to Leibniz''. Cambridge Univ. Press.</ref> and create a ''[[calculus ratiocinator]]'' that would make all arguments "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."<ref>Wiener, Philip, 1951. ''Leibniz: Selections''. Scribner.</ref>

[[Joseph Diaz Gergonne|Gergonne]] (1816) said that reasoning does not have to be about objects about which one has perfectly clear ideas, because algebraic operations can be carried out without having any idea of the meaning of the symbols involved.<ref>''Essai de dialectique rationelle'', 211n, quoted in Bochenski p. 277.</ref> [[Bernard Bolzano|Bolzano]] anticipated a fundamental idea of modern proof theory when he defined logical consequence or "deducibility" in terms of variables:<ref>{{cite book |author-last=Bolzano |author-first=Bernard |url=https://books.google.com/books?id=oA1NDDirneQC&q=%22deducible%20from%20propositions%22&pg=PA209 |title=The Theory of Science: Die Wissenschaftslehre oder Versuch einer Neuen Darstellung der Logik |date=1972 |publisher=[[University of California Press]] |isbn=978-0-52001787-0 |editor-last=George |editor-first=Rolf |page=209 |translator-last=Rolf |translator-first=George}}</ref><blockquote>Hence I say that propositions <math>M</math>, <math>N</math>, <math>O</math>,... are ''deducible'' from propositions <math>A</math>, <math>B</math>, <math>C</math>, <math>D</math>,... with respect to variable parts <math>i</math>, <math>j</math>,..., if every class of ideas whose substitution for <math>i</math>, <math>j</math>,... makes all of <math>A</math>, <math>B</math>, <math>C</math>, <math>D</math>,... true, also makes all of <math>M</math>, <math>N</math>, <math>O</math>,... true. Occasionally, since it is customary, I shall say that propositions <math>M</math>, <math>N</math>, <math>O</math>,... ''follow'', or can be ''inferred'' or ''derived'', from <math>A</math>, <math>B</math>, <math>C</math>, <math>D</math>,.... Propositions <math>A</math>, <math>B</math>, <math>C</math>, <math>D</math>,... I shall call the ''premises'', <math>M</math>, <math>N</math>, <math>O</math>,... the ''conclusions.''</blockquote>This is now known as [[semantic validity]].

===Algebraic period===
[[File:George Boole color.jpg|thumb|140px|George Boole]]
Modern logic begins with what is known as the "algebraic school", originating with Boole and including [[Charles Sanders Peirce|Peirce]], [[William Stanley Jevons|Jevons]], [[Ernst Schröder (mathematician)|Schröder]], and [[John Venn|Venn]].<ref>See e.g. Bochenski p. 296 and ''passim''</ref> Their objective was to develop a calculus to formalise reasoning in the area of classes, propositions, and probabilities. The school begins with Boole's seminal work ''Mathematical Analysis of Logic'' which appeared in 1847, although [[Augustus De Morgan|De Morgan]] (1847) is its immediate precursor.<ref>Before publishing, he wrote to [[Augustus De Morgan|De Morgan]], who was just finishing his work ''Formal Logic''.  De Morgan suggested they should publish first, and thus the two books appeared at the same time, possibly even reaching the bookshops on the same day. cf. Kneale p. 404</ref> The fundamental idea of Boole's system is that algebraic formulae can be used to express logical relations. This idea occurred to Boole in his teenage years, working as an usher in a private school in [[Lincoln, Lincolnshire]].<ref>Kneale p. 404</ref> For example, let x and y stand for classes, let the symbol ''='' signify that the classes have the same members, xy stand for the class containing all and only the members of x and y and so on. Boole calls these ''elective symbols'', i.e. symbols which select certain objects for consideration.<ref name="Kneale p. 407">Kneale p. 407</ref> An expression in which elective symbols are used is called an ''elective function'', and an equation of which the members are elective functions, is an ''elective equation''.<ref>Boole (1847) p. 16</ref> The theory of elective functions and their "development" is essentially the modern idea of [[truth-function]]s and their expression in [[disjunctive normal form]].<ref name="Kneale p. 407"/>

Boole's system admits of two interpretations, in class logic, and propositional logic. Boole distinguished between "primary propositions" which are the subject of syllogistic theory, and "secondary propositions", which are the subject of propositional logic, and showed how under different "interpretations" the same algebraic system could represent both. An example of a primary proposition is "All inhabitants are either Europeans or Asiatics." An example of a secondary proposition is "Either all inhabitants are Europeans or they are all Asiatics."<ref>Boole 1847 pp. 58–59</ref> These are easily distinguished in modern predicate logic, where it is also possible to show that the first follows from the second, but it is a significant disadvantage that there is no way of representing this in the Boolean system.<ref>Beaney p. 11</ref>

In his ''Symbolic Logic'' (1881), [[John Venn]] used diagrams of overlapping areas to express Boolean relations between classes or truth-conditions of propositions. In 1869 Jevons realised that Boole's methods could be mechanised, and constructed a "logical machine" which he showed to the [[Royal Society]] the following year.<ref name="Kneale p. 407"/> In 1885 [[Allan Marquand]] proposed an electrical version of the machine that is still extant ([https://web.archive.org/web/20080908073359/http://finelib.princeton.edu/instruction/wri172_demonstration.php picture at the Firestone Library]).
[[File:Charles Sanders Peirce.jpg|left|thumb|160px|Charles Sanders Peirce]]
The defects in Boole's system (such as the use of the letter ''v'' for existential propositions) were all remedied by his followers. Jevons published ''Pure Logic, or the Logic of Quality apart from Quantity'' in 1864, where he suggested a symbol to signify [[exclusive or]], which allowed Boole's system to be greatly simplified.<ref>Kneale p. 422</ref> This was usefully exploited by Schröder when he set out theorems in parallel columns in his ''Vorlesungen'' (1890–1905). Peirce (1880) showed how all the Boolean elective functions could be expressed by the use of a single primitive binary operation, "[[Logical NOR|neither ... nor ...]]" and equally well "[[Sheffer stroke|not both ... and ...]]",<ref>Peirce, "A Booli<!-- sic! -->an Algebra with One Constant", 1880 MS, ''Collected Papers'' v. 4, paragraphs 12–20, reprinted ''Writings'' v. 4, pp. 218–221. Google [https://archive.org/details/writingsofcharle0002peir <!-- quote=378 Winter. --> Preview].</ref> however, like many of Peirce's innovations, this remained unknown or unnoticed until [[Henry M. Sheffer|Sheffer]] rediscovered it in 1913.<ref>''Trans. Amer. Math. Soc., xiv (1913)'', pp. 481–488. This is now known as the [[Sheffer stroke]]</ref> Boole's early work also lacks the idea of the [[logical sum]] which originates in Peirce (1867), [[Ernst Schröder (mathematician)|Schröder]] (1877) and Jevons (1890),<ref>Bochenski 296</ref> and the concept of [[Inclusion (logic)|inclusion]], first suggested by Gergonne (1816) and clearly articulated by Peirce (1870).
[[File:Boolean multiples of 2 3 5.svg|alt=Coloured diagram of 4 interlocking sets|right|thumb|250px|Boolean multiples]]
The success of Boole's algebraic system suggested that all logic must be capable of algebraic representation, and there were attempts to express a logic of relations in such form, of which the most ambitious was Schröder's monumental ''Vorlesungen über die Algebra der Logik'' ("Lectures on the Algebra of Logic", vol iii 1895), although the original idea was again anticipated by Peirce.<ref>See CP III</ref>

Boole's unwavering acceptance of Aristotle's logic is emphasized by the historian of logic [[John Corcoran (logician)|John Corcoran]] in an accessible introduction to ''[[The Laws of Thought|Laws of Thought]].''<ref>[[George Boole]]. 1854/2003. The Laws of Thought, facsimile of 1854 edition, with an introduction by J. Corcoran. Buffalo: Prometheus Books (2003). Reviewed by James van Evra in Philosophy in Review. 24 (2004) 167–169.</ref> Corcoran also wrote a point-by-point comparison of ''Prior Analytics'' and ''Laws of Thought''.<ref>JOHN CORCORAN, Aristotle's Prior Analytics and Boole's Laws of Thought, History and Philosophy of Logic, vol. 24 (2003), pp. 261–288.</ref> According to Corcoran, Boole fully accepted and endorsed Aristotle's logic. Boole's goals were "to go under, over, and beyond" Aristotle's logic by 1) providing it with mathematical foundations involving equations, 2) extending the class of problems it could treat—from assessing validity to solving equations—and 3) expanding the range of applications it could handle—e.g. from propositions having only two terms to those having arbitrarily many.

More specifically, Boole agreed with what [[Aristotle]] said; Boole's 'disagreements', if they might be called that, concern what Aristotle did not say. 
First, in the realm of foundations, Boole reduced the four propositional forms of Aristotelian logic to formulas in the form of equations—by itself a revolutionary idea. 
Second, in the realm of logic's problems, Boole's addition of equation solving to logic—another revolutionary idea—involved Boole's doctrine that Aristotle's rules of inference (the "perfect syllogisms") must be supplemented by rules for equation solving. 
Third, in the realm of applications, Boole's system could handle multi-term propositions and arguments whereas Aristotle could handle only two-termed subject-predicate propositions and arguments. For example, Aristotle's system could not deduce "No quadrangle that is a square is a rectangle that is a rhombus" from "No square that is a quadrangle is a rhombus that is a rectangle" or from "No rhombus that is a rectangle is a square that is a quadrangle".

===Logicist period===
[[File:Young frege.jpg|thumb|160px|Gottlob Frege.]]
After Boole, the next great advances were made by the German mathematician [[Gottlob Frege]]. Frege's objective was the program of [[Logicism]], i.e. demonstrating that arithmetic is identical with logic.<ref name="k435">Kneale p. 435</ref> Frege went much further than any of his predecessors in his rigorous and formal approach to logic, and his calculus or [[Begriffsschrift]] is important.<ref name="k435"/> Frege also tried to show that the concept of [[number]] can be defined by purely logical means, so that (if he was right) logic includes arithmetic and all branches of mathematics that are reducible to arithmetic. He was not the first writer to suggest this. In his pioneering work {{Lang|de|Die Grundlagen der Arithmetik}} (The Foundations of Arithmetic), sections 15–17, he acknowledges the efforts of Leibniz, [[J. S. Mill]] as well as Jevons, citing the latter's claim that "algebra is a highly developed logic, and number but logical discrimination."<ref>Jevons, ''The Principles of Science'', London 1879, p. 156, quoted in ''Grundlagen'' 15</ref>

Frege's first work, the ''Begriffsschrift'' ("concept script") is a rigorously axiomatised system of propositional logic, relying on just two connectives (negational and conditional), two rules of inference (''modus ponens'' and substitution), and six axioms. Frege referred to the "completeness" of this system, but was unable to prove this.<ref>Beaney p. 10 – the completeness of Frege's system was eventually proved by [[Jan Łukasiewicz]] in 1934</ref> The most significant innovation, however, was his explanation of the [[Quantifier (logic)|quantifier]] in terms of mathematical functions. Traditional logic regards the sentence "Caesar is a man" as of fundamentally the same form as "all men are mortal." Sentences with a proper name subject were regarded as universal in character, interpretable as "every Caesar is a man".<ref>See for example the argument by the medieval logician [[William of Ockham]] that singular propositions are universal, in [[Summa Logicae]] III. 8 (??)</ref> At the outset Frege abandons the traditional "concepts ''subject'' and ''predicate''", replacing them with ''argument'' and ''function'' respectively, which he believes "will stand the test of time. It is easy to see how regarding a content as a function of an argument leads to the formation of concepts. Furthermore, the demonstration of the connection between the meanings of the words ''if, and, not, or, there is, some, all,'' and so forth, deserves attention".<ref>{{harvnb |Frege |1879}} in {{harvnb |van Heijenoort |1967 |p=7}}</ref> Frege argued that the quantifier expression "all men" does not have the same logical or semantic form as "all men", and that the universal proposition "every A is B" is a complex proposition involving two ''functions'', namely ' – is A' and ' – is B' such that whatever satisfies the first, also satisfies the second. In modern notation, this would be expressed as

: <math>\forall \; x \big( A(x) \rightarrow B (x) \big)</math>

In English, "for all x, if Ax then Bx". Thus only singular propositions are of subject-predicate form, and they are irreducibly singular, i.e. not reducible to a general proposition. Universal and particular propositions, by contrast, are not of simple subject-predicate form at all. If "all mammals" were the logical subject of the sentence "all mammals are land-dwellers", then to negate the whole sentence we would have to negate the predicate to give "all mammals are ''not'' land-dwellers". But this is not the case.<ref>"On concept and object" p. 198; Geach p. 48</ref> This functional analysis of ordinary-language sentences later had a great impact on philosophy and [[linguistics]].

This means that in Frege's calculus, Boole's "primary" propositions can be represented in a different way from "secondary" propositions. "All inhabitants are either men or women" is
[[File:BS-13-Begriffsschrift Quantifier2-svg.svg|130px|alt=Straight line with bend; text "x" over bend; text "F(x)" to the right of the line.|thumb|[[Frege]]'s "Concept Script"]]
: <math>\forall \; x \Big( I(x) \rightarrow \big( M(x) \lor W(x) \big) \Big) </math>

whereas "All the inhabitants are men or all the inhabitants are women" is

: <math>\forall \; x \big( I(x) \rightarrow M(x) \big) \lor \forall \;x \big( I(x) \rightarrow W(x) \big)</math>

As Frege remarked in a critique of Boole's calculus:

: "The real difference is that I avoid [the Boolean] division into two parts ... and give a homogeneous presentation of the lot. In Boole the two parts run alongside one another, so that one is like the mirror image of the other, but for that very reason stands in no organic relation to it."<ref>BLC p. 14, quoted in Beaney p. 12</ref>

As well as providing a unified and comprehensive system of logic, Frege's calculus also resolved the ancient [[problem of multiple generality]]. The ambiguity of "every girl kissed a boy" is difficult to express in traditional logic, but Frege's logic resolves this through the different scope of the quantifiers. Thus

:<math>\forall \; x \Big( G(x) \rightarrow  \exists \; y \big( B(y) \land K(x,y) \big) \Big)</math>
[[File:Giuseppe_Peano.jpg|thumb|120px|Peano]]
means that to every girl there corresponds some boy (any one will do) who the girl kissed. But

:<math>\exists \;x \Big( B(x) \land \forall \;y \big( G(y) \rightarrow K(y, x) \big) \Big)</math>

means that there is some particular boy whom every girl kissed. Without this device, the project of logicism would have been doubtful or impossible. Using it, Frege provided a definition of the [[ancestral relation]], of the [[Injective function|many-to-one relation]], and of [[mathematical induction]].<ref>See e.g. [http://www.utm.edu/research/iep/f/frege.htm The Internet Encyclopedia of Philosophy], article "Frege"</ref>
[[File:Ernst Zermelo 1900s.jpg|thumb|left|130px|Ernst Zermelo]]
This period overlaps with the work of what is known as the "mathematical school", which included [[Richard Dedekind|Dedekind]], [[Moritz Pasch|Pasch]], [[Giuseppe Peano|Peano]], [[David Hilbert|Hilbert]], [[Ernst Zermelo|Zermelo]], [[Edward Vermilye Huntington|Huntington]], [[Oswald Veblen|Veblen]] and [[Arend Heyting|Heyting]]. Their objective was the axiomatisation of branches of mathematics like geometry, arithmetic, analysis and set theory. Most notable was [[Hilbert's Program]], which sought to ground all of mathematics to a finite set of axioms, proving its consistency by "finitistic" means and providing a procedure which would decide the truth or falsity of any mathematical statement. The standard [[axiomatization]] of the [[natural number]]s is named the [[Peano axioms]] eponymously. Peano maintained a clear distinction between mathematical and logical symbols. While unaware of Frege's work, he independently recreated his logical apparatus based on the work of Boole and Schröder.<ref>Van Heijenoort 1967, p. 83</ref>

The logicist project received a near-fatal setback with the discovery of a paradox in 1901 by [[Bertrand Russell]]. This proved Frege's [[naive set theory]] led to a contradiction. Frege's theory contained the axiom that for any formal criterion, there is a set of all objects that meet the criterion. Russell showed that a set containing exactly the sets that are not members of themselves would contradict its own definition (if it is not a member of itself, it is a member of itself, and if it is a member of itself, it is not).<ref>See e.g. Potter 2004</ref> This contradiction is now known as [[Russell's paradox]]. One important method of resolving this paradox was proposed by [[Ernst Zermelo]].<ref>Zermelo 1908</ref> [[Zermelo set theory]] was the first [[axiomatic set theory]]. It was developed into the now-canonical [[Zermelo–Fraenkel set theory]] (ZF). Russell's paradox symbolically is as follows:

:<math>\text{Let } R = \{ x \mid x \not \in x \} \text{, then } R \in R \iff R \not \in R</math>
The monumental [[Principia Mathematica]], a three-volume work on the [[foundations of mathematics]], written by Russell and [[Alfred North Whitehead]] and published 1910–1913 also included an attempt to resolve the paradox, by means of an elaborate [[system of types]]: a set of elements is of a different type than is each of its elements (set is not the element; one element is not the set) and one cannot speak of the "[[set of all sets]]". The ''Principia'' was an attempt to derive all mathematical truths from a well-defined set of [[axiom]]s and [[inference rule]]s in [[Mathematical logic|symbolic logic]].

===Metamathematical period===
[[File:Kurt gödel.jpg|thumb|130px|right|Kurt Gödel]]
The names of [[Kurt Gödel|Gödel]] and [[Alfred Tarski|Tarski]] dominate the 1930s,<ref>Feferman 1999 p. 1</ref> a crucial period in the development of [[metamathematics]]—the study of mathematics using mathematical methods to produce [[metatheory|metatheories]], or mathematical theories about other mathematical theories. Early investigations into metamathematics had been driven by Hilbert's program. Work on metamathematics culminated in the work of Gödel, who in 1929 showed that a given [[first-order logic|first-order sentence]] is [[Provability logic|deducible]] if and only if it is logically valid—i.e. it is true in every [[structure (mathematical logic)|structure]] for its language. This is known as [[Gödel's completeness theorem]]. A year later, he proved two important theorems, which showed Hibert's program to be unattainable in its original form. The first is that no consistent system of axioms whose theorems can be listed by an [[Effective method|effective procedure]] such as an [[algorithm]] or computer program is capable of proving all facts about the [[natural number]]s. For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second is that if such a system is also capable of proving certain basic facts about the natural numbers, then the system cannot prove the consistency of the system itself. These two results are known as [[Gödel's incompleteness theorems]], or simply ''Gödel's Theorem''. Later in the decade, Gödel developed the concept of [[set-theoretic constructibility]], as part of his proof that the [[axiom of choice]] and the [[continuum hypothesis]] are consistent with [[Zermelo–Fraenkel set theory]].
<!-- Commented out: [[File:Alonzo Church.jpg|thumb|left|130px|Alonzo Church]] -->
In [[proof theory]], [[Gerhard Gentzen]] developed [[natural deduction]] and the [[sequent calculus]]. The former attempts to model logical reasoning as it 'naturally' occurs in practice and is most easily applied to [[intuitionistic logic]], while the latter was devised to clarify the derivation of logical proofs in any formal system. Since Gentzen's work, natural deduction and sequent calculi have been widely applied in the fields of proof theory, mathematical logic and computer science. Gentzen also proved normalization and cut-elimination theorems for intuitionistic and classical logic which could be used to reduce logical proofs to a normal form.<ref>{{cite book |author-last1=Girard |author-first1=Jean-Yves |url=https://archive.org/details/proofstypes0000gira |title=Proofs and Types |author-last2=Taylor |first2=Paul |author-last3=Lafont |author-first3=Yves |date=1990 |publisher=Cambridge University Press (Cambridge Tracts in Theoretical Computer Science, 7) |isbn=0-521-37181-3 |author-link1=Jean-Yves Girard |orig-date=1989 |url-access=registration}}</ref>
[[File:AlfredTarski1968.jpeg|right|200px|alt=Balding man, with bookshelf in background|thumb|Alfred Tarski]]
[[Alfred Tarski]], a pupil of [[Jan Łukasiewicz|Łukasiewicz]], is best known for his definition of truth and [[logical consequence]], and the semantic concept of [[Open sentence|logical satisfaction]]. In 1933, he published (in Polish) ''The concept of truth in formalized languages'', in which he proposed his [[semantic theory of truth]]: a sentence such as "snow is white" is true if and only if snow is white. Tarski's theory separated the [[metalanguage]], which makes the statement about truth, from the object language, which contains the sentence whose truth is being asserted, and gave a correspondence (the [[T-schema]]) between phrases in the object language and elements of an [[interpretation (logic)|interpretation]]. Tarski's approach to the difficult idea of explaining truth has been enduringly influential in logic and philosophy, especially in the development of [[model theory]].<ref>Feferman and Feferman 2004, p. 122, discussing "The Impact of Tarski's Theory of Truth".</ref> Tarski also produced important work on the methodology of deductive systems, and on fundamental principles such as [[completeness (logic)|completeness]], [[decidability (logic)|decidability]], [[consistency]] and [[Structure (mathematical logic)|definability]]. According to Anita Feferman, Tarski "changed the face of logic in the twentieth century".<ref>Feferman 1999, p. 1</ref>

[[Alonzo Church]] and [[Alan Turing]] proposed formal models of computability, giving independent negative solutions to Hilbert's ''[[Entscheidungsproblem]]'' in 1936 and 1937, respectively. The ''Entscheidungsproblem'' asked for a procedure that, given any formal mathematical statement, would algorithmically determine whether the statement is true. Church and Turing proved there is no such procedure; Turing's paper introduced the [[halting problem]] as a key example of a mathematical problem without an algorithmic solution.

Church's system for computation developed into the modern [[λ-calculus]], while the [[Turing machine]] became a standard model for a general-purpose computing device. It was soon shown that many other proposed models of computation were equivalent in power to those proposed by Church and Turing. These results led to the [[Church–Turing thesis]] that any deterministic [[algorithm]] that can be carried out by a human can be carried out by a Turing machine. Church proved additional undecidability results, showing that both [[Peano arithmetic]] and [[first-order logic]] are [[Undecidable problem|undecidable]]. Later work by [[Emil Post]] and [[Stephen Cole Kleene]] in the 1940s extended the scope of computability theory and introduced the concept of [[degrees of unsolvability]].

The results of the first few decades of the twentieth century also had an impact upon [[analytic philosophy]] and [[philosophical logic]], particularly from the 1950s onwards, in subjects such as [[modal logic]], [[temporal logic]], [[deontic logic]], and [[relevance logic]].

===Logic after WWII===
[[File:Kripke.JPG|alt=Man with a beard and straw hat on a beach|thumb|[[Saul Kripke]]]]

After World War II, [[mathematical logic]] branched into four inter-related but separate areas of research: [[model theory]], [[proof theory]], [[computability theory]], and [[set theory]].<ref>See e.g. Barwise, ''Handbook of Mathematical Logic''</ref>

In set theory, the method of [[Forcing (mathematics)|forcing]] revolutionized the field by providing a robust method for constructing models and obtaining independence results. <!-- Kunen p. 235ff, Kanamori p. 114ff --> [[Paul Cohen]] introduced this method in 1963 to prove the independence of the [[continuum hypothesis]] and the [[axiom of choice]] from [[Zermelo–Fraenkel set theory]].<ref>{{cite journal | jstor=72252 | last1=Cohen | first1=Paul J. | title=The Independence of the Continuum Hypothesis, II | journal=Proceedings of the National Academy of Sciences of the United States of America | date=1964 | volume=51 | issue=1 | pages=105–110 | doi=10.1073/pnas.51.1.105 | pmid=16591132 | pmc=300611 | bibcode=1964PNAS...51..105C | doi-access=free }}</ref> His technique, which was simplified and extended soon after its introduction, has since been applied to many other problems in all areas of mathematical logic.

Computability theory had its roots in the work of Turing, Church, Kleene, and Post in the 1930s and 40s. It developed into a study of abstract computability, which became known as [[recursion theory]].<ref>Many of the foundational papers are collected in ''The Undecidable'' (1965) edited by Martin Davis</ref> The [[Turing degree|priority method]], discovered independently by [[Albert Muchnik]] and [[Richard Friedberg]] in the 1950s, led to major advances in the understanding of the [[degrees of unsolvability]] and related structures. <!-- cooper 246 --> Research into higher-order computability theory demonstrated its connections to set theory. <!-- sacks "higher recursion theory" --> The fields of [[constructive analysis]] and [[computable analysis]] were developed to study the effective content of classical mathematical theorems; these in turn inspired the program of [[reverse mathematics]]. A separate branch of computability theory, [[computational complexity theory]], was also characterized in logical terms as a result of investigations into [[descriptive complexity]].

Model theory applies the methods of mathematical logic to study models of particular mathematical theories. Alfred Tarski published much pioneering work in the field, which is named after a series of papers he published under the title ''Contributions to the theory of models''. <!-- "alfred tarski's work in model theory", vaught, JSL, https://www.jstor.org/stable/2273900 --> In the 1960s, [[Abraham Robinson]] used model-theoretic techniques to develop calculus and analysis based on [[non-standard analysis|infinitesimals]], a problem that first had been proposed by Leibniz. <!-- Keisler, fundamentals of model theory, HML, p. 48 -->

In proof theory, the relationship between classical mathematics and intuitionistic mathematics was clarified via tools such as the [[realizability]] method invented by [[Georg Kreisel]] and Gödel's [[Dialectica interpretation|''Dialectica'' interpretation]]. This work inspired the contemporary area of [[proof mining]]. The [[Curry–Howard correspondence]] emerged as a deep analogy between logic and computation, including a correspondence between systems of natural deduction and [[typed lambda calculus|typed lambda calculi]] used in computer science. As a result, research into this class of formal systems began to address both logical and computational aspects; this area of research came to be known as modern type theory. Advances were also made in [[ordinal analysis]] and the study of independence results in arithmetic such as the [[Paris–Harrington theorem]].

This was also a period, particularly in the 1950s and afterwards, when the ideas of mathematical logic begin to influence philosophical thinking. For example, [[tense logic]] is a formalised system for representing, and reasoning about, propositions qualified in terms of time. The philosopher [[Arthur Prior]] played a significant role in its development in the 1960s. [[Modal logic]]s extend the scope of formal logic to include the elements of [[Linguistic modality|modality]] (for example, [[Logical possibility|possibility]] and [[Necessary and sufficient conditions#Necessary conditions|necessity]]). The ideas of [[Saul Kripke]], particularly about [[possible world]]s, and the formal system now called [[Kripke semantics]] have had a profound impact on [[analytic philosophy]].<ref>Jerry Fodor, "[http://www.lrb.co.uk/v26/n20/jerry-fodor/waters-water-everywhere Water's water everywhere]", ''London Review of Books'', 21 October 2004</ref> His best known and most influential work is ''[[Naming and Necessity]]'' (1980).<ref>See ''Philosophical Analysis in the Twentieth Century: Volume 2: The Age of Meaning'', Scott Soames: "''Naming and Necessity'' is among the most important works ever, ranking with the classical work of Frege in the late nineteenth century, and of Russell, Tarski and Wittgenstein in the first half of the twentieth century". Cited in Byrne, Alex and Hall, Ned. 2004. 'Necessary Truths'. ''Boston Review'' October/November 2004</ref> [[Deontic logic]]s are closely related to modal logics: they attempt to capture the logical features of [[obligation]], [[Permission (philosophy)|permission]] and related concepts. Although some basic novelties [[syncretism|syncretizing]] mathematical and philosophical logic were shown by [[Bernard Bolzano#Metaphysics|Bolzano]] in the early 1800s, it was [[Ernst Mally]], a pupil of [[Alexius Meinong]], who was to propose the first formal deontic system in his ''Grundgesetze des Sollens'', based on the syntax of Whitehead's and Russell's [[propositional calculus]].

Another logical system founded after World War II was [[fuzzy logic]] by Azerbaijani mathematician [[Lotfi Asker Zadeh]] in 1965.

==See also==
{{Portal|Philosophy}}
* [[History of deductive reasoning]]
* [[History of inductive reasoning]]
* [[History of abductive reasoning]]
* [[History of the function concept]]
* [[History of mathematics]]
* [[Philosophy#History|History of Philosophy]]
* [[Plato's beard]]
* [[Timeline of mathematical logic]]

==Notes==
{{Reflist|2}}

==References==
; Primary Sources
* [[Alexander of Aphrodisias]], ''In Aristotelis An. Pr. Lib. I Commentarium'', ed. Wallies, Berlin, C.I.A.G. vol. II/1, 1882.
* Avicenna, ''Avicennae Opera'' Venice 1508.
* [[Boethius]] ''Commentary on the Perihermenias'', Secunda Editio, ed. Meiser, Leipzig, Teubner, 1880.
* [[Bernard Bolzano|Bolzano, Bernard]] ''Wissenschaftslehre'', (1837) 4 Bde, Neudr., hrsg. W. Schultz, Leipzig I–II 1929, III 1930, IV 1931 (''Theory of Science'', four volumes, translated by Rolf George and Paul Rusnock, New York: Oxford University Press, 2014).
* Bolzano, Bernard ''Theory of Science'' (Edited, with an introduction, by Jan Berg. Translated from the German by Burnham Terrell – [[D. Reidel Publishing Company]], Dordrecht and Boston 1973).
* [[George Boole|Boole, George]] (1847) ''The Mathematical Analysis of Logic'' (Cambridge and London); repr. in ''Studies in Logic and Probability'', ed. [[Rush Rhees|R. Rhees]] (London 1952).
* Boole, George (1854) ''The Laws of Thought'' (London and Cambridge); repr. as ''Collected Logical Works''. Vol. 2, (Chicago and London: [[Open Court Publishing Company|Open Court]], 1940).
* [[Epictetus]], ''Epicteti Dissertationes ab Arriano digestae'', edited by Heinrich Schenkl, Leipzig, Teubner. 1894.
* Frege, G., ''Boole's Logical Calculus and the Concept Script'', 1882, in ''Posthumous Writings'' transl. P. Long and R. White 1969, pp.&nbsp;9–46.
* [[Joseph Diaz Gergonne|Gergonne, Joseph Diaz]], (1816) ''Essai de dialectique rationelle'', in [[Annales de mathématiques pures et appliquées]] 7, 1816/1817, 189–228.
* Jevons, W. S. ''The Principles of Science'', London 1879.
* ''Ockham's Theory of Terms'': Part I of the [[Summa Logicae]], translated and introduced by Michael J. Loux (Notre Dame, IN: [[University of Notre Dame Press]] 1974). Reprinted: South Bend, IN: St. Augustine's Press, 1998.
* ''Ockham's Theory of Propositions'': Part II of the Summa Logicae, translated by Alfred J. Freddoso and Henry Schuurman and introduced by Alfred J. Freddoso (Notre Dame, IN: University of Notre Dame Press, 1980). Reprinted: South Bend, IN: St. Augustine's Press, 1998.
* [[Charles Sanders Peirce|Peirce, C. S.]], (1896), "The Regenerated Logic", ''The Monist'', [https://books.google.com/books?id=pa0LAAAAIAAJ vol. VII], No. 1, p [https://books.google.com/books?id=pa0LAAAAIAAJ&pg=PA19 pp. 19]–40, The Open Court Publishing Co., Chicago, IL, 1896, for the Hegeler Institute. Reprinted (CP 3.425–455). ''Internet Archive'' [https://archive.org/details/monistquart07hegeuoft ''The Monist'' 7].
* [[Sextus Empiricus]], ''Against the Logicians''. (Adversus Mathematicos VII and VIII). Richard Bett (trans.) Cambridge: Cambridge University Press, 2005. {{ISBN|0-521-53195-0}}.
* {{cite journal |author-link=Ernst Zermelo |author-first=Ernst |author-last=Zermelo |date=1908 |title=Untersuchungen über die Grundlagen der Mengenlehre I |journal=Mathematische Annalen |volume=65 |pages=261–281 |doi=10.1007/BF01449999 |issue=2 |s2cid=120085563 |url=http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0065&DMDID=DMDLOG_0018&L=1 |access-date=2013-09-30  |archive-date=2017-09-08  |archive-url=https://web.archive.org/web/20170908192040/http://gdz.sub.uni-goettingen.de/index.php?id=11&PPN=PPN235181684_0065&DMDID=DMDLOG_0018&L=1 |url-status=dead }} English translation in {{cite book |author-link=Jean van Heijenoort |author-first=Jean |author-last=van Heijenoort |date=1967 |title=From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931 |series=Source Books in the History of the Sciences |chapter=Investigations in the foundations of set theory |publisher=Harvard Univ. Press |pages=199–215 |isbn=978-0-674-32449-7}}.
* {{cite book |last=Frege |first=Gottlob |year=1879 |title=Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought}} translated in van Heijenoort 1967.

; Secondary Sources
* [[Jon Barwise|Barwise, Jon]], (ed.), ''Handbook of Mathematical Logic'', Studies in Logic and the Foundations of Mathematics, Amsterdam, North Holland, 1982 {{ISBN|978-0-444-86388-1}} .
* Beaney, Michael, ''The Frege Reader'', London: Blackwell 1997.
* [[Józef Maria Bocheński|Bochenski]], I. M., ''A History of Formal Logic'', Indiana, Notre Dame University Press, 1961.
* [[Philotheus Boehner|Boehner, Philotheus]], ''Medieval Logic'', Manchester 1950.
* {{citation |first=C.B. |last=Boyer |author-link=Carl Benjamin Boyer |title=A History of Mathematics |edition=2nd |place=New York |publisher=Wiley |year=1991 |orig-year=1989 |isbn=978-0-471-54397-8 |url=https://archive.org/details/historyofmathema00boye}}
* Buroker, Jill Vance (transl. and introduction), A. Arnauld, P. Nicole ''Logic or the Art of Thinking'', [[Cambridge University Press]], 1996, {{ISBN|0-521-48249-6}}.
* [[Alonzo Church|Church, Alonzo]], 1936–1938. "A bibliography of symbolic logic". ''Journal of Symbolic Logic 1'': 121–218; ''3'':178–212.  
* [[Everard de Jong|de Jong, Everard]] (1989), ''[[Galileo Galilei]]'s "Logical Treatises" and [[Giacomo Zabarella]]'s "Opera Logica": A Comparison'', PhD dissertation, Washington, DC: Catholic University of America.
* Ebbesen, Sten "Early supposition theory (12th–13th Century)" ''Histoire, Épistémologie, Langage'' 3/1: 35–48 (1981).
*  Farrington, B., ''The Philosophy of [[Francis Bacon]]'', Liverpool 1964.
* Feferman, Anita B. (1999). "Alfred Tarski". ''[[American National Biography]]''. 21. [[Oxford University Press]]. pp.&nbsp;330–332. {{ISBN|978-0-19-512800-0}}.
* {{cite book |author-last1=Feferman |author-first1=Anita B. |author-first2=Solomon |author-last2=Feferman |author-link2=Solomon Feferman |title=Alfred Tarski: Life and Logic |url=https://archive.org/details/alfredtarskilife0000fefe |url-access=registration |date=2004 |publisher=[[Cambridge University Press]] |isbn=978-0-521-80240-6 |oclc=54691904 |ref=F-F}}
* [[Dov Gabbay|Gabbay, Dov]] and [[John Woods (logician)|John Woods]], eds, ''Handbook of the History of Logic'' 2004. 1. Greek, Indian and Arabic logic; 2. Mediaeval and Renaissance logic; 3. The rise of modern logic: from Leibniz to Frege; 4. British logic in the Nineteenth century; 5. Logic from Russell to Church; 6. Sets and extensions in the Twentieth century; 7. Logic and the modalities in the Twentieth century; 8. The many-valued and nonmonotonic turn in logic; 9. Computational Logic; 10. Inductive logic; 11. Logic: A history of its central concepts; [[Elsevier]], {{ISBN|0-444-51611-5}}.
* Geach, P. T. ''Logic Matters'', Blackwell 1972.
* Goodman, Lenn Evan (2003). ''Islamic Humanism''. Oxford University Press, {{ISBN|0-19-513580-6}}.
* Goodman, Lenn Evan (1992). ''Avicenna''. Routledge, {{ISBN|0-415-01929-X}}.
* [[Ivor Grattan-Guinness|Grattan-Guinness, Ivor]], 2000. ''The Search for Mathematical Roots 1870–1940''. [[Princeton University Press]].
* Gracia, J. G. and Noone, T. B., ''A Companion to Philosophy in the Middle Ages'', London 2003.
* [[Leila Haaparanta|Haaparanta, Leila]] (ed.) 2009. ''The Development of Modern Logic'' Oxford University Press.
* [[T. L. Heath|Heath, T. L.]], 1949. ''Mathematics in Aristotle'', Oxford University Press.
* Heath, T. L., 1931, ''A Manual of Greek Mathematics'', Oxford ([[Clarendon Press]]).
* Honderich, Ted (ed.). [[The Oxford Companion to Philosophy]] (New York: Oxford University Press, 1995) {{ISBN|0-19-866132-0}}.
* [[William Kneale (logician)|Kneale, William]] and Martha, 1962. ''The development of logic''. Oxford University Press, {{ISBN|0-19-824773-7}}.
* [[Jan Łukasiewicz|Lukasiewicz]], ''Aristotle's Syllogistic'', Oxford University Press 1951.
* Potter, Michael (2004), ''[https://books.google.com/books?id=FxRoPuPbGgUC&q=logic Set Theory and its Philosophy]'', Oxford University Press.

==External links==
* [https://www.historyoflogic.com The History of Logic from Aristotle to Gödel] with annotated bibliographies on the history of logic
* {{cite SEP |url-id=logic-ancient |title=Ancient Logic |author-last=Bobzien |author-first=Susanne}}
* {{Cite IEP|url-id=av-logic |title=Avicenna (Ibn Sina): Logic |author-first=Saloua |author-last=Chatti}}
* {{cite SEP |url-id=peter-spain |title=Peter of Spain |author-last=Spruyt |author-first=Joke}}
* [http://pvspade.com/Logic/docs/thoughts1_1a.pdf Paul Spade's "Thoughts Words and Things"] – An Introduction to Late Mediaeval Logic and Semantic Theory (PDF)
* [http://humbox.ac.uk/5497/ Open Access pdf download; Insights, Images, Bios, and links for 178 logicians] by David Marans

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