Editing History of logic (section)

==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>