Editing History of logic (section)

===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>