Editing Corollary

{{short description|Secondary statement which can be readily deduced from a previous, more notable statement}}
In [[mathematics]] and [[logic]], a '''corollary''' ({{IPAc-en|US|ˈ|k|ɒr|ə|ˌ|l|e@r|i}} {{respell|KORR|ə|lair|ee}}, {{IPAc-en|uk|k|@|ˈ|r|ɒ|l|ər|i}} {{respell|kər|OL|ər|ee}}) is a [[theorem]] of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a [[proposition]] which is incidentally proved while proving another proposition;<ref>{{Cite web|url=https://www.dictionary.com/browse/corollary|title=Definition of corollary|website=www.dictionary.com|language=en|access-date=2019-11-27}}</ref> it might also be used more casually to refer to something which naturally or incidentally accompanies something else.<ref>{{Cite web|url=https://www.merriam-webster.com/dictionary/corollary|title=Definition of COROLLARY|website=www.merriam-webster.com|language=en|access-date=2019-11-27}}</ref><ref>{{Cite web|url=https://dictionary.cambridge.org/dictionary/english/corollary|title=COROLLARY|website=dictionary.cambridge.org|language=en|access-date=2019-11-27}}</ref>

==Overview==
In [[mathematics]], a corollary is a theorem connected by a short proof to an existing theorem. The use of the term ''corollary'', rather than ''[[proposition]]'' or ''theorem'', is intrinsically subjective. More formally, proposition ''B'' is a corollary of proposition ''A'', if ''B'' can be readily deduced from ''A'' or is self-evident from its proof. 

In many cases, a corollary corresponds to a special case of a larger theorem,<ref>{{Cite web|url=https://www.mathwords.com/c/corollary.htm|title=Mathwords: Corollary|website=www.mathwords.com|access-date=2019-11-27}}</ref> which makes the theorem easier to use and apply,<ref>{{Cite web|url=http://mathworld.wolfram.com/Corollary.html|title=Corollary|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-11-27}}</ref> even though its importance is generally considered to be secondary to that of the theorem. In particular, ''B'' is unlikely to be termed a corollary if its mathematical consequences are as significant as those of ''A''. A corollary might have a proof that explains its derivation, even though such a derivation might be considered rather self-evident in some occasions<ref>{{Cite book|url=https://books.google.com/books?id=6WIMAAAAYAAJ&pg=PA260|title=Chambers's Encyclopaedia|date=1864|publisher=Appleton|volume=3|pages=260|language=en}}</ref> (e.g., the [[Pythagorean theorem]] as a corollary of [[law of cosines]]<ref>{{Cite web|url=https://www.mathwords.com/c/corollary.htm|title=Mathwords: Corollary|website=www.mathwords.com|access-date=2019-11-27}}</ref>).

==Peirce's theory of deductive reasoning<!--'Peirce's theory of deductive reasoning' redirects here-->==
[[Charles Sanders Peirce]] held that the most important division of kinds of [[deductive reasoning]] is that between corollarial and theorematic. He argued that while all deduction ultimately depends in one way or another on mental experimentation on schemata or diagrams,<ref name=minute>Peirce, C. S., from section dated 1902 by editors in the "Minute Logic" manuscript, ''[[Charles Sanders Peirce bibliography#CP|Collected Papers]]'' v. 4, paragraph 233, quoted in part in "[http://www.helsinki.fi/science/commens/terms/corollarial.html Corollarial Reasoning]" in the ''Commons Dictionary of Peirce's Terms'', 2003–present, Mats Bergman and Sami Paavola, editors, University of Helsinki.</ref> in corollarial deduction:

"It is only necessary to imagine any case in which the premises are true in order to perceive immediately that the conclusion holds in that case"

while in theorematic deduction:

"It is necessary to experiment in the imagination upon the image of the premise in order from the result of such experiment to make corollarial deductions to the truth of the conclusion."<ref>Peirce, C. S., the 1902 Carnegie Application, published in ''[[Charles Sanders Peirce bibliography#NEM|The New Elements of Mathematics]]'', Carolyn Eisele, editor, also transcribed by [[Joseph Morton Ransdell|Joseph M. Ransdell]], see <!-- NEXT TWO HYPHENS IN TEXT ARE NEEDED FOR BROWSER SEARCH AT LINKED SITE -->"From Draft A – MS L75.35–39" in [http://www.cspeirce.com/menu/library/bycsp/l75/ver1/l75v1-06.htm#m19 Memoir 19] (once there, scroll down).</ref>

Peirce also held that corollarial deduction matches Aristotle's conception of direct demonstration, which Aristotle regarded as the only thoroughly satisfactory demonstration, while theorematic deduction is:

# The kind more prized by mathematicians
# Peculiar to mathematics<ref name="minute" /> 
# Involves in its course the introduction of a [[Lemma (mathematics)|lemma]] or at least a definition uncontemplated in the thesis (the proposition that is to be proved), in remarkable cases that definition is of an abstraction that "ought to be supported by a proper postulate."<ref>Peirce, C. S., 1901 manuscript "On the Logic of Drawing History from Ancient Documents, Especially from Testimonies', ''[[Charles Sanders Peirce bibliography#EP|The Essential Peirce]]'' v. 2, see p. 96. See quote in "[http://www.helsinki.fi/science/commens/terms/corollarial.html Corollarial Reasoning]" in the ''Commens Dictionary of Peirce's Terms''.</ref>

==See also==
{{wiktionary}}
* [[Lemma (mathematics)]]
* [[Porism]]
*[[Proposition]]
* [[Lodge Corollary]] to the [[Monroe Doctrine]]
* [[Roosevelt Corollary]] to the Monroe Doctrine

==References==
{{Reflist}}

==Further reading==
* [https://www.cut-the-knot.org/pythagoras/corollary.shtml Cut the knot: Sample corollaries of the Pythagorean theorem]
* [https://www.geeksforgeeks.org/corollaries-binomial-theorem/ Geeks for geeks: Corollaries of binomial theorem]

[[Category:Mathematical terminology]]
[[Category:Theorems]]
[[Category:Statements]]