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In mathematics and logic, a corollary (US: /ˈkɒrəˌlɛəri/ KORR-ə-lair-ee, UK: /kəˈrɒləri/ 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;^[1] it might also be used more casually to refer to something which naturally or incidentally accompanies something else.^[2]^[3]

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,^[4] which makes the theorem easier to use and apply,^[5] 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^[6] (e.g., the Pythagorean theorem as a corollary of law of cosines^[7]).

Peirce's theory of deductive reasoning

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,^[8] 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."^[9]

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^[8]
Involves in its course the introduction of a 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."^[10]

References

^ "Definition of corollary". www.dictionary.com. Retrieved 2019-11-27.
^ "Definition of COROLLARY". www.merriam-webster.com. Retrieved 2019-11-27.
^ "COROLLARY". dictionary.cambridge.org. Retrieved 2019-11-27.
^ "Mathwords: Corollary". www.mathwords.com. Retrieved 2019-11-27.
^ Weisstein, Eric W. "Corollary". mathworld.wolfram.com. Retrieved 2019-11-27.
^ Chambers's Encyclopaedia. Vol. 3. Appleton. 1864. p. 260.
^ "Mathwords: Corollary". www.mathwords.com. Retrieved 2019-11-27.
^ ^a ^b Peirce, C. S., from section dated 1902 by editors in the "Minute Logic" manuscript, Collected Papers v. 4, paragraph 233, quoted in part in "Corollarial Reasoning" in the Commons Dictionary of Peirce's Terms, 2003–present, Mats Bergman and Sami Paavola, editors, University of Helsinki.
^ Peirce, C. S., the 1902 Carnegie Application, published in The New Elements of Mathematics, Carolyn Eisele, editor, also transcribed by Joseph M. Ransdell, see "From Draft A – MS L75.35–39" in Memoir 19 (once there, scroll down).
^ Peirce, C. S., 1901 manuscript "On the Logic of Drawing History from Ancient Documents, Especially from Testimonies', The Essential Peirce v. 2, see p. 96. See quote in "Corollarial Reasoning" in the Commens Dictionary of Peirce's Terms.

@@ Line 1: / Line 1: @@
+{{short description|Secondary statement which can be readily deduced from a previous, more notable statement}}
-A '''corollary''' ({{IPAc-en|ˈ|k|ɒr|ə|ˌ|l|ɛr|i}} {{respell|KORR|ə|lerr|ee}}, {{IPAc-en|uk|k|ɒ|ˈ|r|ɒ|l|ər|i}} {{respell|korr|OL|ər|ee}}) is a statement that [[logical consequence|follows readily]] from a previous 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.<ref>{{cite book|last=Wolfram|first=Stephen|title=A New Kind of Science|publisher=Wolfram Media, Inc.|year=2002|page=1176|isbn=1-57955-008-8|url-access=registration|url=https://archive.org/details/newkindofscience00wolf}}</ref> The use of the term ''corollary'', rather than ''[[proposition]]'' or ''theorem'', is intrinsically subjective. Proposition ''B'' is a corollary of proposition ''A'' if ''B'' can be readily deduced from ''A'' or is self-evident from its proof. The importance of the corollary is often considered secondary to that of the initial theorem; ''B'' is unlikely to be termed a corollary if its mathematical consequences are as significant as those of ''A''. Sometimes a corollary has a proof that explains the derivation; sometimes the derivation is considered self-evident.
+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 finally all deduction 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> still 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",  whereas theorematic deduction "is deduction in which 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> He held that corollarial deduction matches Aristotle's conception of direct demonstration, which Aristotle regarded as the only thoroughly satisfactory demonstration, while theorematic deduction (A) is the kind more prized by mathematicians, (B) is peculiar to mathematics,<ref name=minute/> and (C) 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>
+[[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==
@@ Line 11: / Line 26: @@
 * [[Lemma (mathematics)]]
 * [[Porism]]
+*[[Proposition]]
 * [[Lodge Corollary]] to the [[Monroe Doctrine]]
 * [[Roosevelt Corollary]] to the Monroe Doctrine
-==Notes==
+==References==
 {{Reflist}}
-==References==
+==Further reading==
+* [https://www.cut-the-knot.org/pythagoras/corollary.shtml Cut the knot: Sample corollaries of the Pythagorean theorem]
-*{{MathWorld|urlname=Corollary|title=Corollary}}
+* [https://www.geeksforgeeks.org/corollaries-binomial-theorem/ Geeks for geeks: Corollaries of binomial theorem]
-*[http://dictionary.reference.com/browse/corollary ''corollary''] at dictionary.com
-*[[Chambers's Encyclopaedia]]. Volume 3, Appleton 1864, p.&nbsp;260 ({{Google books|6WIMAAAAYAAJ|online copy|page=260}})
 [[Category:Mathematical terminology]]