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{{short description|Secondary statement that can be readily deduced from a previous, more notable statement or its proof}}
{{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 that can be readily deduced from a previous, more notable statement or from the proof of that other 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>
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==
==Overview==
In [[mathematics]], a corollary is a theorem connected by a short proof to an existing theorem or proof. 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. (The latter case is relevant when, for example, the theorem itself is not enough to imply the corollary, but the argument made to prove the theorem actually shows something more.)
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>).
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>).
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[[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:
[[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"
"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:
while in theorematic deduction:

Latest revision as of 10:46, 28 July 2024

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

[edit]

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

[edit]

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:

  1. The kind more prized by mathematicians
  2. Peculiar to mathematics[8]
  3. 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]

See also

[edit]
Look up corollary in Wiktionary, the free dictionary.

References

[edit]
  1. ^ "Definition of corollary". www.dictionary.com. Retrieved 2019-11-27.
  2. ^ "Definition of COROLLARY". www.merriam-webster.com. Retrieved 2019-11-27.
  3. ^ "COROLLARY". dictionary.cambridge.org. Retrieved 2019-11-27.
  4. ^ "Mathwords: Corollary". www.mathwords.com. Retrieved 2019-11-27.
  5. ^ Weisstein, Eric W. "Corollary". mathworld.wolfram.com. Retrieved 2019-11-27.
  6. ^ Chambers's Encyclopaedia. Vol. 3. Appleton. 1864. p. 260.
  7. ^ "Mathwords: Corollary". www.mathwords.com. Retrieved 2019-11-27.
  8. ^ 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.
  9. ^ 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).
  10. ^ 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.

Further reading

[edit]
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