In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
, a corollary ( , ) is a
theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...
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; it might also be used more casually to refer to something which naturally or incidentally accompanies something else (e.g., violence as a corollary of revolutionary social changes).
Overview
In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a corollary is a theorem connected by a short proof to an existing theorem. The use of the term ''corollary'', rather than ''
proposition
In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...
'' 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, which makes the theorem easier to use and apply, 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 (e.g., the
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
as a corollary of
law of cosines
In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states ...
).
Peirce's theory of deductive reasoning
Charles Sanders Peirce
Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American philosopher, logician, mathematician and scientist who is sometimes known as "the father of pragmatism".
Educated as a chemist and employed as a scientist for t ...
held that the most important division of kinds of
deductive reasoning
Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. if it is impossible for the premises to be true and the conclusion to be fals ...
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,
[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. 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."
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
# Involves in its course the introduction of a
lemma
Lemma may refer to:
Language and linguistics
* Lemma (morphology), the canonical, dictionary or citation form of a word
* Lemma (psycholinguistics), a mental abstraction of a word about to be uttered
Science and mathematics
* Lemma (botany), a ...
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."
[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''.
See also
*
Lemma (mathematics)
In mathematics, informal logic and argument mapping, a lemma (plural lemmas or lemmata) is a generally minor, proven proposition which is used as a stepping stone to a larger result. For that reason, it is also known as a "helping theorem" or an ...
*
Porism
A porism is a mathematical proposition or corollary. It has been used to refer to a direct consequence of a Mathematical proof, proof, analogous to how a corollary refers to a direct consequence of a theorem. In modern usage, it is a relationship ...
*
Proposition
In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...
*
Lodge Corollary The Lodge Corollary was a corollary to the Monroe Doctrine. Proposed by Henry Cabot Lodge and ratified by the United States Senate, U.S. Senate in 1912, it forbade any foreign power or foreign interest of any kind from acquiring sufficient territory ...
to the
Monroe Doctrine
The Monroe Doctrine was a United States foreign policy position that opposed European colonialism in the Western Hemisphere. It held that any intervention in the political affairs of the Americas by foreign powers was a potentially hostile ac ...
*
Roosevelt Corollary
In the history of United States foreign policy, the Roosevelt Corollary was an addition to the Monroe Doctrine articulated by President Theodore Roosevelt in his State of the Union address in 1904 after the Venezuelan crisis of 1902–1903. Th ...
to the Monroe Doctrine
References
{{Reflist
Further reading
Cut the knot: Sample corollaries of the Pythagorean theoremGeeks for geeks: Corollaries of binomial theorem
Mathematical terminology
Theorems
Statements