View source for Converse (logic)

{{Short description|Reverse of a categorical or hypothetical proposition}}
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In [[logic]] and [[mathematics]], the '''converse''' of a categorical or implicational statement is the result of reversing its two constituent statements. For the [[Material conditional|implication]] ''P'' → ''Q'', the converse is ''Q'' → ''P''. For the [[categorical proposition]] ''All S are P'', the converse is ''All P are S''. Either way, the truth of the converse is generally independent from that of the original statement.<ref name="Audi">Robert Audi, ed. (1999), ''The Cambridge Dictionary of Philosophy'', 2nd ed., Cambridge University Press: "converse".</ref>

==Implicational converse==
[[File:Venn1101.svg|220px|thumb|[[Venn diagram]] of <math>P \leftarrow Q</math> <br> <small>The white area shows where the statement is false.</small>]]

Let ''S'' be a statement of the form ''P implies Q'' (''P'' → ''Q''). Then the ''converse'' of ''S'' is the statement ''Q implies P'' (''Q'' → ''P''). In general, the truth of ''S'' says nothing about the truth of its converse,<ref>{{Cite web|url=https://www.thoughtco.com/converse-contrapositive-and-inverse-3126458|title=What Are the Converse, Contrapositive, and Inverse?|last=Taylor|first=Courtney|website=ThoughtCo|language=en|access-date=2019-11-27}}</ref> unless the [[Antecedent (logic)|antecedent]] ''P'' and the [[consequent]] ''Q'' are logically equivalent.

For example, consider the true statement "If I am a human, then I am mortal." The converse of that statement is "If I am mortal, then I am a human," which is not necessarily [[logical truth|true]].

However, the converse of a statement with mutually inclusive terms remains true, given the truth of the original proposition. This is equivalent to saying that the converse of a definition is true. Thus, the statement "If I am a triangle, then I am a three-sided polygon" is logically equivalent to "If I am a three-sided polygon, then I am a triangle," because the definition of "triangle" is "three-sided polygon".

A truth table makes it clear that ''S'' and the converse of ''S'' are not logically equivalent, unless both terms imply each other:

{{2-ary truth table|A=P|B=Q
|1|1|0|1|<math> P \rightarrow Q</math>
|thick
|1|0|1|1|<math> P \leftarrow Q</math> (converse)
}}

Going from a statement to its converse is the fallacy of [[affirming the consequent]]. However, if the statement ''S'' and its converse are equivalent (i.e., ''P'' is true [[iff|if and only if]] ''Q'' is also true), then affirming the consequent will be valid.

Converse implication is logically equivalent to the disjunction of <math>P</math> and <math>\neg Q</math>

{| style="text-align: center; border: 1px solid darkgray;"
|-
| <math>P \leftarrow Q</math>
| <math>\Leftrightarrow</math>  
| <math>P</math>
| <math>\lor</math>
| <math>\neg Q</math>
|-
| [[File:Venn1101.svg|50px]]
| <math>\Leftrightarrow</math>  
| [[File:Venn0101.svg|50px]]
| <math>\lor</math>
| [[File:Venn1100.svg|50px]]
|}

In natural language, this could be rendered "not ''Q'' without ''P''".

===Converse of a theorem===
In mathematics, the converse of a theorem of the form ''P'' → ''Q'' will be ''Q'' → ''P''. The converse may or may not be true, and even if true, the proof may be difficult. For example, the [[four-vertex theorem]] was proved in 1912, but its converse was proved only in 1997.<ref>{{Cite web|url=https://www.math.colostate.edu/~clayton/research/talks/FourVertexPrint.pdf|title=The Four Vertex Theorem and its Converse|last=Shonkwiler|first=Clay|date=October 6, 2006|website=math.colostate.edu|access-date=2019-11-26}}</ref>

In practice, when determining the converse of a mathematical theorem, aspects of the antecedent may be taken as establishing context. That is, the converse of "Given P, if Q then R''"'' will be "Given P, if R then Q''"''. For example, the [[Pythagorean theorem]] can be stated as:

<blockquote>
''Given'' a triangle with sides of length ''<math>a</math>'', ''<math>b</math>'', and ''<math>c</math>'', ''if'' the angle opposite the side of length ''<math>c</math>'' is a right angle, ''then'' <math>a^2 + b^2 = c^2</math>'''.'''
</blockquote>

The converse, which also appears in [[Euclid's Elements|Euclid's ''Elements'']] (Book I, Proposition 48), can be stated as:

<blockquote>
''Given'' a triangle with sides of length ''<math>a</math>'', ''<math>b</math>'', and ''<math>c</math>'', ''if'' <math>a^2 + b^2 = c^2</math>, ''then'' the angle opposite the side of length ''<math>c</math>'' is a right angle.
</blockquote>

===Converse of a relation===
[[File:An example of converse property.png|thumb|Converse a simple mathematical relation]]
If <math>R</math> is a [[binary relation]] with <math>R \subseteq A \times B,</math> then the [[converse relation]] <math>R^T = \{ (b,a) : (a,b) \in R \}</math> is also called the ''transpose''.<ref>[[Gunther Schmidt]] & Thomas Ströhlein (1993) ''Relations and Graphs'', page 9, [[Springer books]]</ref>

==Notation==
The converse of the implication ''P'' → ''Q'' may be written ''Q'' → ''P'', <math>P \leftarrow Q</math>, but may also be notated <math>P \subset Q</math>, or "B''pq''" (in [[Józef Maria Bocheński|Bocheński notation]]).{{cn|date=March 2020}}

==Categorical converse==
{{see also|Categorical proposition#Conversion}}

In traditional logic, the process of switching the subject term with the predicate term is called ''conversion''. For example, going from "No ''S'' are ''P"'' to its converse "No ''P'' are ''S"''. In the words of [[Asa Mahan]]: <blockquote>"The original proposition is called the exposita; when converted, it is denominated the converse. Conversion is valid when, and only when, nothing is asserted in the converse which is not affirmed or implied in the exposita."<ref>[[Asa Mahan]] (1857) ''The Science of Logic: or, An Analysis of the Laws of Thought'', [https://books.google.com/books?id=J_wtAAAAMAAJ&pg=PA82 p. 82].</ref> </blockquote>The "exposita" is more usually called the "convertend". In its simple form, conversion is valid only for '''E''' and '''I''' propositions:<ref>William Thomas Parry and Edward A. Hacker (1991), ''Aristotelian Logic'', SUNY Press, [https://books.google.com/books?id=3Sg84H6B-m4C&pg=PA207 p. 207].</ref>

{| class="wikitable"
! Type || Convertend || Simple converse || Converse ''per accidens'' (valid if P exists)
|-
|  '''A''' || All S are P || ''not valid'' || Some P is S
|-
|  '''E''' || No S is P || No P is S || Some P is not S
|-
|  '''I''' || Some S is P || Some P is S || –
|-
|  '''O''' || Some S is not P || ''not valid'' || –
|}

The validity of simple conversion only for '''E''' and '''I''' propositions can be expressed by the restriction that "No term must be distributed in the converse which is not distributed in the convertend."<ref>[[James H. Hyslop]] (1892), ''The Elements of Logic'', C. Scribner's sons, p. 156.</ref> For '''E''' propositions, both subject and predicate are [[Distribution of terms|distributed]], while for '''I''' propositions, neither is.

For '''A''' propositions, the subject is distributed while the predicate is not, and so the inference from an '''A''' statement to its converse is not valid. As an example, for the '''A''' proposition "All cats are mammals", the converse "All mammals are cats" is obviously false. However, the weaker statement "Some mammals are cats" is true. Logicians define conversion ''per accidens'' to be the process of producing this weaker statement. Inference from a statement to its converse ''per accidens'' is generally valid. However, as with [[syllogism]]s, this switch from the universal to the particular causes problems with empty categories: "All unicorns are mammals" is often taken as true, while the converse ''per accidens'' "Some mammals are unicorns" is clearly false.

In [[First-order logic|first-order predicate calculus]], ''All S are P'' can be represented as <math>\forall x. S(x) \to P(x)</math>.<ref>Gordon Hunnings (1988), ''The World and Language in Wittgenstein's Philosophy'', SUNY Press, [https://books.google.com/books?id=5XXz7B2PLRsC&pg=PA42 p. 42].</ref> It is therefore clear that the categorical converse is closely related to the implicational converse, and that ''S'' and ''P'' cannot be swapped in ''All S are P''.

==See also==
{{Portal|Philosophy}}
* [[Aristotle]]
* [[Contraposition]]
* [[Inverse (logic)]]
* [[Logical connective]]
* [[Obversion]]
* [[Term logic]]
* [[Transposition (logic)]]

==References==
{{reflist}}

==Further reading==
* [[Aristotle]]. ''Organon''.
* [[Irving Copi|Copi, Irving]]. ''Introduction to Logic''.  MacMillan, 1953.
* Copi, Irving. ''Symbolic Logic''.  MacMillan, 1979, fifth edition.
* [[Susan Stebbing|Stebbing, Susan]]. ''A Modern Introduction to Logic''. Cromwell Company, 1931.

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