Editing Converse (logic) (section)

==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 of 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>