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HandWiki. Converse (Logic). Encyclopedia. Available online: https://encyclopedia.pub/entry/31404 (accessed on 23 June 2024).

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HandWiki. "Converse (Logic)." *Encyclopedia*. Web. 26 October, 2022.

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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 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.

implicational
logic
mathematics

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,^{[1]} unless the 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 true.

On the other hand, 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:

[math]\displaystyle{ P }[/math] | [math]\displaystyle{ Q }[/math] | [math]\displaystyle{ P \rightarrow Q }[/math] | [math]\displaystyle{ P \leftarrow Q }[/math] (converse) |

True | True | True | True |

True | False | False | True |

False | True | True | False |

False | False | True | True |

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 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]\displaystyle{ P }[/math] and [math]\displaystyle{ \neg Q }[/math]

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

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.^{[2]}

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:

Givena triangle with sides of length[math]\displaystyle{ a }[/math],[math]\displaystyle{ b }[/math], and[math]\displaystyle{ c }[/math],ifthe angle opposite the side of length[math]\displaystyle{ c }[/math]is a right angle,then[math]\displaystyle{ a^2 + b^2 = c^2 }[/math].

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

Givena triangle with sides of length[math]\displaystyle{ a }[/math],[math]\displaystyle{ b }[/math], and[math]\displaystyle{ c }[/math],if[math]\displaystyle{ a^2 + b^2 = c^2 }[/math],thenthe angle opposite the side of length[math]\displaystyle{ c }[/math]is a right angle.

If [math]\displaystyle{ R }[/math] is a binary relation with [math]\displaystyle{ R \subseteq A \times B, }[/math] then the converse relation [math]\displaystyle{ R^T = \{ (b,a) : (a,b) \in R \} }[/math] is also called the **transpose**.^{[3]}

The converse of the implication *P* → *Q* may be written *Q* → *P*, [math]\displaystyle{ P \leftarrow Q }[/math], but may also be notated [math]\displaystyle{ P \subset Q }[/math], or "B*pq*" (in Bocheński notation).

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:

"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."

^{[4]}

The "exposita" is more usually called the "convertend." In its simple form, conversion is valid only for **E** and **I** propositions:^{[5]}

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."^{[6]} For **E** propositions, both subject and predicate are 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 syllogisms, 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 predicate calculus, *All S are P* can be represented as [math]\displaystyle{ \forall x. S(x) \to P(x) }[/math].^{[7]} 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*.

- Taylor, Courtney. "What Are the Converse, Contrapositive, and Inverse?" (in en). https://www.thoughtco.com/converse-contrapositive-and-inverse-3126458.
- Shonkwiler, Clay (October 6, 2006). "The Four Vertex Theorem and its Converse". https://www.math.colostate.edu/~clayton/research/talks/FourVertexPrint.pdf.
- Gunther Schmidt & Thomas Ströhlein (1993) Relations and Graphs, page 9, Springer books
- Asa Mahan (1857) The Science of Logic: or, An Analysis of the Laws of Thought, p. 82. https://books.google.com/books?id=J_wtAAAAMAAJ&pg=PA82
- William Thomas Parry and Edward A. Hacker (1991), Aristotelian Logic, SUNY Press, p. 207. https://books.google.com/books?id=3Sg84H6B-m4C&pg=PA207
- James H. Hyslop (1892), The Elements of Logic, C. Scribner's sons, p. 156.
- Gordon Hunnings (1988), The World and Language in Wittgenstein's Philosophy, SUNY Press, p. 42. https://books.google.com/books?id=5XXz7B2PLRsC&pg=PA42

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