# Converse (logic)

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.[1][2]

## Implicational converse

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][3], 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:

P | Q | P → Q | Q → P (converse) |
---|---|---|---|

T | T | T | T |

T | F | F | T |

F | T | T | F |

F | F | T | T |

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 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.[4]

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,, and,ifthe angle opposite the side of lengthis a right angle,then.

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

Givena triangle with sides of length,, and,if,thenthe angle opposite the side of lengthis a right angle.

### Converse of a relation

If is a binary relation with then the converse relation is also called the **transpose**.[5]

## Categorical converse

In traditional logic, the process of going from "All *S* are *P"* to its converse "All *P* are *S"* is called **conversion**. 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."[6]

The "exposita" is more usually called the "convertend." In its simple form, conversion is valid only for **E** and **I** propositions:[7]

Type | Convertend | Simple converse | Converse per accidens |
---|---|---|---|

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."[8] 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 .[9] 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

## References

- "The Definitive Glossary of Higher Mathematical Jargon — Converse".
*Math Vault*. 2019-08-01. Retrieved 2019-11-27. - Robert Audi, ed. (1999),
*The Cambridge Dictionary of Philosophy*, 2nd ed., Cambridge University Press: "converse". - Taylor, Courtney. "What Are the Converse, Contrapositive, and Inverse?".
*ThoughtCo*. Retrieved 2019-11-27. - Shonkwiler, Clay (October 6, 2006). "The Four Vertex Theorem and its Converse" (PDF).
*math.colostate.edu*. Retrieved 2019-11-26. - 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. - William Thomas Parry and Edward A. Hacker (1991),
*Aristotelian Logic*, SUNY Press, p. 207. - 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.

## Further readings

- Aristotle.
*Organon*. - Copi, Irving.
*Introduction to Logic*. MacMillan, 1953. - Copi, Irving.
*Symbolic Logic*. MacMillan, 1979, fifth edition. - Stebbing, Susan.
*A Modern Introduction to Logic*. Cromwell Company, 1931.