2 b , is a right angle, then B ⊆ In the words of Asa Mahan: "The original proposition is called the exposita; when converted, it is denominated the converse. 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. b For example, the converse of "If it is raining then the grass is wet" is "If the grass is wet then it is raining." Converse implication is logically equivalent to the disjunction of , if R P , or "Bpq" (in Bocheński notation). a ( ) Converse. "[8] For E propositions, both subject and predicate are distributed, while for I propositions, neither is. { {\displaystyle a} T × Example: "if you are a dog then you bark". a In traditional logic, the process of going from "All S are P" to its converse "All P are S" is called conversion. In practice, when determining the converse of a mathematical theorem, aspects of the antecedent may be taken as establishing context. [5], The converse of the implication P → Q may be written Q → P, Switching the hypothesis and conclusion of a conditional statement. The converse, which also appears in Euclid's Elements (Book I, Proposition 48), can be stated as: Given a triangle with sides of length then the converse relation R {\displaystyle a} 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. + , then the angle opposite the side of length For example, the Pythagorean theorem can be stated as: Given a triangle with sides of length Logicians define conversion per accidens to be the process of producing this weaker statement. 2 On the other hand, the converse of a statement with mutually inclusive terms remains true, given the truth of the original proposition. 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". {\displaystyle P\subset Q} c Then the converse of S is the statement Q implies P (Q → P). a : That is, the converse of "Given P, if Q then R" will be "Given P, if R then Q". R {\displaystyle \neg Q}. {\displaystyle R\subseteq A\times B,} ∀ , and S {\displaystyle c} is also called the transpose. ⊂ , if the angle opposite the side of length c , In Mathematical Geometry, a Converse is defined as the inverse of a conditional statement. 2 a , } b P ¬ The converse of that statement is "If I am mortal, then I am a human," which is not necessarily true. = . In logic and mathematics, the converse of a categorical or implicational statement is the result of reversing its two constituent statements. c {\displaystyle R^{T}=\{(b,a):(a,b)\in R\}} {\displaystyle c} Note: As in the example, a proposition may be true but have a false converse. ( Learn what is converse. Consider the statement, If it is raining, then the grass is wet. Q "[6], The "exposita" is more usually called the "convertend." If 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. A conditional statement ("if ... then ...") made by swapping the "if" and "then" parts of another statement. {\displaystyle P} Conversion is valid when, and only when, nothing is asserted in the converse which is not affirmed or implied in the exposita. ( ← 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. This is equivalent to saying that the converse of a definition is true. is a binary relation with {\displaystyle a^{2}+b^{2}=c^{2}} ∈ b However, the weaker statement "Some mammals are cats" is true.