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logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
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 In logic, a categorical proposition, or categorical statement, is a proposition that asserts or denies that all or some of the members of one category (the ''subject term'') are included in another (the ''predicate term''). The study of arguments ...
''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.Robert Audi, ed. (1999), ''The Cambridge Dictionary of Philosophy'', 2nd ed., Cambridge University Press: "converse".


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, unless the antecedent ''P'' and the
consequent A consequent is the second half of a hypothetical proposition. In the standard form of such a proposition, it is the part that follows "then". In an implication, if ''P'' implies ''Q'', then ''P'' is called the antecedent and ''Q'' is called ...
''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 True most commonly refers to truth, the state of being in congruence with fact or reality. True may also refer to: Places * True, West Virginia, an unincorporated community in the United States * True, Wisconsin, a town in the United States * ...
. 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: 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 In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicond ...
''Q'' is also true), then affirming the consequent will be valid. Converse implication is logically equivalent to the disjunction of P and \neg Q 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 The four-vertex theorem of geometry states that the curvature along a simple, closed, smooth plane curve has at least four local extrema (specifically, at least two local maxima and at least two local minima). The name of the theorem derives from ...
was proved in 1912, but its converse was proved only in 1997. 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 In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite ...
can be stated as:
''Given'' a triangle with sides of length ''a'', ''b'', and ''c'', ''if'' the angle opposite the side of length ''c'' is a right angle, ''then'' a^2 + b^2 = c^2.
The converse, which also appears in Euclid's ''Elements'' (Book I, Proposition 48), can be stated as:
''Given'' a triangle with sides of length ''a'', ''b'', and ''c'', ''if'' a^2 + b^2 = c^2, ''then'' the angle opposite the side of length ''c'' is a right angle.


Converse of a relation

If R is a binary relation with R \subseteq A \times B, then the converse relation R^T = \ is also called the transpose.


Notation

The converse of the implication ''P'' → ''Q'' may be written ''Q'' → ''P'', P \leftarrow Q, but may also be notated P \subset Q, or "B''pq''" (in Bocheński notation).


Categorical converse

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 Asa Mahan (; November 9, 1799April 4, 1889) was a U.S. Congregational clergyman and educator and the first president of both the Oberlin Collegiate Institute (later Oberlin College) and Adrian College. He described himself as "a religious teacher ...
:
"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."
The "exposita" is more usually called the "convertend." In its simple form, conversion is valid only for E and I propositions: 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." For E propositions, both subject and predicate are
distributed Distribution may refer to: Mathematics *Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations *Probability distribution, the probability of a particular value or value range of a varia ...
, 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 A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true. ...
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 predicate calculus First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
, ''All S are P'' can be represented as \forall x. S(x) \to P(x).Gordon Hunnings (1988), ''The World and Language in Wittgenstein's Philosophy'', SUNY Press
p. 42
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

* Aristotle * Categorical proposition#Conversion * Contraposition * Converse (semantics) *
Inference Inferences are steps in reasoning, moving from premises to logical consequences; etymologically, the word ''infer'' means to "carry forward". Inference is theoretically traditionally divided into deduction and induction, a distinction that in ...
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Inverse (logic) In logic, an inverse is a type of conditional sentence which is an immediate inference made from another conditional sentence. More specifically, given a conditional sentence of the form P \rightarrow Q , the inverse refers to the sentence \neg P ...
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Logical connective In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary co ...
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Obversion In traditional logic, obversion is a "type of immediate inference in which from a given proposition another proposition is inferred whose subject is the same as the original subject, whose predicate is the contradictory of the original predicate, ...
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Syllogism A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true. ...
*
Term logic In philosophy, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to formal logic that began with Aristotle and was developed further in ancient history mostly by his followers, t ...
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Transposition (logic) In propositional logic, transposition is a valid rule of replacement that permits one to switch the antecedent with the consequent of a conditional statement in a logical proof if they are also both negated. It is the inference from the tru ...


References


Further reading

* 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. {{Logical connectives Logical connectives Immediate inference