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In logic, the converse of a categorical or implicational statement is the result of reversing its two parts. 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. In neither case does the converse necessarily follow from the original statement.^[1]

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 verity of S says nothing about the verity of its converse, 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., if 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. If true, the proof may be difficult. For example, the Four-vertex theorem was proved in 1912, but its converse only in 1998.

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:

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 a² + b² = c².

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 a² + b² = c², then the angle opposite the side of length c is a right angle.

Converse of a relation

If R is a binary relation, $R\subset A\times B,$ then the inverse relation $R^{T}=\{(b,a):(a,b)\in R\}$ is also called the converse or transpose.^[2]

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."^[3] The "exposita" is more usually called the "convertend." In its simple form, conversion is valid only for E and I propositions:^[4]

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."^[5] 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 $\forall x.S(x)\to P(x)$ .^[6] 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.

Material or logical implication

In applied logics, the symbolic notation Q → P is known to be equivalent to P <-- Q, and in an if and only if statement it is said that Q exists only if P exists.
In the aristotelian logic and its applied metaphisics, the two notations have a different meaning, e.g. in the two following statements:

if the creation exists, then God exists,
if God exists, then the creation exists.

If the first statement (P → Q) is believed to be true, and the second (Q → P) is false, in respect of boolean logic terminology we are concluding that creation is a sufficient but not necessary condition of the existence of God.

Instead of this, the logical implication (Q<--P, the knowledge of Q follows the knowledge of P: if the creation is or is in a such way, then we shall demonstrate and know the existence of God) is far different from the material implication (Q &rarr P, the real or measurable existence of P begins after the existence of Q: the creation is the cause of God), which implies the effect begins to exist in a time after the existence of its cause (God after the creation). The logical status of those two propositions is called contingency.

Referring to the history and the so called chronological snobbery, a relevant example of that two tipes of implication, is the following: if the present knows the past and would not repeat the same errors, humans are standing on the shoulders of giants.

References

^ Robert Audi, ed. (1999), The Cambridge Dictionary of Philosophy, 2nd ed., Cambridge University Press: "converse".
^ 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.

@@ Line 62: / Line 62: @@
 In [[First-order logic|first-order predicate calculus]], ''All S are P'' can be represented as <math>\forall x. S(x) \to P(x)</math>.<ref>Gordon Hunnings (1988), ''The World and Language in Wittgenstein's Philosophy'', SUNY Press, [https://books.google.com/books?id=5XXz7B2PLRsC&pg=PA42 p. 42].</ref> 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''.
+== Material or logical implication ==
+In applied logics, the symbolic notation ''Q'' &rarr; ''P'' is known to be equivalent to ''P'' <-- ''Q'', and in an [[if and only if]] statement it is said that ''Q'' exists only if ''P'' exists. <br />
+In the [[term logic|aristotelian logic]] and its applied [[metaphisics]], the two notations have a different meaning, e.g. in the two following statements:
+# if the [[Ex nihilo|creation]] exists, then [[Existence of God|God exists]],
+# if God exists, then the creation exists.
+If the first statement (''P'' &rarr; ''Q'') is believed to be true, and the second (''Q'' &rarr; ''P'') is false, in respect of boolean logic terminology we are concluding that creation is a [[Necessity and sufficiency|sufficient but not necessary]] condition of the existence of God.
+Instead of this, the [[logical consequence|''logical'' implication]] (''Q''<--''P'', the ''knowledge'' of ''Q'' follows the ''knowledge'' of ''P'': if the creation is or is in a such way, then we shall demonstrate and know the existence of God) is far different from the [[Material implication (rule of inference)|material implication]] (''Q'' &rarr ''P'', the real or measurable existence of ''P'' begins after the existence of ''Q'': the creation is the [[causality|cause]] of God), which implies the effect begins to exist in a time after the existence of its cause (God after the creation). The logical status of those two propositions is called [[Contingency (philosophy)|contingency]].
+Referring to the history and the so called [[chronological snobbery]], a relevant example of that two tipes of implication, is the following: if the present knows the past and would not repeat the same errors, humans are [[Standing on the shoulders of giants|standing on the shoulders of giants]].
 ==See also==