Editing Converse (logic) (section)

==Categorical converse==
{{see also|Categorical proposition#Conversion}}

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]]: <blockquote>"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."<ref>[[Asa Mahan]] (1857) ''The Science of Logic: or, An Analysis of the Laws of Thought'', [https://books.google.com/books?id=J_wtAAAAMAAJ&pg=PA82 p. 82].</ref> </blockquote>The "exposita" is more usually called the "convertend". In its simple form, conversion is valid only for '''E''' and '''I''' propositions:<ref>William Thomas Parry and Edward A. Hacker (1991), ''Aristotelian Logic'', SUNY Press, [https://books.google.com/books?id=3Sg84H6B-m4C&pg=PA207 p. 207].</ref>

{| class="wikitable"
! 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."<ref>[[James H. Hyslop]] (1892), ''The Elements of Logic'', C. Scribner's sons, p. 156.</ref> For '''E''' propositions, both subject and predicate are [[Distribution of terms|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 [[syllogism]]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 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''.