Conversion (logic)

Conversion ( inversion ) is an expression of both traditional logic and modern relational logic.

Conversion in the sense of traditional logic

In traditional logic , conversion refers to an immediate conclusion by interchanging the subject and the predicate in categorical sentences (statements, judgments).

A distinction is made between simple and restricted conversion , whereby the canonical theorems of Aristotelian logic are used.

Easy conversion

The simple conversion of a categorical sentence arises from the fact that subject and predicate are swapped in a categorical sentence.

According to the laws of conversion , this leads to valid conclusions for the E and I sentences:

(1) S e P P e S (No S is P. No P is S.) ${\ displaystyle \ Leftrightarrow}$ ${\ displaystyle \ Leftrightarrow}$

Example: No human is a bird. No bird is human. ${\ displaystyle \ Leftrightarrow}$

(2) S i P P i S (Some S are P. Some P are S.) ${\ displaystyle \ Leftrightarrow}$ ${\ displaystyle \ Leftrightarrow}$

Example: Some people have a bird. Some who have a bird are humans. ${\ displaystyle \ Leftrightarrow}$

Limited conversion

With restricted conversion (also: Konversion per accidens , accidental conversion ), the subject and the predicate are exchanged in a general categorical sentence and the quantity of the sentence is changed without impairing its quality. The "a" is replaced by "i" and "e" by "o".

(1) S a P ⇒ P i S.

Example: All whales are mammals. ⇒ Some mammals are whales.

The restricted conversion from an A-sentence is only permitted if the subject of the sentence is not empty.

(2) S e P ⇒ P o S.

Example: Nobody is a fish (= all people are not fish) ⇒ Some fish are not people.

The restricted conversion from an E sentence is only permitted if the predicate of the sentence is not empty.

No permitted conversion from O-sentences

No conversion law applies to a particular negative sentence (O-sentence).

Conversion in the sense of the logic of relations

In the logic of relations, the conversion of a relation (also: the reversed relation to relation R, the converse , the inverse relation) is a relation that exists between objects b and a when the objects are in relation a and b to each other.

The converse is symbolized differently, among other things by "R´" or by " " and can then be defined symbolized by: ${\ displaystyle R ^ {- 1}}$

yR´x = xRy.

Or - more mathematically -:

for a relation the converse is defined as ${\ displaystyle R \ subseteq A \ times B}$

{\ displaystyle R ^ {- 1} = \ {(b, a) \ in B \ times A \ mid (a, b) \ in R \}}

.

Example 1: The conversion of the relation of "husband of" is the relation "wife of".

Example 2: The conversion of the relation "less than" is the converse relation "greater than".

Individual evidence

^ Borkowski, Ludwik: Formal logic. Akademie Verlag, Berlin 1976, p. 386

[1] Borkowski, Ludwik: Formal logic. Akademie Verlag, Berlin 1976, p. 386