Contravalence

Venn diagram of The contravalence is that or with excluded and . Among the set operations, this join corresponds to the union with excluded intersection .

{\ displaystyle A \; \; \! \! {\ dot {\ lor}} \; \; \! \! B}

Contra is in the classical logic and mathematics the name for the connection of two statements by the two- connective , of either - or , exclusive or as well Kontravalentor is.

Synonymous with contravalence, the terms exclusive disjunction (also complete or antivalent disjunction ), bisubtraction , exclusive or, antivalence, contradictory opposition , contrajunction or alternation are used. In switching algebra one speaks of the exclusive-or gate (XOR gate), in propositional logic it is called XOR operation.

Definition and characteristics

Defines the contra is the truth function of their Junktors: A contra is only true if both different by it related statements truth values have, that is, if either the one or the other is true, but if not both true or both false at the same time are. The Latin expression for this exclusive or as "either - or" is "aut - aut".

A truth table (matrix) gives the aut function as a truth value function of contravalence as follows:

A.	B.	${\ displaystyle A \; \; \! \! {\ dot {\ lor}} \; \; \! \! B}$
true	true	not correct
true	not correct	true
not correct	true	true
not correct	not correct	not correct

The contravalence is associative and commutative . It is also self-inverse and distributive with respect to logical AND , but not with respect to OR :

It always applies , ${\ displaystyle A \ land (B \; \; \! \! {\ dot {\ lor}} \; \; \! \! C) = (A \ land B) \; \; \! \! { \ dot {\ lor}} \; \; \! \! (A \ land C)}$
however, is only if is wrong. ${\ displaystyle A \ lor (B \; \; \! \! {\ dot {\ lor}} \; \; \! \! C) = (A \ lor B) \; \; \! \! { \ dot {\ lor}} \; \; \! \! (A \ lor C)}$ ${\ displaystyle A}$
It is self-inverse because: . ${\ displaystyle A \; {\ dot {\ lor}} \; B \; {\ dot {\ lor}} \; B = A}$

Differentiation and similarities

The difference to the non-exclusive or (in the narrower sense the disjunction) consists in the “tightened information” that “it is certain from the start that one of the two alternatives must be true”, that is, not only at least but also at most one of the two Facts exist.

Equivalences of contravalence, i.e. formulas with other connectors that have the same truth value sequence, are:

Negation of the biconditional (negation of equivalence ) ${\ displaystyle \ neg (A \ leftrightarrow B)}$
${\ displaystyle (A \ vee B) \ wedge \ neg (A \ wedge B)}$ or
${\ displaystyle (A \ vee B) \ wedge (\ neg A \ vee \ neg B)}$ or
${\ displaystyle (A \ wedge \ neg B) \ vee (\ neg A \ wedge B)}$ .

Importance and practical application

The importance of contravalence is rather small in modern logic, "since it allows relatively few connections to be formulated". In switching algebra, however, it is of great importance as an XOR link. The property that the double application of the XOR operation corresponds to the identity, i.e. This means that it is self-inverse, is used in cryptography - where it enables the same function to be used for encryption and decryption - as well as in the RAID system.

Notation and pronunciation

Symbols of the contravalentor include:

${\ displaystyle {\ dot {\ lor}}}$
⌴ half a square open at the top.
XOR
${\ displaystyle \ nleftrightarrow}$
"> - <"
"> <"
${\ displaystyle \ oplus}$

The way of speaking for the junction also varies: ${\ displaystyle A \; \; \! \! {\ dot {\ lor}} \; \; \! \! B}$

"A versus B"
"A or (but) B"
"Either A, or B"
"A, except that B"
"A, except that B"
"A, unless B"
"A if and only if not B"

Colloquially, the contravalentor is paraphrased as "either - or". But this expression can also have other colloquial meanings that do not correspond to contravalence; For example, “Either Emil or I will pick you up” can also be understood as true if both of them pick up the conversation partner.

Individual evidence

↑ See Lorenz: Disjunction. In: Jürgen Mittelstraß (Hrsg.): Encyclopedia Philosophy and Philosophy of Science. 2nd edition 2005.
In a different meaning also the truth value function that interprets this junctor
↑ z. B. Lorenz: Disjunction. In: Mittelstraß (Hrsg.): Encyclopedia Philosophy and Philosophy of Science, 2nd edition 2005
↑ z. B: Paul Lorenzen : Logic, 4th ed. (1970), p. 48 (to avoid the word "disjunction")
↑ z. B. Menne: Logic, 6th edition (2001), p. 39
↑ Strobach: Introduction to Logic (2005), p. 22: sometimes, but not well according to the Latin meaning
↑ ^a ^b ^c Essler / Martínez: Grundzüge der Logik I, 4th ed. (1991), p. 51
↑ Schülerduden, Philosophie, 2nd ed. (2002), Disjunction
↑ Hilbert / Ackermann: Grundzüge, 6th edition (1972), p. 6; Reichenbach: Fundamentals of symbolic logic (1999), p. 33
↑ Essler / Martínez: Grundzüge der Logic I, 4th ed. (1991), p. 98 fn. 33
↑ Lorenzen: Logic. 4th ed. (1970), p. 39.
↑ Menne: Logic, 6th edition (2001), p. 39
↑ Essler / Martínez: Grundzüge der Logic I, 4th ed. (1991), p. 51
↑ Detel: Basic Course Philosophy I: Logic (2007), p. 71
^ ^A ^b ^c Wilhelm K. Essler: Introduction to Logic (= Kröner's pocket edition . Volume 381). 2nd, expanded edition. Kröner, Stuttgart 1969, DNB 456577998 , p. 96.
↑ Spies: Introduction to Logic (2004), p. 13.
↑ Rosenkranz: Introduction to Logic (2006), p. 81.

[1] See Lorenz: Disjunction. In: Jürgen Mittelstraß (Hrsg.): Encyclopedia Philosophy and Philosophy of Science. 2nd edition 2005.
In a different meaning also the truth value function that interprets this junctor

[2] z. B. Lorenz: Disjunction. In: Mittelstraß (Hrsg.): Encyclopedia Philosophy and Philosophy of Science, 2nd edition 2005

[3] z. B: Paul Lorenzen : Logic, 4th ed. (1970), p. 48 (to avoid the word "disjunction")

[4] z. B. Menne: Logic, 6th edition (2001), p. 39

[5] Strobach: Introduction to Logic (2005), p. 22: sometimes, but not well according to the Latin meaning

[Essler51-6] Essler / Martínez: Grundzüge der Logik I, 4th ed. (1991), p. 51

[7] Schülerduden, Philosophie, 2nd ed. (2002), Disjunction

[8] Hilbert / Ackermann: Grundzüge, 6th edition (1972), p. 6; Reichenbach: Fundamentals of symbolic logic (1999), p. 33

[9] Essler / Martínez: Grundzüge der Logic I, 4th ed. (1991), p. 98 fn. 33

[10] Lorenzen: Logic. 4th ed. (1970), p. 39.

[11] Menne: Logic, 6th edition (2001), p. 39

[12] Essler / Martínez: Grundzüge der Logic I, 4th ed. (1991), p. 51

[13] Detel: Basic Course Philosophy I: Logic (2007), p. 71

[Essler96-14] A ^b ^c Wilhelm K. Essler: Introduction to Logic (= Kröner's pocket edition . Volume 381). 2nd, expanded edition. Kröner, Stuttgart 1969, DNB 456577998 , p. 96.

[15] Spies: Introduction to Logic (2004), p. 13.

[16] Rosenkranz: Introduction to Logic (2006), p. 81.