View source for Conditioned disjunction

{{Infobox logical connective
| title        = Conditioned disjunction
| other titles = 
| Venn diagram = Venn 0100 0111.svg
| definition   = <math>(q \to p) \land (\neg q \to r)</math>
| truth table  = <math>(0100 0111)</math>
| logic gate   = 
| DNF          = <math>\overline p \overline q r + p \overline q r + pq\overline r + pqr</math>
| CNF          = <math>(\overline q + p) (q + r)</math>
| Zhegalkin    = <math>qp \oplus qr \oplus r</math>
| 0-preserving = yes
| 1-preserving = yes
| monotone     = no
| affine       = no
| self-dual    = no
}}

In logic, '''conditioned disjunction''' (sometimes called '''conditional disjunction''') is a [[Ternary operation|ternary]] [[logical connective]] introduced by [[Alonzo Church|Church]].<ref>{{cite book |title=Introduction to Mathematical Logic |last=Church |first=Alonzo |author-link=Alonzo Church |year=1956 |publisher=Princeton University Press |url=https://archive.org/details/dli.ernet.449121}}</ref><ref>{{cite journal |title=Conditioned disjunction as a primitive connective for the propositional calculus |last=Church |first=Alonzo |author-link=Alonzo Church |year=1948 |journal=Portugaliae Mathematica |volume=7 |pages=87–90 |url=https://purl.pt/2171/1/ |lang=en}}</ref> Given operands ''p'', ''q'', and ''r'', which represent [[truth-value]]d [[proposition]]s, the meaning of the conditioned disjunction {{nowrap|[''p'', ''q'', ''r'']}} is given by
: <math>[p, q, r] \Leftrightarrow (q \to p) \land (\neg q \to r).</math>
In words, {{nowrap|[''p'', ''q'', ''r'']}} is equivalent to: "if ''q'', then ''p'', else ''r''", or "''p'' or ''r'', according as ''q'' or not ''q''". This may also be stated as "''q'' implies ''p'', and not ''q'' implies ''r''". So, for any values of ''p'', ''q'', and ''r'', the value of {{nowrap|[''p'', ''q'', ''r'']}} is the value of ''p'' when ''q'' is true, and is the value of ''r'' otherwise.

The conditioned disjunction is also equivalent to
: <math>(q \land p) \lor (\neg q \land r)</math>
and has the same truth table as the [[ternary conditional operator]] <code>?:</code> in many programming languages (with <math>[b, a, c]</math> being equivalent to <code>a ? b : c</code>). In electronic logic terms, it may also be viewed as a single-bit [[multiplexer]].

In conjunction with truth constants denoting each truth-value, conditioned disjunction is [[Truth function#Functional completeness|truth-functionally complete]] for [[classical logic]].<ref>{{cite journal |author=Wesselkamper, T. C. |title=A sole sufficient operator |journal=Notre Dame Journal of Formal Logic |volume=XVI |number=1 |year=1975 |pages=86–88 |doi=10.1305/ndjfl/1093891614}}</ref> There are other truth-functionally complete ternary connectives.

== Truth table ==
The [[truth table]] for <math>[p,q,r]</math>:

{| class="wikitable" style="text-align:center"
! <math>p</math> || <math>q</math> || <math>r</math> || <math>[p, q, r]</math>
|-
| {{yes2|True}} || {{yes2|True}} || {{yes2|True}} || {{yes2|True}}
|-
| {{yes2|True}} || {{yes2|True}} || {{no2|False}} || {{yes2|True}}
|-
| {{yes2|True}} || {{no2|False}} || {{yes2|True}} || {{yes2|True}}
|-
| {{yes2|True}} || {{no2|False}} || {{no2|False}} || {{no2|False}}
|-
| {{no2|False}} || {{yes2|True}} || {{yes2|True}} || {{no2|False}}
|-
| {{no2|False}} || {{yes2|True}} || {{no2|False}} || {{no2|False}}
|-
| {{no2|False}} || {{no2|False}} || {{yes2|True}} || {{yes2|True}}
|-
| {{no2|False}} || {{no2|False}} || {{no2|False}} || {{no2|False}}
|}

==References==
{{reflist}}

==External links==
* {{Commonscatinline}}

[[Category:Logical connectives]]
[[Category:Ternary operations]]


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