Conditioned disjunction

In logic, conditioned disjunction (sometimes called conditional disjunction) is a ternary logical connective introduced by Church. Given operands p, q, and r, which represent truth-valued propositions, the meaning of the conditioned disjunction Template:Nowrap is given by

$[p,q,r]\iff (q\to p)\wedge (\mathrm{\neg }q\to r).$

In words, Template:Nowrap 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 Template:Nowrap is the value of p when q is true, and is the value of r otherwise.

The conditioned disjunction is also equivalent to

$(q\wedge p)\vee (\mathrm{\neg }q\wedge r)$

and has the same truth table as the ternary conditional operator ?: in many programming languages (with $[b,a,c]$ being equivalent to a ? b : c). 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-functionally complete for classical logic. There are other truth-functionally complete ternary connectives.

Truth table

The truth table for $[p,q,r]$:

$p$	$q$	$r$	$[p,q,r]$
Template:Yes2	Template:Yes2	Template:Yes2	Template:Yes2
Template:Yes2	Template:Yes2	Template:No2	Template:Yes2
Template:Yes2	Template:No2	Template:Yes2	Template:Yes2
Template:Yes2	Template:No2	Template:No2	Template:No2
Template:No2	Template:Yes2	Template:Yes2	Template:No2
Template:No2	Template:Yes2	Template:No2	Template:No2
Template:No2	Template:No2	Template:Yes2	Template:Yes2
Template:No2	Template:No2	Template:No2	Template:No2

External links

• Template:Commonscatinline

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