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 [p, q, r] is given by:
In words, [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 [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:
and has the same truth table as the "ternary" (?:) operator in many programming languages.
In conjunction with truth constants denoting each truth-value, conditioned disjunction is truth-functionally complete for classical logic. Its truth table is the following:
There are other truth-functionally complete ternary connectives.