Conditioned disjunction

In logic, conditioned disjunction (sometimes called conditional disjunction) is a ternary logical connective introduced by Church.^[1][2] Given operands p, q, and r, which represent truth-valued propositions, the meaning of the conditioned disjunction [p, q, r] is given by

${\displaystyle [p,q,r]\Leftrightarrow (q\to p)\land (\neg q\to r).}$

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

${\displaystyle (q\land p)\lor (\neg q\land r)}$

and has the same truth table as the ternary conditional operator ?: in many programming languages (with ${\displaystyle [b,a,c]}$ 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.^[3] There are other truth-functionally complete ternary connectives.