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 [p, q, r] is given by
 * $$[p, q, r] \Leftrightarrow (q \to p) \land (\neg q \to r).$$

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
 * $$(q \land p) \lor (\neg q \land r)$$

and has the same truth table as the ternary conditional operator  in many programming languages (with $$[b, a, c]$$ being equivalent to  ). 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 $$[p,q,r]$$: