# Conditioned disjunction

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

${\displaystyle [p,q,r]~\leftrightarrow ~(q\rightarrow p)\land (\neg q\rightarrow r)}$

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" (?:) operator in many programming languages.

In conjunction with truth constants denoting each truth-value, conditioned disjunction is truth-functionally complete for classical logic.[2] Its truth table is the following:

Conditioned disjunction
p q r [p,q,r]
T T T T
T T F T
T F T T
T F F F
F T T F
F T F F
F F T T
F F F F

There are other truth-functionally complete ternary connectives.

## References

1. ^ Church, Alonzo (1956). Introduction to Mathematical Logic. Princeton University Press.
2. ^ Wesselkamper, T., "A sole sufficient operator", Notre Dame Journal of Formal Logic, Vol. XVI, No. 1 (1975), pp. 86-88.