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:

$[p, q, r] ~\leftrightarrow~(q \rightarrow p) \and (\neg q \rightarrow r)$

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:

$(q \and p) \or (\neg q \and r)$

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.