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:
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 conditional operator ?: in many programming languages (with 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.
^Church, Alonzo (1956). Introduction to Mathematical Logic. Princeton University Press.
^Church, Alonzo (1948). "Conditioned disjunction as a primitive connective for the propositional calculus". Portugaliae Mathematica, vol. 7, pp. 87-90.
^Wesselkamper, T., "A sole sufficient operator", Notre Dame Journal of Formal Logic, Vol. XVI, No. 1 (1975), pp. 86-88.
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