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". 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.

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.

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