Conditioned disjunction

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:

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

Conditioned disjunction

Definition	${\displaystyle (q\rightarrow p)\land (\neg q\rightarrow r)}$
Truth table	${\displaystyle (01000111)}$
Normal forms
Disjunctive	${\displaystyle {\overline {p}}{\overline {q}}r+p{\overline {q}}r+pq{\overline {r}}+pqr}$
Conjunctive	${\displaystyle ({\overline {q}}+p)(q+r)}$
Zhegalkin polynomial	${\displaystyle p\oplus qr\oplus r}$
Post's lattices
0-preserving	yes
1-preserving	yes
Monotone	no
Affine	no

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 conditional operator ?: in many programming languages (with ${\displaystyle [b,a,c]}$ 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.