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# Conditioned disjunction

Definition ${\displaystyle (q\rightarrow p)\land (\neg q\rightarrow r)}$ ${\displaystyle (01000111)}$ ${\displaystyle {\overline {p}}{\overline {q}}r+p{\overline {q}}r+pq{\overline {r}}+pqr}$ ${\displaystyle ({\overline {q}}+p)(q+r)}$ ${\displaystyle p\oplus qr\oplus r}$ yes yes no no .mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}.mw-parser-output .infobox .navbar{font-size:100%}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}

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:

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

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 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] 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. ^ Church, Alonzo (1948). "Conditioned disjunction as a primitive connective for the propositional calculus". Portugaliae Mathematica, vol. 7, pp. 87-90.
3. ^ Wesselkamper, T., "A sole sufficient operator", Notre Dame Journal of Formal Logic, Vol. XVI, No. 1 (1975), pp. 86-88.
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