Definition | |
---|---|

Truth table | |

Normal forms | |

Disjunctive | |

Conjunctive | |

Zhegalkin polynomial | |

Post's lattices | |

0-preserving | yes |

1-preserving | yes |

Monotone | no |

Affine | no |

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" (?:) 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:

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

**^**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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