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Conditioned disjunction

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Conditioned disjunction
Venn diagram of Conditioned disjunction
Definition
Truth table
Normal forms
Disjunctive
Conjunctive
Zhegalkin polynomial
Post's lattices
0-preservingyes
1-preservingyes
Monotoneno
Affineno
Self-dualno

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.

Truth table

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The truth table for :

True True True True
True True False True
True False True True
True False False False
False True True False
False True False False
False False True True
False False False False

References

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  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. 7: 87–90.
  3. ^ Wesselkamper, T. C. (1975). "A sole sufficient operator". Notre Dame Journal of Formal Logic. XVI (1): 86–88. doi:10.1305/ndjfl/1093891614.
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