Conditioned disjunction
 | |
| Definition | [math]\displaystyle{ (q \rightarrow p) \land (\neg q \rightarrow r) }[/math] |
|---|---|
| Truth table | [math]\displaystyle{ (0100 0111) }[/math] |
| Normal forms | |
| Disjunctive | [math]\displaystyle{ \overline p \overline q r + p \overline q r + pq\overline r + pqr }[/math] |
| Conjunctive | [math]\displaystyle{ (\overline q + p) (q + r) }[/math] |
| Zhegalkin polynomial | [math]\displaystyle{ p \oplus qr \oplus r }[/math] |
| white;">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:
- [math]\displaystyle{ [p, q, r] ~\leftrightarrow~(q \rightarrow p) \land (\neg q \rightarrow r) }[/math]
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:
- [math]\displaystyle{ (q \land p) \lor (\neg q \land r) }[/math]
and has the same truth table as the ternary conditional operator ?: in many programming languages (with [math]\displaystyle{ [b,a,c] }[/math] 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
The truth table for [math]\displaystyle{ [p,q,r] }[/math]:
| [math]\displaystyle{ p }[/math] | [math]\displaystyle{ q }[/math] | [math]\displaystyle{ r }[/math] | [math]\displaystyle{ [p,q,r] }[/math] |
| 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
- ↑ 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.
External links
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