In logic, conditioned disjunction (sometimes called conditional disjunction) is a
ternary
Ternary (from Latin ''ternarius'') or trinary is an adjective meaning "composed of three items". It can refer to:
Mathematics and logic
* Ternary numeral system, a base-3 counting system
** Balanced ternary, a positional numeral system, usef ...
logical connective
In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary ...
introduced by
Church
Church may refer to:
Religion
* Church (building), a building for Christian religious activities
* Church (congregation), a local congregation of a Christian denomination
* Church service, a formalized period of Christian communal worship
* Ch ...
. Given operands ''p'', ''q'', and ''r'', which represent
truth-value
In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values (''true'' or ''false'').
Computing
In some progr ...
d
proposition
In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, "meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...
s, the meaning of the conditioned disjunction is given by:
:
In words, 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 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. In electronic logic terms, it may also be viewed as a single-bit
multiplexer
In electronics, a multiplexer (or mux; spelled sometimes as multiplexor), also known as a data selector, is a device that selects between several analog or digital input signals and forwards the selected input to a single output line. The sel ...
.
In conjunction with truth constants denoting each truth-value, conditioned disjunction is
truth-functionally complete for
classical logic.
[Wesselkamper, T., "A sole sufficient operator", ''Notre Dame Journal of Formal Logic'', Vol. XVI, No. 1 (1975), pp. 86-88.] Its
truth table
A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra (logic), Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expression (mathematics) ...
is the following:
There are other truth-functionally complete ternary connectives.
References
External links
*
Logical connectives
Ternary operations
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