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, useful ...

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 co ...

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 * C ...

. 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 progra ...

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: :

~\leftrightarrow~(q \rightarrow p) \land (\neg q \rightarrow r)

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: :

(q \land p) \lor (\neg q \land r)

and has the same truth table as the

ternary conditional operator In computer programming, the ternary conditional operator is a ternary operator that is part of the syntax for basic conditional expressions in several programming languages. It is commonly referred to as the conditional operator, ternary if, or ...

?: 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 Classical logic (or standard logic or Frege-Russell logic) is the intensively studied and most widely used class of deductive logic. Classical logic has had much influence on analytic philosophy. Characteristics Each logical system in this class ...

.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, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional argumen ...

is the following: There are other truth-functionally complete ternary connectives.

References

External links

* Logical connectives Ternary operations {{logic-stub