Conditioned disjunction
Encyclopedia
In logic, conditioned disjunction (sometimes called conditional disjunction) is a ternary
logical connective
introduced by Church
.. Given operands p, q, and r, which represent truth-valued proposition
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". 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.
In conjunction with truth constants denoting each truth-value, conditioned disjunction is truth-functionally complete for classical logic
. Its truth table
is the following:
There are other truth-functionally complete ternary connectives.
Ternary operation
In mathematics, a ternary operation is an n-ary operation with n = 3. A ternary operation on a set A takes any given three elements of A and combines them to form a single element of A. An example of a ternary operation is the product in a heap....
logical connective
Logical connective
In logic, a logical connective is a symbol or word used to connect two or more sentences in a grammatically valid way, such that the compound sentence produced has a truth value dependent on the respective truth values of the original sentences.Each logical connective can be expressed as a...
introduced by Church
Alonzo Church
Alonzo Church was an American mathematician and logician who made major contributions to mathematical logic and the foundations of theoretical computer science. He is best known for the lambda calculus, Church–Turing thesis, Frege–Church ontology, and the Church–Rosser theorem.-Life:Alonzo Church...
.. Given operands p, q, and r, which represent truth-valued proposition
Proposition
In logic and philosophy, the term proposition refers to either the "content" or "meaning" of a meaningful declarative sentence or the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence...
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". 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.
In conjunction with truth constants denoting each truth-value, conditioned disjunction is truth-functionally complete for classical logic
Classical logic
Classical logic identifies a class of formal logics that have been most intensively studied and most widely used. The class is sometimes called standard logic as well...
. Its truth table
Truth table
A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—to compute the functional values of logical expressions on each of their functional arguments, that is, on each combination of values taken by their...
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.