In formal logic and related branches of mathematics, a functional predicate, or function symbol, is a logical symbol that may be applied to an object term to produce another object term. Functional predicates are also sometimes called mappings, but that term has other meanings as well. In a model, a function symbol will be modelled by a function.
Specifically, the symbol F in a formal language is a functional symbol if, given any symbol X representing an object in the language, F(X) is again a symbol representing an object in that language. In typed logic, F is a functional symbol with domain type T and codomain type U if, given any symbol X representing an object of type T, F(X) is a symbol representing an object of type U. One can similarly define function symbols of more than one variable, analogous to functions of more than one variable; a function symbol in zero variables is simply a constant symbol.
Now consider a model of the formal language, with the types T and U modelled by sets and and each symbol X of type T modelled by an element in . Then F can be modelled by the set
which is simply a function with domain and codomain . It is a requirement of a consistent model that = whenever = .
Read more about Functional Predicate: Introducing New Function Symbols, Doing Without Functional Predicates
Famous quotes containing the words functional and/or predicate:
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—Melinda M. Marshall (20th century)
“The only thing that one really knows about human nature is that it changes. Change is the one quality we can predicate of it. The systems that fail are those that rely on the permanency of human nature, and not on its growth and development. The error of Louis XIV was that he thought human nature would always be the same. The result of his error was the French Revolution. It was an admirable result.”
—Oscar Wilde (1854–1900)