Functional predicate

In <a href="/facts/Formal_logic/SaI7Y3wN">formal logic</a> and related branches of <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, 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 <a href="/facts/Mapping_(mathematics)/FABF1l2f">additional meanings in mathematics</a>.
In a <a href="/facts/Model_(logic)/GsMx1fEb">model</a>, a function symbol will be modelled by a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a>.
Specifically, the symbol F in a <a href="/facts/Formal_language/crDTyP8q">formal language</a> is a functional symbol if, <a href="/facts/Given_any/agNmvJKN">given any</a> symbol X representing an object in the language, F(X) is again a symbol representing an object in that language.
In <a href="/facts/Typed_logic/xj0xzVqI">typed logic</a>, 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 <a href="/facts/0_(number)/4NBRtKHc">zero</a> variables is simply a <a href="/facts/Logical_constant/bvTHIVCM">constant</a> symbol.
Now consider a model of the formal language, with the types T and U modelled by <a href="/facts/Set_(mathematics)/BfucfHMq">sets</a> [T] and [U] and each symbol X of type T modelled by an element [X] in [T].
Then F can be modelled by the set

[
        F
        ]
        :=
        
          
            {
          
        
        (
        [
        X
        ]
        ,
        [
        F
        (
        X
        )
        ]
        )
        :
        [
        X
        ]
        ∈
        [
        
          T
        
        ]
        
          
            }
          
        
        ,
      
    
    {\displaystyle [F]:={\big \{}([X],[F(X)]):[X]\in [\mathbf {T} ]{\big \}},}

which is simply a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> with domain [T] and codomain [U].
It is a requirement of a consistent model that [F(X)] = [F(Y)] whenever [X] = [Y].

Functional predicate open-in-new

Functional predicate