Predicate (logic)

In <a href="/facts/Mathematical_logic/SaI7Y3wN">logic</a>, a predicate is a symbol that represents a property or a relation. For instance, in the <a href="/facts/First_order_logic/lcISqWIg">first-order formula</a> 
 
 
 
 P
 (
 a
 )
 
 
 {\displaystyle P(a)}
 
, the symbol 
 
 
 
 P
 
 
 {\displaystyle P}
 
 is a predicate that applies to the <a href="/facts/Individual_constant/SJU82QQR">individual constant</a> 
 
 
 
 a
 
 
 {\displaystyle a}
 
. Similarly, in the formula 
 
 
 
 R
 (
 a
 ,
 b
 )
 
 
 {\displaystyle R(a,b)}
 
, the symbol 
 
 
 
 R
 
 
 {\displaystyle R}
 
 is a predicate that applies to the individual constants 
 
 
 
 a
 
 
 {\displaystyle a}
 
 and 
 
 
 
 b
 
 
 {\displaystyle b}
 
. 
According to <a href="/facts/Gottlob_Frege/FXyStVYj">Gottlob Frege</a>, the meaning of a predicate is exactly a function from the domain of objects to the <a href="/facts/Truth_value/SY4e2w8l">truth values</a> "true" and "false".
In the <a href="/facts/Semantics_of_logic/VdAKHszK">semantics of logic</a>, predicates are interpreted as <a href="/facts/Relation_(mathematics)/HjblDaMy">relations</a>. For instance, in a standard semantics for first-order logic, the formula 
 
 
 
 R
 (
 a
 ,
 b
 )
 
 
 {\displaystyle R(a,b)}
 
 would be true on an <a href="/facts/Interpretation_(logic)/goPmjjQG">interpretation</a> if the entities denoted by 
 
 
 
 a
 
 
 {\displaystyle a}
 
 and 
 
 
 
 b
 
 
 {\displaystyle b}
 
 stand in the relation denoted by 
 
 
 
 R
 
 
 {\displaystyle R}
 
. Since predicates are <a href="/facts/Non-logical_symbol/SJU82QQR">non-logical symbols</a>, they can denote different relations depending on the interpretation given to them. While <a href="/facts/First-order_logic/lcISqWIg">first-order logic</a> only includes predicates that apply to individual objects, other logics may allow predicates that apply to collections of objects defined by other predicates.

Predicate (logic) open-in-new

Predicate (logic)