First-order formula in the context of "Finitary relation"

In logic, a predicate is a non-logical symbol that represents a property or a relation, though, formally, does not need to represent anything at all. For instance, in the first-order formula ${\displaystyle P(a)}$, the symbol ${\displaystyle P}$ is a predicate that applies to the individual constant ${\displaystyle a}$ which evaluates to either true or false. Similarly, in the formula ${\displaystyle R(a,b)}$, the symbol ${\displaystyle R}$ is a predicate that applies to the individual constants ${\displaystyle a}$ and ${\displaystyle b}$. Predicates are considered a primitive notion of first-order, and higher-order logic and are therefore not defined in terms of other more basic concepts.

The term derives from the grammatical term "predicate", meaning a word or phrase that represents a property or relation.

In logic, a quantifier is an operator that specifies how many individuals in the domain of discourse satisfy an open formula. For instance, the universal quantifier ${\displaystyle \forall }$ in the first-order formula ${\displaystyle \forall xP(x)}$ expresses that everything in the domain satisfies the property denoted by ${\displaystyle P}$. On the other hand, the existential quantifier ${\displaystyle \exists }$ in the formula ${\displaystyle \exists xP(x)}$ expresses that there exists something in the domain which satisfies that property. A formula where a quantifier takes widest scope is called a quantified formula. A quantified formula must contain a bound variable and a subformula specifying a property of the referent of that variable.

The most commonly used quantifiers are ${\displaystyle \forall }$ and ${\displaystyle \exists }$. These quantifiers are standardly defined as duals; in classical logic: each can be defined in terms of the other using negation. They can also be used to define more complex quantifiers, as in the formula ${\displaystyle \neg \exists xP(x)}$ which expresses that nothing has the property ${\displaystyle P}$. Other quantifiers are only definable within second-order logic or higher-order logics. Quantifiers have been generalized beginning with the work of Andrzej Mostowski and Per Lindström.