A predicate symbol or predicate variable is a type of variable that stands for some predicate in a sentence.

Predicate symbols are usually coupled with one or more quantified variables or constants, which stand for the objects and/or subjects of the sentence.

Examples

For example, we may use the predicate variable R(1) — the (1) signifies how many places the predicate symbol needs, and we may also write Rx to reserve a place for the object of the expression — to represent the concept “is rich” in the sentence “Wanda is rich.”, and assign the constant w to represent “Wanda”. The sentence “Wanda is rich” would therefore be rendered Rw.

Predicate symbols may have multiple variables attached to accomodate compound predicates. For instance, the phrase “x likes y” would require two variables, giving us L(2) Lxy. If we want to say that “Wanda likes George”, we may add a constant, g, to refer to George, and express Lwg.

Predicate symbols are not truth-functional and have no logical value in themselves. They may be combined with other logical symbols to form formulae, with a predicate expression taking the place of a propositional variable in an expression. For instance, we may want to express “If Wanda is rich, then George likes Wanda”. To do so, we may formally write:

Rw → Lgw

Predicates and quantifiers

Predicate symbols may give information about variables rather than constants, if those variables are given some quantification — either universal or existential.

For instance, if we want to say that everybody likes Wanda, we could say “for every x, x likes Wanda, or, formally:

∀xLxw

Differences in notation

Some logical systems will use parentheses to indicate the variables and constants attached to predicate variables. In those systems, Rw would become R(w), and Lxy would become L(x, y). This convention is not generally used on this site unless it would be confusing to remove the parentheses.