Concept

Predicate (logic)

Overview

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 the semantics of logic, predicates are interpreted as relations. For instance, in a standard semantics for first-order logic, the formula

(

)

{\displaystyle R(a,b)}

would be true on an interpretation if the entities denoted by

{\displaystyle a}

and

{\displaystyle b}

stand in the relation denoted by

{\displaystyle R}

. Since predicates are non-logical symbols, they can denote different relations depending on the interpretation given to them. While first-order logic only includes predicates that apply to individual objects, other logics may allow predicates that apply to collections of objects defined by other predicates.

Strictly speaking, a predicate does not need to be given any interpretation, so long as its syntactic properties are well-defined. For example, equality may be understood solely through its reflexive and substitution properties (cf. Equality (mathematics) § Axioms). Other properties can be derived from these, and they are sufficient for proving theorems in mathematics. Similarly, set membership can be understood solely through the axioms of Zermelo–Fraenkel set theory.

Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).

In the original texts

Passages from public-domain books where Predicate (logic) is discussed.

Every proposition consists of three parts: the Subject, the Predicate, and the Copula. The predicate is the name denoting that which is affirmed or denied. The subject is the name denoting the person or thing which something is affirmed or denied of.
John Stuart Mill, A System of Logic
names the idea; states what it is; opens on it
Read the chapter · Open in the reader
169.] The abstract terms of the judgment, 'The individual is the universal,' present the subject (as negatively self-relating) as what is immediately _concrete,_ while the predicate is what is _abstract,_ indeterminate, in short, the universal. But the two elements are connected together by an 'is': and thus the predicate (in its universality) must also contain the speciality of the subject, must, in short, have…
Georg Wilhelm Friedrich Hegel, The Logic of Hegel
names the idea; states what it is; opens on it
Read the chapter · Open in the reader
If the similarity between the scientific judgment and the more immediate and simple judgment is to be maintained, it is clear that the predicate of the simple judgment must be of like nature. The structure of the two varieties of judgment differs only in the degree of explicitness which the hypothesis acquires. That is, the predicate of a judgment, as such, is ideal; it is meaning, significant quality. If conditions…
John Dewey, Studies in Logical Theory
names the idea; states what it is; opens on it
Read the chapter · Open in the reader
We first of all separate a predicate of a thing from the conception of the thing, and afterwards connect with this predicate its opposite, and hence do not establish any contradiction with the subject, but only with its predicate, which has been conjoined with the subject synthetically—a contradiction, moreover, which obtains only when the first and second predicate are affirmed in the same time. If I say: “A man…
Immanuel Kant, The Critique of Pure Reason
names the idea; states what it is; opens on it
Read the chapter · Open in the reader
When the name which is the subject of the proposition is a general name, we may intend to affirm or deny the predicate, either of all the things that the subject denotes, or only of some. When the predicate is affirmed or denied of all and each of the things denoted by the subject, the proposition is universal; when of some undefined portion of them only, it is particular. Thus, All men are mortal; Every man is…
John Stuart Mill, A System of Logic, Ratiocinative and Inductive
names the idea; states what it is; opens on it
Read the chapter · Open in the reader

Predicate (logic)

Overview

In the original texts

Philosophers who discussed Predicate (logic)

Related ideas

Place in the tradition