Definition:Predicate

Definition

The predicate of a simple statement in logic is the part of the statement which defines what is being said about the subject.

It is a word or phrase which, when combined with one or more names of object, turns into a meaningful sentence.

The predicates of a simple statements are atomic in predicate logic.

The subject and predicate of a simple statement are referred to as its terms.

Predicate Symbol

Let $P$ be some property.

Suppose $x$ is an object which has the property $P$.

Then we write $P \left({x}\right)$ to mean:

$x$ has the property $P$

or, more compactly:

$x$ has $P$.

The symbol $P$ in this context is called a predicate symbol.

Compare propositional function, which is an extension of this concept.

Linguistic Interpretation: The Meaning of Is

There are two basic types of sentences, namely, assertions of belonging:

$x \in A$

and assertions of equality:

$A = B$

-- Paul Halmos in Naive Set Theory (1960), $\S 2$: The Axiom of Specification.

The Is of Predication

Consider the statement:

Socrates is a man.

This means:

The object named Socrates has the property of being a man.

Thus we see that is here means has the property of being.

In this context, is here is called the is of predication.

The Is of Identity

Compare this with the sentence:

Socrates is the philosopher who taught Plato.

We could of course reword this as:

The object named Socrates has the property of being the philosopher who taught Plato.

However, the meaning that is really being conveyed here is that of:

The object named Socrates is the same object as the object which is the philosopher who taught Plato.

In this context, is is not being used in the same way as the is of predication.

When being used to indicate that one object is the same object as another object, is is called the is of identity.

In this context, is means the same as equals.

Quote

It depends on what the meaning of the word 'is' is. -- W.J. Clinton

Sources

• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 2$: The Axiom of Specification
• Alan G. Hamilton: Logic for Mathematicians (1978): $\S 1.1$
• H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability (1996): $\S 2.1$