# 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 objects, turns into a meaningful sentence.

The predicates of simple statements are atomic in predicate logic.

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

## Linguistic Interpretation: The Meaning of Is

*There are two basic types of sentences, namely, assertions of belonging:*- $x \in A$

*and assertions of equality:*- $A = B$

-- 1960: Paul R. Halmos: *Naive Set Theory*: $\S 2$: The Axiom of Specification.

### The Is of Predication

Consider the statement:

**Socrates is a man.**

This means:

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

Consider the sentence:

**Socrates is the philosopher who taught Plato.**

This could be reworded as:

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.*-- William 'B.J.' Clinton

## Examples

### Favourite Music

The string **Prime.mover** is a **constant**.

The statement:

*$x$ is one of***Prime.mover'**s favourite pieces of music

is a predicate which involves a variable and a constant.

## Sources

- 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 2$: The Axiom of Specification - 1973: Irving M. Copi:
*Symbolic Logic*(4th ed.) ... (previous) ... (next): $4$: Propositional Functions and Quantifiers: $4.1$: Singular Propositions and General Propositions - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 1$: Some mathematical language: Variables and quantifiers - 1980: D.J. O'Connor and Betty Powell:
*Elementary Logic*... (previous) ... (next): $\S \text{III}$: The Logic of Predicates $(1): \ 2$: Predicate expressions - 1988: Alan G. Hamilton:
*Logic for Mathematicians*(2nd ed.) ... (previous) ... (next): $\S 1$: Informal statement calculus: $\S 1.1$: Statements and connectives - 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): $\S 2.1$: Introduction - 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): Chapter $1$: Elementary, my dear Watson: $\S 1.1.1$: 'You talking to me?': Definition $1.1.4$