The identity predicate is a special two-place predicate in predicate logic that represents equality or sameness between two terms.

Motivation

The need for an identity predicate arises from situations where standard predicate logic falls short in expressing certain concepts. Consider the following examples:

Expressing Equality

For the following argument

Intuitively, this is a valid argument. However, if we translate that into predicate logic language as

it becomes invalid.

A better approach is to use a two-place predicate, but even then the argument remains invalid:

Counting Distinct Objects

We may come up translations such as or , but both will be true even when there is one dog!

We need a new construct to express those ideas.

The Identity Predicate:

  • Syntactically, is like any other two-place predicate.
  • Semantically, is special: in every model, it represents the identity relation.
  • is considered part of the logical vocabulary, similar to quantifiers and connectives.

We can write to mean "", but we more often use the abbreviations . And we use to represent .

Properties

Since the identity predicate represents the identity relation, it inherits all the properties of an identity relation:

The Identity Predicate and English

Return to our motivational questions.

can be translated as

And

can be translated as

Tip

While the English work “is” sometimes mean the identity predicate, it is more often to just mean a unary predicate. For example,

  • In “Clark Kent is a reporter,” “is” represents a unary predication ().
  • In “Clark Kent is the Superman,” “is” represents identity ().

Further Examples

”Alice Is the Tallest person”

Glossary

  • : Alice
  • : is a person
  • : is taller than

Note the the first conjunct (“Alice is a person”) is implied from the sentence

”Many Owns Something that Someone Else want”

Glossary

  • : is a person
  • : Mary
  • : owns
  • : wants

We can understand the sentence as “there is some x such that (Mary own x and someone else want x).” Then the whole sentence become

”If Mary Owns a Beagle, then no One Else does”

Glossary

  • : is a person
  • : Mary
  • : owns
  • : is a beagle

References

  • Logic : The Laws of Truth Chapter 13