The identity predicate
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
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
Properties
Since the identity predicate represents the identity relation, it inherits all the properties of an identity relation:
- Reflexive:
- Symmetric:
- Transitive:
- Leibniz’s law:
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
”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