Law of Identity

By Daniel Machado Updated: 09/30/2026

The law of identity is a fundamental axiom of classical logic stating that every proposition is identical to itself. It is formally expressed as a logical implication or as an equivalence:

p → p

p ≡ p

The formula is read: "if p, then p," or "p is equivalent to p." This means that the truth value of proposition p is exactly the same as that of p, leaving no possibility for change or ambiguity within the same context.

Proof

The formula p → p is a tautology. Its truth is independent of the specific content of proposition p, as confirmed by its truth table:

p	p → p
T	T
F	T

As shown, the column p → p is always true (T).

The law of identity is one of the three pillars of classical propositional logic, alongside:

The law of non-contradiction: ¬(p ∧ ¬p)
The law of excluded middle: p ∨ ¬p

Although seemingly simple and even trivial, this principle provides the foundation that ensures stability and consistency in reasoning. It establishes a fixed point of reference: any well-formed proposition maintains its truth value within a given context, enabling the construction of coherent arguments.

Examples

This principle appears in everyday and formal reasoning:

In factual statements: "If today is Monday, then today is Monday." The truth of the antecedent necessarily entails the truth of the identical consequent.
In mathematical properties: "If the number 4 is even, then the number 4 is even." The property of being even applies identically to the subject of the statement.
In physical states: "If the door is closed, then the door is closed." The described state is self-referential.
In value judgments: "If this painting is beautiful, then this painting is beautiful."
In logical relations: "If A equals B, then A equals B."

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