// laws and rules · Logical equivalence
Idempotent law_
Idempotence expresses something that looks obvious but is worth stating formally: asserting the same thing twice adds no information. Both p ∧ p and p ∨ p are equivalent to p, a property that sets logic apart from arithmetic, where a + a is not a.
Example
(p ∧ p) ⇔ p
What the variables mean
- ▸ p: “The server is down”
In plain words
“The server is down and the server is down” says no more than “the server is down”.
Truth table
| p | p ∧ p | (p ∧ p) ⇔ p★ |
|---|---|---|
| T | T | T |
| F | F | T |
Classification: Tautology · 2 rows
Statement
The law states that p ∧ p ≡ p, and its dual that p ∨ p ≡ p. With only one variable involved, the table has just 2 rows.
An operator is idempotent when applying it to an element and itself returns that element. Conjunction and disjunction are; XOR is not, because p ⊕ p is always false.
Why it holds: reading the table
Row 1 (p = T): T ∧ T = T, which matches p. Row 2 (p = F): F ∧ F = F, which matches as well. Since both rows agree, the biconditional is a tautology.
Intuitively: conjunction demands that both members be true, but here both members are the same proposition, so the demand collapses to the demand on p.
How it is used
It shows up constantly as an intermediate step when simplifying: after applying a distributive law you often get repeated terms, which idempotence collapses into one.
It also justifies why any exact duplicate can be removed from a list of conditions without changing the outcome.
Examples
Everyday: on a list of requirements, asking for “photo ID” twice does not make the requirement any stricter.
Programming: `if (active && active)` is exactly `if (active)`. The concept also appears in idempotent HTTP methods: repeating a PUT leaves the resource in the same state.
Relation to other laws
It is an ingredient in the proof of absorption, where p ∨ p collapses after applying distributivity.
Do not confuse it with p ∧ ¬p, which is a contradiction, nor with p ∨ ¬p, which is a tautology: idempotence repeats the same proposition, not its negation.
Try it yourself
Edit the expression in the calculator and watch how every step of the table changes.
Open in the calculator →Related operators
Frequently asked questions
Why does the table have only 2 rows? ▼
Because there is a single propositional variable, p, and the row count is 2ⁿ: 2¹ = 2.
Are all operators idempotent? ▼
No. ∧ and ∨ are, but ⊕ is not: p ⊕ p is always false. Neither is the conditional in the usual sense, even though p ⇒ p is a tautology.
Is it the same as the identity law? ▼
No. Identity relates p to the constants (p ∧ 1 ≡ p, p ∨ 0 ≡ p); idempotence relates p to itself.