// laws and rules · Fundamental law
Law of excluded middle_
The law of excluded middle states that every proposition is either true or false, with no third possibility. Formally, p ∨ ¬p is a tautology: the most frequently cited example of a formula that is true in every row of its truth table.
Example
p ∨ ¬p
What the variables mean
- ▸ p: “There is life on Mars”
In plain words
“There is life on Mars or there is no life on Mars” is true regardless of the fact that we still do not know which alternative holds.
Truth table
| p | ¬p | p ∨ ¬p★ |
|---|---|---|
| T | F | T |
| F | T | T |
Classification: Tautology · 2 rows
Statement
The law states that p ∨ ¬p is true for any proposition p. It is one of the three classical laws of thought, alongside identity and non-contradiction.
Its name comes from excluding a “third” option: between being true and being false there is no intermediate state.
Why it holds: reading the table
The table has only two rows. Row 1 (p = T): the disjunction is T because its first member is. Row 2 (p = F): ¬p is T, so the disjunction is T again.
Since no row yields F, the calculator classifies the expression as a tautology. It is the canonical example used to introduce that very concept.
How it is used
It enables reasoning by cases: because p ∨ ¬p is always true, you may prove a conclusion by establishing it under the assumption p and again under ¬p.
Together with double negation it underpins proof by contradiction as practised in classical mathematics.
The intuitionistic objection
Brouwer and the intuitionist school reject the principle for statements about infinite sets: asserting “an object with that property exists or it does not” without being able to decide which branch holds strikes them as illegitimate.
This does not mean they consider p ∨ ¬p false, only that they do not accept it as a universal axiom. In many-valued or fuzzy logics the principle also fails, since they admit intermediate degrees of truth.
Relation to other laws
It is the dual of the law of non-contradiction, ¬(p ∧ ¬p). Applying De Morgan to one yields the other, so they are two faces of the same idea.
It is also related to Peirce's law, another classical tautology that fails intuitionistically and can be used to recover it.
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
Is it the same as bivalence? ▼
They are closely related but not identical: bivalence is a semantic claim about truth values; excluded middle is a formula of the language that happens to be a tautology.
Does it always hold in practice? ▼
In classical logic, yes. In systems admitting indeterminate values — three-valued or fuzzy logics, databases with NULL — it fails.
How does it relate to non-contradiction? ▼
They are duals: ¬(p ∧ ¬p) follows from p ∨ ¬p by De Morgan and double negation.