Law of Excluded Middle

Proof Rule

The is a valid argument in certain types of logic dealing with disjunction $\lor$ and negation $\neg$.

This includes classical propositional logic and predicate logic, and in particular natural deduction, but for example not intuitionistic propositional logic.

Proof Rule

$\phi \lor \neg \phi$ for all statements $\phi$.

Sequent Form

The can be symbolised by the sequent:

$\vdash p \lor \neg p$

Explanation

The can be expressed in natural language as:

Every statement is either true or false.

This is one of the Aristotelian principles upon which rests the whole of classical logic, and the majority of mainstream mathematics.

Intuitionist Perspective

The is rejected by the intuitionistic school, which rejects the existence of an object unless it can be constructed within an axiomatic framework which does not include the .

This is because from the perspective of intuitionism:

an object cannot be known to exist unless it can be constructed in a finite number of steps

a statement cannot be known to be true if its proof needs an argument requiring an infinite number of steps.

Thus:

while it is sufficient to prove a statement is not true by demonstrating that it is not true

it is insufficient to prove a statement is true by demonstrating it is not false.

Also known as

The is otherwise known as:

• (Principium) tertium non datur, Latin for third not given, that is, a third possibility is not possible
• Principium tertii exclusi, Latin for the Principle of the Excluded Third (PET).
• the principle of bivalence.

Also see

• Definition:Disjunction
• Definition:Logical Not

• Principle of Non-Contradiction

Sources

• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.1$: The need for logic
• 1988: Alan G. Hamilton: Logic for Mathematicians (2nd ed.) ... (previous) ... (next): $\S 1$: Informal statement calculus: $\S 1.2$: Truth functions and truth tables: Example $1.6 \ \text{(a)}$
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): excluded middle
• 1993: M. Ben-Ari: Mathematical Logic for Computer Science ... (previous) ... (next): Chapter $1$: Introduction: $\S 1.4$: Non-standard logics
• 1993: Richard J. Trudeau: Introduction to Graph Theory ... (previous) ... (next): $2$. Graphs: Paradox
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): bivalence (principle of)
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): excluded middle, law or principle of
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): intuitionism
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): bivalence, principle of
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): excluded middle, law or principle of the
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): intuitionism
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): excluded middle