# Law of Excluded Middle

## Contents

## Proof Rule

The **law of (the) excluded middle** is a valid deduction sequent in propositional logic.

### Proof Rule

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

### Sequent Form

The Law of Excluded Middle can be symbolised by the sequent:

- $\vdash p \lor \neg p$

## Explanation

The **law of (the) excluded middle** can be expressed in natural language as:

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

The **LEM** 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 LEM.

## Also known as

The **law of (the) excluded middle** 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)**.

## Also see

## Sources

- 1978: Alan G. Hamilton:
*Logic for Mathematicians*... (previous) ... (next): $\S 1.2$: Truth functions and truth tables: Example $1.6 \ \text{(a)}$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 1.1$: The need for logic - 1993: M. Ben-Ari:
*Mathematical Logic for Computer Science*(1st ed.) ... (previous) ... (next): $\S 1.4$: Non-standard logics