Law of Excluded Middle
Proof Rule
The law of (the) excluded middle 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 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
- 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): excluded middle, law or principle of
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): excluded middle, law or principle of the
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): excluded middle