Law of Excluded Middle/Proof Rule

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.

As a proof rule it is expressed in the form:

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

It can be written:

$\ds {{} \over \phi \lor \neg \phi} \textrm{LEM} \qquad \text { or } \qquad {\top \over \phi \lor \neg \phi} \textrm{LEM}$

where the symbol $\top$ (top) signifies tautology.

Tableau Form

Let $\phi$ be a well-formed formula.

The Law of Excluded Middle is invoked in the following manner:

			Pool:					None
			Formula:					$\phi \lor \neg \phi$
			Description:					Law of Excluded Middle
			Depends on:					Nothing
			Abbreviation:					$\text{LEM}$

Explanation

The law of (the) excluded middle 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.

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).
• the principle of bivalence.

Also see

• This is a rule of inference of the following proof systems:
  • Definition:Natural Deduction

Technical Note

When invoking Law of Excluded Middle in a tableau proof, use the {{ExcludedMiddle}} template:

{{ExcludedMiddle|line|statement}}

or:

{{ExcludedMiddle|line|statement|comment}}

where:

line is the number of the line on the tableau proof where the Law of Excluded Middle is to be invoked

statement is the statement of logic that is to be displayed in the Formula column, without the $ ... $ delimiters

comment is the (optional) comment that is to be displayed in the Notes column.

Sources

• 2000: Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems ... (previous) ... (next): $\S 1.2.2$: Derived rules