Law of Excluded Middle/Proof Rule

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Proof Rule

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

As a proof rule it is expressed in the form:

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


It can be written:

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

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


Tableau Form

Let $\phi$ be a propositional 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.


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).


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