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Law of Excluded Middle

Law of Excluded Middle

A fundamental principle of classical logic stating that for any proposition, either that proposition is true or its negation is true, with no third possibility.

Law of Excluded Middle

The Law of Excluded Middle (LEM), also known as the principle of excluded middle or tertium non datur, stands as one of the fundamental axioms of classical logic. First formally articulated by Aristotle in his Metaphysics, this principle asserts that for any meaningful proposition P, either P is true or its negation (not-P) is true, with no third or middle option possible.

Formal Expression

In formal logical notation, the law can be expressed as:

  • P ∨ ¬P (read as "P or not P")

For example:

  • Either it is raining, or it is not raining
  • Either a number is even, or it is not even
  • Either a statement is true, or it is false

Relationship to Other Logical Principles

The Law of Excluded Middle forms part of the three classic laws of thought:

  1. law of identity
  2. law of non-contradiction
  3. Law of Excluded Middle

Together, these principles form the foundation of classical logic and binary logic, where every proposition must be either true or false.

Challenges and Criticisms

Several philosophical traditions and logical systems challenge or reject LEM:

  1. Intuitionistic logic rejects the universal application of LEM, particularly in infinite domains
  2. Many-valued logic systems allow for additional truth values beyond true and false
  3. Quantum mechanics suggests some physical properties might exist in superposition states

Applications and Importance

The Law of Excluded Middle plays crucial roles in:

In Mathematics

In mathematical reasoning, LEM enables powerful proof techniques like:

Historical Development

The principle's evolution spans multiple traditions:

  1. Ancient Greek origins with Aristotle
  2. Medieval scholastic development
  3. Modern formal logical treatment
  4. Contemporary challenges and modifications

Contemporary Relevance

Despite challenges, LEM remains central to:

The principle continues to spark philosophical debate about the nature of truth, logic, and reasoning, particularly in contexts where binary distinctions may not fully capture reality's complexity.

See Also