law of excluded middle
C1/C2Formal, Technical, Academic
Definition
Meaning
A fundamental principle in classical logic stating that for any proposition, it is either true or its negation is true; there is no third, middle, or intermediate possibility.
The principle is central to bivalent logic systems and underpins methods of reasoning like proof by contradiction. In philosophy, it is debated in contexts of vagueness, future contingents, or certain interpretations of quantum mechanics.
Linguistics
Semantic Notes
Not a law in the legal sense. It is an axiom or principle of logical systems. Often abbreviated as LEM. The 'middle' refers to a truth value other than true or false.
Dialectal Variation
British vs American Usage
Differences
No significant lexical or grammatical differences. The concept is identical in logic and philosophy on both sides of the Atlantic.
Connotations
In both dialects, it carries the same technical, academic connotation. It is not used in everyday conversation.
Frequency
Equally rare outside of logic, mathematics, computer science, and analytic philosophy circles in both regions.
Vocabulary
Collocations
Grammar
Valency Patterns
The [argument/proof] relies on the law of excluded middle.In [system/logic] S, the law of excluded middle [holds/fails].Vocabulary
Synonyms
Strong
Neutral
Weak
Vocabulary
Antonyms
Phrases
Idioms & Phrases
- “No idioms contain this phrase.”
Usage
Context Usage
Business
Never used.
Academic
Primary context. Used in logic, mathematics, computer science (theory), and analytic philosophy to describe a foundational axiom or point of debate.
Everyday
Virtually never used.
Technical
Used precisely as in academic contexts, especially in formal specifications, logical foundations of programming, and theoretical discussions.
Examples
By Part of Speech
verb
British English
- To argue, one must often law-of-excluded-middle the possibilities.
- The system does not allow one to law-of-excluded-middle vague predicates. (Note: This is highly non-standard; the phrase is almost exclusively a noun phrase.)
American English
- You cannot simply law-of-excluded-middle your way out of this paradox.
- Formalists law-of-excluded-middle all statements in the classical system. (Note: This is highly non-standard.)
adverb
British English
- He reasoned law-of-excluded-middle, leaving no room for ambiguity. (Rare/constructed)
- The proposition was treated law-of-excluded-middle. (Rare/constructed)
American English
- She argued law-of-excluded-middle, rejecting any third option. (Rare/constructed)
- The problem must be considered law-of-excluded-middle. (Rare/constructed)
adjective
British English
- A law-of-excluded-middle framework underpins his reasoning.
- They adopted a law-of-excluded-middle approach to the problem. (Note: Hyphenated adjectival use is rare but possible in technical prose.)
American English
- The proof requires a law-of-excluded-middle assumption.
- This is a classic law-of-excluded-middle logic system.
Examples
By CEFR Level
- (Not applicable for this C2-level term.)
- (Not applicable for this C2-level term.)
- The philosopher explained that the law of excluded middle means a statement is either true or not true.
- In basic logic, we often use the law of excluded middle without thinking about it.
- The validity of the proof by contradiction rests entirely on the acceptance of the law of excluded middle.
- Intuitionistic mathematicians reject the law of excluded middle for statements about infinite sets.
- Debates about future contingents often question the applicability of the law of excluded middle.
Learning
Memory Aids
Mnemonic
Think of a light switch: it's either ON (true) or OFF (false). The 'excluded middle' is a dim, half-on state—classical logic says that state doesn't exist for logical propositions.
Conceptual Metaphor
LOGICAL TRUTH IS A BINARY STATE (like a switch). PROPOSITIONS ARE LOCATIONS WITH ONLY TWO POSSIBLE DESTINATIONS (true or false).
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- Avoid translating 'law' as 'закон' in the sense of legal statute. The correct equivalent is 'закон' in the logical/mathematical sense (e.g., закон исключённого третьего).
- Do not confuse with 'middle ground' or 'compromise'—this is not about negotiation but absolute logical dichotomy.
- The word 'excluded' here means 'not allowed as an option', not 'physically removed'.
Common Mistakes
- Using it in non-logical contexts (e.g., 'We need a law of excluded middle for this political debate').
- Confusing it with the 'law of non-contradiction' (which states a proposition cannot be both true and false).
- Misspelling as 'law of excluded middle'.
Practice
Quiz
In which of these fields is the 'law of excluded middle' a foundational and often uncontroversial principle?
FAQ
Frequently Asked Questions
No. They are related but distinct. The law of non-contradiction states that a proposition cannot be both true and false at the same time. The law of excluded middle states that a proposition must be either true or false. Together, they define classical bivalent logic.
Yes. Intuitionistic logic, many multi-valued logics (e.g., fuzzy logic), and some paraconsistent logics explicitly reject or do not include the law of excluded middle as a valid principle for all propositions.
Consider the proposition 'It is raining.' The law of excluded middle asserts that either 'It is raining' is true, or 'It is not raining' is true. A third option ('It is neither raining nor not raining') is excluded.
The 'middle' refers to a possible third truth value or state between 'true' and 'false'. The law states that this middle option is 'excluded' or not permitted in the classical system.