indirect proof
LowFormal, Academic
Definition
Meaning
A method of mathematical or logical proof where a statement is proven true by demonstrating that assuming it to be false leads to a contradiction.
More broadly, any argument or reasoning that establishes the truth of a proposition by showing the absurdity or impossibility of its negation. Used in philosophy, law, and general discourse to describe evidence or reasoning that is not direct or immediately obvious.
Linguistics
Semantic Notes
The term is most precise and most frequently used in mathematics and formal logic. In informal contexts, it can sometimes be used to describe any evidence that is circumstantial or not straightforward.
Dialectal Variation
British vs American Usage
Differences
No significant lexical or grammatical differences. The terminology is identical in academic settings.
Connotations
No differential connotations.
Frequency
Slightly more frequent in American academic discourse due to common references in logic and computer science curricula.
Vocabulary
Collocations
Grammar
Valency Patterns
[Subject] constructs/provides/uses an indirect proof of [Proposition].An indirect proof that [Proposition] is true proceeds by assuming its negation.Vocabulary
Synonyms
Strong
Neutral
Weak
Vocabulary
Antonyms
Phrases
Idioms & Phrases
- “To prove something by the back door (informal parallel).”
Usage
Context Usage
Business
Rare. Might be used metaphorically in strategic planning, e.g., 'Our indirect proof that the market will fail is that all optimistic scenarios lead to impossible resource demands.'
Academic
Standard in mathematics, logic, philosophy, and theoretical computer science. Core concept in formal reasoning.
Everyday
Virtually never used. A layperson would say 'proving by showing the opposite is impossible' or 'arguing from a contradiction.'
Technical
Precise term in formal logic and mathematics. Key method for proving existential statements, uniqueness, and irrationality (e.g., proof that √2 is irrational).
Examples
By Part of Speech
verb
British English
- The theorem was proven indirectly.
- She argued indirectly for the policy by showing the flaws in all alternatives.
American English
- The theorem was proven indirectly.
- He argued indirectly for the proposal by demonstrating the failures of opposing views.
adverb
British English
- The result was proved indirectly.
- He approached the problem indirectly.
American English
- The result was proven indirectly.
- She solved the issue indirectly.
adjective
British English
- An indirect method of proof was employed.
- He offered indirect evidence for his claim.
American English
- An indirect method of proof was used.
- She provided indirect evidence supporting her theory.
Examples
By CEFR Level
- We didn't see him take it, but we have indirect proof he was there.
- Sometimes you can prove something indirectly.
- The teacher explained that an indirect proof starts by assuming the opposite is true.
- If assuming something false creates a problem, that's an indirect proof it must be true.
- In geometry, we often use indirect proof to establish that lines must be parallel.
- The philosopher's argument was essentially an indirect proof of the soul's existence, derived from the contradictions inherent in materialism.
- Euclid's elegant indirect proof of the infinitude of primes is a cornerstone of number theory.
- The legal team constructed an intricate indirect proof of their client's alibi by systematically invalidating the prosecution's timeline.
Learning
Memory Aids
Mnemonic
Think: INdirect proof is like proving you DIDN'T take the cookie by showing that IF you had taken it, the dog would have barked (contradiction to known fact).
Conceptual Metaphor
PROOF IS A JOURNEY; indirect proof is taking a detour to arrive at the destination by showing all other roads are blocked.
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- Avoid confusing with 'косвенное доказательство' (indirect evidence) in a legal or everyday sense. The mathematical term is 'доказательство от противного' (proof by contradiction).
Common Mistakes
- Using 'indirect proof' to mean weak or suggestive evidence.
- Confusing it with 'proof by contrapositive', which is a specific type of indirect proof for conditional statements.
Practice
Quiz
Which of the following best describes the first step in a standard indirect proof?
FAQ
Frequently Asked Questions
In precise terms, 'proof by contradiction' is a specific and most common type of indirect proof. All proofs by contradiction are indirect, but some indirect proofs (like proof by contrapositive for conditional statements) have a slightly different formal structure.
No. In formal logic and mathematics, an indirect proof is just as rigorous and valid as a direct proof. It is a fundamental and accepted method of reasoning.
Yes. Euclid's proof that there are infinitely many prime numbers is a famous and ancient indirect proof. It assumes a finite number of primes and then constructs a new number that must be prime but isn't on the list, leading to a contradiction.
Essentially, yes. 'Reductio ad absurdum' (Latin for 'reduction to absurdity') is the classical name for the argumentative strategy underlying proof by contradiction, which is the primary form of indirect proof.