by contradiction

USAGE SUMMARY

"by contradiction" is correct and usable in written English.
It is typically used to refer to a contradiction between two or more ideas or concepts. For example, "We can prove this statement to be true by contradiction, because it eliminates any other possible outcomes."

✓ Grammatically correct

Science

Academia

Human-verified examples from authoritative sources

Exact Expressions

60 human-written examples

We will show that \(f\) is surjective by contradiction.

Cornell University

by contradiction.

Boundary Value Problems

(iii) Proceed by contradiction.

Fixed Point Theory and Applications

By contradiction, assume.

Boundary Value Problems

We prove by contradiction.

Data Science and Engineering

We follow by contradiction.

Advances in Difference Equations

Proof by contradiction.

EURASIP Journal on Wireless Communications and Networking

Proof Argue by contradiction.

Advances in Difference Equations

We argue by contradiction.

Bulletin of Mathematical Sciences

Suppose by contradiction that for.

Advances in Difference Equations

Proof We argue by contradiction.

Journal of Inequalities and Applications

Expert writing Tips

Best practice

When using "by contradiction", ensure that the contradiction you arrive at is a direct consequence of the initial assumption you are trying to disprove.

Common error

Avoid starting a proof "by contradiction" without clearly stating the initial assumption you intend to disprove. A poorly defined starting point can lead to invalid conclusions.

Linguistic Context

The phrase "by contradiction" functions as an adverbial phrase, modifying a verb or clause to indicate the method or approach used in an argument or proof. Ludwig AI confirms its correctness and usability in written English. It specifies how something is demonstrated or proven.

Expression frequency: Very common

Frequent in

Science

75%

Academia

25%

Formal & Business

0%

Less common in

News & Media

0%

Encyclopedias

0%

Wiki

0%

Ludwig's WRAP-UP

In summary, "by contradiction" is a grammatically correct and frequently used prepositional phrase, mainly in formal academic and scientific writing. According to Ludwig AI, it effectively indicates a specific method of logical proof where the opposite of a statement is assumed to derive a contradiction, thereby validating the original statement. While "by contradiction" is a common and valid method, remember to ensure that your starting premise is well-defined to avoid faulty reasoning.

More alternative expressions

Phrases that express similar concepts, ordered by semantic similarity:

proof by negation

This alternative specifies the 'proof' nature explicitly, while "by contradiction" implies it.

indirect proof

This term refers to the general strategy of disproving a statement by showing its contrary leads to absurdity.

reductio ad absurdum

This is a more formal, often Latin, term emphasizing reducing the premise to an absurd conclusion.

through negation

This phrase focuses more on the act of negating rather than reaching a contradictory conclusion.

reasoning by inconsistency

Highlights that the method relies on creating an inconsistency.

establishing a contradiction

Focuses on the action of establishing the conflict to prove a point.

arguing from opposites

This phrase presents a simplified version of the original phrase, by using the argument and opposites.

disproof by assumption

This term highlights that one begins by assuming the opposite of what one hopes to prove.

showing an absurdity

This term focuses on the result of the process which is exposing the flaw or the absurdity.

proof of impossibility

This alternative phrase highlights that the main intention is to prove that something is impossible.

FAQs

How is "by contradiction" used in mathematical proofs?

In mathematical proofs, "by contradiction" is a method where you assume the opposite of what you want to prove, and then show that this assumption leads to a logical impossibility. This proves that the original statement must be true.

What's an example of an argument "by contradiction"?

For instance, to prove that there is no largest integer, you might assume there is a largest integer, N. But then N+1 would be larger, which contradicts the assumption. Thus, the initial assumption is false.

Are there situations where using "by contradiction" is not the best approach?

Yes, sometimes a direct proof is more straightforward and easier to understand. Over-reliance on proof by contradiction can make an argument more complex than necessary. Sometimes "indirect proof" is also not the best solution.

What are some alternative ways to phrase "by contradiction" in academic writing?

Depending on the context, you could use phrases like "proof by negation", "reductio ad absurdum", or "establishing a contradiction". See also "proof by negation".