## What makes a statement a contradiction?

A logical contradiction is **the conjunction of a statement S and its denial not-S**. In logic, it is a fundamental law- the law of non contradiction- that a statement and its denial cannot both be true at the same time. Here are some simple examples of contradictions.

## How do you prove a statement by contradiction?

**The steps taken for a proof by contradiction (also called indirect proof) are:**

- Assume the opposite of your conclusion. …
- Use the assumption to derive new consequences until one is the opposite of your premise. …
- Conclude that the assumption must be false and that its opposite (your original conclusion) must be true.

## What is the contradiction of an implication?

To prove a statement of the form P ⇒ Q by contradiction, assume the assumption, P, is true, but the conclusion, Q, is false, and derive from this assumption a contradiction, i.e., **a statement such as “0 = 1” or “0 ≥ 1” that is patently false**: Assume P is true, and that Q is false. …

## What is the contradiction rule?

The contradiction rule is **the basis of the proof by contradiction method**. The logic is simple: given a premise or statement, presume that the statement is false. If this presumption leads to a contradiction, then the given statement must be true.

## Why is a contradiction problematic?

Contradictions are problematic in these theories because **they cause the theories to explode**—if a contradiction is true, then every proposition is true. The classical way to solve this problem is to ban contradictory statements, to revise the axioms of the logic so that self-contradictory statements do not appear.

## What is a contradiction example?

A contradiction is a situation or ideas in opposition to one another. **Declaring publicly that you are an environmentalist but never remembering to take out the recycling** is an example of a contradiction. A “contradiction in terms” is a common phrase used to describe a statement that contains opposing ideas.

## Why is proof by contradiction valid?

Proof by contradiction is valid only under certain conditions. The main conditions are: – **The problem can be described as a set of (usually two) mutually exclusive propositions**; – These cases are demonstrably exhaustive, in the sense that no other possible proposition exists.