Why is this Statement correct: G implies ¬Contradiction?


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:

  1. Assume the opposite of your conclusion. …
  2. Use the assumption to derive new consequences until one is the opposite of your premise. …
  3. 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.

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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.