How to prove anything from a contradiction?

To prove something by contradiction, we assume that what we want to prove is not true, and then show that the consequences of this are not possible. That is, the consequences contradict either what we have just assumed, or something we already know to be true (or, indeed, both) – we call this a contradiction.

What follows from a contradiction?

In classical logic, intuitionistic logic and similar logical systems, the principle of explosion (Latin: ex falso [sequitur] quodlibet, ‘from falsehood, anything [follows]’; or ex contradictione [sequitur] quodlibet, ‘from contradiction, anything [follows]’), or the principle of Pseudo-Scotus, is the law according to …

What is explosion in logic?

Explosion is a valid principle of classical logic. It states that an inconsistent set of propositions entails any proposition whatsoever. However, ordinary agents presumably do — occasionally, at least — have inconsist- ent belief sets.

Why does anything follow from a contradiction?

The reason that anything (or something if you prefer) follows from a contradiction, is that all communication is an expression of logical functioning in accordance with the 3 Laws of Logic. A contradiction consists of a pairing of Identity Statements. (The Identity Statements are called Premises or Warrants).

How do you prove Contrapositive?

In mathematics, proof by contrapositive, or proof by contraposition, is a rule of inference used in proofs, where one infers a conditional statement from its contrapositive. In other words, the conclusion “if A, then B” is inferred by constructing a proof of the claim “if not B, then not A” instead.

How do you prove negation?

Proof of negation is an inference rule which explains how to prove a negation:

1. To prove ¬ϕ , assume ϕ and derive absurdity.
2. To prove ϕ , assume ¬ϕ and derive absurdity.
3. “Suppose ϕ . Then … bla … bla … bla, which is a contradiction. QED.”
4. “Suppose ¬ϕ . Then … bla … bla … bla, which is a contradiction. QED.”

Can contradictions be true?

More precisely, it is the belief that there can be a true statement whose negation is also true. Such statements are called “true contradictions”, dialetheia, or nondualisms. Graham Priest defines dialetheism as the view that there are true contradictions.

Can a false statement prove anything?

4 Answers. If we accept a false statement, can we prove everything? If by “false” we mean contradictory, then the answer is yes, by the principle of explosion. For example, begin with the Peano axioms and adjoin the sentence 1=0.

Can a contrapositive be false?

Truth. If a statement is true, then its contrapositive is true (and vice versa). If a statement is false, then its contrapositive is false (and vice versa). If a statement (or its contrapositive) and the inverse (or the converse) are both true or both false, then it is known as a logical biconditional.

What is a negation example?

A negation is a refusal or denial of something. If your friend thinks you owe him five dollars and you say that you don’t, your statement is a negation. “I didn’t kill the butler” could be a negation, along with “I don’t know where the treasure is.” The act of saying one of these statements is also a negation.