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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. [15] 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.
Negation elimination states that anything follows from an absurdity. Sometimes negation elimination is formulated using a primitive absurdity sign . In this case the rule says that from and follows an absurdity. Together with double negation elimination one may infer our originally formulated rule, namely that anything follows from an absurdity.
Going from a statement to its converse is the fallacy of affirming the consequent.However, if the statement S and its converse are equivalent (i.e., P is true if and only if Q is also true), then affirming the consequent will be valid.
The inverse and the converse of a conditional are logically equivalent to each other, just as the conditional and its contrapositive are logically equivalent to each other. [1] But the inverse of a conditional cannot be inferred from the conditional itself (e.g., the conditional might be true while its inverse might be false [2]). For example ...
Boolean negation A form of negation where the negation of a non-true proposition is true, and the negation of a non-false proposition is false. [34] [35] [36] Boolean operator An operator used in logic and computer science that performs logical operations on its operands, such as AND, OR, and NOT. borderline case
It is an application of the general truth that if a statement is true, then so is its contrapositive. The form shows that inference from P implies Q to the negation of Q implies the negation of P is a valid argument. The history of the inference rule modus tollens goes back to antiquity. [4]
In contrast, proof of negation and principle of noncontradiction are both intuitionistically valid. Brouwer–Heyting–Kolmogorov interpretation of proof by contradiction gives the following intuitionistic validity condition: if there is no method for establishing that a proposition is false, then there is a method for establishing that the ...
This is the contrapositive of the first statement, and it must be true if and only if the original statement is true. Example 2. If an animal is a dog, then it has four legs. My cat has four legs. Therefore, my cat is a dog.