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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.
In logic, negation, also called the logical not or logical complement, is an operation that takes a proposition to another proposition "not ", written , , ′ [1] or ¯. [citation needed] It is interpreted intuitively as being true when is false, and false when is true.
Two transformation rules stating that the negation of a conjunction is the disjunction of the negations, and the negation of a disjunction is the conjunction of the negations. denotation The direct reference or literal meaning of a word or phrase, as opposed to its connotation or implied meaning. dense
Modus tollens (also known as "the law of contrapositive") is a deductive rule of inference. It validates an argument that has as premises a conditional statement (formula) and the negation of the consequent ( ¬ Q {\displaystyle \lnot Q} ) and as conclusion the negation of the antecedent ( ¬ P {\displaystyle \lnot P} ).
In logic, a set of symbols is commonly used to express logical representation. The following table lists many common symbols, together with their name, how they should be read out loud, and the related field of mathematics.
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]
A double negation does not affirm the law of the excluded middle ; while it is not necessarily the case that PEM is upheld in any context, no counterexample can be given either. Such a counterexample would be an inference (inferring the negation of the law for a certain proposition) disallowed under classical logic and thus PEM is not allowed ...
In logic, the law of excluded middle or the principle of excluded middle states that for every proposition, either this proposition or its negation is true. [1] [2] It is one of the three laws of thought, along with the law of noncontradiction, and the law of identity; however, no system of logic is built on just these laws, and none of these laws provides inference rules, such as modus ponens ...