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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.
One obtains the rules for intuitionistic negation the same way but by excluding double negation elimination. Negation introduction states that if an absurdity can be drawn as conclusion from P {\displaystyle P} then P {\displaystyle P} must not be the case (i.e. P {\displaystyle P} is false (classically) or refutable (intuitionistically) or etc.).
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 ...
The negation "must not" has a stronger meaning (the effect is to apply the logical negation to the following infinitive rather than applying it to the full clause with must). For more details and other similar cases, see the relevant sections of English modal verbs .
Syntactically, (1) and (2) are derivable from each other via the rules of contraposition and double negation. Semantically, (1) and (2) are true in exactly the same models (interpretations, valuations); namely, those in which either Lisa is in Denmark is false or Lisa is in Europe is true. (Note that in this example, classical logic is assumed.
Subsumed within the auspice of Indian logic, 'Buddhist logic' has been particularly focused in its employment of the fourfold negation, as evidenced by the traditions of Nagarjuna and the Madhyamaka, particularly the school of Madhyamaka given the retroactive nomenclature of Prasangika by the Tibetan Buddhist logico-epistemological tradition ...
An expression in a formal language that contains free variables, which does not denote a specific object or truth value until the variables are instantiated. o-proposition In traditional logic, a particular negative categorical proposition, stating that some members of the subject class are not members of the predicate class. [2] [218] or
In propositional logic, the double negation of a statement states that "it is not the case that the statement is not true". In classical logic, every statement is logically equivalent to its double negation, but this is not true in intuitionistic logic; this can be expressed by the formula A ≡ ~(~A) where the sign ≡ expresses logical equivalence and the sign ~ expresses negation.