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If a statement's inverse is false, then its converse is false (and vice versa). If a statement's negation is false, then the statement is true (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.
Since an inverse is the contrapositive of the converse, inverse and converse are logically equivalent to each other. [1] For example, substituting propositions in natural language for logical variables, the inverse of the following conditional proposition "If it's raining, then Sam will meet Jack at the movies." would be
In logic and mathematics, the converse of a categorical or implicational statement is the result of reversing its two constituent statements. For the implication P → Q, the converse is Q → P. For the categorical proposition All S are P, the converse is All P are S. Either way, the truth of the converse is generally independent from that of ...
In propositional logic, affirming the consequent (also known as converse error, fallacy of the converse, or confusion of necessity and sufficiency) is a formal fallacy (or an invalid form of argument) that is committed when, in the context of an indicative conditional statement, it is stated that because the consequent is true, therefore the ...
converse The statement formed by reversing the antecedent and consequent of a conditional statement, not necessarily maintaining logical equivalence. converse domain In set theory and logic, the set of all elements that are related to any element of a given set under a specific relation. [72] converse barcan formula
Given a type A statement, "All S are P.", one can make the immediate inference that "All non-P are non-S" which is the contrapositive of the given statement. Given a type O statement, "Some S are not P.", one can make the immediate inference that "Some non-P are not non-S" which is the contrapositive of the given statement.
Given the conditional proposition , one can form truth tables of its converse, its inverse (), and its contrapositive (). Truth tables can also be defined for more complex expressions that use several propositional connectives.
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]