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Contraposition. In logic and mathematics, contraposition, or transposition, refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as § Proof by contrapositive. The contrapositive of a statement has its antecedent and consequent inverted and flipped.
More broadly, proof by contradiction is any form of argument that establishes a statement by arriving at a contradiction, even when the initial assumption is not the negation of the statement to be proved. In this general sense, proof by contradiction is also known as indirect proof, proof by assuming the opposite, [2] and reductio ad impossibile.
An immediate inference is an inference which can be made from only one statement or proposition. [1] For instance, from the statement "All toads are green", the immediate inference can be made that "no toads are not green" or "no toads are non-green" (Obverse). There are a number of immediate inferences which can validly be made using logical ...
Modus tollens is a mixed hypothetical syllogism that takes the form of "If P, then Q. Not Q. Therefore, not P." 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.
Proof by contraposition infers the statement "if p then q" by establishing the logically equivalent contrapositive statement: "if not q then not p". For example, contraposition can be used to establish that, given an integer , if is even, then is even: Suppose is not even.
Tautology (logic) In mathematical logic, a tautology (from Ancient Greek: ταυτολογία) is a formula that is true regardless of the interpretation of its component terms, with only the logical constants having a fixed meaning. For example, a formula that states, "the ball is green or the ball is not green," is always true, regardless of ...
v. t. e. 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 ...
Inverse (logic) In logic, an inverse is a type of conditional sentence which is an immediate inference made from another conditional sentence. More specifically, given a conditional sentence of the form , the inverse refers to the sentence . Since an inverse is the contrapositive of the converse, inverse and converse are logically equivalent to ...