|
In logic, contraposition is a law that says that a conditional statement is logically equivalent to its contrapositive. The contrapositive of the statement has its antecedent and consequent inverted and flipped: the contrapositive of is thus . For instance, the proposition "''All bats are mammals''" can be restated as the conditional "''If something is a bat, then it is a mammal''". Now, the law says that statement is identical to the contrapositive "''If something is not a mammal, then it is not a bat''." The contrapositive can be compared with three other relationships between conditional statements: *Inversion (the inverse): . "''If something is not a bat, then it is not a mammal''." Unlike the contrapositive, the inverse's truth value is not at all dependent on whether or not the original proposition was true, as evidenced here. The inverse here is clearly not true. *Conversion (the converse): . "''If something is a mammal, then it is a bat''." The converse is actually the contrapositive of the inverse and so always has the same truth value as the inverse, which is not necessarily the same as that of the original proposition. *Negation: . "''There exists a bat that is not a mammal''. " If the negation is true, the original proposition (and by extension the contrapositive) is false. Here, of course, the negation is false. Note that if is true and we are given that ''Q'' is false, , it can logically be concluded that ''P'' must be false, . This is often called the ''law of contrapositive'', or the ''modus tollens'' rule of inference. ==Intuitive explanation== Consider the Euler diagram shown. According to this diagram, if something is in A, it must be in B as well. So we can interpret "all of A is in B" as: : It is also clear that anything that is not within B (the white region) cannot be within A, either. This statement, : is the contrapositive. Therefore, we can say that :. Practically speaking, this may make life much easier when trying to prove something. For example, if we want to prove that every girl in the United States (A) is blonde (B), we can either try to directly prove by checking all girls in the United States to see if they are all blonde. Alternatively, we can try to prove by checking all non-blonde girls to see if they are all outside the US. This means that if we find at least one non-blonde girl within the US, we will have disproved , and equivalently . To conclude, for any statement where A implies B, then ''not B'' always implies ''not A''. Proving or disproving either one of these statements automatically proves or disproves the other. They are fully equivalent. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Contraposition」の詳細全文を読む スポンサード リンク
|