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Concept
Contraposition
Summary
In logic and mathematics, contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as proof by contraposition. The contrapositive of a statement has its antecedent and consequent inverted and flipped.
Conditional statement . In formulas: the contrapositive of is .
If P, Then Q. — If not Q, Then not P. "If it is raining, then I wear my coat" — "If I don't wear my coat, then it isn't raining."
The law of contraposition says that a conditional statement is true if, and only if, its contrapositive is true.
The contrapositive () can be compared with three other statements:
Inversion (the inverse), "If it is not raining, then I don't wear my coat." 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.
Conversion (the converse), "If I wear my coat, then it is raining." The converse is actually the contrapositive of the inverse, and so always has the same truth value as the inverse (which as stated earlier does not always share the same truth value as that of the original proposition).
Negation (the logical complement), "It is not the case that if it is raining then I wear my coat.", or equivalently, "Sometimes, when it is raining, I don't wear my coat. " If the negation is true, then the original proposition (and by extension the contrapositive) is false.
Note that if is true and one is given that is false (i.e., ), then it can logically be concluded that must be also false (i.e., ). This is often called the law of contrapositive, or the modus tollens rule of inference.
In the Euler diagram shown, 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 blue region) cannot be within A, either. This statement, which can be expressed as:
is the contrapositive of the above statement. Therefore, one can say that
In practice, this equivalence can be used to make proving a statement easier.
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Ontological neighbourhood
Related publications (7)
Related concepts (16)
Transposition (logic)
In propositional logic, transposition is a valid rule of replacement that permits one to switch the antecedent with the consequent of a conditional statement in a logical proof if they are also both negated. It is the inference from the truth of "A implies B" to the truth of "Not-B implies not-A", and conversely. It is very closely related to the rule of inference modus tollens. It is the rule that where "" is a metalogical symbol representing "can be replaced in a proof with".
Logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or logical truths. It studies how conclusions follow from premises due to the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. It examines arguments expressed in natural language while formal logic uses formal language.
Antecedent (logic)
An antecedent is the first half of a hypothetical proposition, whenever the if-clause precedes the then-clause. In some contexts the antecedent is called the protasis. Examples: If , then . This is a nonlogical formulation of a hypothetical proposition. In this case, the antecedent is P, and the consequent is Q. In an implication, if implies then is called the antecedent and is called the consequent. Antecedent and consequent are connected via logical connective to form a proposition. If is a man, then is mortal.