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.

Conditional statement $P\rightarrow Q$ . In formulas: the contrapositive of $P\rightarrow Q$ is $\neg Q\rightarrow \neg P$ .^[1]

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.^[2]

The contrapositive ( $\neg Q\rightarrow \neg P$ ) can be compared with three other statements:

Note that if $P\rightarrow Q$ is true and one is given that $Q$ is false (i.e., $\neg Q$ ), then it can logically be concluded that $P$ must be also false (i.e., $\neg P$ ). This is often called the law of contrapositive, or the modus tollens rule of inference.^[3]

Proofs[edit]

Simple proof by definition of a conditional[edit]

In first-order logic, the conditional is defined as:

The contrapositive is "If an object does not have color, then it is not red." This follows logically from our initial statement and, like it, it is evidently true.

The inverse is "If an object is not red, then it does not have color." An object which is blue is not red, and still has color. Therefore, in this case the inverse is false.

The converse is "If an object has color, then it is red." Objects can have other colors, so the converse of our statement is false.

The negation is "There exists a red object that does not have color." This statement is false because the initial statement which it negates is true.

In nonclassical logics[edit]

Intuitionistic logic[edit]

In intuitionistic logic, the statement $P\to Q$ cannot be proven to be equivalent to $\lnot Q\to \lnot P$ . We can prove that $P\to Q$ implies $\lnot Q\to \lnot P$ , but the reverse implication, from $\lnot Q\to \lnot P$ to $P\to Q$ , requires the law of the excluded middle or an equivalent axiom.

Subjective logic[edit]

Contraposition represents an instance of the subjective Bayes' theorem in subjective logic expressed as:

where $(\omega _{Q|P}^{A},\omega _{Q|\lnot P}^{A})$ denotes a pair of binomial conditional opinions given by source $A$ . The parameter $a_{P}$ denotes the base rate (aka. the prior probability) of $P$ . The pair of derivative inverted conditional opinions is denoted $(\omega _{P{\tilde {|}}Q}^{A},\omega _{P{\tilde {|}}\lnot Q}^{A})$ . The conditional opinion $\omega _{Q|P}^{A}$ generalizes the logical statement $P\to Q$ , i.e. in addition to assigning TRUE or FALSE the source $A$ can assign any subjective opinion to the statement. The case where $\omega _{Q\mid P}^{A}$ is an absolute TRUE opinion is equivalent to source $A$ saying that $P\to Q$ is TRUE, and the case where $\omega _{Q\mid P}^{A}$ is an absolute FALSE opinion is equivalent to source $A$ saying that $P\to Q$ is FALSE. In the case when the conditional opinion $\omega _{Q|P}^{A}$ is absolute TRUE the subjective Bayes' theorem operator ${\widetilde {\phi \,}}$ of subjective logic produces an absolute FALSE derivative conditional opinion $\omega _{P{\widetilde {|}}\lnot Q}^{A}$ and thereby an absolute TRUE derivative conditional opinion $\omega _{\lnot P{\widetilde {|}}\lnot Q}^{A}$ which is equivalent to $\lnot Q\to \lnot P$ being TRUE. Hence, the subjective Bayes' theorem represents a generalization of both contraposition and Bayes' theorem.^[17]