In predicate logic, the contrapositive (or transposition) of the statement "p implies q" is "not-q implies not-p." A statement and its contrapositive are always logically equivalent, unlike a statement's inverse or its converse. ... In symbolic logic, transposition is the rule of inference that permits one to infer from the truth of A implies B the truth of Not-B implies not-A, and conversely. ... In propositional calculus, or logical calculus in mathematics, the logical conditional is a binary logical operator connecting two statements, if p then q where p is a hypothesis (or antecedent) and q is a conclusion (or consequent). ... In logic, statements p and q are logically equivalent if they have the same logical content. ... In logic, if S is a statement of the form P implies Q then the inverse of S is a statement of the form (not P) implies (not Q). ... In logic, if S is a statement of the form P implies Q, then the converse of S is a statement of the form Q implies P. In general, the verity of S says nothing about the verity of its converse. ...

One can informally convince oneself of this equivalence by examining examples from ordinary English. Consider the statement, "If there is fire here, then there is oxygen here." The contrapositive would be, "If there is no oxygen here, then there is no fire here." If the statement and its contrapositive are indeed logically equivalent, then these sentences should either both be true or both false. But they are indeed both true. (See combustion.) Thus logical equivalence holds, at least in this case. The phrase ordinary English, like ordinary language, is often used in philosophy and logic to distinguish between ordinary, unsurprising uses of terms and their more specialized uses in theorizing, or jargon. ... Combustion or burning is an exothermic reaction between a substance (the fuel) and a gas (the oxidizer), usually O2, to release heat. ...

Note that while a statement is logically equivalent to its contrapositive (where the two statements are both negated and "swapped"), it is not logically equivalent to its converse (with the two statements "swapped", but not negated). In logic, if S is a statement of the form P implies Q, then the converse of S is a statement of the form Q implies P. In general, the verity of S says nothing about the verity of its converse. ...

Proofs of logical equivalence

That a statement and its contrapositive are always logically equivalent can be proven rigorously using formal logic. There are two basic methods for doing this: deriving the equivalence from the axioms of the propositional calculus, or with a truth table. The propositional calculus is a formal deduction system whose atomic formulas are propositional variables. ... Truth tables are a type of mathematical table used in logic to determine whether an expression is true or whether an argument is valid. ...

Via the propositional calculus

The logical equivalence of a statement and its contrapositive is not one of the axioms of the propositional calculus as defined in this work. However, the equivalence can be proved as follows: The propositional calculus is a formal deduction system whose atomic formulas are propositional variables. ...

First, we can prove that p → q entails ¬q → ¬p:

1. Suppose p → q. Assuming this, we can reason as follows:
   1. Suppose ¬q. Assuming this, we can reason as follows:
      1. Suppose p. Assuming this, we can reason as follows:
         1. q (Modus ponens, lines 1 and 1.1.1)
         2. ¬q (Copying from above)
         3. q and ¬q (Conjunction introduction)
      2. Since this is a contradiction, then ¬p (Reductio ad absurdum)
   2. Thus ¬q → ¬p (Conditional proof)
2. Thus (p → q) → (¬q → ¬p) (Conditional proof)

A very similar proof will show that ¬q → ¬p also entails p → q. Combined, these facts show the two statements to be logically equivalent. Modus ponens (Latin: mode that affirms) is a valid, simple argument form (often abbreviated to MP): If P, then Q. P. Therefore, Q. or in logical operator notation: where represents the logical assertion. ... Conjunction introduction is the inference that, if p is true, and q is true, then the conjunction p and q is true. ... Reductio ad absurdum (Latin for reduction to the absurd, traceable back to the Greek ἡ εις άτοπον απαγωγη, reduction to the impossible, often used by Aristotle) is a type of logical argument where we assume a claim for the sake of argument, arrive at an absurd result, and then conclude the original assumption must... Conditional proof is a proof that takes the form of asserting a conditional, and proving that the premise or antecedent of the conditional necessarily leads to the conclusion. ... Conditional proof is a proof that takes the form of asserting a conditional, and proving that the premise or antecedent of the conditional necessarily leads to the conclusion. ...

Via a truth table

Alternatively, logical equivalence can be proved using the following truth table: Truth tables are a type of mathematical table used in logic to determine whether an expression is true or whether an argument is valid. ...

The first two columns can be taken as given. The third follows from the first two by the truth table definition of the logical conditional. The fourth and fifth follow from the first two by negation. The sixth follows from the fourth and fifth, again by the definition of the logical conditional. In propositional calculus, or logical calculus in mathematics, the logical conditional is a binary logical operator connecting two statements, if p then q where p is a hypothesis (or antecedent) and q is a conclusion (or consequent). ... Negation, in its most basic sense, changes the truth value of a statement to its opposite. ... In propositional calculus, or logical calculus in mathematics, the logical conditional is a binary logical operator connecting two statements, if p then q where p is a hypothesis (or antecedent) and q is a conclusion (or consequent). ...

Since the third and sixth columns have the same truth values for all values of p and q, the two are logically equivalent. Also, you must memorize the patterns of the true-false, true-false statements.

In Aristotelian logic

In Aristotelian logic (or categorical logic), moreover, categorical propositions can have contrapositives. Aristotelian logic, also known as syllogistic logic, is the particular type of logic created by Aristotle, primarily in his works Prior Analytics and De Interpretatione. ... Categorical logic is a branch of category theory within mathematics, adjacent to mathematical logic but in fact more notable for its connections to theoretical computer science. ... A categorical proposition is a proposition that affirms or denies a predicate of a subject. ...

• The contrapositive of "All S is P" is "All non-P is non-S." (These are "A" propositions.)
• The contrapositive of "No S is P" is "No non-P is non-S." (These are "E" propositions.)
• The contrapositive of "Some S is P" is "Some non-P is non-S." (These are "I" propositions.)
• The contrapositive of "Some S is not P" is "Some non-P is not non-S." (These are "O" propositions.)

So-called "E" and "I" propositions are not logically equivalent to their contrapositives. For example, we cannot infer from "no bachelors are women" to "no non-women are non-bachelors" and from "some dogs are flea-bitten animals" to "some animals which are not flea-bitten are non-dogs"

However, so-called "A" and "O" propositions are logically equivalent to their contrapositives. For example, from "all violins are musical instruments," we can infer "All non-musical instruments are non-violins." Similarly, from "some plants are not trees," we can infer "some non-trees are not non-plants."

See also: Transposition, Conversion, Obversion In symbolic logic, transposition is the rule of inference that permits one to infer from the truth of A implies B the truth of Not-B implies not-A, and conversely. ... In traditional logic Conversion is a form of immediate inference in which from a given categorical proposition another proposition is inferred which has as its subject the predicate of the original proposition, and has as its predicate the subject of the original proposition, with the quality of the proposition remaining... Obversion is a process in logic where the quality (affirmation or negation) of the subject term in a categorical statement is changed and the predicate term is replaced by its compliment: All S are P → No S are non-P All cats are animals → No cats are non-animals Some...