Contraposition

This is often called the law of contrapositive, or the modus tollens rule of inference.

by checking that all girls in the United States do indeed have brown hair, or try to prove

One statement is the contrapositive of the other only when its antecedent is the negated consequent of the other, and vice versa.

The principle was stated as a theorem of propositional logic by Russell and Whitehead in Principia Mathematica as where

Therefore, B must be true: Combining the two proved statements together, we obtain the sought-after logical equivalence between a conditional and its contrapositive: Logical equivalence between two propositions means that they are true together or false together.

To prove that contrapositives are logically equivalent, we need to understand when material implication is true or false.

): This reads "It is not the case that (R is true and S is false)", which is the definition of a material conditional.

, we then obtain the desired contrapositive: In Hilbert-style deductive systems for propositional logic, only one side of the transposition is taken as an axiom, and the other is a theorem.

We describe a proof of this theorem in the system of three axioms proposed by Jan Łukasiewicz: (A3) already gives one of the directions of the transposition.

In other words, the contrapositive is logically equivalent to a given conditional statement, though not sufficient for a biconditional.

Since the statement and the converse are both true, it is called a biconditional, and can be expressed as "A polygon is a quadrilateral if, and only if, it has four sides."

In some cases, contraposition involves a change of the former's quality (i.e. affirmation or negation).

Contraposition also has philosophical application distinct from the other traditional inference processes of conversion and obversion where equivocation varies with different proposition types.

In traditional logic, the process of contraposition is a schema composed of several steps of inference involving categorical propositions and classes.

For "E" statements, partial contraposition can be obtained by additionally making a change in quantity.

The successive applications of conversion and obversion within the process of contraposition may be given by a variety of names.

In regard to the "A" proposition this is circumvented in the symbolism of modern logic by the rule of transposition, or the law of contraposition.

In this sense the use of the term "contraposition" is usually referred to by "transposition" when applied to hypothetical propositions or material implications.

The symbol for material implication signifies the proposition as a hypothetical, or the "if–then" form, e.g. "if P, then Q".

The symbol for the biconditional ("↔") signifies the relationship between the propositions is both necessary and sufficient, and is verbalized as "if and only if", or, according to the example "If P, then Q 'if and only if' if not Q, then not P".

Necessary and sufficient conditions can be explained by analogy in terms of the concepts and the rules of immediate inference of traditional logic.

In the categorical proposition "All S is P", the subject term S is said to be distributed, that is, all members of its class are exhausted in its expression.

The obverse is then converted, resulting in "No non-P is S", maintaining distribution of both terms.

Since nothing is said in the definition of contraposition with regard to the predicate of the inferred proposition, it is permissible that it could be the original subject or its contradictory.

This is in contradistinction to the form of the propositions of transposition, which may be material implication, or a hypothetical statement.

This can be shown by setting √N equal to the rational expression a/b with a and b being positive integers with no common prime factor, and squaring to obtain N = a2/b2 and noting that since N is a positive integer b=1 so that N = a2, a square number.

More often than not, this approach is preferred if the contrapositive is easier to prove than the original conditional statement itself.

Logically, the validity of proof by contrapositive can be demonstrated by the use of the following truth table, where it is shown that p → q and

of subjective logic produces an absolute FALSE derivative conditional opinion

[17] Contraposition represents an instance of Bayes' theorem which in a specific form can be expressed as: In the equation above the conditional probability