contraposition

For contraposition in the field of traditional logic, see Contraposition (traditional logic).

For contraposition in the field of symbolic logic, see Transposition (logic).

In propositional logic, contraposition is a logical relationship between two propositions of material implication. A proposition Q (e.g. "Socrates is human") is materially implicated by a proposition P (e.g. "Socrates is a man") when the following relationship holds:

In vernacular terms, this states "If P then Q", or, "If Socrates is a man then Socrates is human." In a conditional such as this, P is called the antecedent and Q the consequent. One statement is the contrapositive of the other just when its antecedent is the negated consequent of the other, and vice-versa. The contrapositive of the given example statement would be:

That is, "If not-Q then not-P", or more clearly, "If Q is not the case, then P is not the case." Using our example, this is rendered "If Socrates is not human, then Socrates is not a man." This statement is said to be contraposed to the original, and is logically equivalent to it. Due to their logical equivalence, stating one is effectively the same as stating the other, and where one is true, the other is also true (likewise with falsity).

Strictly, a contraposition can only exist in the above form of two simple conditionals. However, it is common to call two more complex statements contraposed if they are the same apart from containing a contraposition. Thus, , or "All P's are Q's" is contraposed to , or "All non-Q's are non-P's".

Equivalence of contrapositives

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 is only false when P is true and Q is false. Therefore, we can reduce this proposition to the statement "False when P and not-Q", i.e. "True when it is not the case that (P and not-Q)", i.e.:

The elements of a conjunction can be reversed with no effect:

We define R as equal to "", and S as equal to (from this, is equal to , which is equal to just P). Making these substitutions we get:

This reads "It is not the case that (R is true and S is false)", which is the definition of a material conditional - we can thus make this substitution:

Swapping back our definitions of R and S, we arrive at:

Comparisons

Examples

Take the statement "All red objects have color." This can be equivalently expressed as "If an object is red, then it has color."

• 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." Again, an object which is blue is not red, and still has color. Therefore the inverse is false.
• The converse is "If an object has color, then it is red." Objects can have other colors, of course, so, the converse of our statement is false.
• The contradiction is "There exists a red object that does not have the properties of color". If the contradiction were true, then both the converse and the inverse would be correct in exactly that case where the shade of red is not a color. However, in our world this statement is entirely untrue (and therefore false).

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

Similarly, take the statement "All quadrilaterals have four sides," or equivalently expressed "If a shape is a quadrilateral, then it has four sides."

• The contrapositive is "If a shape does not have four sides, then it is not a quadrilateral." This follows logically, and as a rule, contrapositives share the truth value of their conditional.
• The inverse is "If a shape is not a quadrilateral, then it does not have four sides." In this case, unlike the last example, the inverse of the argument is true.
• The converse is "If a shape has four sides, then it is a quadrilateral." Again, in this case, unlike the last example, the converse of the argument is true.
• The contradiction is "There is at least one quadrilateral that does not have four sides."

Since the statement and the converse are both true, it is called a biconditional, and can be expressed as "A shape is a quadrilateral if, and only if, it has four sides." That is, having four sides is both necessary to be a quadrilateral, and alone sufficient to deem it a quadrilateral.

Truth

• If a statement is true, then its contrapositive is always true (and vice versa).
• If a statement is false, its contrapositive is always false (and vice versa).
• If a statement's inverse is true, its converse is always true (and vice versa).
• If a statement's inverse is false, its converse is always false (and vice versa).
• If a statement's contradiction is false, then the statement is true.
• If a statement (or its contrapositive) and the inverse (or the converse) are both true or both false, it is known as a logical biconditional.

Application

Because the contrapositive of a statement always has the same truth value (truth or falsity) as the statement itself, it can be a powerful tool for proving mathematical theorems via proof by contradiction, as in the proof of the irrationality of the square root of 2. By the definition of a rational number, the statement can be made that "If is rational, then it can be expressed as an irreducible fraction". This statement is true because it is a restatement of a true definition. The contrapositive of this statement is "If cannot be expressed as an irreducible fraction, then it is not rational". This contrapositive, like the original statement, is also true. Therefore, if it can be proven that cannot be expressed as an irreducible fraction, then it must be the case that is not a rational number.