Contraposition
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- For contraposition in the field of traditional logic, see Contraposition (traditional logic).
Contraposition is a logical relationship between two propositions of material implication. One proposition is the contrapositive of the other just when its antecedent is the negated consequent of the other, and vice-versa, resulting in two statements that are logically equivalent. Strictly, a contraposition can only exist between two statements each of which is no more complex than involving the same two propositions materially implicated. However, it is common to see two statements called contrapositives just when the statements each contain a material conditional, and are precisely the same apart from one of these implications being the contrapositive of the other (in the strict sense).
In propositional logic, a proposition Q is materially implicated by a proposition P when the following relationship holds:
- <math>(P \to Q)</math>
In vernacular terms, this states "If P then Q". The contrapositive of this statement would be:
- <math>(\neg Q \to \neg P)</math>
That is, "If not-Q then not-P", or more clearly, "If Q is not the case, then P is not the case." The two above statements are said to be contraposed. 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). Any propositions containing the first statement (e.g. <math>\forall{x}(P{x} \to Q{x})</math>, "All P's are Q's") are likewise contraposed in the non-strict sense to a duplicate proposition that involves the second statement (e.g. <math>\forall{x}(\neg Q{x} \to \neg P{x})</math>, "All non-Q's are non-P's").
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[edit] 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.
- <math>(P \to Q)</math>
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.:
- <math>\neg(P \and \neg Q)</math>
The elements of a conjunction can be reversed with no effect:
- <math>\neg(\neg Q \and P)</math>
We define <math>R</math> as equal to "<math>\neg Q</math>", and <math>S</math> as equal to <math>\neg P</math> (from this, <math>\neg S</math> is equal to <math>\neg\neg P</math>, which is equal to just <math>P</math>). Making these substitutions we get:
- <math>\neg(R \and \neg S)</math>
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:
- <math>(R \to S)</math>
Swapping back our definitions of R and S, we arrive at:
- <math>(\neg Q \to \neg P)</math>
[edit] Comparisons
| name | form | description |
|---|---|---|
| implication | if P then Q | first statement implies truth of second |
| inverse | if not P then not Q | negation of both statements |
| converse | if Q then P | reversal of both statements |
| contrapositive | if not Q then not P | negation of reversal of both statements |
[edit] Example
Take the statement "All Red things have Color". This can be equivalently expressed as "If an object is Red, then it has color".
- The converse is false in this case. "If an object has Color, then it is Red". Because of the vast possibilities of other Colors, the Converse of our statement is nonsensical and False.
- The inverse is "If an object is not Red, then it does not have Color". The inverse, by the nature of the relationship between Color and Red is also nonsense, utterly False.
- The contradiction is "There exists a shade of Red 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).
- The contrapositive is "If an object does not have Color, then it is not Red". This is the only other true statement that follows by logical validity from "All Red things have Color," and would therefore be the shortest route to proving our statement to be sound.
In other words, the contrapositive is the logically valid equivalent of a given Conditional statement, though not necessarily for a Biconditional.
[edit] 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.
[edit] 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 <math>\sqrt{2}</math> 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 <math>\sqrt{2}</math> 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 <math>\sqrt{2}</math> cannot be expressed as an irreducible fraction, then it must be the case that <math>\sqrt{2}</math> is not a rational number.
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