Contraposition

For contraposition in the field of traditional logic, see Contraposition (traditional logic). In Traditional logic, contraposition is a form of Immediate inference in which from a given Proposition another is inferred having for its subject

Contraposition is a logical relationship between two propositions of material implication. In Logic and Philosophy, proposition refers to either (a the content or Meaning of a meaningful Declarative sentence The material conditional, also known as the material implication or truth functional conditional, expresses a property of certain Conditionals in Logic 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. In Logic and Mathematics, negation or not is an operation on Logical values for example the logical value of a Proposition A consequent is the second half of a hypothetical Proposition. In Logic, statements p and q are logically equivalent if they have the same logical content 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:

$(P \to Q)$

In vernacular terms, this states "If P then Q". This is a technical mathematical article about the area of mathematical logic variously known as "propositional calculus" or "propositional logic" Vernacular refers to the Native language of a country or a locality The contrapositive of this statement would be:

$(\neg Q \to \neg P)$

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. $\forall{x}(P{x} \to Q{x})$ , "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. $\forall{x}(\neg Q{x} \to \neg P{x})$ , "All non-Q's are non-P's").

1 Equivalence of contrapositives
2 Comparisons
- 2.1 Example
- 2.2 Truth
3 Application

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. In Logic, statements p and q are logically equivalent if they have the same logical content

$(P \to Q)$

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. :

$\neg(P \and \neg Q)$

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

$\neg(\neg Q \and P)$

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

$\neg(R \and \neg S)$

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:

$(R \to S)$

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

$(\neg Q \to \neg P)$

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	reversal of negation of both statements

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 contrapositive is "If an object does not have color, then it is not red". This is follows logically from our initial statement and, like it, it is evidently true.
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 inverse is "If an object is not red, then it does not have color. Logic is the study of the principles of valid demonstration and Inference. " Again, an object which is blue is not red, and still has color. Therefore the inverse is also false.
The contradiction is "There exists a shade of red that does not have the properties of color". In Classical logic, a contradiction consists of a logical incompatibility between two or more Propositions It occurs when the propositions taken together yield 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 necessarily for a biconditional. In Logic and Mathematics, logical biconditional (sometimes also known as the material biconditional) is a Logical operator connecting two statements

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. In Logic and Mathematics, logical biconditional (sometimes also known as the material biconditional) is a Logical operator connecting two statements

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. In Logic and Mathematics, a logical value, also called a truth value, is a value indicating the extent to which a Proposition is true In Mathematics, a theorem is a statement proven on the basis of previously accepted or established statements Reductio ad absurdum ( Latin for "reduction to the absurd" also known as an apagogical argument, reductio ad impossibile In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction By the definition of a rational number, the statement can be made that "If $\sqrt{2}$ is rational, then it can be expressed as an irreducible fraction". In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions An irreducible fraction (or fraction in lowest terms or reduced form) is a Vulgar fraction in which the Numerator and Denominator This statement is true because it is a restatement of a true definition. The contrapositive of this statement is "If $\sqrt{2}$ 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 $\sqrt{2}$ cannot be expressed as an irreducible fraction, then it must be the case that $\sqrt{2}$ is not a rational number.

Dictionary

-noun

(logic) The statement of the form "if not Q then not P", given the statement "if P then Q".

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