## English

### Noun

- The statement of the form "if not Q then not P", given the statement "if P then Q".

- The statement of the form "if not Q then not P", given the statement "if P then Q".

logical statement

- Czech: obměna

- 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:

- (P \to Q)

In vernacular terms, this states
"If P then Q". 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(P \to Q), "All Ps are Qs") are likewise
contraposed in the non-strict sense to a duplicate proposition that
involves the second statement (e.g. \forall(\neg Q \to \neg P),
"All non-Qs are non-Ps").

- (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)

- 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." 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". 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.

- 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.

contraposition in Danish: Kontraposition

contraposition in German: Kontraposition

contraposition in Spanish: Contraposición
lógica

contraposition in French: Proposition
contraposée

contraposition in Japanese: 対偶 (論理学)

contraposition in Russian: Закон
контрапозиции

contraposition in Swedish:
Kontraposition

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swimming upstream, traversal, undercurrent

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