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In mathematics, proof by contrapositive, or proof by contraposition, is a rule of inference used in proofs, where one infers a conditional statement from its contrapositive. [15] In other words, the conclusion "if A , then B " is inferred by constructing a proof of the claim "if not B , then not A " instead.
The thing of importance is that the dog detects or does not detect an intruder, not whether there is one.) Example 1: If I am the burglar, then I can crack a safe. I cannot crack a safe. Therefore, I am not the burglar. Example 2: If Rex is a chicken, then he is a bird. Rex is not a bird. Therefore, Rex is not a chicken.
In logic, an inverse is a type of conditional sentence which is an immediate inference made from another conditional sentence. More specifically, given a conditional sentence of the form P → Q {\displaystyle P\rightarrow Q} , the inverse refers to the sentence ¬ P → ¬ Q {\displaystyle \neg P\rightarrow \neg Q} .
Given a type A statement, "All S are P.", one can make the immediate inference that "All non-P are non-S" which is the contrapositive of the given statement. Given a type O statement, "Some S are not P.", one can make the immediate inference that "Some non-P are not non-S" which is the contrapositive of the given statement.
There are many places to live in California other than San Diego. On the other hand, one can affirm with certainty that "if someone does not live in California" (non-Q), then "this person does not live in San Diego" (non-P). This is the contrapositive of the first statement, and it must be true if and only if the original statement is true ...
Wherever logic is applied, especially in mathematical discussions, it has the same meaning as above: it is an abbreviation for if and only if, indicating that one statement is both necessary and sufficient for the other. This is an example of mathematical jargon (although, as noted above, if is more often used than iff in statements of definition).
For example, if the formula () stands for the sentence "Socrates is a banker" then the formula articulates the sentence "It is possible that Socrates is a banker". [127] To include these symbols in the logical formalism, modal logic introduces new rules of inference that govern what role they play in inferences.
For example, going from "No S are P" to its converse "No P are S". In the words of Asa Mahan: "The original proposition is called the exposita; when converted, it is denominated the converse. Conversion is valid when, and only when, nothing is asserted in the converse which is not affirmed or implied in the exposita." [5]