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The false positive rate (FPR) is the proportion of all negatives that still yield positive test outcomes, i.e., the conditional probability of a positive test result given an event that was not present. The false positive rate is equal to the significance level. The specificity of the test is equal to 1 minus the false positive rate.
If a statement's inverse is false, then its converse is false (and vice versa). If a statement's negation is false, then the statement is true (and vice versa). If a statement (or its contrapositive) and the inverse (or the converse) are both true or both false, then it is known as a logical biconditional.
Then if is true, that rules out the first disjunct, so we have . In short, P → Q {\displaystyle P\to Q} . [ 3 ] However, if P {\displaystyle P} is false, then this entailment fails, because the first disjunct ¬ P {\displaystyle \neg P} is true, which puts no constraint on the second disjunct Q {\displaystyle Q} .
In statistical hypothesis testing, a type I error, or a false positive, is the rejection of the null hypothesis when it is actually true. A type II error, or a false negative, is the failure to reject a null hypothesis that is actually false. [1] Type I error: an innocent person may be convicted. Type II error: a guilty person may be not convicted.
The assertion that Q is necessary for P is colloquially equivalent to "P cannot be true unless Q is true" or "if Q is false, then P is false". [9] [1] By contraposition, this is the same thing as "whenever P is true, so is Q". The logical relation between P and Q is expressed as "if P, then Q" and denoted "P ⇒ Q" (P implies Q).
The proposition to be proved is P. We assume P to be false, i.e., we assume ¬P. It is then shown that ¬P implies falsehood. This is typically accomplished by deriving two mutually contradictory assertions, Q and ¬Q, and appealing to the law of noncontradiction. Since assuming P to be false leads to a contradiction, it is concluded that P is ...
material conditional (material implication) implies, if P then Q, it is not the case that P and not Q propositional logic, Boolean algebra, Heyting algebra: is false when A is true and B is false but true otherwise.
In most logical systems, one proves a statement of the form "P iff Q" by proving either "if P, then Q" and "if Q, then P", or "if P, then Q" and "if not-P, then not-Q". Proving these pairs of statements sometimes leads to a more natural proof, since there are not obvious conditions in which one would infer a biconditional directly.