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In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement : "If P then Q ", Q is necessary for P , because the truth of Q is guaranteed by the truth of P .
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In propositional logic, affirming the consequent (also known as converse error, fallacy of the converse, or confusion of necessity and sufficiency) is a formal fallacy (or an invalid form of argument) that is committed when, in the context of an indicative conditional statement, it is stated that because the consequent is true, therefore the ...
Logical consequence (also entailment or logical implication) is a fundamental concept in logic which describes the relationship between statements that hold true when one statement logically follows from one or more statements.
Sufficiency: the solution pair , (,) satisfies the KKT conditions, thus is a Nash equilibrium, and therefore closes the duality gap. Necessity: any solution pair x ∗ , ( μ ∗ , λ ∗ ) {\displaystyle x^{*},(\mu ^{*},\lambda ^{*})} must close the duality gap, thus they must constitute a Nash equilibrium (since neither side could do any ...
For this reason, or perhaps for their familiarity and simplicity, necessity and possibility are often casually treated as the subject matter of modal logic. Moreover, it is easier to make sense of relativizing necessity, e.g. to legal, physical, nomological, epistemic, and so on, than it is to make sense of relativizing other notions.
Sufficiency finds a useful application in the Rao–Blackwell theorem, which states that if g(X) is any kind of estimator of θ, then typically the conditional expectation of g(X) given sufficient statistic T(X) is a better (in the sense of having lower variance) estimator of θ, and is never worse.
For instance, "If it's bound, it is a book; if it's not a book, it's not bound" and vice versa. (() ()) ("if not-A implies both B and its negation not-B, then not-A must be false, then A must be true"), which is the principle known as reductio ad absurdum. For instance, "If it's not bound, we know it's a book, if it's not bound, we know it's ...