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The de nition (P Q) ( P Q), together with DeMorgan s law for negating conjunctions, yields the following fact: (P Q) ( Q P). The implication If not Q, then not P is called the contrapositive of If P then Q. In mathematics, it s often easier to discover rules of inference that validate the contrapositive form of an implication, and doing so is called proof by contrapositive.
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According to the de nition of logical implication, the statement P Q holds except when P is true and Q is false. In particular, it holds whenever P is false, regardless of the truth value of Q. As a result, some if-then statements are logically true but may sound false or seem puzzling. For example, these propositions are both true: If 1=0, the moon is made of cheese. If the real number x is negative and positive, then x equals 11.
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In both propositions, the if part of the implication is false, so the entire if-then statement is true. Because implications gure prominently in logical inference, we re accustomed to encountering implications in a context where the if part is true, and the implication allows the then part to be deduced. This isn t the case in the preceding statements. No inference is possible, and the statements provide no information about the truth value of the then part. An implication P Q is called vacuously true if P is false. Similarly, the quanti ed statement x U (P(x) Q(x)) is called vacuously true if P(x) is false for all values of x in its domain of discourse. The reason for this terminology is simple: the statement x U (P(x) Q(x)) asserts that Q(x) holds whenever P(x) holds. If P(x) never holds, the statement asserts nothing at all.
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Quanti cation
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Statements that assert either the universality or the existence of some fact over a universe of discourse are called quanti ed statements. Here s an example of each kind. The words in italic are the ones that indicate quanti cation. Universally quanti ed statement Existentially quanti ed statement symphony. The Philharmonic has performed every Haydn symphony. The Philharmonic Orchestra has performed a Haydn
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Negating Quanti ed Statements
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The ability to negate quanti ed statements is a valuable skill for programmers, especially SQL programmers. As Itzik shows later in this book, some problems are easier to solve when analyzed using reverse logic. Instead of nding all the answers to a question, nd everything that isn t not an answer. Earlier in the chapter, I warned you that to negate a proposition, you can t simply negate everything in sight. The logical opposite of an advertising claim that all our books are discounted is not all our books are not discounted, nor is it none of our books are discounted, nor is it all our nonbooks are discounted. The actual logical opposite which expresses simply that the claim is false is it is not true that all our books are discounted, or equivalently, at least one of our books is not discounted. While we might also say this more simply as not all our books are discounted, this use of not all invites misinterpretation or at least mistranslation when translated into a computer program. Two general principles concern the negation of quanti ed statements. Universally quanti ed statements are false if there is one exception to the universal claim they make. Existentially quanti ed statements are false if there are no examples of the existence they claim. Generally, universal statements may be hard to prove (because their validity must be veri ed universally) but easy to disprove (because one exception violates the universality). On the other hand, existential statements may be easy to prove (only one valid example is enough) but hard to disprove (because everything must be proven invalid).
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