GMAT考试-Testprep数学精解(8)

“If I eat nuts， then I break out in hives.” This in turn can be symbolized² a

　　s N——>H.

　　Next， we interpret the clause “there is a blemish³ on my hand” to mean “hives

　　，“ which we symbolize¹ as H. Substituting these symbolssintosthe argument yie

　　lds the following diagram：

　　N——>H

　　H

　　Therefore， N

　　The diagram clearly shows that this argument has the same structure as the g

　　iven argument. The answer， therefore， is （B）。

　　Denying the Premise⁴ Fallacy

　　A——>B

　　~A

　　Therefore， ~B

　　The fallacy of denying the premise occurs when an if-then statement is prese

　　nted， its premise denied， and then its conclusion wrongly negated⁵.

　　Example： （Denying the Premise Fallacy）

　　The senator will be reelected only if he opposes the new tax bill. But he wa

　　s defeated. So he must have supported the new tax bill.

　　The sentence “The senator will be reelected only if he opposes the new tax b

　　ill“ contains an embedded⁶ if-then statement： ”If the senator is reelected， t

　　hen he opposes the new tax bill.“ （Remember： ”A only if B“ is equivalent to

　　“If A， then B.”） This in turn can be symbolized as R——>~T. The sentence “But

　　the senator was defeated“ can be reworded as ”He was not reelected，“ which

　　in turn can be symbolized as ~R. Finally， the sentence “He must have support

　　ed the new tax bill“ can be symbolized as T. Using these symbols the argumen

　　t can be diagrammed as follows

R——>~T

　　~R

　　Therefore， T

　　[Note： Two negatives make a positive， so the conclusion ~（~T） was reduced to

　　T.] This diagram clearly shows that the argument is committing the fallacy

　　of denying the premise. An if-then statement is made； its premise is negated

　　； then its conclusion is negated.

　　Transitive Property

　　A——>B

　　B——>C

　　Therefore， A——>C

　　These arguments are rarely difficult， provided you step back and take a bir

　　d's-eye view. It may be helpful to view this structure as an inequality in m

　　athematics. For example， 5 > 4 and 4 > 3， so 5 > 3.

　　Notice that the conclusion in the transitive property is also an if-then sta

　　tement. So we don't know that C is true unless we know that A is true. Howev

　　er， if we add the premise “A is true” to the diagram， then we can conclude t

　　hat C is true：

　　A——>B

　　B——>C

　　A

　　Therefore， C

　　As you may have anticipated， the contrapositive can be generalized to the tr

　　ansitive property：

　　A——>B

　　B——>C

　　~C

　　Therefore， ~A

　　Example： （Transitive Property）

　　If you work hard， you will be successful in America. If you are successful i

　　n America， you can lead a life of leisure. So if you work hard in America， y

　　ou can live a life of leisure