SDCAR2010【逻辑入门】（十三）Formal Logic (1) - ChaseDream

If-Then Statements

Example: If you run a red light in Beijing, then youwill get a ticket for 300 RMB.

To diagram this statement, let’s shorten the original statement byrepresenting each clause with one letter:

If R, then T.

R stands for “you run a red light in Beijing”

T stands for “you will get a ticket for 300 RMB”

From the original statement, we can infer that:

If not T, then not R.

In other words, if someone has never gotten a ticket (not T), then thatperson must not have run a red light in Beijing (not R). Basically, the newstatement switches the clauses and then negates both variables. This newif-then statement or inference is called a “contrapositive.” If the originalstatement is true, then contrapositive must also be true. Because both theoriginal and the contrapositive statements are logically equivalent, thecontrapositive is just another way of stating the original statement.

CommonMistakes

The problem is that many people apply only one of the steps above. Theyonly switch or they only negate. Neither of these two operations on theoriginal statement will produce an equivalent of the original one.

Original: If you run a red light in Beijing, then youwill get a ticket for 300 RMB. (If R, then T.)

Mistake one: If you got a ticket for 300 RMB, then you rana red light in Beijing. (If T, then R.)

The problem is you might be fined because you were speeding, not becauseyou were running a red light. So we cannot conclude that “you ran a red light”simply because “you got a ticket.”

Mistake two: If you did not run a red light in Beijing, then youwill not get a ticket for 300 RMB.

Again, this statement is obviously wrong since you could get a ticketfor speeding.

Negating And and Or

When you negate and, it becomes or.And when you negate or, it becomes and.

Example:

Statement: If you eat a poison,you will get sick and call your mom.

If P, then S and M

Contrapositive:

If you did not get sick or did not call your mom, you did not eat a poison.

If not S or not M, then not P.

It is important to use or herebecause using and wouldgo too far.

Only If

Like the words if and then,the phrase onlyif can also create andif-then relationship. But it can be confusing. Even though onlyif ends with if it does not introduce the if-clause.In fact, onlyif introduce the then-clause;whatever comes immediately afteronly if is then then-clause. The rest of thestatement is the if-clause.

Example:

Ming attends the meeting only if Hua attends.

Translation: If Ming attends the meeting, then Hua attends.

Only if you wear a shirt will you enter this restaurant.

Translation: If you enter this restaurant, then you were a shirt.

If andOnly If

The phrase if and only if actually introduces two rules. Consider thisexample:

The Shanghai Shark will win the tournament if, and only if, it has YaoMing as its center.

In this sentence, both if and onlyif introduce the lastclause—“it has Yao Ming as its center.” Yet if introducesif-clause and only if introducesthen-clause.

Translation: “If Shanghai Shark has Yao Ming as its center, then it willwin the tournament” and “If the Shanghai Shark won the tournament, then it hadYao Ming as its center.”

SDCAR2010【逻辑入门】（十二）Principles

SDCAR2010【逻辑入门】（十四）Formal Logic (2)

不好意思呀...我还是在搞...接着问您...><看了6楼的“充分条件和必要条件”，瞬间好像明白了点。也就是说，我们看到的传统的if开头的句子，是表达“充分条件”意思的，only if呢就是表达“必要条件”的。你提到的if and only if就是“充分必要"的...
那你文中提到的if-clause和then-clause是指句子里的if引导的部分和then引导的部分嘛？那我如果没有看到only if如何分辨呢？就比如：If you run a red light in Beijing, then you will get a ticket for 300RMB. 这个我根据意思就是会读成充分条件（R=>T），但是这里面不是同时有if和then了么？如果把刚刚那题的D直接改成if...then...的形式，我读出来还会是充分条件(afford=>high profits)...

额...NN能明白我意思么...我觉得我今天是绕进去了...再有问题我一定明天问！让它自然沉淀下...囧
谢谢！

-- by 会员 ffen (2011/9/18 21:58:22)