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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) |
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发表于 2011-7-13 12:06:28
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