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【邏輯入門】Formal Logic
If-Then Statements
Example: If you run a red light in Beijing, then you will get a ticket for 300 RMB.
To diagram this statement, let’s shorten the original statement by representing 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 that person must not have run a red light in Beijing (not R). Basically, the new statement switches the clauses and then negates both variables. This new if-then statement or inference is called a “contrapositive.” If the original statement is true, then contrapositive must also be true. Because both the original and the contrapositive statements are logically equivalent, the contrapositive is just another way of stating the original statement.
Common Mistakes
The problem is that many people apply only one of the steps above. They only switch or they only negate. Neither of these two operations on the original statement will produce an equivalent of the original one.
Original: If you run a red light in Beijing, then you will get a ticket for 300 RMB. (If R, then T.)
Mistake one: If you got a ticket for 300 RMB, then you ran a red light in Beijing. (If T, then R.)
The problem is you might be fined because you were speeding, not because you 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 you will not get aticket for 300 RMB.
Again, this statement is obviously wrong since you could get a ticket for 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 here because using and would go too far.
Only If
Like the words if and then, the phrase only if can also create and if-then relationship. But it can be confusing. Even though only if ends with if it does not introduce the if-clause. In fact, only if introduce the then-clause; whatever comes immediately after only if is then then-clause. The rest of the statement 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 and Only If
The phrase if and only if actually introduces two rules. Consider this example:
The Shanghai Shark will win the tournament if, and only if, it has Yao Ming as its center.
In this sentence, both if and only if introduce the last clause—“it has Yao Ming as its center.” Yet if introduces if-clause and only if introduces then-clause.
Translation: “If Shanghai Shark has Yao Ming as its center, then it will win the tournament” and “If the Shanghai Shark won the tournament, then it had Yao Ming as its center.”
Unless
The word unless can also create an if-then relationship. But it can be the most confusing and counterintuitive “logic” word. Consider this example:
Your CR score will not be high unless you study formal logic.
There are two ways to translate this statement into if-then clauses:
1) If your CR score is high, then you have studied formal logic.
2) If you did not study formal logic, then your CR score will not be high.
The second method is the foolproof way, which replaces unless with if not. The preferred way is the first method, which negate the clause before unless and cap it with if, and replace unless with then. Basically the clause after unless is the necessary condition which must happen for the negated form of the other clause. Back to the example we have here, “study formal logic” is a necessary step for one to get high score in CR. Without “studying formal logic”, one would not be able to score a high mark in CR. But “studying formal logic” alone might not be sufficient to help you score high in CR.
When the unless-clause comes at the beginning of the sentence, everything between the word unless and the comma is the unless-clause.
Either
The word either can also create an if-then relationship. Consider this example:
Either Peking University or Tshinghua University is on the list of my dream schools.
Given this rule, if PKU is not on my list, then THU is on my list because one of them must be. Further, the rule does not exclude the possibility that both schools are on my list. Therefore, the correct way to say the same thing using if-then clauses is:
If PKU is NOT on my list, then THU is.
Notice the word NOT is added to the if-clause, not the then-clause. Otherwise, we would make the mistaken assumption that both schools cannot be on the list together, which is not necessarily true—at least on the GMAT or LSAT.
Hidden if-then statements
Many if-then statements on the test are hidden because they do not use if or then. Instead, they use words like all, any, when, must and so on. Consider this example:
All Chinese students are diligent.
Translation: If you are a Chinese student, then you are diligent. (I truly hope so!!)
The trick here is that all means if.
There are also words that mean then. Here is another example:
Reading SDCAR’s posts on CR requires good understanding of English.
Translation: If you can read SDCAR’s posts on CR, then you have good understanding of English. (Pat yourself on the back, please!!)
Here are more words you can use to find hidden if-then statements:
If: All, always, any, each, every, in order to, invariably, no, none, things that, those who, to, when.
Then: Depends on, essential, must, necessary, needs, only, only if, only when, prerequisite, requires.
Unless (if not): Except, until, without.
No
When you see no at the beginning of a sentence, change no to if and negate the other clause, which is your then clause. Example:
No one who has a cold should go outside. (No X is Y.)
Translation: If you have a cold, then you should NOT go outside. (If X, then NOT Y.)
Most, some, and not all
Most means more than half. Most could be all.
Some means at least one. Some = Many. Some could be most, could be all.
Not all means some did not. Not all could be none.
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