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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 a ticket 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.” |
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