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