Implication is a logical operation on two variables. "P implies Q" is usually written as "P → Q".
Implication, the result of "P → Q" is defined by the following table:
In other words, if P is true, then Q must also be true.
Somewhat counterintuitively, if P is false, then P → Q is true. To illustrate why this makes sense, imagine a teacher who tells her class that any student who gets 100% on the final exam will pass the class. In other words,
- P: A student gets 100% on the final
- Q: That student passes the class
- P → Q: A student gets 100% on the final → That student passes the class
Now imagine a student who did poorly on the final exam, but did well enough on other exams to pass the course. Did the teacher lie?
No. She said nothing about students who do not get 100% on the final (i.e., the case where P is false). Unless there is a student who both got 100% on the final and did not pass the course, the teacher told the truth.
For this reason, "P → Q" can be restated as "¬(P ∧ ¬ Q)" or "not (P and not Q)", or "it is not the case that P is true and Q is false".