Implication is a logical operation on two statements, typically represented by the variables P and Q. "P implies Q" is written symbolically as "P → Q". Equivalent statements include "if P then Q", "P is sufficient for Q", and "Q is necessary for P".
Implication, the result of "P → Q" is defined by the following table:
Note that if P is true, then Q must also be true for the implication to hold (be true). However, if P is false, Q may or may not be true and the implication still holds.
To illustrate the latter fact, consider 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: If a student gets 100% on the final, then that student passes the class.
Now consider the case in which two students do poorly on the final exam; one of them did well enough on the other exams to pass the course, but the other did not. 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".
In summary, the following statements are equivalent:
|In symbols||In English|
|P → Q||P implies Q; if P then Q; P sufficient for Q; Q necessary for P|
|¬(P ∧ ¬Q)||not(P and not Q)|
|¬P ∨ Q||(not P) or Q [equivalent by De Morgan's laws]|
|¬Q → ¬P||not Q implies not P; (not P) unless Q|
The statement "P only if Q" is equivalent to "if P then Q" (P → Q), not "if Q then P".
To see why this is so, consider the statement:
- I will pass the course only if I study hard.
What this means is:
- If I don't study hard, I won't pass the course.
- P: I will pass the course.
- Q: I study hard.
- ¬ Q → ¬ P: If I don't study hard, I won't pass the course.
But this is equivalent to:
- P → Q: If I will pass the course then I study hard. (Or, rephrased slightly: If I pass the course then I must have studied hard.)
If and only if
The phrase "if and only if" means "is logically equivalent to", or "is a necessary and sufficient condition for". It is often abbreviated iff.
- P if and only if Q
means the same thing as:
- P implies Q, and Q implies P.