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".
The statement "if P then Q" is called a conditional statement; P is known as the antecedent and Q the consequent.
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|
Unless and only if
The statement "P unless Q" can be translated using implication as "if not Q, then P". It might be more familiar when stated with a negated antecedent: "not P unless Q" means "if not Q then not P".
The statement "P only if Q" is equivalent to "if not Q, then not P", or just "if P then Q". Note that it is not equivalent to "P if Q", which means "if Q then P".
To illustrate how "only if" and "unless" statements work, consider the statement:
- I will pass the class only if I study hard.
What this means is:
- If I don't study hard, I won't pass the class.
Or, equivalently (as explained above):
- I can't both not study hard and pass the class.
If the last statement were not equivalent to the original, then I might be able to not study and still pass the class — but this conflicts with my original statement that it was only by studying hard that I would pass the class!
Using an "unless" statement to say the same thing:
- I won't pass the class unless I study 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.