Implication (also known as logical consequence, implies, or If ... then) is a logical operation. It is the relationship between statements that holds true when one logically "follows from" one or more others. While a statement of the form "if P then Q" is often written as ${\displaystyle P\to Q}$, the assertion that "Q is a logical consequence P" is often written as ${\displaystyle P\implies Q}$.^[1][2]

Implications take two arguments. It returns false if and only if the first term is true and the second term is false.^[2]

This may be problematic, because it means that from a false proposition, anything can follow.

The following shows a (valid) implication

1. All humans are mortal (they die).
2. Aristotle is human.
3. Therefore, Aristotle is mortal.

On the other hand, the statement I promise that if I am healthy, I will come to class has four possibilities:

1. I am healthy, and I do come to class. I have kept my promise.
2. I am healthy, and I do not come to class. I have not kept my promise.
3. I am not healthy, and I do come to class. I have kept my promise.
4. I am not healthy, and I do not come to class. I have kept my promise.

In the second scenario, the statement is false, since the promise is broken. In other scenarios, the statement is true, since the promise is kept.

• If and only if
• Implication (grammar)
• Material conditional

1. ↑ "Comprehensive List of Logic Symbols". Math Vault. 2020-04-06. Retrieved 2020-09-04.
2. ↑ ^{2.0 2.1} Weisstein, Eric W. "Implies". mathworld.wolfram.com. Retrieved 2020-09-04.