Implication (logic)

logical connective between two assertions, frequently symbolized by a (most often double) arrow to the right

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.

Examples

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.

References

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