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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 [math]P \to Q[/math], the assertion that "Q is a logical consequence P" is often written as [math]P \implies Q[/math].
This may be problematic, because it means that from a false proposition, anything can follow.
The following shows a (valid) implication
On the other hand, the statement I promise that if I am healthy, I will come to class has four possibilities:
- I am healthy, and I do come to class. I have kept my promise.
- I am healthy, and I do not come to class. I have not kept my promise.
- I am not healthy, and I do come to class. I have kept my promise.
- 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.
- "Comprehensive List of Logic Symbols" (in en-US). 2020-04-06. https://mathvault.ca/hub/higher-math/math-symbols/logic-symbols/.
- Weisstein, Eric W.. "Implies" (in en). https://mathworld.wolfram.com/Implies.html.