Implication (logic) facts for kids

Implication is a special idea in logic that helps us understand how statements are connected. You can think of it like an "if...then" rule. When we say "if A, then B," it means that if statement A is true, then statement B must also be true. It's about one thing leading to another.

This idea is also called a logical consequence because if the first part (the "if" part) is true, the second part (the "then" part) is a direct result or consequence.

Implication connects two statements. Let's call them the "first statement" and the "second statement." The only time an implication is considered false is if the first statement is true, but the second statement is false. In all other situations, the implication is true.

Sometimes, this can seem a bit strange. For example, if the first statement is false, the "if...then" rule is always considered true, no matter what the second statement says! This is a key part of how logic works.

Understanding Implication with Examples

Let's look at some examples to make this clearer.

Simple Examples

Here's a classic example of a valid implication:

• All humans are mortal (this means they will die).
• Aristotle is human.
• Therefore, Aristotle is mortal.

In this example, if the first two statements are true, then the conclusion (Aristotle is mortal) must also be true. The conclusion logically follows from the first two statements.

Promises and Consequences

Imagine someone makes this promise: "If I am healthy, then I will come to class." Let's see what happens in different situations:

• Situation 1: The person is healthy, and they do come to class.
  • In this case, the promise was kept. The implication is true.

• Situation 2: The person is healthy, but they do not come to class.
  • Here, the promise was broken. The implication is false because the "if" part was true, but the "then" part was false.

• Situation 3: The person is not healthy, and they do come to class.
  • Even though they weren't healthy, they still came. The promise was kept. The implication is true because the "if" part was false.

• Situation 4: The person is not healthy, and they do not come to class.
  • They didn't come, but they also weren't healthy, so they didn't break the promise. The promise was kept. The implication is true because the "if" part was false.

As you can see, the only time the "if...then" statement is false is when the "if" part is true, but the "then" part doesn't happen.

Related pages

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