Introducing material implication

If "Johnny ate 5 apples", then "Johnny is fat"

Above is known as a conditional statement, or an implication. In math, we may assign the premise (or hypothesis) "Johnny ate 5 apples" to a variable P, and the conclusion (or consequent) "Johnny is fat" to a variable Q .

We may rewrite this in mathematical notation, $P \implies Q$ .

We may find the truth table for this statement here:

$\vdots$

Now, one of the most unintuitive things about this is that when P is false, and Q is true, the conditional statement is still regarded as true, known as vacuously true. Why is that?

Explanation

The statement is still true because P is false. When $P$ is false, that means the premise, or the "cause" of the consequence never occurred in the first place. Johnny never eats the five apples. Because of that, Johnny may or may not be fat. Let's say he is fat. This however, does not falsify the idea that if Johnny ate 5 apples, he would become fat. The statement is only concerned about the consequence of Johnny eating exactly 5 apples. It doesn't make any claim about when Johnny doesn't eat 5 apples. Perhaps he eats 6 apples.

We have to think of P and Q as independent events, and $P \implies Q$ as a statement that claims a causal relationship between the two independent events. $P$ is not the only means of $Q$ being true. Perhaps Johnny ate 10 apples, or an entire cow, and that made him fat. Who knows? The important thing to note is that if $P$ doesn't occur in the first place, we cannot falsify $Q$.

Read also:

• Paradoxes of material implication
• Principle of explosion