Implication

Implication

Implication is the binary operator we ask the learners to spend most time understanding. This request is not because this operation is difficult to define but because it’s the most widely misunderstood and commonly confused operator in real-world situations.

The implication is a binary operation connecting two propositions: the premise or the hypothesis and the conclusion. We denote the implication operator by placing the symbol “$\Rightarrow$” between the premise or hypothesis and the conclusion. Let $p$ and $q$ be two propositions, we can construct a implication $I$ as follows:

$$I: p \Rightarrow q.$$

We can state or express this implication, $I$, in English in the following ways:

• $I$: $p$ implies $q.$

• $I$: If $p$ then $q.$

• $I$: $q$ when $p.$

• $I$: $q$ unless $\neg p.$

• $I$: Under the hypothesis $p$ the conclusion $q$ holds.

We also call a statement containing an implication operator a conditional statement. The reason for calling it a conditional statement is due to it’s “If hypothesis then conclusion” structure.

Let’s carefully understand what an implication means in everyday language. The implication claims that whenever the hypothesis $p$ is true, the conclusion $q$ must be true.

What does it say if the hypothesis $p$ is false? A moment of thought tells us that, in that case, no claim is being made. If the hypothesis is false, the conclusion may or may not be true. Therefore, when $p$ is false, the implication $I$ is true regardless of the truth value conclusion $q$.

However, when $p$ is true, we have to check the truth value of $q$. If the conclusion is true, the implication holds; therefore, $I$ is true. If the conclusion is false, then the implication does not hold and is false.

Let $p$ and $q$ be two propositions and,

$$I: p \Rightarrow q,$$

be an implication. In the compound statement $I$, $p$ is called the premise or the hypothesis, whereas $q$ is called the conclusion.

The truth table below formally defines the implication $I$.

$p$	$q$	$I: p \Rightarrow q$
T	T	T
T	F	F
F	T	T
F	F	T

Note that in the above truth table, only one of the entries in the last column is F. In that row, $p$ is true, and $q$ is false. Therefore, if we want to claim that an implication is false, we must show that the premise is true and the conclusion is false.

Learners have significant difficulty with the last two rows of the above table. Note that when the premise $p$ is false, the implication remains true regardless of the conclusion.

Let’s look at an example:

• $M$: If the moon is made out of cheese

• $H$: The author of this lesson has horns on their head

Now let’s look at the implication:

$$I_1: M \rightarrow H.$$

In plain English, we can write this sentence as:

$I_1$: “If the moon is made out of cheese, then the author of this lesson has horns on their head.”

Now, we ask the question: “Is $I_1$ true?”

We know that the premise $M$ is false (the moon is not made out of cheese). Therefore, we can conclude that $I_1$ should be true. You can reach this conclusion without checking the head of the author!

If an implication’s premise is false, it is true regardless of the conclusion. In such cases, mathematicians say that the implication is vacuously true.

Such truisms often seem counterintuitive—and they are. It takes some time to get used to them.

> Note: If an implication’s premise is false, then the implication is true regardless of the truth value of the conclusion.

All of you might have had a conversation in which a friend claims an astonishing fact that seems unlikely to be true. In that case, some people sarcastically ridicule them by making a vacuously true statement using their claim.

Let’s look at an example:

• Boastful Bill: I ran 100 meters in less than 9 seconds.

• Sarcastic Sam: If you ran 100 meters in less than 9 seconds, then I am the President of the United States.

In the above conversation, Sam tells Bill that his claim cannot possibly be true.

Examples

Let’s look at a few examples of the implication operator at work. We will take a few propositions and then use the implication operator to make new ones.