Learn about conditional statements and the implication operator.

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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 pp and qq be two propositions, we can construct a implication II as follows:

I:pq.I: p \Rightarrow q.

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

  • II: pp implies q.q.

  • II: If pp then q.q.

  • II: qq when p.p.

  • II: qq unless ¬p.\neg p.

  • II: Under the hypothesis pp the conclusion qq 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 pp is true, the conclusion qq must be true.

What does it say if the hypothesis pp 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 pp is false, the implication II is true regardless of the truth value conclusion qq.

However, when pp is true, we have to check the truth value of qq. If the conclusion is true, the implication holds; therefore, II is true. If the conclusion is false, then the implication does not hold and is false.

Let pp and qq be two propositions and,

I:pq,I: p \Rightarrow q,

be an implication. In the compound statement II, pp is called the premise or the hypothesis, whereas qq is called the conclusion.

The truth table below formally defines the implication II.

pp qq I:pqI: p \Rightarrow q

Note that in the above truth table, only one of the entries in the last column is F. In that row, pp is true, and qq 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 pp is false, the implication remains true regardless of the conclusion.

Let’s look at an example:

  • MM: If the moon is made out of cheese

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

Now let’s look at the implication:

I1:MH.I_1: M \rightarrow H.

In plain English, we can write this sentence as:

I1I_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 I1I_1 true?”

We know that the premise MM is false (the moon is not made out of cheese). Therefore, we can conclude that I1I_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.


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.

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