Implication
Learn about conditional statements and the implication operator.
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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 realworld 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.
Consider:
 $E$: Peter is eighteen years old.
 $D$: Peter has permission to drive a car.
Now,
 $I_2: E \Rightarrow D.$
In everyday language, we have:
 $I_2$: If Peter is eighteen years old, then he has permission to drive a car. >Note: We must understand that $I_2$ is only false when $E$ is true and $D$ is false. That is when Peter is eighteen years old, yet does not have permission to drive a car.
For another example, let:
 $W$: The Sun rises from the West.
 $T$: 2+2 = 4.
Now,
 $I_3 : W \Rightarrow T.$
In everyday language, we can write,
 $I_3$: If Sun rises from the West then 2+2 = 4.
> Note: $I_3$ is true because the hypothesis ($W$) is false. We do not need to check the truth value of the conclusion (which is true in this case).
Take another proposition:
 $S$: 2+2 = 7.
Then,
 $I_4 = W \Rightarrow F.$
In everyday language, we have,
 $I_4$: If Sun rises from the West then 2+2 = 7.
> Note: $I_4$ is true because the truth value of hypothesis ($W$) is false. We do not need to check the truth value of the conclusion (which is false in this case).
This point must be drilled in over and over again. In an implication, if the hypothesis (or the premise) is false, the implication is true regardless of the truth value of the conclusion. If a person claims that $I_4$ is true, then he is not wrong simply because the $W$ is false.
Let’s consider the following:
 $E$: Sun rises from the East.
Now,

$I_5 = E \Rightarrow S.$

$I_5$: If Sun rises from the East then 2+2 = 7.
> Note: The truth value of $I_5$ is false because the truth value of hypothesis ($E$) is true and the truth value of conclusion ($S$) is false.
Let’s look at another example.
Consider the following propositions:
 $R$: It is raining.
 $C$: It is cloudy.
Now,
 $I_6= R \Rightarrow C.$
In everyday language, we have,
 $I_6$: If it is raining, then it is cloudy.
$I_6$ can only be false if it is raining and it is not cloudy. In all other cases, $I_6$ will be true.
Properties
Unlike conjunction and disjunction, the implication is not commutative. Let $p$ and $q$ are two propositions and let $I$, be the implication:
 $I: p \Rightarrow q$
Reversing the roles of $p$ and $q$ we get a new implication,
 $C: q \Rightarrow p.$
Note that $I \not \equiv C$
This can be observed by making a truth table.
$p$  $q$  $I: p \Rightarrow q$  $C: q\Rightarrow p$ 

F  F  T  T 
F  T  T  F 
T  F  F  T 
T  T  T  T 
In the above table, consider the entry where $p$ is true and $q$ is false. In this case, $I$ is false, but $C$ is true.
Similarly, in the entry where $p$ is false and $q$ is true, $I$ is true, but $C$ is false.
Note that the notation $\Rightarrow$ has been chosen deliberately for implication. This notation reminds us that an implication and its converse are not equivalent.
To understand associativity, let’s look at the following truth table. Take $p, q$, and $r$ as three propositions:
$p$  $q$  $r$  $p \Rightarrow q$  $\left(p \Rightarrow q\right)\Rightarrow r$  $q \Rightarrow r$  $p \Rightarrow \left(q \Rightarrow r\right)$ 

T  T  T  T  T  T  T 
T  T  F  T  F  F  F 
T  F  T  F  T  T  T 
T  F  F  F  T  T  T 
F  T  T  T  T  T  T 
F  T  F  T  F  F  T 
F  F  T  T  T  T  T 
F  F  F  T  F  T  T 
If we look at the bold entries in the truth table presented above, it is easy to conclude:
$\left(p \Rightarrow q\right)\Rightarrow r \not \equiv p \Rightarrow \left(q\Rightarrow r\right)$
Hence, the implication is not associative.
Quiz
Test your understanding of the implication.
What must be the case to make the implication $q_1 \Rightarrow q_2$ false?
$q_1$ is true and $q_2$ is true.
$q_1$ is false and $q_2$ is false.
$q_1$ is false and $q_2$ is true.
$q_1$ is true and $q_2$ is false.