Basics of Implication and Deduction

What you will get from this section. To gain a basic intuitive understanding of $P$ implies $Q$, as well as related terms such as deduction and implication.

What is an implication of a statement $P$?

An implication $Q$ of a statement $P$ is a conclusion that must follow if or whenever $P$ is true. In other words, once we know that $P$ is true, we are forced to conclude that $Q$ is true as well.

Example 1

Suppose we know: $n$ is a multiple of $10$. This is our statement $P$. What implications or conclusions can we draw?

Well, even though we do not know exactly what number $n$ is, based on $P$, we can conclude:

• $n$ is even.
• $n$ is divisible by $5$.
• $n$ must end in $0$.

These are three separate implications of our statement $P$.

Example 2

Suppose we know: James is in a boat floating on the River Thames in London. This is our statement $P$. What implications or conclusions can we draw?

• James is not on land.
• James is in England.
• There is at least one boat floating on the River Thames.

For both examples, you can see that in each case, multiple implications or conclusions $Q$ were drawn, or deduced, from $P$. It is also possible to draw even more conclusions. In each case, we can write the relationship between $P$ and $Q$ in short as:

The double-lined arrow $\Rightarrow$ is known as the implication arrow symbol. It means the statement the arrow is pointing to is an implication of the statement on the other side of the arrow. Here are some examples:

• $n$ is a multiple of $10$ $\Rightarrow$ $n$ is even.
• $x>2 \Rightarrow x>0$.
• It rained heavily here $\Rightarrow$ the roads got wet.
• $n^2$ is odd $\Leftarrow$ $n$ is odd.
• A triangle is right-angled $\Leftarrow$ its side lengths satisfy $a^2+b^2=c^2$.

An implication is a deduction

At this level, we can treat the words implication and deduction as having the same meaning. Similarly, the verbs imply and deduce are closely related.

For example, we may say:

• $x>2$ implies $x>1$.
• $x>1$ is an implication of $x>2$.
• $x>1$ is a deduction from $x>2$.
• From $x>2$, we deduce $x>1$.

All of the above statements are different ways of saying the same thing, namely:

So you should get used to all of these wordings.

Chained implications or deductions

In mathematical logical arguments, we often write a chain of implications or deductions to form the basis of an argument or proof. An example of this structure in words is:

We know $A$ is true, therefore $B$ is true. Then because $B$ is true, $C$ is true. Then because $C$ is true, $D$ is true.

In symbols:

Here is a mathematical example of a direct proof using chained implications.

Question:

Prove that $x^2+2x+4$ is always positive.

Proof:

We start with an obvious fact that does not require proof:

Then:

Therefore $x^2+2x+4$ is always positive.

Caution 1: $P \Rightarrow Q$ tells us nothing about $Q \Rightarrow P$

In general, just because $P$ implies $Q$, it does not automatically mean that $Q$ implies $P$. This is a common source of mistakes. For example:

• $x>1 \Rightarrow x^2>1$ is true, but $x^2>1 \Rightarrow x>1$ is not true. For example, take $x=-2$. Then $(-2)^2>1$, but $-2$ is not greater than $1$.
• $x=2 \Rightarrow (x-1)x=2(x-1)$ is true, but $(x-1)x=2(x-1) \Rightarrow x=2$ is false. This is because $(x-1)x=2(x-1)$ gives $x=1$ or $x=2$. So saying $x=2$ would mean $x$ must be $2$, which is false.

Caution 2: $P \Rightarrow Q$ tells us nothing about $P$ or $Q$ themselves

This one is a little subtle. The statement $P \Rightarrow Q$ only tells us that whenever $P$ is true, $Q$ must also be true. It does not say anything about the truth value of $P$ or $Q$ by themselves. It is a hypothetical statement.

Here is a mathematical example. Suppose our $P \Rightarrow Q$ is:

This implication is true. However, depending on the actual value of $x$, zero, one, or both of $P$ and $Q$ may be true.

For example, if $x=3$, then both $P$ and $Q$ are true. If $x=1.5$, then only $Q$ is true. If $x=0$, then both $P$ and $Q$ are false. The only combination we cannot have is when $P$ is true and $Q$ is false, but more on this in later sections.

Now for a fun non-mathematical example. Suppose our $P \Rightarrow Q$ is:

This is a hypothetical statement. We could accept it as reasonable: if one day genetically modified pigs grew large, strong wings, then they might really be able to fly. So this $P \Rightarrow Q$ can be true as a hypothetical statement, even though neither $P$, "pigs have strong enough wings", nor $Q$, "pigs can fly", is currently true.

Summary

• $P \Rightarrow Q$ means $P$ implies $Q$.
• "$P$ implies $Q$" and "$Q$ can be deduced from $P$" are the same statement in different words.
• $P \Rightarrow Q$ itself does not tell us anything about the truth of $P$ or $Q$. It is a hypothetical statement: if and when $P$ is true, then $Q$ is true.
• Implications can be chained: $P \Rightarrow Q$ and $Q \Rightarrow R$ give $P \Rightarrow R$.

Worksheets

Put the ideas from this section into practice: reading and writing implications $P \Rightarrow Q$, deciding what can and cannot be deduced from a given statement, and chaining implications together. Each worksheet below comes as a PDF with fully worked solutions.

• Worksheet 1 — Questions · Solutions
• Worksheet 2 — Questions · Solutions