P ⇒ Q: Equivalent Ways to Say It

What you will get from this section. Mathematicians express the single idea $P \Rightarrow Q$ in a surprising number of ways — "if $P$ then $Q$", "$P$ only if $Q$", "$P$ is sufficient for $Q$", "$Q$ is necessary for $P$", and more. They all say the same thing. By the end, given any one of these phrasings, you'll be able to rewrite the statement in all the others.

A quick reminder: $P \Rightarrow Q$ says that $P$ implies $Q$. Every phrasing below is just a different way of saying exactly that. In what follows, I will list several equivalent ways of expressing $P \Rightarrow Q$, in increasing order of difficulty, using the same three examples each time.

"If P then Q"

This is one of the most direct ways to express $P \Rightarrow Q$, clearly expressing the idea as a hypothetical: if, and whenever $P$ is true, then we can definitely conclude $Q$ is true.

• If $x>2$, then $x>1$.
• If Jane is in London, then Jane is in England.
• If my son played on the iPad for 3 hours straight, then the iPad will get hot.

"P is a sufficient condition for Q"

Another relatively direct way to express $P \Rightarrow Q$. Here "sufficient" means enough. To say $P$ is sufficient for $Q$ is to say that $P$ on its own is enough to guarantee $Q$: once you have $P$, you have everything you need for $Q$ to hold. That is precisely $P \Rightarrow Q$.

• $x>2$ is a sufficient condition for $x>1$.
• Jane is in London is a sufficient condition for Jane to be in England.
• My son played on the iPad for 3 hours straight is a sufficient condition for the iPad to get hot.

Notice the "a" here. It is important because $P \Rightarrow Q$ means $P$ is one of possibly many sufficient conditions for $Q$, and not necessarily the only one. For example, $x>3$ is another sufficient condition for $x>1$.

"Q is an inevitable consequence of P"

This is one of the least formal phrasings, but also one of the most intuitive ways to express $P \Rightarrow Q$. To say that $Q$ is an inevitable consequence of $P$ means that once $P$ holds, $Q$ cannot fail to follow. $Q$ is not merely likely or possibly true given $P$; it is forced, with no way out. That force is exactly what the arrow $P \Rightarrow Q$ asserts. This phrasing is more useful for building intuition than for writing formal proofs, but it captures the meaning clearly.

• $x>1$ is an inevitable consequence of $x>2$.
• Jane being in England is an inevitable consequence of Jane being in London.
• The iPad getting hot is an inevitable consequence of my son playing on the iPad for 3 hours straight.

Notice the "an" here. It emphasises the idea that $Q$ may be one of many inevitable consequences of $P$, not necessarily the only one.

"Q is a necessary condition for P"

This follows nicely from the previous phrasing, "$Q$ is an inevitable consequence of $P$". If $Q$ must follow whenever $P$ is true, then $Q$ is necessary for $P$. In other words, without $Q$, $P$ could not be true. We can prove this concretely!

Suppose $Q$ is not true. Then there are two possibilities for $P$: case 1 $P$ is true, or case 2 $P$ is not true.

Case 1: Suppose $P$ is true. Then, by the promise of $P \Rightarrow Q$, we can immediately conclude that $Q$ is true. But this means $Q$ and not $Q$ are true at the same time, which cannot be right. This is a contradiction, therefore case 1 is wrong.

Since case 1 cannot be right, then the only remaining case must be right, that is: $P$ is not true.

Therefore, if $Q$ is not true, then $P$ is not true. This is why $Q$ is a necessary condition for $P$.

• $x>1$ is a necessary condition for $x>2$.
• Jane being in England is a necessary condition for Jane being in London.
• The iPad getting hot is a necessary condition for my son playing on the iPad for 3 hours straight.

"P only if Q"

This is perhaps the phrasing students find most confusing, but it is actually just a rephrasing, or shorter version, of "$Q$ is a necessary condition for $P$". We say "$P$ only if $Q$" to mean that $P$ can be true only if, and when, $Q$ is true. Implicit in this is the idea that $P$ cannot be true if $Q$ is not true, or equivalently, that $Q$ is necessary for $P$.

• $x>2$ only if $x>1$.
• Jane is in London only if Jane is in England.
• My son played on the iPad for 3 hours straight only if the iPad got hot.

The last example is particularly interesting because it is, in fact, a real-life example. My 10-year-old son, who has not been taught logic, once tried desperately to convince my wife that he had not been sneaking off and playing on his iPad for the last three hours. My wife decided not to believe him! In his desperation, he exclaimed that he could not have done it because the iPad was not even warm. This comical example shows that the idea of "$Q$ is necessary for $P$" is in fact innately understandable to most people, and is used in everyday life without any realisation of its formal logical foundations. It shows that these ideas are more natural than people tend to think.

Summary

Each of the following conveys the same idea:

• $P$ implies $Q$, or symbolically $P \Rightarrow Q$.
• If $P$, then $Q$.
• $P$ is a sufficient condition for $Q$.
• $Q$ is an inevitable consequence of $P$.
• $Q$ is a necessary condition for $P$.
• $P$ only if $Q$.

There are, in fact, other formulations of the same idea, which we will go through in later sections.

Worksheets

Practise recognising the many equivalent ways of expressing $P \Rightarrow Q$ — "if $P$ then $Q$", "$P$ only if $Q$", and the language of sufficient and necessary conditions — and translating fluently between them. Each worksheet below comes as a PDF with fully worked solutions.

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