Logical Implication

Implication

Implication is perhaps the most important, but also the most confusing of the logical connectives. In fact it even has a paradox named after it.

The implication of two statements $P$ and $Q$ is the statement is that $Q$ is True whenever $P$ is True. Some ways to phrase this are

P

implies

Q

.

If

P

then

Q

.

P

only if

Q

.

Q

if

P

.

Q

is a necessary condition for

P

.

P

is a sufficient condition for

Q

.

When we use the phrase "If ... then ..." in English it usually means there is some sort of causality going on. For example the statement

If it rains the traffic will be terrible.

somehow contains the idea that the rain will cause the traffic to be terrible. But in terms of logic there doesn't have to be any such connection between the two statement. This is where the paradox, one of the 'paradoxes of material implication', comes in. Namely, if $P$ is a false statement, then the implication $P$ implies $Q$ is true, even if there is no connection between $P$ and $Q$ . For example

If 0=1 then the moon is made of cheese.

is logically true even though whether the moon is made of cheese has nothing to do with whether 0 is equal to 1.

This state of affairs may seem rather strange, which is why it's called a paradox. So perhaps it would help to ask when you can say that the statement $P$ implies $Q$ is False rather than when you can say it's True. Imagine your dentist says to you

If you eat a lot of sugar then you'll get more cavities.

This is an implication between the two statements

You eat a lot of sugar.

and

You'll get more cavities.

Now suppose you want to prove your dentist wrong and say "Ha! You don't know what you're talking about. I shall seek dental care elsewhere." If you stay away from sugar and don't get cavities then your dentist will be right. If you stay away from sugar but get cavities anyway then your dentist can ask "Did you brush after eating?" and you'll say "No," and your dentist will say "There you go!" and will still be right. The only way you can prove your dentist wrong is to eat a lot of sugar but not get cavities.

This fact is actually useful in some situations and since it's logically valid there's nothing wrong with using it in a proof.

Examples:

First statement	Second statement	Implication
The hall was long.	The hall had many doors.	If the hall was long then it had many doors.
Mike's dog has a wet nose.	Mike's dog is healthy.	If Mike's dog has a wet nose then he/she is healthy.
4 is even.	6 is odd.	If 4 is even then 6 is odd.
Triangle ABC is equilateral.	Triangle ABC is isosceles.	If Triangle ABC is equilateral then it is isosceles.

As we've seen, the implication $P$ implies $Q$ is True when $P$ is false. It's also True when $Q$ is True and only false when $P$ is True and $Q$ is False. In tabular form:

$P$	$Q$	$P\Rightarrow Q$
T	T	T
T	F	F
F	T	T
F	F	T

The logical symbol for implication is " $\implies$ ", though " $\supset$ " is sometimes seen instead. so you can write $P\implies Q$ for $P$ implies $Q$ .

Unlike $P$ and $Q$ and $P$ or $Q$ , the value of $P$ implies $Q$ may change if you switch $P$ with $Q$ . In other words

P

implies

Q

is not always the same as

Q

implies

P

.

The two statements are related though and we call the statement

Q

implies

P

the 'converse' of

P

implies

Q

Implication plays an important role since most theorems take on the form of an implication.

Note that while the inverse of $P\implies Q$ (that is, $\neg P\implies \neg Q$ ) does not necessarily have the same truth value as $P\implies Q$ , the contrapositive ( $\neg Q\implies \neg P$ ) does. For example, $x>10\implies x>0$ and its contrapositive, $x\leq 0\implies x\leq 10$ , are logically equivalent, and always have the same truth value for any number $x$ .