2. A only if B

There’s a big difference between saying that A is true IF B is true, and A is true ONLY IF B is true. Let’s look at this difference.

Here’s a conditional expressed using “only if”:

“The match is burning only if there’s oxygen in the room.”

We need to figure which of these is the antecedent, “the match is burning” or “there’s oxygen in the room”.

Given what we did in the last lecture, it’s tempting to just look for the “if” and apply the rule, whatever comes after the “if” is the antecedent, and conclude that “there’s oxygen in the room” is the antecedent.

But that’s wrong. “There’s oxygen in the room” isn’t the antecedent.

If this was the antecedent, then the sentence would be saying that if there’s oxygen in the room then the match will be burning. But if you’re saying that then you’re saying that the presence of oxygen is enough to guarantee that the match is burning.

That’s not what’s being said. What’s being said is that the presence of oxygen in the room is necessary for the match to be burning, it doesn’t say that it will be burning.

This sentence expresses a conditional, but the antecedent of the conditional is, in fact, “the match is burning”.

The “only if” makes a dramatic difference. This sentence is equivalent to the following conditional written in standard form:

“If the match is burning then there’s oxygen in the room.”

The “only if” actually reverses the direction of logical dependency. When you have “only if”, the claim that precedes the “only if’ is antecedent, what follows it is the consequent.

Here’s the “only if” rule:

“A only if B” = “If A then B”

The antecedent doesn’t come after the “if”, the consequent comes after the “if”.

Let’s take away the symbols and compare the “if” and “only if’ rules.

___ if ___

___ only if ___

When you’re given a conditional that uses “if” or “only if”, you look for the “if”, and if the “if” is all by itself, then the antecedent is what immediately follows the “if”.

If the “if” is preceded by “only” then you do the opposite, what follows the “only if” is the consequent, what precedes the “only if” is the antecedent. Once you’ve got that, then the rest is easy:

(consequent) if (antecedent)

(antecedent) only if (consequent)

From here you can easily write the conditional in standard form, “If A then B”.

Examples

Let’s look at some examples. Here are two sentences:

“Our team will kick off if the coin lands heads.”

“I’ll buy you a puppy only if you promise to take care of it.”

They both express conditionals. You need to write these in standard form, in the form “If A then B”. That requires that you identify the antecedent and the conditional in each sentence.

In the first sentence, “Our team will kick off if he coin lands heads”, the “if” appears by itself, so we know that what immediately follows the ‘if” is the antecedent.

So we write the conditional in standard form as follows:

“If the coin lands heads then our team will kick off.”

For the second sentence we have an “only if”, so we know to do precisely the opposite of what we did in the previous case. The antecedent is what precedes the “only if”. So, you write the conditional in standard form like this:

“If I buy you a puppy then you promise to take care of it.”

This conditional doesn’t say that if you promise to take care of it I’ll buy you a puppy. It’s saying that if I buy you a puppy, then you can be sure that you promised to take care of it, because that was a necessary condition for buying the puppy. But merely promising to take care of the puppy doesn’t guarantee that I’ll buy it.

It might be a bit more natural to write it like this: “If I bought you a puppy then you promised to take care of it”.

Sometimes shifting the tenses around can be helpful in expressing conditionals like this in a more natural way. For our purposes they mean the same thing.

Here’s the general rule once again:

“A only if B” = “If A then B”

In the next lecture we’ll look at what happens when you combine the “if” and “only if”.