Tag: consequent

Difficulty: What the heck

In logic, words like “if,” “only if,” “if and only if,” and “unless” express different logical relationships. So if you conflate those terms and symbolize them incorrectly, you end up with confused garbage. Today’s post is on what “if,” “only if,” “if and only if,” and “unless” mean in logic.

“If”

This one is pretty straightforward. When we say “If P, then Q,” we mean that P is a sufficient condition for Q. That means that if P is true, that is enough for us to say that Q is true.

For instance, in the sentence “If Einstein is a parrot, then he is a bird,” Einstein’s being a parrot is sufficient for us to say that he is a bird. We symbolize a statement like this as follows:

P → Q

In this conditional, P is called the antecedent, and Q is called the consequent.

“Only if”

Now, that is logically equivalent to saying, “Einstein is a parrot only if he is a bird.” In ordinary language, whatever immediately follows “only if” is the necessary condition.

So the sentence “Einstein is a parrot only if he is a bird” means that being a bird is necessary for being a parrot. In other words, if it is true that Einstein is a parrot, then it must also be true that Einstein is a bird. So, once again, we symbolize the sentence like this:

P → Q

This is one reason people get confused: the phrase “only if” often makes them want to reverse the conditional. Don’t. The thing after “only if” gives you the necessary condition, not the antecedent.

Also note that this is equivalent to saying, “Only if Einstein is a bird is he a parrot.” That sounds more awkward, but the logical relationship is the same.

“If and only if”

Now consider the following sentence:

I will kick your ass if and only if you kick my ass.

This is what is called a biconditional. It means that each side is both necessary and sufficient for the other. In other words, the sentence can be broken down into two conditionals:

If you kick my ass, I will kick your ass.

If I kick your ass, you will kick my ass.

These can be symbolized as follows:

P → Q

Q → P

And those two together can be symbolized like this:

P ↔ Q

So “if and only if” means both directions hold. That is why it is stronger than plain old “if.”

“Unless”

And then there is “unless,” which seems like a pain in the ass to symbolize, but is not that hard once you get the hang of it. A useful rule of thumb is that “unless” can often be translated as “if not.”

So, “You will die unless you upload your brain to a computer” is logically equivalent to, “If you do not upload your brain to a computer, you will die.” This can be symbolized as:

¬B → D

where ‘¬’ means ‘not’, ‘B’ = ‘you upload your brain to a computer’, and ‘D’ = ‘you will die’.

So you can think of “unless” this way: “Unless B, D” means “If not B, then D.”

One last quick summary

Here is the baby-logic version:

• If = sufficient condition
• Only if = necessary condition
• If and only if = necessary and sufficient condition
• Unless = usually easiest to symbolize as “if not”

If you keep those straight, you will already be ahead of a whole lot of students who mangle conditionals into logical mush.

Apologies for the morbid tone today. Oh well. Until next time, then!