"If" vs "Only if" vs "If and only if"

Recall that in formal logic, your expressions are used as follows:

• A if B means that B implies A
• A only if B means that A implies B
• A if and only if B means that A is equivalent to B.

For example if in the morning I tell my wife:

"I'll buy that shirt we saw yesterday only if it costs less than 40 dollars"

and she sees me wear it in the evening, she can deduce that I paid less than 40 dollars.

So, logically speaking, in your example, the deduction is that if a yell is heard, necessarily you fell: no notion of obligation is involved.

However, human beings are not computer programs and the use of "only if" in real life is more subtle than what logicians might decree. Here is an example:

It would be cruel for a father to tell his daughter "I'll buy you this dress only if you get good grades at the end of the year" and then buy nothing even though she had only A's.

You wouldn't have committed a logical error but would have shown you are a lousy parent!

No :P

"Only If" is not a stronger version of "If" that keeps what it implies.

This is the illustration for the two equivalent statements:

• Yell only if I fall.
• If you yell, I must have fallen. (Notice the subtlety. I didn't say "I fall if you yell". This is logical deduction, not cause-effect.)

Each circle represents a set of instances of "I fall" or "Yell".

Disclaimer: English is not my first language. I'm only talking about the mathematical sense.

If I said

Yell if I fall

then that would mean that I am asking to yell in the event of me falling, but you can still yell for other reasons. I can also flip this. The rather awkward sentence

I fall only if you yell

means the same thing as the former because "A if B" is the same thing as "B only if A", hence the arrow notation B => A and A <= B. If I did in fact fall, then it is guaranteed that you yelled, but if you yelled, then I didn't necessarily have to fall.

Yell only if I fall

This means that the only reason for you to yell is me falling, but if I do fall, then you don't necessarily have to yell. This is the same as saying

I fall if you yell

Meaning that if you yell, then I am guaranteed to fall, but I can still fall without you yelling.

Yell if and only if I fall

This means that the only reason for you to yell is me falling, but you have to yell if I do fall. You are guaranteed to yell if I fall, and I am guaranteed to fall if you yell.

It occurs to me that there is an allowance for ambiguity in the common world. Even if you say if, then one can generally deduce that you do actually mean only if (or maybe even if and only if).