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

If I said:

Yell only if I fall.

Would the person have to yell once I fell?

Sources of confusion

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!

• This is why logicians use iff for 'if and only if'. I think it would be useful in real life, but can't see it catching on. Commented Jul 1, 2011 at 10:01
• In the last of your three logical expressions, I'd suggest replacing "A is equivalent to B" with "A implies B and B implies A". If you said that you'd buy the shirt iff it costs less than \$40, then she'd know you bought the shirt if she saw that its price was less than \$40 OR she'd know it cost less than \$40 if she saw you wearing the shirt. (Well, assuming you'd wear the shirt iff you bought it. ;-) Commented Sep 13, 2011 at 20:10
• Caleb, "A is equivalent to B" means exactly the same as the conjunction of "A implies B" and "B implies A" . Commented Oct 2, 2011 at 21:06
• Georges, just wondering about your last example. Are you implying that getting only A's =/= good grades? (Yes, I'm aware this is old, but I see you're still active) Commented May 12, 2013 at 20:30

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.

• `"Only If" is not a restriction of "If"` - I think you'll find it is (and especially in a mathematical sense)! "Yell if I fall" (note the missing "only") leaves the option for the person being instructed to yell at other things, thus them yelling doesn't necessarily imply you have fallen. This clearly doesn't match your Venn diagram, as the "Yell" would have an area outside of "Fall". Commented Jul 1, 2011 at 8:29
• Oops sorry. I was confused. Thanks DMA57361. What I had in my mind was `"Only If" was not a stronger version of "If" that still kept the meaning of "If" in a stricter sense`.
– Phil
Commented Jul 1, 2011 at 9:01

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).

Question:

Would the person have to yell once I fell?

Yes, the person would yell once you fell, but only if you fell.

"If" and "Only if" used in the same way means the same thing, except that "only if" is more forceful, more compelling.

"If and only if" is the most obligatory of the three, in which the action has been distinguished and emphasised, "If, and only if" It's the most forceful of the three

Practically speaking, it's an ambiguous command, and if you really want to be clear in your request, you would phrase it in an unambiguous way.

Technically speaking... well, who cares? The goal of language is to communicate with humans. There may be some arcane rules of language that provide for a definitive answer to your question, but since few people are familiar with it, knowing the rule won't do you any good.

Edit: First time visiting/posting so I'm not sure how this community leans between looking for technical grammar rules as opposed to practicality. In real life and in a lot of writing I find "Just say what you mean in a way that people will understand you!" to be the underlying answer to a lot of grammatical questions. Since I've been downvoted already, I can see that at least one person prefers the former.

• Yell if I fall.

You want him to yell if you fall. No guarantee he will yell if you fall. You don't mind if he yells for any other reason.

• Yell only if I fall.

You want him only to yell if you fall. No other reason. No guarantee he will yell if you fall, but if he yells, you want him only to have done so if you have fallen. You don't want him to yell for any other reason.

• Yell if and only if I fall.

Same as "only if". Redundant. Why mathematicians/logicians insist on the "if and only if" phrase is paradoxical because it is inelegant (redundant).

• It is only redundant because you misinterpret it. You say it's the same as "only if". But it is not. "Only if", as you say, means "no guarantee he will yell if you fall". The first if provides just that guarantee. In other words, 3 is a combination of 1 and 2, and you simply failed to combine your correct reasoning for 1 and 2 into the correct reasoning for 3. But anyway, all of this has been covered in the top and accepted answer two years ago. Commented Dec 6, 2013 at 13:41
• In formal logic, case 1 is `Fall => Yell` ("If I fall, you must yell"), so a fall must always result in a yell. In strict logical terms, your assertion "No guarantee he will yell if you fall" is not correct, leading you to misunderstand the strict logical meaning of "if and only if". Commented Dec 6, 2013 at 15:33