understanding "only" phrases

[gyoo01]

Hello,

I am a little bit confused about different phrases that contain "only."
For example, in PT34, S2, Q10 (the question about special education/learning disabilities), the answer is "the only students with special educational needs are students with learning disabilities"
Here, 'the only' makes the special educational needs part a sufficient condition.
What I am confused about is when "only" is used as an indicator of a necessary condition. For example, in your LR guide textbook, in p.354 drill #6, "only troublemakers stay after school": here, "only troublemakers" are considered as a necessary condition.
It seems to me that 'the only' and 'only' almost play an opposite role in conditional logic. Could you clarify this?
I know I am trying to make this too rigid and formulaic, but I thought it would be helpful to recognize sufficient/necessary part right after I read something with "only."
The only, only, only if... those are so confusing!!!

Help me please thanks in advance!

[ohthatpatrick]

You are right in almost all regards.

The only, only, and only if ARE annoying.

And it IS dangerous to have an overly rigid formulaic way of thinking about them, because sometimes there will be a weird sentence that forces you to consider on the fly, "Which thing is required or guaranteed in this sentence?"

However, we can get fairly formulaic with it.

"Only" and "Only if" are always indicating something necessary. You can draw a conditional arrow through those words, if you like, so you remind yourself that the arrow will be pointing at whatever idea comes right after the "only" or "only if".

"The only" is more rare, but it indicates a sufficient condition.

If I say,
"The only people who understand quantum mechanics are good at math."
Would you diagram that as
Good at math --> Understand quantum mechanics
or
Understand quantum mechanics --> Good at math

The 2nd one is the correct translation. Knowing QM tells me for sure you're good at math. Being good at math doesn't guarantee me that you get QM.

Now, notice that these two sentences mean the same thing:
"The only people who understand quantum mechanics are good at math."
and
"People who are good at math are the only ones who understand quantum mechanics"

In both cases, "the only" indicates the sufficient condition, "understand quantum mechanics".

In summary, the vast majority of the time we see 'only', it's in the form of "only" or "only if" and acts as necessary.

If you can remember that THE ONLY is a different creature, then you'll react accordingly.

Hope this helps.