# "There is only" in first order logic

I'm trying to translate the statement "There is only three things that are not small" into first order logic. I'm using some software to verify my sentences, but I feel like I don't understand what "There is only" is meant to claim.

I interpreted it as "There are at most", and used the answer here, in particular method 2 for "At most $n$".

The sentence I've produced with this is:

$$\exists x \exists y \exists z \forall w \, (\lnot \text{Small}(w) \to (w = x \lor w = y \lor w = z))$$

Which I understand to mean that there exists up to 3 objects, which, for all objects $w$, if it is not small, it is one of the 3 objects.

This passes 3 of the 4 test worlds for the software, but fails on the last one.

I was hoping someone could help me clarify what is meant by "only". I've spent a decent amount of time Googling, but most results lead to explanations of the biconditional.

Thanks!

• Your statement is a good translation of 'at most 3 things are not small' .... maybe they mean 'exactly 3 things are not small?' Your test worlds should give you a clue ... Aug 10 '18 at 1:47
• I hate these kind of questions. It's not about first order logic, it's about what "there are only" means in ordinary English. In real life, "there are only three" means the same as "there are three and only three" or "there are exactly three". But maybe your instructor or your textbook author thinks it means something else.
– bof
Aug 10 '18 at 1:49
• @AnyAD but the universal says that any non-small object has to be one of those three ... so there really cannot be more than 3 Aug 10 '18 at 1:51
• I would see if the last world demands that there be three. You could add $x \neq y \wedge y \neq z \wedge x \neq z$ to your sentence and see if it works. That would support the theory that only three means exactly three. Aug 10 '18 at 2:05
• @Bram28 Was right, the test worlds seem to indicate that "Exactly 3" was the correct way to read the problem. Aug 10 '18 at 2:14