# Question regarding translating English into first-order logic

Someone asked this question: "A language $L$ that is regular will have the following property: there will be some number $N$ (that depends on $L$) such that if $s$ is a string in $L$ (a $string$ is a sequence of characters) whose length is at least $N$ then $s$ can be written as $xyz$ where $y$ is not the empty string and $xy^iz$ is in the language $L$ for every nonnegative integer $i$."

Translating English into First Order Logic

My question how would one even translate that into first order logic? The user started writing out his attempted approach. But I couldn't quite follow it so I was wondering how one could even write such statements in first order logic. I looked into pumping lemma as one of the comments suggested but it didn't help me much. It got me slightly more confused.

P.S. I'm sorry if I broke any rules in stating this question. I'm new to the site.

Primarily because the quantifier-variable plus connectives representation of statements is wonderfully clear and non-ambiguous about logical structure. Careless or sloppy vernacular talk may fail to distinguish between the propositions represented by $P \lor (Q \land R)$ and $(P \lor Q) \land R$ [Recall the ambiguous invitation: "Bring your partner or come alone and have a good time!"]. Logical notation forces us to distinguish, as it also forces us to distinguish $\forall x \neg Px$ and $\neg\forall xPx$ [compare the ambiguous "Everyone has not yet arrived"], and to distinguish $\forall x\exists yRxy$ and $\exists y\forall xRxy$, and so on and so forth.
$$\exists n\forall s[l(s)\geq n \to \exists x\exists y\exists z(s = x^{\cap}y^{\cap}z \land y \neq 0 \land \forall i\,x^{\cap}y^i{}^{\cap}z \in L )]$$
if we take the quantifiers to be implicitly appropriately sorted so $s, x, y, z$ run over $L$-strings, and $n$ and $i$ over non-negative numbers (and where ${}^{\cap}$ is concatenation). The further finer details (e.g. whether you choose to explicitly type the variables by writing e.g. $\forall x \in L$ where $L$ is the set of $L$-strings) won't matter -- in most contexts -- so much as revealing the overall quantificational structure and scope dependencies.