I don't want to give any offense here. I can only comment as a lay person who has only a lay understanding of university maths.
It is noteworthy that not everyone here believes that first order logic is not Turing complete. A Turing machine is a finite state machine with an infinite tape. It is only an elementary (i.e. doable) computer coding exercise to implement this.
Only a tiny subset of Dartmouth Basic or Applesoft Basic is required for a Turing complete system. In recent years, Terry Tao asked (possibly on this site)if there was a group that was turing complete. Perhaps we could go through a box ticking exercise, noting that any one of the objects we encounter in introductory maths course at university, suffices by itself to build a Turing complete system.
The wikipedia entry for first order logic is quite detailed (many hundreds of lines).
It says: First-order logic is able to formalize many simple quantifier constructions in natural language, such as "every person who lives in Perth lives in Australia". But there are many more complicated features of natural language that cannot be expressed in (single-sorted) first-order logic. "Any logical system which is appropriate as an instrument for the analysis of natural language needs a much richer structure than first-order predicate logic" (Gamut 1991, p. 75).
The wikipedia article also says:
Ordinary first-order interpretations have a single domain of discourse over which all quantifiers range. Many-sorted first-order logic allows variables to have different sorts, which have different domains. ...
When there are only finitely many sorts in a theory, many-sorted first-order logic can be reduced to single-sorted first-order logic. One introduces into the single-sorted theory a unary predicate symbol for each sort in the many-sorted theory, and adds an axiom saying that these unary predicates partition the domain of discourse.
From the above, I am far from convinced that first order logic is unsuitable for a full treatment of natural language. If we follow the Turing doctrine, aren't we forced to agree that we can code a full treatment of natural language using any Turing complete system which I am guessing from the wikipedia article, certainly includes first order logic.
I will follow up the Gamut article in an attempt to shed light on this. But prima facie the Gamut article is incorrect because it begs the question - it says you can't express natural language in single-sorted predicate calculus. But the Wikipedia article makes it clear that this may not be a limitation - at least it shows that a multi-sorted predicate calculus can be coded in single sorted predicate calculus.