Are predicate or function symbols with 3+ places actually used in mathematical logic?

Question

Let $n$ be a nonnegative integer. The language of first-order logic includes the following symbols :

1. predicate symbols with $n$ places: $P^n_0$, $P^n_1$, $P^n_2$, $\dots$
2. function symbols with $n$ places: $f^n_0$, $f^n_1$, $f^n_2$, $\dots$

What is an example of a formal system which actually uses a predicate or function symbol with 3 or more places? I am asking about a specific symbol, not predicate variables. It seems that, in practice, only unary (1-place) and binary (2-place) symbols are used.

Examples.

1. Axiomatic set theory uses one binary predicate symbol (membership) and no function symbols.

2. Formal number theory uses one binary predicate symbol (equality), one unary function symbol (succession) and two binary function symbols (addition, multiplication).

3. Formal group theory uses one binary predicate symbol (equality), one unary function symbol (inversion), and one binary function symbol (multiplication).

*The last two systems use a constant symbol, which may be regarded as a 0-place function symbol.

Answer 1

Tarskian geometry uses 3-place and 4-place predicates (at least I think they are 3-place relations, not functions, I haven't studied it in detail). Here's an example paper on the topic.

Answer 2

What is an example of a formal system which actually uses a predicate or function symbol with 3 or more places? I am asking about a specific symbol, not predicate variables. It seems that, in practice, only unary (1-place) and binary (2-place) symbols are used.

Hmm, I am a little unclear on your question ... but I think your question is about something we use in practice. For example, when doing something with numbers, we might use a 2-place predicate that we want to use for 'Smaller than'. Or, we could use a 1-place predicate 'Even'. And for functions we could use the 1-place function 'successor', or the 2-place function 'addition'. So yes, plenty of 1-place or 2-place relations or functions here. But do we have any 'natural' 3-place predicates or functions when we apply our logic system to some domain?

Well, I suppose something like '$Sum(x,y,z)$', meant to mean: '$x$ is the sum of $y$ and $z$' could work

Or maybe someone could make good use of a $Between(x,y,z)$ ('$X$ is between $y$ and $z$') predicate ... which could be used in all kinds of domains (e.g. not just numbers, but think of objects having some location in some world)