Are predicate or function symbols with 3+ places actually used in mathematical logic? Let $n$ be a nonnegative integer. The language of first-order logic includes the following symbols : 


*

*predicate symbols with $n$ places: $P^n_0$, $P^n_1$, $P^n_2$, $\dots$

*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. 


*

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

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

*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.
 A: 
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)
A: 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.
