I think that you may have replied before I saved my entire response, so I am not sure how much of it you had read before replying yourself.

I have replied to your last statement:

> "you can use the second parameter of a binary function to identify a unary function just as you can use the fourth parameter of a quaternary function to identify a trinary one."

As I have explained above, what you propose does not work. It works in functions with 3 or more parameters, but it does not work in binary functions, because you cannot make binary functions from unary functions (without using some auxiliary binary functions).

> As I have explained above, what you propose does not work. It works in functions with 3 or more parameters, but it does not work in binary functions, because you cannot make binary functions from unary functions (without using some auxiliary binary functions).

I have no idea what you're trying to say. If you can use one parameter to identify a desired function, then obviously you can use a function of arity n+1 to define as many functions of arity n as you want, and it doesn't matter what the value of n is.

For example:

selector(3, "sin") = sin 3

selector(3, "log2") = log₂ 3

This works going from arity 4 to arity 3, and it also works going from arity 2 to arity 1. Your "response" talks about going from arity 1 to arity 2, a non sequitur.