Using separators as functional symbols in first order logic

Question

Suppose we have the following definition of a term:

A $term$ is:

• $x$, where "$x$" is a variable
• $c$, where "$c$" is a constant symbol
• $f(\tau_1,...,\tau_n)$, where "$f$" is a functional symbol with $n$ arguments and $\tau_1,...,\tau_n$ are also terms
• $(\tau)$, where $\tau$ is also a term

You can also use the symbols comma "$,$" left bracket "$($" and right bracket "$)$" as constant or functional symbols.

Prove or disprove that if two terms are symbolically equivalent, then their interpretations are also equivalent regardless what the variable assignment is.

Here are some examples:

$\tau=f((,))$ - this can be interpreted in two ways: $f(c,d)$ where $'(' = c$ and $')'=d$ or $f((c))$ where $',' = c$. This doesn't cause ambiguity since we know how arguments $f$ has.

$\tau=(((x))$ - this also can be interpreted in two ways: $(f(x))$ where $'(' = f$ or $f((x))$ where $'(' = f$. These two terms are structurally different but essentially they refer to the same object after variable assignment.

Answer 1

Consider f((,),(,)) where f is a three-argument function and (, ), and , are also constants.