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.
