In mathematical logic and model theory, one considers *interpretations* of syntactic expressions: terms without free variables are interpreted as elements of some structure, formulas without free variables have truth values, formulas with free variables can be interpreted as relations.

Multiple expressions may have identical interpretations. For example, $\ulcorner 1 + 1\urcorner$ and $\ulcorner 2\urcorner$ are both interpreted as $2$.

**Question:** does anyone ever consider formal languages where terms can have multiple interpretations? Is there some standard approach or framework?

I am thinking about this because i am trying to understand the $\omicron$ and $O$ notation in analysis, like in $$ \ln(x) =\omicron(x),\quad x\to +\infty. $$ Also, when calculating an indefinite integral, on often writes $$ \int 2x\,dx = x^2 + C. $$

**Update.**
I understand that when the equality sign is used with $\omicron$/$O$ notation, it does not represent an equivalence relation.
I also know that $\omicron(g(x))$ can be viewed as a set of functions.
However, this interpretation does not fit my intuition well.
When i write $\sin x = x + \omicron(x^2)$, $x→0$, i do not think about sets of functions, i think that i am replacing an anonymous implicitly understood function with a placeholder.
In other words, the designated object does not change (it is still a function or a number, not a set of functions or set of numbers), only the notation is abbreviated and made less explicit, a bit like when i write "$1 + 2 + 3$" instead of "$((1 + 2) + 3)$".

setsof functions rather than functions (but one term still denotes a unique set), operations such as composition and $O$ do the expected thing, and $t=s$ is interpreted as “every function in the denotation of the LHS equals some function in the denotation of the RHS”, and similarly for other relations like $t\le s$ (unless you consider this invalid notation). $\endgroup$ – Emil Jeřábek Dec 4 '14 at 22:35sets of functions). $\endgroup$ – Alexey Muranov Dec 4 '14 at 23:50