So if n is a member of the set of natural numbers, such that n is greater than 1 and lesser than 3, then n=2 necessarily?
This is more on the lines of commentary on your question, rather than an answer.
If one uses the usual natural numbers then yes. But are there other systems of natural numbers?
There are non-standard integers: The integers are axiomatically defined by the Peano Axioms. We actually obtain them in a model. The question here is are all such models isomorphic - that is categorial. If we use second-order logic, they are; if first-order then not. But the initial segment of any non-standard integer model is isomorphic to the usual ones. So this doesn't work.
We could use topos rather than semantics for the PA axioms, this means the axioms are interpreted in a topos. This means that the topos must have a Natural Number Object. Whether there is a topos, within which the condition you specify fails, is for me, an open question.