The question has already posted here before:
Proving that a set with a ternary logical connective is functionally incomplete (i.e. inadequate)
But I came up with a different proposal for a solution in which I am trying to prove that the given set of connectives is inadequate using a property that is true for every proposition that is built using this set of connectives.
Need to show that the set $\{\lnot ,G\}$ of logical connectives is inadequate where $G$ is a ternary connective that gives $T$ (True) if most of its propositions are $T$.
I noticed that the following property holds :
For every truth assignment $v$ and for every proposition $\phi$ that is built using the given set $\{\lnot ,G\}$ we get that $v(\phi) \neq v^C(\phi)$ when $v^C$ is defined to be the opposite truth assignment
Now I tried to formalize this using structural induction but I got stuck on the inductive step, I'll appreciate any help