# Proving a set of connectives is inadequate

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

• @Bram28 I mean that for any truth assignment of atomic propositions then the $V^C(\phi) != v(\phi)$ holds for any proposition that is built using the given connectives. Commented Apr 20, 2023 at 17:50
• Oh, I see! OK, then yes, that approach totally works! I'll sketch the inductive proof pattern in an Answer momentarily Commented Apr 20, 2023 at 18:43

Here is the proof that for any $$v$$ and $$\phi$$ it is true that $$v^C(\phi) \neq v(\phi)$$ (i.e. that $$v^C(\phi)$$ and $$v(\phi)$$ always have the opposite truth-value).
We take any $$v$$, and now do structural induction on $$\phi$$:
Base: $$\phi = P$$ for some atomic proposition $$P$$. Given that by definition $$v^C(P)$$ is the opposite of $$v(P)$$, we have that $$v^C(\phi)$$ and $$v(\phi)$$ have the opposite truth-value as well.
A. $$\phi = \neg \psi$$, where by inductive hypothesis we have that $$v^C(\psi)$$ and $$v(\psi)$$ have the opposite truth-value. By semantics of $$\neg$$, we have that $$v(\psi)$$ and $$v(\neg \psi)$$ have opposite truth-values, and we also have that $$v^C(\psi)$$ and $$v^C(\neg \psi)$$ have opposite truth-values. So, $$v^C(\phi) = v^C(\neg \psi)$$ has the opposite value of $$v^C(\psi)$$. But since by inductive hypothesis $$v^C(\psi)$$ has the opposite value of $$v(\psi)$$, that means that $$v^C(\phi)$$ has the same truth-value as $$v(\psi)$$ .... but that has the opposite truth-value as $$v(\phi)$$.
B. $$\phi = G(\psi_1, \psi_2,\psi_3)$$, where by inductive hypothesis we have that $$v^C(\psi_i)$$ and $$v(\psi_i)$$ have the opposite truth-value for $$1 \leq i \leq 3$$. Now, $$v^C(\phi)$$ evaluates to True iff the majority of $$v^C(\psi_i)$$ evaluates to True iff (Inductive Hypothesis) the majority of $$v(\psi_i)$$ evaluates to False iff $$v(\phi)$$ evaluates to False. (in that last step, we use the fact that $$G$$ takes an odd number of argument, so there is always either a majority of argument that evaluate to True, or a majority of argument that evaluate to False.). So, $$v^C(\phi)$$ and $$v(\phi)$$ will have opposite truth-values.