Would anyone please give me a hint to prove that if $*$ is a functionally complete binary connective and $@$ is a symbol for tautology, we must always have $ @*@$ is equivalent to contradiction (I mean the statement which is always false ).
Thanks for your help.
I use "$\top$" and "$\perp$" for true and false, respectively.
HINT: If $\{*\}$ is functionally complete, then there must be some expression $t(-)$ only involving $*$ and one sentence variable, $x$, such that $t(x)$ is equivalent to $\neg x$. Now:
Show that since $*$ is a truth functional, either $\top*\top=\top$ or $\top*\top=\perp$.
Along the same lines, show that either $x*x$ is equivalent to $\neg x$ or is equivalent to $x$; or that it is equivalent to either $\top$ or $\perp$.
By induction on the complexity of $t$, why is this a problem if we assume $\top *\top\not=\perp$? (Hint: for the constant-valued possibilities for $x*x$, look at what happens when $x$ is that truth value - e.g. if $x*x$ is equivalent to false, then imagine setting $x=\perp$ in $t(x)$.)