  We ask the following question: If all instantiations of a propositional
formula $A(x_1,...,x_n)$ in $n$ propositional variables are decidable in some
sufficiently strong recursive theory, does it follow that $A$ is tautological
or contradictory? and answer it in the affirmative. We also consider the
following related question: Suppose that for some propositional formula
$A(x_1,...,x_n)$, there is a Turing program $P$ such that
$P([\phi_{1}],...,[\phi_{n}])\downarrow=1$ iff $\mathbb{N}\models
A(\phi_{1},...,\phi_{n})$ and otherwise
$P([\phi_{1}],...,[\phi_{n}])\downarrow=0$ (where $[\phi]$ denotes the G\"odel
number of $\phi$), does it follow that the truth value of
$A(\phi_{1},...,\phi_{n})$ is independent of $\phi_1,...,\phi_{n}$ and hence
that $A$ is tautological or contradictory?
