  In this paper, we will provide a method to compute the density of tautologies
among the set of well-formed formulae consisting of $m$ variables, the negation
symbol and the implication symbol; which has a possibility to be applied for
other logical systems. This paper contains computational numerical values of
the density of tautologies for two, three, and four variable cases. Also, for
certain quadratic systems, we will build a theory of the $s$-cut concept to
make a memory-time trade-off when we compute the ratio by brute-force counting,
and discover a fundamental relation between generating functions' values on the
singularity point and ratios of coefficients, which can be understood as
another interpretation of the Szeg\H{o} lemma for such quadratic systems. With
this relation, we will provide an asymptotic lower bound
$m^{-1}-(7/4)m^{-3/2}+O(m^{-2})$ of the density of tautologies in the logic
system with $m$ variables, the negation, and the implication, as $m$ goes to
the infinity
