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In this paper, we will provide a method to compute the density of tautologiesamong the set of well-formed formulae consisting of $m$ variables, the negationsymbol and the implication symbol; which has a possibility to be applied forother logical systems. This paper contains computational numerical values ofthe density of tautologies for two, three, and four variable cases. Also, forcertain quadratic systems, we will build a theory of the $s$-cut concept tomake a memory-time trade-off when we compute the ratio by brute-force counting,and discover a fundamental relation between generating functions' values on thesingularity point and ratios of coefficients, which can be understood asanother interpretation of the Szeg\H{o} lemma for such quadratic systems. Withthis 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 logicsystem with $m$ variables, the negation, and the implication, as $m$ goes tothe infinity

Computing the density of tautologies in propositional logic by solving system of quadratic equations of generating functions