We consider types and typed lambda calculus over a finite number of ground types. We are
going to investigate the size of the fraction of inhabited types of the given length n against
the number of all types of length n. The plan of this paper is to find the limit of that
fraction when n → ∞. The answer to this question is equivalent to finding the ‘density’ of
inhabited types in the set of all types, or the so-called asymptotic probability of finding an
inhabited type in the set of all types. Under the Curry–Howard isomorphism this means
finding the density or asymptotic probability of provable intuitionistic propositional
formulas in the set of all formulas. For types with one ground type (formulas with one
propositional variable), we prove that the limit exists and is equal to 1/2 + √5/10, which is
approximately 72.36%. This means that a long random type (formula) has this probability
of being inhabited (tautology). We also prove that for every finite number k of ground-type
variables, the density of inhabited types is always positive and lies between
(4k + 1)/(2k + 1)2 and (3k + 1)/(k + 1)2.
Therefore we can easily see that the density is
decreasing to 0 with k going to infinity. From the lower and upper bounds presented we can
deduce that at least 1/3 of classical tautologies are intuitionistic.