In finite-model theory, a zero-one law states that certain properties hold with asymptotic probability of either 0 or 1 in large random finite structures. For classical predicate logic, such a law was first established by Y.Glebskii (et al.) in 1969 [2], and later independently by R.Fagin in 1975 [1]. In the modal setting, J. Halpern and B. Kapron (1994) [3] advanced a purported proof of a zero–one law for propositional modal logic S4 with respect to frames; however, this claim was refuted when J. M. Le Bars (2002) [5] provided a counterexample.

The question of whether S4 admits a genuine zero–one law remained unsettled until recently, when a proof was established confirming the property [6]. This result shows that the asymptotic behaviour of large random S4-frames is subject to a strong probabilistic regularity, paralleling the classical cases. The proof combines methods from model theory and combinatorics, drawing in particular on the asymptotic enumeration of Kripke frames and partial orders [4].

In this talk, we present an exposition of the proof, situate it within the historical development of zero–one laws, and highlight its implications for the study of probabilistic phenomena in modal and non-classical logics.

1. Ronald Fagin, Probabilities on finite models, The Journal of Symbolic Logic, vol.41 (1976), no.1, pp.50–58.
2. Yuri V. Glebskii, Volume and fraction of satisfiability of formulas of the lower predicate calculus, Otdelenie Matematiki, Mekhaniki i Kibernetiki AkademiiNauk Ukrainskol SSR. Kibernetika, vol.2 (1969), no.18, pp.17–27.
3. Joseph Y. Halpern and Bruce Kapron, Zero-one laws for modal logic, Annals of Pure and Applied Logic, vol.69 (1994), no.2, pp.157–193.
4. Daniel Kleitman and Bruce Lee Rothschild, Asymptotic enumeration of partial orders on a finite set, Transactions of the American Mathematical Society, vol.205 (1975), no.1, pp.205–220.
5. Jean-Marc Le Bars, The 0-1 law fails for frame satisfiability of propositional modal logic, Proceedings of the 17th Annual IEEE Symposium on Logic in Computer Science, IEEE Computer Society, 2002, pp. 225–234.
6. Sigfrido D. Ciletti, Guillermo Badia, Marcel Jackson and Tomasz o, Zero-One law for intutitionistic propositional logic, unpublished draft.