Abstract

We proved in [1] that there exist a continuum consistent monotone intuitionistic modal logics which do not admit the law of the excluded middle p ∨ ¬p. Rieger [2] and Nishimura [3] introduced a sequence of formulas ϕ0, ϕ1, . . . , ϕω of one variable p such that for any intuitionistic formula ϕi containing only the variable p there exists a formula ϕi from this sequence equivalent to ϕ in the intuitionistic propositional logic . In [5] V. Tselkov has proved that for each i ≥ 4 there exist at least countably many consistent monotone intuitionistic modal logics which do not admit the formula ϕi. In this paper we strengthen this result by showing that there exist continuum of such logics. We construct an example of monotone intuitionistic modal logic not admitting all formulas ϕi, where i ≥ 4 and i =6 ω. Using similar constructions we prove that there exist at least countably many maximal monotone intuitionistic modal logics. Let us mention that in the classical case Makinson [6] has shown that there are exactly three such logics