The extensions of BAlt3 ? revisited

Interpolation in Normal Extensions of the Brouwer Logic

“…Proof: Suppose, on the contrary, that there is a formula γ, written in one variable p, such that α → γ and γ → β are theorems KTB.Alt (3). Then in each reflexive, symmetric and linear Kripke frame F = W, R , at any point x ∈ W and for all valuations V j we get:…”

“…Theorem 1. [Byrd and Ullrich, 1977;Byrd, 1978] Every normal modal logic which is a proper extension of KTB.Alt(3) has the finite model property and is finitely axiomatizable (and hence -decidable).…”

On Halldén Completeness of Modal Logics Determined by Homogeneous Kripke Frames

“…Since we have defined an anti-chain of Halldén complete logics in N EXT (KTB.alt 3 ) then it might seem to be possible to define a continuum of logics by taking their infinite intersections. But the whole family N EXT (KTB.alt 3 ) is countably infinite only, hence as well the family of Halldén incomplete logics, see [2].…”

Normal Modal Logics Determined by Aligned Clusters

“…It is proved in [2,3] that all logics from NExt(KTBAlt(3)), have also very strong properties. It is easily seen by the above theorem that the cardinality of the class NExt(KTBAlt(3)) is only countably infinite.…”