Degrees of logics with Henkin quantifiers in poor vocabularies

We investigate some logics with Henkin quantifiers. For a given logic L, we consider questions of the form: what is the degree of the set of L–tautologies in a poor vocabulary (monadic or empty)? We prove that the set of tautologies of the logic with all Henkin quantifiers in empty vocabulary L∗∅ is of degree 0 . We show that the same holds also for some weaker logics like L∅(Hω) and L∅(Eω). We show that each logic of the form L (k) ∅ (Q) with the number of variables restricted to k is decidable. Nevertheless – following the argument of M. Mostowski from [Mos89] – for each reasonable set theory no concrete algorithm can provably decide L(k)(Q), for some Q. We improve also some results related to undecidability and expressibility for logics L(H4) and L(F2) of Krynicki and M. Mostowski from [KM92].