Studia Logica 100 (1-2):361-383 (2012)

Abstract
The completeness w.r.t. Kripke frames with equality (or, equivalently, w.r.t. Kripke sheaves, [ 8 ] or [4, Sect. 3.6]) is established for three superintuitionistic predicate logics: ( Q - H + D *), ( Q - H + D *&K), ( Q - H + D *& K & J ). Here Q - H is intuitionistic predicate logic, J is the principle of the weak excluded middle, K is Kuroda’s axiom, and D * (cf. [ 12 ]) is a weakened version of the well-known constant domains principle D . Namely, the formula D states that any individual has ancestors in earlier worlds, and D * states that any individual has $${\neg\neg}$$ -ancestors (i.e., ancestors up to $${\neg\neg}$$ -equality) in earlier worlds. In particular, the logic ( Q - H + D *& K & J ) is the Kripke sheaf completion of ( Q - H + E & K & J ), where E is a version of Markov’s principle (cf. [ 12 ]). On the other hand, we show that the logic ( Q - H + D *& J ) is incomplete w.r.t. Kripke sheaves
Keywords superintuitionistic predicate logics  Kripke semantics  Kripke sheaves  Kripke sheaf completeness
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DOI 10.1007/s11225-012-9382-2
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Directed Frames.Giovanna Corsi & Silvio Ghilardi - 1989 - Archive for Mathematical Logic 29 (1):53-67.
Applications of Trees to Intermediate Logics.Dov M. Gabbay - 1972 - Journal of Symbolic Logic 37 (1):135-138.

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