On the constant domains principle and its weakened versions in the Kripke sheaf semantics

1 We consider the semantics of predicate Kripke frames with equality (called I-frames, for short), which is equivalent to the semantics of Kripke sheaves (see [3]). Namely, a (predicate) Kripke frame is a pairM = (W,U) formed by a posetW with the least element 0W and a domain map U defined on W such that U(u)⊆U(v) for u≤v. An I-frame is a triple M = (W,U, I), in which (W,U) is a Kripke frame and I is a family of equivalence relations Iu on U(u) for u∈W such that Iu⊆Iv for u≤v. A valuation u A (for u∈W and formulas A with parameters replaced by elements of U(u)) satisfies the monotonicity: u ≤ v, u A ⇒ v A and the usual inductive clauses for connectives and quantifiers, e.g. u (B → C) ⇔ ∀v≥u [(v B)⇒ (v C)], u ∀xB(x) ⇔ ∀v≥u∀c∈U(v) [v B(c)], etc. (for the case with equality, a=b is interpreted by aIub in an I-frame and by a=b in a usual Kripke frame, for a, b∈U(u) ). For an I-frame we admit only the valuations preserving Iu (on every U(u), u∈W ), i.e., ∧