
## Appendix B Properties Of Mknf B.1 General Properties

Lemma B.1 (**Models of Positive Sentences**)
Let φ be a positive MKNF sentence, I a propositional interpretation and **M, N**0 ∈ M. If hI, M, N0i |= φ, then hI, M, Ni |= φ for any N ∈ M.

Proof Follows directly from Definition 2.6 **and the fact that the valuation of a positive**
formula in a structure hI, M, Ni **is independent of** N.

Corollary B.2 Let φ be a positive MKNF sentence. Then the MKNF models of φ **are exactly the** subset-maximal S5 models of φ. Proof Follows from Definition 2.6 and Lemma **B.1.** Lemma B.3 Let ≤ be a binary relation defined on the set M of all sets of first-order interpretations for any M, N ∈ M **as follows:**
M ≤ N ⇐⇒ M ⊇ N
Then (M, ≤**) is a complete partial order with the least element** I. Proof Follows from the set-theoretic properties of the subset relation ⊆ **and of the set** intersection ∩. Notice that even subsets of M **that are not directed have their** supremum (intersection) in M. Lemma B.4 Let φ be an first-order sentence and M, N ∈ M be such that M ≤ N**. If** M |= K φ, then also N |= K φ.

## Proof

Suppose M |= K φ and consider some interpretation I ∈ N**. By the assumption**
we obtain I ∈ M and so hI, M, Mi |= K φ. Hence hI, M, Mi |= φ and since φ **is a**
first-order formula, its valuation in the structure hI, M, Mi **doesn't depend on** M,
so hI, N, Ni |= φ. Furthermore, our choice of I **was arbitrary, so we can conclude**
that hI, N, Ni |= φ for all I ∈ N. Consequently, N |= K φ **as desired.**

## B.2 Models Of First-Order Theories
