
## 8 **M. Slota And J. Leite**

2
◦If φ is a first-order formula of the form ¬ψ, then φ is true in hI, M, Ni **if and**
only if ψ is not true in h**I, M, N**i; 3
◦If φ is a first-order formula of the form φ1 ∧ φ2, then φ is true in h**I, M, N**i if and only if φ1 is true in hI, M, Ni and φ2 is true in h**I, M, N**i; 4
◦If φ is a first-order formula of the form (∃x : ψ), then φ is true in h**I, M, N**i if and only if ψ[c/x] is true in hI, M, Ni **for some constant** c ∈ ∆;
5
◦If φ is a formula of the form K ψ, then φ is true in hI, M, Ni **if and only if** ψ is true in hJ, M, Ni **for each** J ∈ M;
6
◦If φ is a formula of the form not ψ, then φ is true in hI, M, Ni **if and only if**
ψ is not true in hJ, M, Ni **for some** J ∈ N.
The fact that φ is true in hI, M, Ni is denoted by hI, M, Ni |= φ**. A formula** φ is false in hI, M, Ni if and only if it is not true in hI, M, Ni, denoted by hI, M, N**i 6|**= φ.

Now we are ready to introduce the notions of MKNF interpretation and model.

## Definition 2.6 (**Mknf Interpretation And Model**)

An MKNF interpretation M **is a non-empty set of first-order interpretations. By**
M = 2I **we denote the set of all MKNF interpretations together with the empty**
set.

Let φ be an MKNF sentence, S an MKNF theory and M ∈ M. We say φ **is true**
in M, denoted by M |= φ, if hI, M, Mi |= φ **for each** I ∈ M.

3 Otherwise φ **is false**
in M, denoted by M 6|= φ. S is true in M, denoted by M |= S, if M |= φ **for each** φ ∈ S. Otherwise, S is false in M**, denoted by** M 6|= S.

If M ∈ M is non-empty4**, then** M is

- an **S5 model of** φ if M |= φ; - an **S5 model of** S if M |= S;
- an MKNF model of φ if M is an S5 model of φ and for every MKNF interpretation M′ ) M **there is some** I
′ ∈ M′**such that** hI
′, M′, M**i 6|**= φ;
- an MKNF model of S if M is an S5 model of S and for every MKNF interpretation M′ ) M **there is some** I
′ ∈ M′ and some φ ∈ S **such that**
hI
′, M′, M**i 6|**= φ.
If there exists the greatest S5 model of φ**, then it is denoted by** mod(φ). If φ has no S5 model, then mod(φ**) denotes the empty set. For the rest of MKNF sentences,** mod(·) stays undefined. If there exists the greatest S5 model of S**, then it is denoted**
by mod(S). If S **has no S5 model, then** mod(S**) denotes the empty set. For the rest**
of MKNF theories, mod(·**) stays undefined.**