(iii) ⊢CSMS ((A ⇔ B1) ⊓ (A ⇔ B1)**· · · ⊓** (A ⇔ Bn)) → (A ⇔ B) from (ii)
by (T6)

Sin e (A ⇔ Bi) ∈ w, we 
on
lude (A ⇔ B) ∈ w 3. immediate by axiom (2).

```
4. By 1, we have that w
                         A ∪{A} is 
onsistent, hen
e is is 
ontained in a maximal
   
    onsistent formula set z ∈ U by lemma 9. We show then that w
                                                                      A is 
                                                                            ontained
                                                                                     
   in a set x ∈ ∆ We show that for al l x ∈ U , if w
                                                        A ⊆ x and (A ⇔ ⊥) ∈ w,
   then R(w, x). We have ∀C ∈ w,(C ⇔ ⊥) ∈ w, be
ause of the reexivity of R.
   By axiom (4) we have (A ⇔ ⊥) → (A ⇔ ¬(C ⇔ ⊥)) ⊔ (¬(C ⇔ ⊥) ⇔ ⊥).
   By axiom (6), we obtain (¬(C ⇔ ⊥) ⇔ ⊥) 6∈ w, sin
e (C ⇔ ⊥) ∈ w.
   Hen
        e (A ⇔ ¬(C ⇔ ⊥)) ∈ w. This entails (C ⇔ ⊥) ∈ w
                                                                    A and therefore
                                                                                     
   (C ⇔ ⊥) ∈ x. This means that we have R(w, x). Hen
e we have x ∈ ∆ and
   w
     A ⊆ x.

We now are in a position to dene the 
                         anoni
                             al model.
                                   

```

Denition 14 (Canoni al Model). Sin e C is not derivable in **CSMS**, ¬C
is onsistent, and so there is a maximal onsistent set z ∈ U su
h that ¬C ∈ z .

We dene the anoni al model MC = (∆,(≺w)w∈∆, .MC ) as fol lows:
 ∆ = ˜z.

 x ≺w y i there exists a formula B ∈ y su
h that for al l formulas A ∈ x,
(A ⇔ B) ∈ w.

 V MC
i = {x ∈ ∆ | Vi ∈ x}, for al l propositional variables Vi .
For A ∈ LCSL , we dene the set of ob je
ts in ∆ 
ontaining A by kAk = {z | A ∈ z ∩ ∆}.

Lemma 15. For ea h w ∈ ∆, ≺w is 
entered and modular.

Proof. 1 ≺w is 
entered. Let be x 6= y. Then there is (i) B ∈ y and B 6∈ x,

i.e. ¬B ∈ x. Let be any LCSL-formula A with A ∈ x. Then A ⊓ ¬B ∈ x from whi h fol lows that (ii)(A ⇔ B) ∈ x by axiom (3). From (i) and (ii) we obtain x ≺x y.

2 ≺w is modular. Let be x ≺w y and suppose there is u ∈ ∆ su
h that x 6≺w u and u 6≺w y. We get then:
(i) ∃B ∈ y su
h that ∀A ∈ x, (A ⇔ B) ∈ w
(ii) ∀C ∈ u ∃A′ ∈ x su
h that ¬(A′ ⇔ C) ∈ w and (iii) ∀B′ ∈ y ∃C
′ ∈ u su
h that ¬(C
′ ⇔ B′) ∈ w Then we have (A′ ⇔ B) ∈ w from (1), ¬(A′ ⇔ C
′) ∈ w from (ii) and
¬(C
′ ⇔ B) ∈ w from (iii). By axiom (1) and (ii), we get (C ⇔ A′) ∈ w and by transitivity (T5) and (i), we obtain (C
′ ⇔ B) ∈ w whi
h 
ontradi
ts the onsisten y of w.

Subsequently, we show a weak variant of the limit assumption for sets of ob je ts satisfying a formula.

Lemma 16. If kAk 6= ∅ then min≺w (kAk) 6= ∅.