iP M is Complete with respect to a class of finite frames

Lemma 3.3.1 In any canonical model: vR¯ ⊆ wR¯ iff w2 ⊆ v2 where x2 denotes the set of boxed formulas inx.

Lemma 3.3.2 14 _{(i) The principle M pcorresponds to theM pproperty:}

∀wvu(wR≤u→ ∃x(wRx∧v≤x≤u∧xR¯ ⊆uR¯). (ii) The logic iP M is canonical.

In the following section we will show that iP M is complete with respect to the class of finite M p

frames. But before we begin to prove this statement, we need to define some important notions to the proof. As we know, the reason why the proof in Proposition 4.5.2 in Iemhoff [2001] did not go through for the finite case is that2A→B does not necessarily belong to the adequate setX if the adequate set is finite. If we had required that the adequate set meet the closure condition that 2A, B ∈ X

implies2A→B ∈X, thenX would explode15 _{So, if we have a mechanism to control the tendency}

to explode, then we will have the hope that the proof in the thesis will go through. In the following, we will show that this is manageable.

Definition 3.3.3 Given any finite setX0that is closed under subformulas and contains>and⊥, we

enumerate all the boxed formulas inX0:2E1,2E2,· · ·,2En, which are calledatomic boxed formulas.

A formulaAis called anM-boxed formulaif it is a conjunction of atomic boxed formulas in which any two atomic conjuncts are different. LetX1 be the union ofX0and the set ofM-boxed formulas. It is

easy to see thatX1 is closed under subformulas and contains >and⊥. X1 can be divided into two

sets: the setXb

1 ofM-boxed formulas and the set X1n of formulas which are not M-boxed formulas.

DefineX2:={E→B|E∈X1b, B∈X1}andX:=X2∪X1. ObviouslyX is closed under subformulas

and contains>and⊥. We call itadequate.

¢

It is easy to see that the set ofM-boxed formulas is finite because it contains less than 2n _elements.

If we assume the size ofX0 is m, then X1 contains at most 2n+m elements. Obviously the size of

X2 is less than (2n+m)×2n. SoX is finite. We define theiP M-canonicalX-model as usual.

One observation: If2A∈X, then 2Ais an atomic boxed formula. 14_{proposition 4.5.2 in Iemhoff [2001]}

15_{This is actually not really true. By Diego’s theorem a finite set of propositional letters generates a finite number of}

equivalent formulas if one only uses→. But this number is superexponential (in Hendriks [1996]) and we prefer to stay in smaller ranges.

Lemma 3.3.4 The iP M-canonicalX-model satisfies theM p-property.

Proof. Take w, v, u∈X. Assume that wRv⊆u. It is easy to see that the following property is a ∗-extendiblew-successor property:

∗(x) : for allB ∈X, ifw`x¤B, thenB∈u.

If we can show that∗(v∪u2), then there is a nodex∈X such that wRx, v⊆x⊆uandxR¯ ⊆uR¯.

So it remains to show that∗(v∪u2), i.e.

ifw`v, u2¤B, thenB∈u.

Assume thatw`v, u2¤B. We may assume thatu2consists of2C1,· · · ,2Cs, all of which are atomic

boxed formulas according to the above observation and are different from each other. We divide the proof into two cases on whetherB is inX1or inX2:

1. LetB∈X1.

w`(v,2C1,· · ·,2Cs)¤B

w`v¤((2C1∧ · · · ∧2Cs)→B) (byM p)

Since (2C1∧ · · · ∧2Cs)∈X1b andB∈X1, (2C1∧ · · · ∧2Cs)→B ∈X2. Of course it is inX.

By the definition of accessibility relation on the canonical model and the assumption thatwRv, (2C1∧ · · · ∧2Cs)→B ∈ v. Of course,(2C1∧ · · · ∧2Cs)→B ∈u. Since 2C1,· · · ,2Cs are

also inu,u`B. B∈ubecauseuis saturated andB∈X.

2. LetB ∈ X2 and B ≡Cb →C1 where Cb ∈ X1b andC1 ∈X1. Define F to be the conjunction

of all the atomic boxed formulas that occur in either2C1∧ · · · ∧2Csor Cb in which any two

conjuncts are different. It is easy to see thatF ∈Xb 1. w`(v,2C1,· · ·,2Cs)¤B w`v¤((2C1∧ · · · ∧2Cs)→B) (byM p) w`v¤((2C1∧ · · · ∧2Cs)→(Cb→C1)) w`v¤(F →C1) SinceF ∈Xb

1 andC1∈X1, F →C1∈X2. ThereforeF →C1∈X. It follows thatF →C1∈v.

Because v ⊆u, it implies that u` (2C1∧ · · · ∧Cs)→ (Cb →C1). Since 2C1,· · ·,2Cs ∈u,

u`Cb→C1. Then we get thatCb →C1∈ubecause Cb→C1∈X2 and henceCb→C1∈X.

SoB∈u.

Conclude: ∗(v∪u2).

qed

Theorem 3.3.5 `iP M A iffAis valid on all finite M pframes.

Proof.The left-to-right direction follows from the correspondence result. Now we just need to consider the other direction. Suppose that6`iP M A. First defineY0 consisting of all sub-formulas ofA,>and

⊥. By using the same construction prescribed in the definition 3.3.3, we get a finite adequate set Y. It is easy to see that there is a saturated setwsuch thatA6∈w(by lemma 2.1.8). Then we construct theiP M-canonicalY modelM as usual. As the above lemma shows,M satisfies theM pproperty.

It remains to show the truth lemma. Since the accessibility relation on the canonical models is defined as usual, we can borrow the proof of truth lemma from the completeness proof ofiP (Th. 2.1.10). It follows thatM, w6|=A. SoA is not valid on all finiteM pframes.

3.3. CONSERVATION OFIP M OVERIK 41