3.2
Completeness Proof for the Static Part
In this chapter, we are going to prove strong completeness of AX’ with respect to the semantics of the language LSt. We will need the following three lemmas:
Lemma 3.2.1 Every AX’-consistent set Γ ⊆ LSt can be extended to a maximal AX’-
consistent set w.r.t. LSt. In addition, if Γ is a maximal AX’-consistent set, then it satisfies
the following properties:
2. ϕ∧ψ∈Γ iffϕ∈Γ and ψ∈Γ.
3. ifϕ and ϕ→ψ are both inΓ, then ψ is inΓ. 4. ifψ∈AX0 thenψ∈Γ.
Proof: It is a classical proof that we don’t give. ¤
Lemma 3.2.2 For all maximal consistent sets V, W, for allk∈Z,
V /Kj ⊆W →(V /Bk
j ⊆W ↔W/Bjk ⊆W)
Proof: V /Kj ⊆W ⇔ for all formulaϕ, Kjϕ∈V →ϕ∈W (*)
We are going to show that under the assumption (*),
Bjkϕ∈V ⇔Bkjϕ∈W for all k, all formulasϕ
• Assume Bk jϕ∈V.
We know that Bk
jϕ→KjBjkϕ∈V because it is an axiom (see lemma 3.2.1).
So KjBjkϕ∈V. Then by (*), Bk jϕ∈W • Assume Bk jϕ∈W. Assume Bk
jϕ /∈V. Then¬Bjkϕ∈V by lemma. But¬Bjkϕ→Kj¬Bkjϕ∈V.
So Kj¬Bjkϕ∈V.
Then ¬Bk
jϕ∈W by (*)
i.e. Bk
jϕ /∈W (by lemma 3.2.1). This is impossible.
So Bk jϕ∈V.
We have provedBk
jϕ∈V ⇔Bjkϕ∈W for all k, all formulas ϕ.
Now, if V /Kj ⊆W
thenBk
jϕ∈V ⇔Bjkϕ∈W for all ϕ
then (Bk
jϕ∈V →ϕ∈W)⇔(Bjkϕ∈W →ϕ∈W) for all formulas ϕ
thenV /Bk j ⊆W ↔W/B k j ⊆W So,V /Kj ⊆W →(V /Bk j ⊆W ↔W/Bjk ⊆W) ¤ Lemma 3.2.3 V /Bk j ⊆V /Bk 0 j for all k≥k 0.
Proof: Letϕ∈V /Bk j. Then Bk jϕ∈V butBk jϕ→Bk 0 j ϕ∈AX0 soBk jϕ→Bk 0 j ϕ∈V by lemma 3.2.1 thenBk0 j ϕ∈V by lemma 3.2.1 again i.e. ϕ∈V /Bk0 j ¤
Theorem 3.2.4 AX’ is a sound and strongly complete axiomatization w.r.t. the semantics of LSt.
Proof: Soundness can easily be checked. We only prove completeness. To prove completeness, we must show that
Every AX’-consistent set Γ of formulas is satisfiable on some belief epistemic modelMc (*).
We prove it using a general technique that works for a wide variety of modal logics. We construct a special structure Mc which is a belief epistemic model called the canonical
structure for AX0. Mc has a state wV corresponding to every maximal AX’-consistent set
V. Then we show:
Mc, w
V |=ϕ iffϕ∈V (**)
That is, we show that a formula is true at a statewV exactly if it is one of the formulas
in V. Note that (**) suffices to prove (*), for by lemma 3.2.1, if Γ is AX’-consistent, then Γ is contained in some maximal AX’-consistent set V. From (**) it follows that Mc, w
V |= Γ,
and so Γ is satisfiable inMc. Therefore, Γ is satisfiable w.r.t. the semantics ofL
St, as desired.
We thus defineMc like that :
Mc= (Wc,∼
j, κj, V)
• Wc ={w
V;V maximal AX’-consistent set }
• ∼j={(wV, wW);V /Kj ⊆W}used before where V /Kj ={ϕ;Kjϕ∈V}
• κj(wW) =min{k;W/Bk
j ⊆W}
• wW ∈V(p) iff p∈W
1. First, we are going to show:
Mc, w
We prove it by induction on ϕ.
The propositional case is straightforward. The knowledge case is classical. We only prove the belief case.
• Assume ϕ=Bk jψ∈V.
i.e. ψ∈V /Bk j
For all maximal consistent sets (m.c.s.) W such that V /Kj ⊆W, V /Bk
j ⊆ W →
ψ∈W becauseψ∈VBk
j.
i.e. For all W such thatV /Kj ⊆W, andV /Bk
j ⊆W,ψ∈W.
i.e. For all W such thatV /Kj ⊆W, andW/Bk
j ⊆W,ψ∈W by lemma 3.2.1.
i.e. For all W such thatwV ∼j wW and κj(wW)≤k,ψ∈W by definition.
i.e. For all wV ∼j wW and κj(wW)≤k,Mc, wW |=ψ by induction hypothesis.
i.e. Mc, wV |=Bjkψ. i.e. Mc, w V |=ϕ. • Assume Mc, w W |=ϕ=Bjkψ VBk j ∪ {¬ψ} is not AX’-consistent.
For suppose otherwise.
There is then a m.c.s W such thatV /Bk
j ∪ {¬ψ} ⊆W If k≥M, thenV /Bk j =V /Kj becauseB k jϕ↔Kjϕ∈V (1). If k < M, thenV /BM j ⊆V /Bjk by lemma 3.2.3 i.e. V /Kj ⊆V /Bk j because of (1) In both cases, V /Kj ⊆V /Bk j Moreover, V /Bk j ⊆W So, V /Kj ⊆W andV /Bk j ⊆W then V /Kj ⊆W andW/Bk j ⊆W by lemma 3.2.2 i.e. wV ∼j wW and κj(wW)≤k
By the induction hypothesis, we have Mc, wW |=¬ψ because¬ψ∈W
So, Mc, w
Since V /Bk
j ∪ {¬ψ} is not AX’-consistent, there must be some finite subset, say
{ϕ1, .., ϕn, ψ} which is not AX consistent. Thus by propositional reasoning, we
have: `ϕ1→(ϕ2 →(...→(ϕn→ψ))) By the axiom ofBk j-generalisation we have `Bk j(ϕ1 →(ϕ2→(...→(ϕn→ϕ)))) By the axiom ofBk
j-distributivity and modus ponens,
`Bk
jϕ1→(Bjkϕ2 →...→(Bkjϕn→Bjkψ)).
So by lemma 3.2.1,
Bk
jϕ1 →(Bjkϕ2→...→(Bjkϕn→Bkjψ))∈V.
Then again by the lemma 3.2.1 because Bk
jϕi∈V for all i, we get
Bk
jψ∈V, i.e. ϕ∈V
2. Now we have to prove that Mc is well-defined and thatMc is a belief epistemic model.
• To prove definedness, we have to show thatκj is well defined. It is the case because
– κj(wW) exists for allwW.
IndeedV /Kj ⊆V by axiom 12
and V /Kj =V /BMj by axiom 16
So {k;V /Bk
j ⊆V} 6=∅
Then the minimum exists.
– κj(wW)≤k⇒ for all k0 ≥k κj(wW)≤k0 i.e. V /Bk j ⊆V ⇒ for all k 0≥k V Bk0 j ⊆ V
This is true by lemma 3.2.3
– 0≤κj(wW)≤M for all wW by the first point.
• The proof that Mc is a belief epistemic model amounts to prove that ∼ j is an
equivalence relation. This is a classical proof that we don’t give, it uses the axioms 12-13-14(and 16).