In this section we will compareAMLQ to the systems we discussed in Chapter 2. We will show thatAMLQcan be seen as a conservative extension of bothIDELandAML.
3.10.1
Inquisitive Dynamic Epistemic Logic
As we have already seen in Chapter 2, IDELcan encode public utterances of questions as well as statements, and in the current chapter it has been our goal to extend this to more private utterances. In this section, we will show thatAMLQis conservative overIDEL: that is, the notion of public utterance inAMLQis no different than it is inIDEL. Let us first see how we would encode a public utterance in anAMLQaction model.
Definition3.10.1. Public utterance in AMLQ
For any formula ϕ∈ LAMLQ, the public utterance of ϕ is modelled as an update with the action modelMϕ, defined as follows:
Mϕ=h{pubϕ},{∼a |a∈ A},conti, wherecont(pubϕ) =ϕand for alla∈ A: pubϕ∼apubϕ.
We can show that updating with this action model is essentially the same as performing an update in IDEL. The resulting updated models are not the same, but this is only because the individual worlds of the models do not have the same name. In other words, we can show that they are isomorphic.
Proposition 3.10.1. Isomorphism between updated models
For any inquisitive epistemic model M and for any ϕ∈ LIDEL and ψ ∈ LAMLQ: if ϕ≡ ψ, thenM0=M ⊗Mψ is isomorphic toMϕ.
Proof: Take anyϕ∈ LIDELandψ∈ LAMLQsuch thatϕ≡ψ. We define the following models: • M =hW,{Σa |a∈ A}, Vi(our original model)
• M0=M⊗Mψ=
hW0,{Σ0
a | a∈ A}, V0i
• Mϕ=hWϕ,{Σϕ
a |a∈ A}, Vϕi
We start by showing that w∈ Wϕ just in case hw,pubψi ∈ W0, by unpacking the definitions ofWϕ andW0: w∈Wϕ ⇐⇒ w∈W∩ |ϕ|M ⇐⇒ w∈W andw∈ |ϕ|M ⇐⇒ w∈W andM, w|=ϕ ⇐⇒ w∈W andM, w|=pre(pubψ) ⇐⇒ hw,pubψi ∈W0
Letf(w) =hw,pubψi. Thenf is a bijection between Wϕand W0. Now let us show
that it is indeed an isomorphism. For this, we need to show two things:
(i) The mapping preserves the structure of the state maps. That is, if we let f(s) be{f(w)|w∈s}, then we have:
s∈Σϕa(w) ⇐⇒ f(s)∈Σ0a(f(w))
(ii) The mapping preserves the valuation: Vϕ(w) =V0(f(w)).
Given the definition of valuation, the latter is trivial, so we only show (i). Take any worldw ∈Wϕ and any state s⊆Wϕ. Let s0 =f(s). Then we need to show that
s∈Σϕ
a(w) ⇐⇒ s0 ∈Σ0a(hw,pub ψ
i). Then the update procedure of IDEL gives us conditions (i) and (iv) of the update procedure of AMLQ, while conditions (ii) and (iii) are trivial given the definition of the action modelMψ. We have:
s∈Σϕa(w) ⇐⇒ s∈Σa(w)∩[ϕ]M ⇐⇒ s∈Σa(w) ands∈[ϕ]M ⇐⇒ π1(f(s))∈Σa(w) andπ1(f(s))∈[ϕ]M ⇐⇒ π1(s0)∈Σa(w) andπ1(s0)|=cont(pubψ) ⇐⇒ s0∈Σ0a(hw,pub ψ i)
Proposition 3.10.2. s[pubψ] is the image of s∩ |ϕ|M under isomorphism f
For every information states: w∈s∩ |ϕ|M ⇐⇒ hw,pubψi ∈s[pubψ]
Proof: We only need to unpack the definition of updated states:
w∈s∩ |ϕ|M ⇐⇒ w∈sandw∈ |ϕ|M
⇐⇒ w∈sandM, w|=pre(pubψ) ⇐⇒ hw,pubψi ∈s[pubψ]
Definition3.10.2. Translation ofIDEL formulas to AMLQ formulas
For everyϕ∈ LIDEL, its translationϕ∗∈ LAMLQ is defined recursively as follows: • p∗=p
• ⊥∗=⊥
• (ϕ◦ψ)∗=ϕ∗◦ψ∗ for◦ ∈ {∧,→,> } • (ϕ)∗=ϕ∗ for∈ {Ka, Ea}
• ([ϕ]ψ)∗= [Mϕ∗,pubϕ∗]ψ∗
Proposition 3.10.3. Every IDEL formula is equivalent to itsAMLQ translation
For everyϕ∈ LIDEL and its translationϕ∗∈ LAMLQ,ϕ≡ϕ∗.
Proof: By induction on the complexity of ϕ. The only step which is not immediate is the step for the dynamic modality.
Suppose ϕ is [ψ]χ. By the induction hypothesis, we have some ψ∗, χ∗ ∈ LAMLQ equivalent to ψ and χ respectively. Take any state s in any inquisitive epistemic model M. Let M0 = M ⊗Mψ∗. Then by Proposition 3.10.1, M0 and Mψ are
isomorphic. Together with Proposition 3.10.2 and the support conditions of both dynamic modalities, we obtain:
M, s|= [ψ]χ ⇐⇒ Mψ, s∩ |ψ|M |=χ
⇐⇒ M0, s[Mψ∗,pubψ∗]|=χ
⇐⇒ M0, s[Mψ∗,pubψ∗]|=χ∗
⇐⇒ M, s|= [Mψ∗,pubψ∗]χ∗
⇐⇒ M, s|= ([ψ]χ)∗
We have thereby shown that AMLQ can express everything that IDEL can and that the interpretation of dynamic modalities of public utterance coincides. We may even define [ϕ]ψ in AMLQ as an abbreviation for [Mϕ,pubϕ]ψ. In that case, we can viewAMLQ as a conservative extension ofIDEL. It follows from Proposition 8.2.6 ofCiardelli(2016) that the result of a public utterance of a statement in AMLQ is the same as the result of a public announcement inPAL.
3.10.2
Action Model Logic
In this section we will show that AMLQ is a conservative extension ofAML. We start by showing that, in terms of updating knowledge, our procedure coincides with the standard one. First, we define a way in which we associate a standard Kripke model with any inquisitive epistemic model.
Definition3.10.3. Kripke model determined by an inquisitive epistemic model
For any inquisitive epistemic model M =hW,{Σa | a∈ A}, Vi, MK is the Kripke model
determined byM, defined by:
MK =hW,{∼
a |a∈ A}, Viwhere∼a={hw, w0i |w0 ∈σa(w)}
Any Kripke modelMK is a standard epistemic model that encodes the same knowledge as
M.
Definition3.10.4. AMLQ variant of standard action model
Let M = hS,{∼a | a ∈ A},prei be a standard action model. Then its AMLQ variant b
As all preconditions in standard action models are formulas ofAML, they are also formulas of AMLQ. Furthermore, as they cannot contain an inquisitive disjunction, they are declaratives and therefore truth-conditional. This makescontandpreinterchangeable.
Proposition 3.10.4. The AMLQupdate procedure is standard in terms of knowl-
edge
For any inquisitive epistemic modelM and any standard action model M: (M ⊗M)b K =
MK⊗M.
Proof: Take any inquisitive epistemic modelM =hW,{Σa |a∈ A}, Viand standard action
modelM. We define the following two standard updated models: • M0= (M⊗ b M)K =hW0,{∼0 a |a∈ A}, V0i • M00=MK⊗M=hW00,{∼00 a |a∈ A}, V00i
From the definitions of both update procedures, it is immediate thatW0=W00 and
V0=V00. It remains to show that ∼0 a=∼00a.
(⇒) Assumehw,xi ∼0
ahw0,x0i.
Then byDefinition 3.10.3, hw0,x0i ∈ σa0(hw,xi). This means that {hw0,x0i} ∈ Σ0a(hw,xi). By condition (i) ofDefinition 3.2.4,{w0} ∈Σa(w) ofM. This means
by the definition ofMK that w∼
aw0. By condition (ii),x∼ax0. Then by the
update procedure of standard action models,hw,xi ∼00
a hw0,x0i.
(⇐) Assumehw,xi ∼00
a hw0,x0i.
Then by the update procedure of standard action models,w∼aw0andx∼a x0.
We want to show that{hw0,x0i} ∈Σ0a(hw,xi), which means we need to check that this state satisfies conditions (i)-(iv) ofDefinition 3.2.4.
As w ∼a w0, by the definition of MK we have {w0} ∈ Σa(w), which means
condition (i) is satisfied. Byx∼a x0 we already have condition (ii). Condition
(iii) is trivial as {hw0,x0i} is a singleton. We have M, w0 |= pre(x0) because otherwise the pairhw0,x0iwould not have been in the domain of the updated model. ByProposition 3.2.1, M, w0 |=cont(x0), which means condition (iv) is
satisfied too.
Because{hw0,x0i} ∈Σa0(hw,xi), byDefinition 3.10.3we havehw,xi ∼0
ahw0,x0i.
We have shown that the update procedure ofAMLQaction models is standard with respect to knowledge. Furthermore, we can check that all connectives of the language ofAMLhave the same truth conditions as those ofAMLQand that allα∈AMLare truth-conditional. It follows that the semantics of these formulas is standard in our setting. Therefore, for every Γ∪ {α} ∈ LAML: Γ|=
AMLα ⇐⇒ Γ|=AMLQ α. This makesAMLQ a conservative extension ofAML.