In Chapter 4, we have already shown how to adapt the proof systems and completeness proofs given in Chapter 3 to turn them into complete proof systems for IAML. We only needed to adapt the clauses for the entertain modality, since this is the only case in which the reduction equivalences ofAMLQandIAMLdiffer. The same is the case when we compare IAMLQ with AML or IAML. Therefore, we can follow the same procedure we followed in
Chapter 4 to obtain two complete axiomatizations for IAMLQ. We skip the proofs in this section.
5.8.1
Completeness via replacement of equivalents
The proof system denoted by`IAMLQRE consists of all the inference rules for IEL(Ciardelli,
2014) and the rules inFigure 5.6. We denote the relation of inter-derivability bya`IAMLQRE.
Using the same proof strategy as in the previous chapter, we can obtain the following theorem.
Theorem 5.8.1. IAMLQRE is sound and complete For any Φ∪ {ψ} ⊆ LIAMLQ,Φ|=ψ ⇐⇒ Φ`
!Atom !∧ !K [x]p pre(x)→p [s](ϕ∧ψ) [s]ϕ∧[s]ψ [x]Kaϕ pre(x)→Ka[δa(x)]ϕ !⊥ !→ !E [x]⊥ ¬pre(x) [x](ϕ→ψ) [x]ϕ→[x]ψ [x]Eaϕ pre(x)→V s∈∆a(x)Ea( V y∈s(pre(y)→cont(y))→[s]ϕ) AUD !> RE [s]α V x∈s[x]α [s](ϕ> ψ) [s]ϕ> [s]ψ ϕ↔ψ χ[ϕ/p]↔χ[ψ/p]
Figure 5.6: The inference rules for dynamic modalities inIAMLQ. The double lines indicate that the inference is allowed in both directions. The rule AUD can only be applied to declarativesα.
5.8.2
Completeness via monotonicity
Once again, we provide an alternative complete proof system forIAMLQusing monotonicity of dynamic modalities. The proof system denoted by`IAMLQ!Monconsists of the inference rules
of`IAMLQRE, with !Mon (Figure 4.6) instead of RE. By adapting the proofs in the previous
chapter slightly, we obtain:
Theorem 5.8.2. IAMLQ!Mon is sound and complete
For any Φ∪ {ψ} ⊆ LIAMLQ,Φ|=ψ ⇐⇒ Φ`
IAMLQ!Monψ.
Like for AMLQ and IAML, we have thereby provided two complete axiomatizations of IAMLQ.
5.9
Comparison
We now briefly compare IAMLQ to the other logics discussed in this thesis. Let us start withAMLQ. It is easy to transform anyAMLQaction model into anIAMLQ action model. We can take over the same actions, and whenever there is uncertainty for some agent which action is the actual one, we always let this be an issue (formally: ∆a(x) ={{y} |x∼a y}∪∅).
Then we have a subset ofIAMLQaction models for which we can show that the two update procedures coincide. We can then give a very straightforward translation in which we only have to translate the dynamic modalities, and we can show thatIAMLQis conservative over AMLQ.
With respect toIAML, it is even easier to see thatIAMLQis a conservative extension: every IAML action model is already anIAMLQ action model, namely one without questions as the contents of actions. This makesLIAML a strict subset of LIAMLQ. Since condition (iii) of the update procedure ofIAMLQ is void whenevery is a statement (because this makes
pre(y)→cont(y) a tautology), the update procedure ofIAMLQ is conservative over that of IAML. Therefore, the logic is also a conservative extension ofIAML.
Since IAMLQ is conservative over AMLQ and IAML, it is also conservative over AML and IDEL. Furthermore, everything we said inSection 4.10, when we comparedIAMLtoELQm, carries over to IAMLQ. We may even add, as an extra argument for IAMLQ over ELQm, that the action of asking a question can be represented in IAMLQas a regular action with a question as its content, rather than as a special action consisting of multiple statements, like inELQm and IAML. As we argued earlier in this chapter, this representation is more natural.
5.10
Conclusion
We have argued that a lot of relevant situations can already be accurately encoded inAMLQ or IAML. The step to IAMLQ is motivated mainly by conceptual considerations. We can profit from both the advantage of AMLQ (asking a question is a regular action, just like uttering a statement) and the advantage ofIAML(we can encode issues explicitly in action models).
It may be seen as an extra advantage ofIAMLQthat we can now also encode situations in which we have uncertain and uninterested agents and one of the possible actions concerns a question. However, our update procedure gives us issues in the updated model that are, arguably, difficult to match with any intuitions.
The most important contribution of this chapter is that we have formulated an update procedure that is conservative over bothAMLQ and IAML. Furthermore, we have checked that all the properties we discussed forAMLQandIAMLcarry over. For the axiomatization, we followed the same strategies as before, but with an altered and more complex reduction axiom for the entertain modality. The length and complexity of this reduction axiom is a nice illustration of how useful the addition of dynamic modalities in fact is.
Conclusions and further work
6.1
Conclusions
Since Chapters 3, 4 and 5 end with their own conclusions with respect to the systems developed in these chapters, in this section we only repeat what we consider to be the most important contributions of this thesis in general.
We started with the observation that inquisitive dynamic epistemic logic, in which we can encode public utterances of statements and questions and compute their results, lacked the means to encode (semi-)private utterances. At the same time, (semi-)private announcements are encoded in dynamic epistemic logic using action models. We identified two different strategies towards a combination of these two insights: adapting the contents of actions and adapting the structure of action models. We developed the former inAMLQ, the latter in IAML, and both strategies together inIAMLQ.
We have seen that if we want to have questions as normal actions in our system, action contents cannot be reduced to preconditions, since preconditions are truth-conditional. In- stead, we generalized this notion of action content, and retrieve preconditions as a derived notion: the presupposition that we associate with the action content.
The combination of inquisitive logic and dynamic modalities gives a new perspective on complex dynamic modalities, built up out of more than one action. While they are commonly interpreted in terms of non-deterministic actions, we interpret them in terms of partial information about the action taking place. In this way, we can make sense of complex dynamic modalities in combination with questions.
We have provided two complete axiomatizations for each of the systems. Both strategies rely on a reduction toIEL. Interestingly, we have found that formulas of the form [s](ϕ→ψ) cannot be elegantly reduced. However, we have found a way around this using the normal form result familiar from inquisitive logics.
The update procedures ofAMLQ,IAMLandIAMLQare all standard with respect to knowl- edge. This means they differ only in the issues that result from updates. These differences are reflected in their axiomatizations: they differ only in the reduction rule for formulas of the form [s]Eaϕ, which express which issues an agent entertains after an update.
IAMLQbroadens the scope of epistemic situations that can be modelled. The possibility to encode semi-private utterances of statements carries over fromAML. However, the range of statements is extended with statements about issues (a entertains ϕ) and statements that embed questions (aknows whetherµ), which are not available in AML.
Furthermore, while we can encode only the knowledge agents have about epistemic actions inAML, inIAMLQwe can also encode their issues: this means that we can allow uncertainty about epistemic actions to raise issues for agents, which reflect the extent to which they are interested in which action is occurring.
An even more important addition is the possibility to encode semi-private questions in IAMLQ. While we already had public questions inIDEL, we could not yet model the act of asking a question in a situation where agents are not necessarily certain about its content. At least, not the range of questions that we know from inquisitive logics, which include conditional questions and questions with presuppositions.
IAMLQcan be regarded as conservative over bothAMLandIDEL. We can viewAMLaction models as IAMLQ action models and there is a straightforward way of encoding a public utterance inIAMLQ. Formulas ofAML are formulas of IAML, and their semantics remain standard. Formulas ofIDELcan be defined as abbreviations inIAMLQ.
This thesis is accompanied by a computational tool using which product updates inIAMLQ can be calculated. Its source code has been made available to the community (see Ap- pendix A).