We will now evaluate both solutions to the syntactic expressivity issue that have been proposed in the previous two subsections. First, note that both solutions indeed allow us to explicitly define the setsEi[w] in the (extended) language,
and thus also to express an [EXP]-reduction axiom forPi(ϕ)≥k:
1. [EXP]Pi(ϕ)≥k ↔ ↓x.Pi([EXP]ϕ∧Qixˆ )≥kPi( ˆQix) 2. [EXP]Pi(ϕ)≥k ↔ α∧Pi([EXP]ϕ∧αi)≥kPi(αi)
∨ ¬αi∧Pi([EXP]ϕ∧ ¬αi)≥kPi(¬αi) It is clear that —at least on an intuitive level— there is a structural analogy between both reduction axioms. First of all, note that ˆQixand (¬)αiplay simi- lar roles, viz. defining the current experiment cell. That these axioms truly talk about thecurrentexperiment cell is ensured in the first solution by means of the binder↓, whereas in the second solution it is ensured by making a case distinc- tion (about which is the current experiment cell) which is explicitly represented in the reduction axiom.
Solution 1. The first solution involves introducing the Qi-operator. This is not a major disadvantage, since it is merely an extension of the analogy —that was already discussed above— between our approach to experiments and the dynamic epistemic logic of questions (this operator is also introduced in [45]). The major advantage of the first solution is that it places no restrictions on the generality of the experiments: experiments continue to correspond toany kind of partition, no matter how fine/coarse (i.e. no restriction to binary partitions only).
However, the first solution also has some serious problems. Technically
speaking, the move to hybrid logic (and in particular, the introduction of the binder↓) negatively influences the metaproperties of the resulting system. For example, adding ‘merely’ the binder↓already leads to the undecidability of the satisfiability problem [9, Theorem 5.1].
Furthermore, the first solution has some broadmethodologicalissues as well. Hybrid logic provides us with nice technical tools to ensure the definability of certain sets, but by themselves, these tools seem to lack any intuitive interpreta- tion. So far, all operators of the object language had relatively straightforward intuitive meanings (knowledge before/after the experiment, subjective proba- bility, etc.), but this is not the case for the binder ↓ and state variables x. Furthermore, note that hybrid logic can be seen asyet another conceptual tool- box, next to the ones of basic modal logic and probabilistic reasoning that are already being used. From this perspective, the system as a whole tends to be- come rather chaotic: it starts to look like a ‘patchwork’ that consists of several bits and pieces —each needed to solve a particular problem—, but that lacks overall coherence.
Solution 2. The second solution does not have any of these problems. Techni- cally speaking, the metatheoretical properties are not hurt by introducing the special proposition lettersαi(more substance will be given to this claim in Sec- tion 5.4). Methodologically speaking, the second approach has a clear intuitive interpretation: agent i performs a binary experiment, i.e. she asks the yes-no question ‘isαithe case?’ (cf.supra). As such, the second solution is based on a particular case of the general analogy between experiments and questions that was already discussed earlier, and that was already being used in the previous chapters.4 The second solution is thus not a threat to the ‘unity’ of the system
as a whole: it leaves the system as coherent as it was before.
Because of these remarks, the second solution seems to be preferable. One might object at this point that restricting to only binary experiments is too drastic. We have three replies to this objection.
First of all, we reiterate the remark with which we started Section 5.1: until now, our modeling of the experiments has been fully general. All the agreement 4Obviously, moving from arbitrary to binary experiments is a severerestriction of general-
ity. Note, however, that this restriction does not hurt theintuitive interpretation. We restrict from arbitrary experiments (arbitrary questions) to binary experiments (yes/no questions), but the intuitive interpretation (the analogy experiment/question) remains intact.
theorems in Chapter 4 were proved in this general context. Restricting to binary experiments is only necessary when one decides to focus on syntactic issues.
Second, it would technically not be difficult to allow for ternary or quaternary experiments. All the conceptual issues, however, arise already at the level of binary experiments, and therefore we have chosen to stick to binary experiments. Third, in the beginning of this thesis we noted that because the experiment relation can be arbitrarily fine-grained, it can also be used to modelsequencesof experiments (cf. Remark 13). This perspective can be reversed. The experiment relation is now only binary, but every finitary experiment can be represented as a finite sequence of binary experiments.5 Of course, this requires a way of
formally representing sequences of experiments. We will return to this in the final chapter.
Conclusion. We conclude this subsection by reiterating that of the two solu- tions to the syntactic expressivity issue that were proposed in Subsections 5.1.1 and 5.1.2, the second one is to be preferred on both technical and methodolog- ical grounds. Furthermore, the shortcomings of the second approach are not so bad as they might first look. Therefore we will henceforth fully and uniformly adopt the second solution (binary experiments).