3.3 Inquisitive Plausibility Models
3.3.2 Conditional Belief
We define conditional belief on ipms as a straightforward generalisation of conditional belief with respect to pms to interrogatives as well as declaratives.
Definition 3.3.15(Truth for (conditional) belief).
M; wBa 'iff8t 2Minw
aŒŒ ; M; t'
Definition 3.3.16(Support for (conditional) belief).
M; sBa 'iff8w2sW 8t 2Minw
aŒŒ ; M; t'
Indeed, as a corollary of following proposition shows, the semantic interpretation of belief and considers will coincide if an agent has no epistemic goals.10
Proposition 3.3.17. M; wBaˇiff8v2Minw
aj j; M; vˇ. Proof. By definition,M; wBaˇiff8t 2Minw
aŒŒ ; M; t ˇ. By proposition 1.2.15
this is the case iff8v 2 jMinw
aŒŒ j; M; v ˇ, iff8v 2 Minwaj j; M; v ˇ, by
proposition 3.3.6.
Analogous to the reasoning with respect to considers, above, this means ipm belief is a conservative extension of pm belief, with respect to declaratives.
Still, the pm interpretation of (conditional) belief as truth in the most plausible worlds cannot be straightforwardly applied to interrogatives, as it relies on the fact that declaratives are truth-conditional. One way to interpret belief is as a special case of the considers modality, where only the conditional doxastic goals of the agent are taken into account, parallel to the conditional entertains modality. However, we choose to interpret conditional belief independently of an agent’s inquisitive goals, and instead interpret belief to capture relations between propositions subject to an agent’s doxastic state. Indeed, this perspective can was briefly raised with respect toCDLbelief. Thus,
Ba'reads ‘on the basis of the agent’s current beliefs, the agent holds to entail'.’
Interpreted with respect to declaratives on pms, a declarative formula entails another just in case all the worlds the agent believes as good candidates for the actual world given the truth of the former proposition are worlds in which the latter is true. This is exactly what belief boils down to in the special case of declaratives.
M; wBa˛ˇiff8v2Minw
aj˛j; M; vˇ
Corollary 3.3.18. Ca˛ˇBa˛ˇ
It is when conditionalising on an interrogative that the generalised concept of en- tailment is apposite for interpreting conditional belief. For, by proposition 3.3.6 the states quantified over by conditional belief are the result of intersecting the agent’s dox- astic state, conditionalised on the informative content of , with . Therefore, the modality quantifies over the resolutions to , given the agent’s beliefs. More precisely, if8t 2Minw
aŒŒ ; M; t ', this means that, given the agent’s beliefs given the infor-
mative content of , for every resolution of , some resolution of'is supported. This is captured by the following reduction, proved as part of theorem 5.2.1, in section 4.3.11 Fact 3.3.19.
M; wBa'iff8˛2R. /; ifM; w²BaŠ :˛;9ˇ2R.'/WM; wBa˛ˇ
In other words, from the agent’s perspective, given their current beliefs, resolving would resolve'. Alternatively, the issue of implies the issue of'.
Similarly, if is a declarativeˇand'an interrogative, then the agent’s learns a singular piece of information, and so ifis resolved, then the agent’s epistemic state must support a specific resolution of. We have the following reduction:
10As, in such a case˙
a.w/D}.a.w//.
11The fact is a natural language statement of one of the axiom which allows the reduction of conditional belief to the considers modality. Namely,Ba'$V˛2R. /.:Ba:˛!Wˇ2R.'/Ba˛ˇ /.
Fact 3.3.20. M; wBaˇiff9˛2R./W 8v2Minw ajˇj; M; v˛; iff9˛2R./WM; wB ˇ ˛ p q r s (a)Ba‹p‹q p q r s (b)Ba‹fp;q;sg‹f:p;:q;:sg Figure 3.7
Unlike knowledge, conditional belief generalised to interrogatives is not distributive in general, as can be seen from the examples in figure 3.7. Formally, the distributivity of knowledge is guaranteed by the fact that its semantic clause evaluates what is supported by the agent’s epistemic state, which is in turn an information state. Indeed, an analogous clause for belief would evaluate what is supported by an agent’s doxastic state.
Therefore, for such a clause would readM; w Ba 'iffM;Minw
aj j '. But
note that by persistence, ifM;Minw
aj j', then8s2Minwa}.j j/; M; s'and
if for somev2Minw
aj j; M; v ²', then there exists somes2 Minwa}.j j/such
thatM; s²'.
These two facts entail thatM;Minw
aj j ' iff8s 2 Minwa}.j j/; M; s
'. Therefore, as j j D jŠ j, by corollary 1.2.18, andŒŒŠ D }.jŠ j/, by proposi- tion 1.2.19, this meansM;Minw
aj j'iff8s2MinwaŒŒŠ ; M; s'. So, one can
evaluate the propositions supported by the doxastic state of an agent as a special case of conditional belief, by conditionalising on the proposition expressed by the informative content of the proposition conditionalised on.
Furthermore, through complex formulas containing the believes modality we can distinguish certain aspects of an agent’s epistemic state. For example,B
a^ :BaŠ
expresses that the agent can resolvegiven a resolution of, but not given the presup- position of. Note, this is only possible for interrogatives,; .
Similarly, the added expressive power of the considers modality allows us to capture when an agent believes an issue on their epistemic agenda is resolved by conditionalising on a given proposition. Here, we are interested in capturing when conditionalising on would lead to a resolution of, i.e. that some resolution ofwould be believed given .12 Formally:
M; wRa´Wa^W˛2R./Ba˛
At this point we can observe an important distinction between support and truth con- ditions.
For, when evaluated at a worldW˛2R./Ba ˛reads, by interpreting disjunction as
existential quantification, as there being some resolution˛ofwhich the agent believes, conditional on . However, an agent may consider different resolutions ofat different
12A weaker condition may define when an issue is resolved by the definition: M; w R
a ´
Wa^Ba. This stipulates that, conditional on there issomeresolution oftrue at the most plausi- ble worlds, but does not guarantee the agent is in a position to identify a single resolution. Here, then, conditionalising on would not lead to a resolution of, but a guarantee thatcould in principle be resolved, given additional information.
worlds. For example, it may be the case thatM;fw; vgW˛2R./Ba˛, whileM; w
Ba˛andM; vBaˇ, for˛¤ˇ2R./. This coincides with an interpretation of the
support condition as evaluating an agent’s epistemic state from a position of incomplete information. More generally, every world in a state may agree on the agent’s current beliefs, but differ with respect to how those beliefs will be revised if the agent were to learn new information.
Finally, we can generalise the reduction of knowledge to considers to knowledge of both declaratives and interrogatives by the following condition.
Ka'´W˛2R.'/Ca:˛?
This reduction mirrors the reduction of knowledge to conditional belief inCDL, with the proviso includes for interrogatives that at least one of the resolutions for'is known. Note the reduction can be recast in terms of conditional belief, as in the case of CDL, given the connexion between considers and conditional belief observed in axiom 7, be- low.