Given we have established completeness ofIPLwith respect to ipnms we can use semantic reasoning to prove some additional theorems ofIPL.
Proposition 7.8.1(Further theorems ofIPL). The initial three theorems ofIPLare notable as being adaptations of axioms ofCO. (Cf. Boutilier (1994, p. 101) axioms S. and H. and K0. respectively.) The final theorem partially characterises the interaction between the two plausibility modalities. 1. '!Ea—hEai' 2. .hEai.Ea'^Ea— /_ h—ai.Ea'^Ea— //! .Ea. W ˛2R.'/ W ˇ2R. /.˛_ˇ//^Ea—.W˛2R.'/ W ˇ2R. /.˛_ˇ/// 3. Ea—? !.Ea'!Ea'/ 4. hEa—i' !EahEa—i'
Proof. We show each theorem is valid on an arbitrary state in an arbitrary ipnm. All but the first theorems are declaratives, and therefore we use proposition 1.2.15 to simply the proofs.
1.'!Ea—hEai'
SupposeM; s ' butM; s ² Ea—hEai'. The latter is shorthand forM; s ²
Ea—:Ea:', which by proposition 1.2.15 entails that for somev2sit is the case that
M; v ² Ea—:Ea:'. So,9t 2 ˙—a.v/; M; t ² :Ea:'. By proposition 1.2.15 it
follows that for some worldu2t; M; u²:Ea:', whence we inferM; uEa:'.
From this it follows that8t02˙
a.u/; M; t0:'.
Furthermore, asu2—a.v/we know thata.v/a.u/.
For, asu2—a.v/we know that—a.u/—a.v/, by condition 2 on ipnms. Therefore,
ifv2—a.u/it must be the case thatv2—a.v/. However, we know thatv2a.v/
by condition 3, and thata.v/\—a.v/D ;by condition 4. This is a contradiction,
whencev…—a.u/, but then asu2—a.v/anda.v/Da.v/[—a.v/by condition 5
on ipnms it follows thatu 2 a.v/. So, by condition 6 we knowa.u/ D a.v/.
So, using conditions 4 and 5 again, we can infer using the fact thatv … —a.u/, that
v 2 a.u/. So, from this it follows via condition 1 that˙a.v/ ˙a.u/, whence
a.v/a.u/.
We observed above thatv2
a.v/, and so from this and the previous observation we
know thatv 2
a.u/. Now, we know thatv 2 s, andM; s ', whenceM; v'
by persistence. But we also know that8t02 ˙a.u/; M; t0 :', and asv2a.u/,
fvg 2˙a.u/, whenceM; v:'. Thus we have derived a contradiction. 2..hEai.Ea'^Ea— /_ h—ai.Ea'^Ea— //!
.Ea.W˛2R.'/Wˇ2R. /.˛_ˇ//^Ea—.W˛2R.'/Wˇ2R. /.˛_ˇ///
Suppose it is the case thatM; w hEai.Ea'^Ea— /_ h—ai.Ea'^Ea— /. As
worlds behave classically this entails that either M; w hEai.Ea' ^ Ea— / or
M; w h—ai.Ea' ^Ea— /. Without loss of generality let us assume the former
is the case.
This is shorthand forM; w :Ea:.Ea' ^Ea— /. So,9t 2 ˙a.w/such that
M; t ° :.Ea'^Ea— /. By proposition 1.2.15 it then follows that for somev 2
t; M; vEa'^Ea— .
We now need to show:
i..Ea.W˛2R.'/Wˇ2R. /.˛_ˇ///and ii.Ea—.W˛2R.'/
W
ˇ2R. /.˛_ˇ////
We know from the above reasoning that for somefvg 2 ˙a.w/; M; v Ea' ^ Ea— . Asfvg 2˙a.w/it follows thatv 2a.w/. So, by condition 5 on ipnms we
knowv2a.w/and therefore by the condition of introspection 1 on ipnms we know
˙a.v/ D ˙a.w/. Using this we showM; w Ea.W˛2R.'/
W
ˇ2R. /.˛_ˇ////.
From this and an application of axiom 7 the desired result will follow.
So, lett 2 ˙a.w/be arbitrary. We know˙a.w/ D ˙a.v/, and sot 2 ˙a.v/. Let
u 2 t be arbitrary, and note that ast 2 ˙a.v/; u 2 a.v/. Now, asu 2 a.v/we
know it is the case thatu2
a.v/oru2 —a.v/by condition 5 on ipnms. Without
M; u'. So, by theorem 6.2.7 we know thatM; u˛for some˛2R.'/, whence
M; uW˛2R.'/
W
ˇ2R. /.˛_ˇ/.
Asu2a.v/was arbitrary we know8u2a.v/; M; uW˛2R.'/
W
ˇ2R. /.˛_ˇ/.
So, by proposition 1.2.15 we have8t a.v/; M; t W˛2R.'/
W
ˇ2R. /.˛_ˇ/.
Therefore, via persistence and the fact that˙a.v/ }.a.v//we know that8t 2
˙a.v/; M; t W˛2R.'/
W
ˇ2R. /.˛_ˇ/. So, given we have established˙a.v/ D
˙a.w/it is immediate thatM; wEa.W˛2R.'/
W
ˇ2R. /.˛_ˇ//
3.Ea—? !.Ea'!Ea'/
SupposeM; w Ea—?andM; w Ea'. By the former assumption it must be
the case that˙—a.w/ D f;g, whence—a.w/ D ;and by condition 5 this entails
a.w/ D a.w/and so by condition 7 it follows that˙a.w/ D ˙a.w/. So, as
˙a.w/ŒŒ',˙a.w/ŒŒ', whenceM; wEa'.
4.hEa—i'!EahEa—i'
SupposeM; w hEa—i' whileM; w ² EahEa—i'. Then, by the latterM; w
hEaiEa—:'. And, by the former, we have that9t 2 ˙a.w/; M; t ' and by the
previous inference we have that9v 2a.w/,M; v Ea—:', whence8t02 ˙—a.v/,
M; t0:'.
Asv2a.w/we know by condition 1 on ipnms that˙a.v/˙a.w/, whence it is
an easy argument to observe that˙—a.w/ ˙—a.v/, given the conditions placed on
ipnms. Therefore8t02˙—
a.v/,M; t0:'while for somet 2˙—a.v/,M; t ', so given persistence we have derived a contradiction.
Chapter 8
Conclusion
In this thesis we have introduced and axiomatised two extensions of propositional in- quisitive semantics: inquisitive conditional-doxastic logic and inquisitive plausibility logic. Moreover, we have shown both are sound and complete with respect to the same class of models; inquisitive plausibility models, which allow for an intuitive interpreta- tion of the two logics.
The primary focus of the thesis is conditional-doxastic logic. This generalises con- ditional-doxastic logic, enabling the study of new propositional attitudes, grounded in familiar assumptions about the process of conditionalisation. For as we observed in chapter 3 (cf. corollary 3.3.11) the interpretation of conditionalisation captured by in- quisitive conditional-doxastic logic occurs wholly at the level of declaratives. Therefore, no additional assumptions about the process of conditionalisation are made with the introduction of issues as our core semantic notion. We take the fact that the process of conditionalisation captured by conditional-doxastic logic can be straightforwardly gen- eralised to an inquisitive setting to be the core conceptual achievement of this thesis.
In chapter 3 we gave a preliminary (formal) analysis of the modal operator termed ‘considering,’ and of the behaviour of (conditional) belief when extended to issues on in- quisitive plausibility models. With the formal properties of these modalities established, we hope to explore their potential as formal represent propositional attitudes in future work, and other applications of the logic. We would also like to look at axiomatising the modalities under different assumptions about the semantic properties used to interpret the modalities (e.g., lifting the assumption of negative introspection).
Moreover, as a generalisation of conditional-doxastic logic, inquisitive conditional- doxastic logic forms a basis for extending the standard account of belief revision to an inquisitive setting, where agents may conditionalise on both inquisitive content as well as informative content.
However, the thesis has been largely technical, and we have not explored in any de- tail the consequences of this result, nor in general applications ofICDL. Still, following research presented by Ciardelli and Roelofsen (2014b) future work may build onICDL
to model issues in epistemic change, allowing greater expressive power when modelling, for example, the process ofcontraction—determining the information to give up when conditionalising on information which is inconsistent with an agents’ current doxastic state. Appendix A shows how to axiomatise public announcements with respect toICDL. One aspect ofCDLnot touched upon in detail is its relationship to the AGM principles of belief revision (cf. Baltag and Smets 2006, §3). So, given the relationship between
tic counterpart to syntactic theory of belief revision extended to interrogatives, akin to AGM theory.
However, in chapter 5 we observed that there are limitations toICDL, in particular, while the logic is able to capture the issues agents consider when conditionalising on information, it is unable to capture the issues an agent holds unconditionally as the en- tertains modality of inquisitive epistemic logic is not definable in terms of the considers modality of ICDL. So, in order to obtain a logic in which the interaction between the issues an agent entertains, and those they consider conditional on further issues or in- formation we explored inquisitive plausibility logic.
Inquisitive plausibility logic has sufficient expressive power to express both the en- tertains modality ofIELand the considers modality ofICDL, while being sound and com- plete to the same class of models as inquisitive conditional-doxastic logic. Moreover, we observed howIPLallows the expression of binary plausibility (or preference) operators, allowing one, for example, to express when an agent would prefer to resolve one inter- rogative over the other. However, as in the case ofICDLwe did not consider any specific applications ofIPL, which we leave to future work.
Furthermore, while inquisitive plausibility logic contained the expressive power we desired with respect to inquisitive plausibility models, we did not show thatIPLwas the
weakestlogic with this property. And, given significant distinction in expressive power betweenIPLandICDLit seems safe to conjecture that there are weaker logics between
IPLandICDLwhose axiomatisation leads to a more explicit account of the interaction between the entertains and considers modality. We also leave this to future work.
Below in figure 8.1 we diagram the logics mentioned in this thesis, in terms of their ‘expressive power.’ Given the length of this thesis we have left proofs establishing that logics with greater expressive power are conservative extensions of weaker logics with respect to the language of those logics.
CPC EL InqD CDL IEL ICDL IPL
Appendix A
Dynamics
We briefly show thatICDLandIPLcan be extended to include dynamic modalities for public announcements. The central formal innovations required and used in this ap- pendix can be found in Ciardelli (2015, Chap. 8), in particular §8.3.
The core insight from the chapters of conditional doxastic logic—that the funda- mentals of belief revision takes place at the level of declaratives—is shown to continue, as the interrogative content of a proposition can be factored out of the revision process in the case of ICDL, and equivalent rational applies toIPL. This suggests thatICDLand
IPLcan be straightforwardly extended with other dynamic operations familiar from the transition of dynamic epistemic logic, in particular soft upgrades.
A.1 Updates on ipms
Definition A.1.1(Update).The update of an ipmM D hW;fwagaw22AW;f˙aga2A; Vi with a formula' is the model: M' ´ hW';fw
agwa22AW;f˙ ' aga2A; V'i, defined as follows: 1. W'DW \ j'j M 2. wa D wa \.W'W'/ 3. For everyw2W',˙a'.w/D˙a.w/\ŒŒ'M 4. V'DVW'
As in ipmsa.w/´ fvjvwa ufor someug.
Proposition A.1.2. For any modelM, agenta, formula', and worldw2W'we have:
a'.w/Da.w/\ j'jM
Proof. We observefv j v wa ufor someug D .fv j v wa ufor someug \ j'jM/.
For, ifv 2 fv j v wa ufor someugthenv 2 W \ j'jM andv 2 fv j v wa
ufor someug, whencev 2 fv j v aw ufor someug \ j'jM. Similarly, ifv 2 fv j
v wa ufor someug \ j'jM thenv wa vandv2 W', whencehv; vi 2 W'W',
and sovwa v, establishingv2 fvjvwa ufor someug.
Proof. First we observe that˙a'.w/is a non-empty downward closed set of states. This
is because˙a'.w/D˙a.w/\ŒŒ'M, and the intersection of two non-empty downward
closed set of states is itself a non-empty downward closed set of states.
Second, thatwa is a well-preorder follows from the fact thatwa is a restriction of
the well-preorderwa to the elements ofW'.
We now show that˙a'.w/is an issue overa'.w/ D fv j v wa ufor someug.
Observefv j v wa ufor someug D fv j v wa ufor someug \ j'jM. Therefore,
asS˙a.w/ D fv j v wa ufor someug, we know
S
˙a.w/\ j'jM D fv j v wa
ufor someug \ j'jM by proposition A.1.2. So,S˙a'.w/D fvjvwa ufor someug.
Factivity follows from the fact thatw
a is a restriction ofwa to the elements ofW',
as does introspection 2.
Finally we need to check that the updated maps˙a'satisfy the condition of intro-
spection 1. So, supposev 2 a'.w/. Therefore,v 2 a.w/, by proposition A.1.2,
whence˙a.w/D˙a.v/. From this it follows that˙a.w/\ŒŒ'M D˙a.v/\ŒŒ'M,
and so˙a'.w/D˙a'.v/.