Abstract
The aim of the chapter is to provide a critical assessment of Krister Segerberg’s solution to the problems of introspective belief change. We present three alternative ways in which the paradoxes may be avoided. The first is a solution due to Lindström and Rabinowicz, using a two-dimensional semantics for DDL. The second is found in a logic for belief change suggested by Bonanno, in which the operator for belief is replaced by a class of operators for belief, each supplied with a temporal index. The third solution consists in a logic for belief change due to van Benthem, founded on the method of Dynamic Epistemic Logic in which the dynamics is modelled by operations on entire models, rather than on some structure within the models. We argue that, while there are some differences between these approaches, there is a strong structural similarity between them, and they avoid the paradoxes of DDL in essentially the same way. Furthermore, this way of avoiding the paradoxes is both different from and, we think, more natural than Segerberg’s own solution.
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1 Introduction
Theories of rational belief change [1, 4, 5] are traditionally presented in a semi-formalized manner. While a formalized language is used to speak about the content of a state of belief, the theory of belief revision is, like most mathematical theories, presented in mathematical English rather than a formal language in the strict sense.
It is possible to formulate axioms of belief change, like the well-known AGM postulates [1], in a fully formalized language. This is the purpose of the socalled Dynamic Doxastic Logic(henceforth DDL, or “full” DDL) developed by Krister Segerberg [10, 12], in which epistemic states are modelled using modal operators of belief in the style of Jaakko Hintikka’s classic [7], and belief changes are modelled using dynamic operators reminiscent of those studied in propositional dynamic logic [6].
Reasoning about belief in a formal language has the advantage of added expressive strength. Rather than just speaking about beliefs about the external world, we can now also reason about introspective beliefs, i.e. beliefs that an agent has about her own state of belief. For instance, I can believe that the world is round, which presumably means that I don’t believe it is flat. Suppose now that someone asks me whether I believe the Earth is round; I answer that I do believe it. In these circumstances I am apparently aware that I believe the Earth is round, that is, I believe that I believe that the world is round. In the same manner, I might be asked whether I believe the Earth is flat, and I answer that I do not believe that. In this case, I have revealed that I believe that I do not believe that the Earth is flat. If \(r\) stands for “the Earth is round” and \(f\) for “the Earth is flat”, we can formalize these beliefs as
and
respectively.
In the case of DDL, where we have the capacity to speak about not only beliefs but also belief change, it turns out that this added expressive power comes with a price: given that we adopt the AGM postulate known as Vacuity, we arrive at some disturbing paradoxes of introspective belief change. These paradoxes are discussed at length by Sten Lindström and Wlodek Rabinowicz in [8], where a modification of the semantics of DDL is presented as a solution to the problem.
In this chapter, we present and criticize Krister Segerberg’s own solution to this problem. We present three alternative ways that the paradoxes of introspective belief change may be avoided: the first is a solution due to Sten Lindström and Wlodek Rabinowicz, using a two-dimensional semantics for DDL. The second solution is found in a logic for belief change suggested by Giacomo Bonanno, in which the operator for belief is replaced by a class of operators for belief, each supplied with a temporal index [3]. The third solution we present is a logic for belief change due to Johan van Benthem [14], founded on the method of Dynamic Epistemic Logic where dynamics is modelled by operations on entire models, rather than some structure within the models. We shall argue that, while there are some differences between these approaches, there is a strong structural similarity between them, and the they avoid the paradoxes of DDL in essentially the same way. Furthermore, the way that these logics avoid the paradoxes is both different from and, we think, more natural than Segerberg’s own solution.
Throughout the discussion we presuppose familiarity with the AGM model of belief revision, as well as the basics of modal logic.
2 DDL and the Paradoxes
We begin by introducing the system DDL and the paradoxes it gives rise to. Throughout the chapter, we work with a fixed, countably infinite supply of propositional variables \(Prop\). The language of DDL is then defined in Backus-Naur form as follows, where \(p\in Prop\):
Classical connectives \(\wedge ,\rightarrow ,\leftrightarrow \) are defined as usual. Informally, \(B\alpha \) means “the agent believes \(\alpha \)”, and \([*\alpha ]\beta \) means “after revision by \(\alpha \), it will be the case that \(\beta \)”.
We now provide semantics for this language. Throughout the chapter, given a binary relation \(R\) over a set \(W\) and given an element \(u\in W\), we use the notation
The logic of revision inherent in the semantics will be rather minimal, since the details of belief revision are irrelevant to the problem we address and its solutions. All we shall require of revision in this semantics, and in the other semantics presented in the chapter, are the following conditions:
-
after revision by \(\alpha \), the agent believes \(\alpha \)
-
revision by any consistent sentence results in a consistent belief state and
-
Some semantic version of the Vacuity postulate holds.
We recall that, in the standard AGM framework for belief revision, the Vacuity postulate is:
where \(Cn\) is the logical closure operator of the propositional logic underlying the epistemic states. This postulate says that if some input proposition is consistent with the agent’s beliefs, then revision by that proposition amounts to simply adding the proposition to the initial stock of beliefs and forming the logical closure of the results; in other words, no information is lost in consistent revision.
Semantics for \({\mathcal {L}}_{DDL}\) is given as follows.
Definition 1
A revision model is a structure
defined as follows: \(B\) is a binary relation over \(W\), and \(R^*: 2^W \rightarrow 2^{W \times W}\) is a function from subsets of \(W\) (sometimes called propositions) to relations over \(W\). Furthermore we require that for each \(X\subseteq W\), if \(v R^*(X) w\) then
-
1.
\(B(w)\subseteq X\)
-
2.
if \(X\ne \emptyset \) then \(B(w) \ne \emptyset \)
-
3.
if \(B(v) \cap X \ne \emptyset \) then \(B(w) = B(v)\cap X\)
Finally, \(V : Prop \rightarrow 2^W\) is an evaluation function in the usual sense. A pointed revision model is a pair \((\mathfrak {A},u)\) where \(\mathfrak {A}\) is a revision model and \(u\in W\).
The reader should note that the last item on the list in this definition is the obvious way to formulate the Vacuity postulate in the present framework. The truth definition for formulas of \({\mathcal {L}}_{DDL}\) in a pointed revision model is given as follows:
-
\((\mathfrak {A},u)\vDash p\) iff \(u\in V(p)\)
-
standard clauses for Boolean connectives
-
\((\mathfrak {A},u) \vDash B\alpha \) iff \((\mathfrak {A},v)\vDash \alpha \) for each \(v\) such that \(u B v\)
-
\((\mathfrak {A},u) \vDash [*\alpha ]\beta \) iff \((\mathfrak {A},v)\vDash \beta \) for each \(v\) such that \(u R^*(\left\| \alpha \right\| ) v\)
Here, \(\left\| \alpha \right\| \) denotes the set
From this semantics we define the consequence relation \(\vDash _{DDL}\) over \({\mathcal {L}}_{DDL}\) by setting, for all sets of formulas \(\Gamma \cup \{\alpha \}\), \(\Gamma \vDash _{DDL}\alpha \) iff
for any pointed revision model \((\mathfrak {A},u)\). Here, \((\mathfrak {A},u)\vDash \Gamma \) means that \((\mathfrak {A},u)\vDash \beta \) for each \(\beta \in \Gamma \).
We will need to be precise about what we mean by a logical system in this chapter. Formally, a logic will here be taken to be a pair \(({\mathcal {L}},\vDash )\) where \({\mathcal {L}}\) is a set containing the set of variables \(Prop\) and \(\vDash \subseteq 2^{\mathcal {L}} \times {\mathcal {L}}\). Thus, \(({\mathcal {L}}_{DDL},\vDash _{DDL})\) is a logical system, which we denote by \(S_{DDL}\).
To see why \(S_{DDL}\) is paradoxical, we ask the reader to verify that the following validity holds:
and, furthermore, that we have the following validity:
The former validity is called Preservation by Lindström and Rabinowicz, and the latter validity is called Success. The validity of Preservation is a direct consequence of the fact that the Vacuity postulate is built into the semantics. From Preservation, in turn, we derive the paradoxes: let \(\alpha ,\beta \) be any formulas. Then as an instance of Preservation we have
On the other hand, from Success follows trivially by classical logic that:
But clearly the operator \([*\alpha ]\) is normal, so that we have
By classical logic we can now derive:
This is the formula deemed paradoxical by Lindström and Rabinowicz, and it would be hard to deny that it is quite bizarre. To see why, toss a coin, without looking at it when it lands. Presumably, given that the coin is fair, you now have no opinion at all on whether the coin landed heads or tails. Let \(\alpha \) stand for the proposition that the coin landed heads. Since you have no opinion on whether the toss came out heads or tails, you do not believe that the coin did not land heads. That is, your current belief state satisfies the condition \(\lnot B\lnot \alpha \). But you do not believe that the coin did land heads, and we think that you have the required powers of introspection to be aware of this fact. Thus, your current belief state also satisfies the condition \(B\lnot B\alpha \). But then, according to DDL, the condition \([*\alpha ](B\alpha \wedge B\lnot B\alpha )\) should also be true. This means that if you were to take a look at the coin and learn that it did in fact land heads, as a result you should believe that the coin landed heads, but at the same time you should believe that you do not believe it. Under perfectly ordinary circumstances, revision of beliefs has lead to a curious, or even incoherent, state of belief.
If we simply dropped the Vacuity postulate, then the problem would disappear. But for those who are strongly convinced of the validity of Vacuity, the more attractive route would be to try and retain some semantic version of the Vacuity postulate, while employing some strategy to avoid the paradoxes. In the following section, we present Segerberg’s own strategy for doing so.
3 Segerberg’s Solution
Segerberg treats the paradoxes of introspective belief change, which he refers to as “Moore problems”, in a paper from 2006 [11]. In this paper, he proposes a solution based on Sorensen’s notion of a blindspot from his 1988 book [13].
In Segerberg’s terminology, an agent has a Moore problem (of rank \(0\)) if \(B(\phi \wedge \lnot B\phi )\) or \(B(\phi \wedge B \lnot \phi )\) is true (in a certain situation and with respect to his beliefs). In the former case, the problem is said to be acute, in the latter grave. More generally, the agent has a Moore problem of rank \(n\), where \(n\) is a nonnegative integer, if, for some formula \(\phi \), either \(B^n(\phi \wedge \lnot B\phi )\) or \(B^n(\phi \wedge B \lnot \phi )\), where \(B^n\) abbreviates
Segerberg is very clear on the desirability of avoiding Moore problems:
It is probably impossible to compile a complete list of all the ways in which a doxastic agent may be incoherent or exhibit some degree of inconsistency, but certainly an agent with a Moore problem of any rank is not perfect. Doxastically ambitious agents will stay clear of Moore problems as far as possible! ([11], p.96)
Segerberg’s solution seems radical on first sight: he proposes to reject the assumption that the star operator correctly formalizes revision. Revision by \(\phi \) is not to be formalized as \(*\phi \) but rather as
As Segerberg points out, the Preservation and Success conditions are not affected by this definition, meaning that the derivations of the Moorean sentences are still valid inferences. Yet, the conclusions are no longer “an embarrassment”:
[...] for the fact that in a certain possible situation, a star change leads to a Moore problem is not embarrassing, however plausible the situation – why would one want to perform a star change anyway? ([11], p. 101).
What would be troublesome is if the corresponding sentences could be derived for revision, i.e. if we could derive
and
But these sentences are not derivable. Hence his new definition of revision avoids the Moore problems of rank \(0\). However, as Segerberg shows, some new problems crop up in their stead. Suppose \(\phi \) is such that before revision by \(\phi \), \(\lnot B\lnot (\phi \wedge B\phi )\) is true, and that, before revision, either \(B\lnot BB\phi \) or \(BB\lnot B\phi \) or \(BBB \lnot \phi \) is true. Then it follows, using Preservation and Success, that after revision by \(\phi \), on the new understanding of revision, at least one of \(B(B\phi \wedge \lnot BB \phi )\) or \(BB(\phi \wedge \lnot B\phi )\) or \(BB(\phi \wedge B \lnot \phi )\) is true. Thus the agent is confronted with a Moore problem of rank \(1\).
How can this situation be avoided? Segerberg’s main idea is that the predicament can be avoided by making the problematic sets of sentences inconsistent, “for inconsistent sets describe (what according to the logic) are impossible situations, and it is of no concern that Moore problems arise in impossible situations” ([11], p. 100). In the present case, this strategy translates into finding a plausible underlying logic that makes each of the following sets inconsistent: \(\{\lnot B \lnot (\phi \wedge B\phi ), B\lnot BB\phi \}\), \(\{\lnot B \lnot (\phi \wedge B\phi ), BB\lnot B \phi \}\) and \(\{\lnot B \lnot (\phi \wedge B\phi ),BBB\lnot \phi \}\). Segerberg notes that the weakest normal logic satisfying this condition is the normal extension of K by the following schemata:
- (1A) :
-
\(B\lnot BB\phi \rightarrow B\lnot (\phi \wedge B\phi )\)
- (1B) :
-
\(BB \lnot B\phi \rightarrow B\lnot (\phi \wedge B\phi )\)
- (1C) :
-
\(BBB\lnot \phi \rightarrow B\lnot (\phi \wedge B\phi )\)
Segerberg shows that all three are derivable, for instance, in KD4 which “is a favorite with many doxastic logicians” ([11], p. 102). He then goes on to generalize this approach to Moore problems at rank \(n\) and of rank \(\omega \), showing that the problematic situations can be excluded by a reasonable choice of underlying doxastic logic. Finally, Segerberg connects his approach to Sorensen’s concept of a blindspot by defining a blindspot as a sentence \(\phi \) such that either \(\phi \) is not entertainable or revision by it leads to an inconsistent state and showing that the following principle comes out as valid on his approach: revision by an entertainable proposition leads to a consistent doxastic state if and only if the sentence in question is not a blindspot. Since
and
are theorems in all logics recommended by Segerberg, in those logics the original Moore sentences \(\phi \wedge \lnot B\phi \) and \(\phi \wedge B\lnot \phi \) are blindspots.
This is certainly an impressive treatment of the Moore problems, especially considering the proposal, which we will grant, that the Moore problems arise in impossible situations where what is impossible or not is defined in a principled manner relative to logical frameworks that have an independent standing in the literature. Segerberg can hardly be accused of adhockery in that respect. However, Segerberg’s strategy may still be ad hoc in another regard. Consider again Segerberg’s new definition of revision by \(\phi \), i.e. \( {\mathbf {R}}\phi =_{df.} *(\phi \wedge B\phi )\). First of all, it surely is less simple and striking than the old one. But second and more important, Segerberg does not give any independent motivation for his new definition of revision. Certainly, defining revision in this way does the job of providing a framework within which Moore problems can be avoided, but apart from this fact little speaks in favor of the new definition. And, one might ask, why should every revision by \(\phi \) be, as it were, accompanied by a revision by \(B\phi \)? Suppose \(\phi \) is an object level sentence such as “it is raining”. Why should updating by “it is raining” involve updating by “I believe that it is raining”? Of course, it may often be the case that these two propositions are accommodated in one swoop, but it is less clear that it has to be that way. For certain kinds of introspective agents the new definition of revision may be fine. But what about agents that adopt beliefs routinely without reflecting on those beliefs at the time of adoption? So there is still a sense in which Segerberg’s approach is, at least to some extent, ad hoc.
Another way of putting it is that Segerberg gives but a partial solution to the Moore problems, a solution that takes care of those problems for reflective agents (by which we mean agents for which an update by \(\phi \) is always accompanied by an update by \(B\phi \)), but that he has little to say about the prospects of dealing with those problems from the perspective of unreflective agents.
In the light of these remarks, it is natural to ask whether there is some other way to treat the paradoxes of full DDL. In the next section, we present three different logics for belief revision that can be found in the literature, each of which can be shown to avoid the paradoxes of introspective belief change. We shall begin by introducing each variant formally, and then discuss what we believe is the common structure behind each approach.
4 Three Alternative Solutions
4.1 First Solution: Two-dimensional DDL
The first solution we consider is due to Sten Lindström and Wlodek Rabinowicz. The approach suggested by Lindström and Rabinowicz is to adopt a modified, two-dimensional semantics for DDL in which formulas are no longer evaluated at single worlds, but rather at pairs of worlds. Here, the idea is that in an evaluation point \((u,v)\), the left component \(u\) serves as a point of reference, while \(v\) functions as a point of evaluation. In addition, rather than an accessibility relation \(B\) over the universe of a model, a class of accessibility relations is used, one relative to each world in the universe. Each accessibility relation \(B_v\), where \(v\in W\), represents the agent’s beliefs about the point of reference \(v\).
Formally, the definition of a model from the system \(S_{DDL}\) is modified as follows:
Definition 2
A two-dimensional revision model is a structure
defined as follows: for each \(u\in W\), \(B_u\) is a binary relation over \(W\). For each \(u\in W\) \(R^*_u : 2^W \rightarrow 2^{W \times W}\) is a function from propositions to relations between possible worlds, such that for each \(X\subseteq W\), if \(v R^*_u(X) w\) then
-
1.
\(B_u(w)\subseteq X\)
-
2.
if \(X\ne \emptyset \) then \(B_u(w) \ne \emptyset \)
-
3.
if \(B_u(v) \cap X \ne \emptyset \) then \(B_u(w) = B_u(v)\cap X\)
A pointed two-dimensional revision model is a triple \((\mathfrak {A},u,v)\) where \(\mathfrak {A}\) is a two-dimensional revision model and \(u,v\in W\).
To speak about these models, we use an extension of the language \({\mathcal {L}}_{DDL}\). Formally, the language \({\mathcal {L}}_{2D}\) is given by the following definition where, again, \(p\in Prop\):
The truth definition for formulas is given as follows:
-
\((\mathfrak {A},u,v)\vDash p\) iff \(v\in V(p)\)
-
standard Boolean clauses
-
\((\mathfrak {A},u,v) \vDash B\alpha \) iff \((\mathfrak {A},u,w)\vDash \alpha \) for each \(w\) such that \(v B_u w\)
-
\((\mathfrak {A},u,v) \vDash [*\alpha ]\beta \) iff \((\mathfrak {A},u,w)\vDash \beta \) for each \(w\) such that \(v R^*_u(\left\| \alpha \right\| _u) w\)
-
\((\mathfrak {A},u,v) \vDash \dagger \;\alpha \) iff \((\mathfrak {A},v,v)\vDash \alpha \)
The consequence relation \(\vDash _{2D}\) is defined from this semantics as before, and we let \(S_{2D}\) denote the logical system \((\mathcal {L}_{2D},\vDash _{2D})\). The new component of the language of this logic is the \(\dagger \)-operator, although the meanings of the modal operators present in \(\mathcal {L}_{DDL}\) have changed. This operator has the effect of making the current point of evaluation the current point of reference as well. The formula \(\dagger \;\alpha \) can informally be interpreted as saying that \(\alpha \) is true about the present point of evaluation.
How does this avoid the paradoxes of DDL? As noted by Lindström and Rabinowicz, for each formula \(\alpha \) the paradoxical formula of \(S_{DDL}\) which we recall was:
is still valid in this semantics. But, as we said, the meaning of the connectives has changed. Consider an evaluation point \((u,u)\) in a model \(\mathfrak {A}\) (here, the point of reference and the point of evaluation is the same). Suppose that
so that the agent does not disbelieve \(\alpha \) at \((u,u)\) and she believes that she does not believe \(\alpha \). According to the validity of the previously deemed paradoxical formula, we must have
This means that
But this is not incoherent, since the righthand conjunct here means that at the present point of evaluation (\(v\)), the agent believes that the condition \(\lnot B \alpha \) holds for the point of evaluation prior to revision by \(\alpha \) (\(u\)). By contrast, the formula
is paradoxical, since the beliefs described as resulting after revision by \(\alpha \) are now beliefs about the point of evaluation after revision, not the one prior to it. But this formula is not valid in \(S_{2D}\). Thus, the system \(S_{2D}\) is free of this paradoxical feature of \(S_{DDL}\).
4.2 Second Solution: Temporally Indexed Beliefs
The second solution, present in Giacomo Bonanno’s “simple” modal logic for belief revision, consists in letting a model represent an \(\omega \)-ordered discrete time-line and use belief operators supplied with indexes representing points in the succession of time. Each move forward in time corresponds to an act of revision by some new piece of information. The expression \(B_n\alpha \), where \(n\in \omega \), says that the agent believes \(\alpha \) at the \(n\)th point in the succession of time.
Formally, the language \({\mathcal {L}}_{Temp}\) for Bonanno’s system is defined as follows, where \(n\) is any natural number:
The new operator \(I_n\) is interpreted so that \(I_n\alpha \) means, informally, that \(\alpha \) is the last piece of information received at the \(n\)th point in time, or that \(\alpha \) it is the input of the revision that results in the belief state at time \(n\).
Semantics for \({\mathcal {L}}_{Temp}\) is given by the following definitions:
Definition 3
A temporal belief model is a structure
such that:
-
1.
\(B_{n}(u) \subseteq I_{n}(u)\)
-
2.
if \(I_n(u)\ne \emptyset \) then \(B_n(u)\ne \emptyset \)
-
3.
if \(B_n(u) \cap I_{n+1}(u)\ne \emptyset \) then \(B_{n+1}(u) = B_n(u)\cap I_{n+1}(u)\)
A pointed temporal belief model is a pair \((\mathfrak {A},u)\) where \(\mathfrak {A}\) is a temporal belief model and \(u\in W\).
Truth definitions of formulas in a pointed temporal belief model are:
-
\((\mathfrak {A},u)\vDash p\) iff \(u\in V(p)\)
-
standard clauses for Boolean connectives
-
\((\mathfrak {A},u) \vDash B_n\alpha \) iff \((\mathfrak {A},v)\vDash \alpha \) for each \(v\) such that \(u B_n v\)
-
\((\mathfrak {A},u) \vDash I_n \alpha \) iff \(I_n(u) = \left\| \alpha \right\| \)
Here, as before, \(\left\| \alpha \right\| = \{v\in W : (\mathfrak {A},v)\vDash \alpha \}\). From this semantics we define the consequence relation \(\vDash _{Temp}\) as before, and we define \(S_{Temp}\) to be the logical system \((\mathcal {L}_{Temp}, \vDash _{Temp})\). To get a grasp of how the language works, consider the syntactic form of the Success postulate in this system; this is captured by the validity
This says that if the belief state at time \(n\) is the result of the revision of a prior belief state by \(\alpha \), then \(\alpha \) is believed at time \(n\).
The way that the paradoxes of DDL are avoided in this system is simple. We do have a form of the Preservation formula valid in \(S_{Temp}\), namely:
That is, if \(\alpha \) is consistent with the agent’s beliefs at time \(n\), and the next revision at time \(n+1\) has the input formula \(\alpha \), then everything the agent believes at time \(n\) she believes at time \(n+1\) also. But we cannot derive any paradoxes from this formula, since belief operators come with temporal indexes. To see this, let’s try to derive a paradox in the same manner as before. Consider any formula \(\alpha \). As an instance of the previous validity we get
From this, using the \(S_{Temp}\)-version of Success above together with classical logic, we can derive
The informal content of this formula looks a lot like that of the paradoxical formula we derived in \(S_{DDL}\). But of course, it is not paradoxical. It says that if \(\alpha \) is consistent with the agents beliefs at time \(n\), and the agent is aware that she does not believe \(\alpha \) at time \(n\), then after revision by \(\alpha \) at time \(n+1\), she believes \(\alpha \) and believes that she did not believe it at time \(n\). By contrast, the formula
is paradoxical but not valid.
4.3 Third Solution: The DEL Method
The third alternative way of getting out of the paradoxes of DDL we consider in this chapter is found in Johan van Benthem’s dynamic logic for belief revision. The system is built on a method used in Dynamic Epistemic Logic (DEL), a framework for studying dynamics of multi-agent epistemic scenarios. The relevant aspect of DEL here is not the multi-agent feature, but rather the way in which dynamics is modelled semantically and reasoned about syntactically.
The method can be described in this way: to model changes of some type of states, one should first develop a static base language for reasoning about the states and provide it with a semantics, i.e. define models for it. Then, changes of states are modelled as operations on models for the static base language, which is extended with dynamic operators to reason about these operations. If the static base logic is rich enough in expressive strength, then it is often possible to translate any dynamic formula into a semantically equivalent formula of the static base logic via socalled reduction axioms.
For brevity we will present the static and the dynamic part of van Benthem’s system all in one swoop. For a gentler presentation of the system we refer to [14]. For an introduction to DEL, see [15].
We begin by defining the models for the static part of the logic:
Definition 4
A conditional belief model is a structure
defined as follows. For each \(u\in W\), \(\sigma _u : 2^W \rightarrow 2^W\) is called a selection function and satisfies the following properties:
-
1.
\(\sigma _u(X) \subseteq X\)
-
2.
\(X\ne \emptyset \) implies \(\sigma _u(X)\ne \emptyset \)
-
3.
if \(\sigma _u(X)\cap Y\ne \emptyset \) then \(\sigma _u(X\cap Y) = \sigma _u(X)\cap Y\)
\(V\) is a valuation function as before, and pointed conditional belief models are defined as before.
The central component of these models is the set of selection functions, which can be thought of as encoding the conditional beliefs of the agent. The intuitive explanation is that, for each proposition \(X\subseteq W\), the set \(\sigma _u(X)\) consists of the “most plausible” worlds from the agent’s point of view at the world \(u\). Actual beliefs of the agent are defined as beliefs conditional on the trivial proposition. That is, the set of possible worlds compatible with the agent’s actual beliefs at the world \(u\) is the set \(\sigma _u(W)\). The semantics presented in [14] is a bit different from the presentation here and uses orders of plausibility rather than selection functions, but this is irrelevant to the current issue.
To model dynamics of the models, we will use an operation that van Benthem calls lexicographic upgrade. Or, rather, we use a version of this operation, adapted to the semantics here based on selection functions which is slightly more general than van Benthem’s semantics. Consider a proposition \(X\subseteq W\) in a model \(\mathfrak {A}\); we want a way to revise the selection function \(u\) at a world \(u\) by \(X\). This is provided by the following definition:
Definition 5
Given a conditional belief model \(\mathfrak {A}= \langle W, \{\sigma _u\}_{u\in W}, V\rangle \) and \(X\subseteq W\), we define the revised model \(\mathfrak {A}\Uparrow X\) by
We leave it to the reader to check that this is always a well defined conditional belief model. Notice that we have
With the definition of actual beliefs as beliefs conditional on the trivial proposition, this property can be seen as a semantic formulation of the Vacuity postulate.
Turning to the syntactic side of the system, we define the language \({\mathcal {L}}_{DEL}\):
Here, \(B(\alpha \mid \beta )\) says that \(\alpha \) is believed conditionally on \(\beta \), and \([\Uparrow \alpha ]\beta \) says that the condition \(\beta \) will hold after revision by \(\alpha \). The operator \(A\) is the global necessity operator (see [2]). \(A\alpha \) means that \(\alpha \) holds in all possible worlds of a model; it can be thought of as expressing logical necessity. A static formula of \({\mathcal {L}}_{DEL}\) is a formula without any occurrences of the dynamic operators. We define an operator for actual beliefs by \(B\alpha =_{df.} B(\alpha \mid p\vee \lnot p)\), where \(p\) is a propositional variable.
Truth definitions for formulas in a pointed model are:
-
\((\mathfrak {A},u)\vDash p\) iff \(u\in V(p)\)
-
standard clauses for Boolean connectives
-
\((\mathfrak {A},u)\vDash B(\alpha \mid \beta )\) iff \(\sigma _u(\left\| \beta \right\| )\subseteq \left\| \alpha \right\| \)
-
\((\mathfrak {A},u)\vDash A\alpha \) iff \((\mathfrak {A},v)\vDash \alpha \) for each \(v\in W\)
-
\((\mathfrak {A},u)\vDash [\Uparrow \alpha ]\beta \) iff \((\mathfrak {A}\Uparrow \left\| \alpha \right\| ,u)\vDash \beta \)
The consequence relation \(\vDash _{DEL}\) and the system \(S_{DEL}\) are now defined as before.
It is instructive to look at the reduction axioms for \(S_{DEL}\). These are as follows (we follow van Benthem’s axiomatization almost without any modification):
- \(\Uparrow \)1::
-
\(\left[ \Uparrow \!\gamma \right] q \leftrightarrow q\), \(q\) a propositional atom
- \(\Uparrow \)2::
-
\([\Uparrow \gamma ]\lnot \alpha \leftrightarrow \lnot [\Uparrow \gamma ]\alpha \)
- \(\Uparrow \)3::
-
\([\Uparrow \gamma ](\alpha \vee \beta ) \leftrightarrow ([\Uparrow \gamma ]\alpha \vee [\Uparrow \gamma ]\beta )\)
- \(\Uparrow \)4::
-
\([\Uparrow \gamma ]A\alpha \leftrightarrow A[\Uparrow \gamma ]\alpha \)
- \(\Uparrow \)5::
-
\([\Uparrow \gamma ]B(\alpha \mid \beta ) \leftrightarrow \)
\(\leftrightarrow (E(\gamma \wedge [\Uparrow \gamma ]\beta )\wedge B([\Uparrow \gamma ]\beta \rightarrow [\Uparrow \gamma ]\alpha \mid \gamma )\vee \)
\(\vee (\lnot E(\gamma \wedge [\Uparrow \gamma ]\beta )\wedge B([\Uparrow \gamma ]\alpha \mid [\Uparrow \gamma ]\beta ))\)
The reader can check that these axioms are sound in the semantics for \(S_{DEL}\). These axioms can be thought of as providing recursive definitions of the truth conditions of dynamic formulas in terms of static formulas. Together with a suitable set of complete axioms for the static fragment of \(S_{DEL}\) and a rule for substitution of equivalents, they provide a complete axiomatization of \(S_{DEL}\). To prove this result, one exploits the soundness of the reduction axioms to prove the following proposition as a lemma. The proof is excluded here.
Proposition 1
There exists a function \(\rho : {\mathcal {L}}_{DEL} \rightarrow {\mathcal {L}}_{DEL}\) such that for each formula \(\alpha \in {\mathcal {L}}_{DEL}\), the formula \(\rho (\alpha )\) is a static formula and, furthermore, for each pointed conditional belief model \((\mathfrak {A},u)\),
To get a feel for the system, let us look at some validities. Here, \(p,q\) are two propositional variables and \(\perp \) is any tautological contradiction. First, revision by \(p\) leads the agent to believe \(p\):
Second, revision by a consistent sentence results in a consistent belief state:
What about Preservation? We do indeed have a form of the Preservation principle valid in this system:
Now, if validity in \(S_{DEL}\) were closed under substitutions for propositional variables (as is the case in most logics), then obviously we could derive a paradox in the same manner as in \(S_{DDL}\). However, this is not the case, and it is in fact this feature of \(S_{DEL}\) that makes it non-paradoxical. In particular, the following substition instance of \((i)\):
is invalid. This is exactly the formula that would be required to derive a paradox in \(S_{DEL}\). By contrast, the following formula is valid:
Now, what does this formula say? it says that, if \(\lnot B\lnot \alpha \) and \(B\lnot B\alpha \) are true at some world in a model, than from the point of view of the agent in that world, \(\lnot B\alpha \) will be true in the most plausible worlds where \(\alpha \) is true. Now, the most plausible worlds where \(\alpha \) is true, prior to revision by \(\alpha \), are exactly those worlds that are compatible with the agent’s actual beliefs after revision. But since the truth values of formulas involving beliefs will change at every world in a model through the act of revision by \(\alpha \), it does not follow from this that \(\lnot B \alpha \) will be true at all worlds that are compatible with the agent’s beliefs after the revision. This is why \((ii)\) fails to be valid.
5 Comparison of the Solutions
5.1 What the Three Approaches Have in Common
The three solutions we have just presented are, we think, essentially one and the same. All three of them are based on making a distinction between two different perspectives, the state of affairs prior to revision and the one after revision. This is perhaps clearest in Lindström and Rabinowicz’s system; it is embodied quite explicitly in the distinction between the point of reference (typically being the state prior to revision) and the point of evaluation (typically the state after revision).
But we see the same distinction very clearly in Bonanno’s temporal system of belief revision, although in a different form. Here, it turns up through the temporally indexed belief operators. In particular, in the formula
which is provable in \(S_{Temp}\), the state prior to revision corresponds to the time-point represented by the number \(n\), and the state after revision corresponds to \(n+1\).
It is perhaps a bit less obvious how van Benthem’s system \(S_{DEL}\) fits into this picture, but we think it does. We postpone the task of explaining this to Sect. 5.3, where we will be better prepared to do so. The fact that the same solution to the paradoxes can be found in three seemingly rather different frameworks for belief revision counts, we think, as evidence in favor of this approach as a particularly natural way to resolve the paradoxes. Think of it in analogy with the case of various definitions of computable functions, for example in terms of recursive functions or in terms of Turing machines. The wellknown fact that these definitions turn out to be equivalent speaks strongly in favor of the idea that they all capture the pre-formal notion of computability in a natural way. The present situation, where three different formalisms can be seen to resolve the paradoxes of DDL in the same way, is similar.
To strengthen these claims, we shall establish a formal correspondence between the three logical systems \(S_{2D}, S_{Temp}\) and \(S_{DEL}\). More specifically, we shall show that the system \(S_{2D}\) can in a precise sense be interpreted in \(S_{Temp}\), and in turn, \(S_{DEL}\) can be interpreted in \(S_{2D}\). From this will follow that \(S_{DEL}\) can be interpreted in \(S_{Temp}\) also. These interpretation results will help to clarify the deeper connection that we think exists between the different systems, particularly with respect to how they handle the paradoxes of DDL. In order to formally prove these claims, we need to make precise what it means that a logical system can be interpreted in another. This is captured by the following definition.
Definition 6
Given logical systems \(S_1 = ({\mathcal {L}}_1,\vDash _1)\) and \(S_2 = ({\mathcal {L}}_2,\vDash _2)\), an interpretation of \(S_1\) in \(S_2\) is any function \(F : \mathcal {L}_1 \rightarrow \mathcal {L}_2\) such that \(F(p) = p\) for any \(p\in Prop\). The interpretation \(F\) is said to be a sound interpretation of \(S_1\) if, for all sets of formulas \(\Gamma \cup \{\alpha \}\subseteq \mathcal {L}_1\), we have
So a sound interpretation of a logical system \(S_1\) in \(S_2\) is a translation that maps sentences of \(S_1\) to sentences of \(S_2\) in a way that preserves logically valid consequences. Just like when we interpret a logical system in a semantics, we might consider the question of whether an interpretation is complete in addition to being sound. We could say that an interpretation \(F\) of \(S_1\) in \(S_2\) is sound and complete if, for all \(\Gamma \cup \{\alpha \}\subseteq \mathcal {L}_1\), we have
The issue of completeness will not concern us in this chapter. Rather, we will focus on soundness. Completeness is a welcome property of any interpretation of a logical system, but soundness is absolutely crucial. If an interpretation is not sound, it is doubtful whether it can be called a proper interpretation at all. Also, as we shall see in the next section, the soundness property of the interpretations we provide is enough to make the correspondence quite enlightening.
5.2 Interpreting \(S_{2D}\) in \(S_{Temp}\)
Our first result is that, in the sense of Definition 6, there exists a sound interpretation \(F\) of \(S_{2D}\) in \(S_{Temp}\). First, by induction over the complexity of formulas, we define the class of functions
where \(n,m\in \omega \) as follows:
-
1.
\(\tau _{n,m}(p) = p\)
-
2.
\(\tau _{n,m} (\lnot \alpha ) = \lnot \tau _{n,m}(\alpha )\)
-
3.
\(\tau _{n,m}(\alpha \vee \beta ) = \tau _{n,m}(\alpha )\vee \tau _{n,m}(\beta )\)
-
4.
\(\tau _{n,m} (B\alpha ) = B_m \tau _{n,n}(\alpha )\)
-
5.
\(\tau _{n,m} ([*\alpha ]\beta ) = I_{m+1}\tau _{n,n}(\alpha )\rightarrow \tau _{n,m+1}(\beta )\)
-
6.
\(\tau _{n,m} (\dagger \;\alpha ) = \tau _{m,m}(\alpha )\)
We then set \(F =_{df.} \tau _{0,0}\). For this mapping \(F\) we have the following result, proved in Appendix A.1:
Theorem 1
The translation \(F\) constitutes a sound interpretation of the system \(S_{2D}\) in the system \(S_{Temp}\).
To see how this interpretation relates the two systems to each other, let us consider the interpretation of the formula
given by \(F\). This formula is an instance of the paradoxical schema we derived in \(S_{DDL}\). As we mentioned earlier, the formula is valid in \(S_{2D}\) also, and therefore by soundness its interpretation under \(F\) is valid in \(S_{Temp}\). Now, Lindström and Rabinowicz claim that this formula is not paradoxical under the interpretation given to it in two-dimensional semantics. Then, its interpretation under \(F\) had better not be paradoxical either!
Indeed it is not. For the formula
is identical to
which is perfectly fine. We can derive this as follows: first, recalling that \(F = \tau _{0,0}\) and using translation clauses for Boolean connectives, atomic formulas and \(B\), the formula \((1)\) becomes
Carrying out the translation further, we get
Applying the function \(\tau _{0,1}\) to its arguments here, we get
But \(\tau _{0,0}(\lnot B p) = \lnot B_0 p\), so now we arrive at \((2)\) as desired.
By contrast, let’s look at the translation of the formula
which is paradoxical. Applying the translation \(F\) to this formula, instead of \((2)\) we will get the formula
which is indeed paradoxical, and not valid in \(S_{Temp}\). To see what happens here, we can carry out the translation step by step and check that we eventually arrive at the formula
Applying the translation clause for \(\dagger \), this becomes
But
and so we arrive at \((3)\).
5.3 Interpreting \(S_{DEL}\) in \(S_{2D}\)
We now show how to interpret \(S_{DEL}\) in \(S_{2D}\). The central observation here is that, since we know that there is a translation \(\rho \) that sends every formula \(\alpha \) to an equivalent static formula \(\rho (\alpha )\), it suffices to interpret the static formulas of \(S_{DEL}\) in order to get a full interpretation of \(S_{DEL}\) in \(S_{2D}\).
Formally, we define the mapping \(\tau \) as follows:
-
1.
\(\tau (p) = p\)
-
2.
\(\tau (\lnot \alpha ) = \lnot \tau (\alpha )\)
-
3.
\(\tau (\alpha \vee \beta ) = \tau (\alpha )\vee \tau (\beta )\)
-
4.
\(\tau (A\alpha ) = [*\lnot \tau (\alpha )]B \perp \)
-
5.
\(\tau (B(\alpha \mid \beta )) = [*\tau (\beta )] B \tau (\alpha )\)
Clearly, every static formula of \(\mathcal {L}_{DEL}\) receives an interpretation by this mapping. Letting \(\rho \) be any translation function as specified in Proposition 1, we define an interpretation \(F : {\mathcal {L}}_{DEL} \rightarrow \mathcal {L}_{2D}\) by setting
for each \(\alpha \in {\mathcal {L}}_{DEL}\). As before, we have the following soundness result for this interpretation:
Theorem 2
The translation \(F\) constitutes a sound interpretation of the system \(S_{DEL}\) in the system \(S_{2D}\).
The proof of this result is in Appendix A.2 Furthermore, the composition of two sound interpretations (whenever it is well defined) is obviously a sound interpretation. So by the existence of a sound interpretation of \(S_{DEL}\) in \(S_{2D}\) and a sound interpretation of \(S_{2D}\) in \(S_{Temp}\), we get:
Corollary 1
There exists a sound interpretation of \(S_{DEL}\) in \(S_{Temp}\).
An interesting aspect of the translation \(F\) presented in this section is that, clearly, for any \(\mathcal {L}_{DEL}\)-formula \(\alpha \), the corresponding \(\mathcal {L}_{2D}\)-formula \(F(\alpha )\) will never contain any occurrence of the operator \(\dagger \). Our analysis of this state of affairs is this: consider a formula
contrasted with
What is the difference in meaning between these two formulas? We think it can be understood in terms of Lindström and Rabinowicz’s distinction between point of evaluation and point of reference. Both \((A)\) and \((B)\) can be thought of as expressing that the formula \(\alpha \) is believed after revision by \(\beta \), but the formula \(\alpha \) has different meaning in the two cases. In \((A)\), the point of reference is held fixed, while in \((B)\), the formula \(\alpha \) is evaluated against a different point of reference than \(\beta \). However, since the interpretation \(F\) takes a detour through the static fragment of the system \(S_{DEL}\), in which no formulas of the form \((B)\) occur, it makes sense that the operator \(\dagger \) does not occur in the interpretation of any formulas: it has exactly the effect of changing the point of reference.
Thus, by extracting this insight from our interpretation of \(S_{DEL}\) in \(S_{2D}\), we have managed to show how \(S_{DEL}\) also fits into the picture we described earlier. The distinction between a perspective corresponding to the states of affairs before and after revision, respectively, is mirrored in \(S_{DEL}\) by the distinction between expressions of the form \((A)\) and \((B)\). Expressions of the first kind describe our revised beliefs about the state prior to revision, and expressions of the second kind describe our revised beliefs about the state of affairs after revision. Really, we do not have three different solutions; we have three different logical systems, each of which solves the problem with full DDL in one and the same way.
6 Discussion
We have argued that the systems \(S_{2D}\), \(S_{Temp}\) and \(S_{DEL}\) all solve the problems of full DDL by distinguishing between two perspectives, expressed most explicitly in Lindström and Rabinowicz’s two-dimensional approach. Given this, it is striking to find that Segerberg himself has suggested a two-dimensional approach to resolve another well-known paradox, namely Fitch’s paradox (in a paper from 1994 with Rabinowicz [9]). Given the obvious similarities between Fitch’s paradox and the paradoxes of full DDL, and given that Segerberg argued for a two-dimensional approach to the former, one would have expected him to embrace a two-dimensional approach to the latter as well. Thus it is surprising that Segerberg instead bases his solution on Sorensen’s notion of a blindspot, which is essentially unrelated to the two-dimensional approach.
In fact it is not only surprising but, we think, it is questionable from a methodological point of view. Given the affinities between these paradoxes it would be desirable to treat them in a uniform fashion. Thus, for Segerberg, who is associated with two-dimensional semantics and the blindspot approach, the following uniform approaches suggests themselves:
-
(1)
Treating both paradoxes as involving blindspots
-
(2)
Treating both paradoxes in a two-dimensional semantics
By contrast, the following would seem less attractive from a systematic perspective:
-
(3)
Treating Fitch’s paradox in a two-dimensional semantics and Moore’s paradox as involving blindspots
-
(4)
Treating Fitch’s paradox as involving blindspots and Moore’s paradox in a two-dimensional semantics
And yet, as we saw, Segerberg’s published responses to the paradoxes correspond to option (3), a suboptimal strategy from a systematic perspective. Finally, the result of the present article suggests that option (2) is, in a sense, considerably more plausible than meets the eye. More precisely, (2) is but a specific variant of a more general approach:
- (2\(^\prime \)):
-
Treating both paradoxes as arising from failure to distinguish between different perspectives
As we have argued, two-dimensional DDL, Bonanno’s temporal system and van Benthem’s DEL-style system are all instances of (2\(^\prime \)). They all resolve the paradoxes by distinguishing between two different perspectives, in the two-dimensional case the point of reference and the point of evaluation, in Bonanno’s case the time before and after revision, and in van Benthem’s logic between conditional beliefs and beliefs after revision. Thus, the main competitor to the blindspot approach, as things must look from Segerberg’s point of view, is more widely adopted, and thus has a stronger standing in the research community, than the apparent diversity could lead one to believe. Perhaps even more important is the fact that the main competitor—the perspectival strategy—is a very natural way of dealing with the problems, or else researchers with widely different starting points would not have converged on it. Furthermore, to reconnect with our discussion of Segerberg’s own solution, the perspectival strategy is perfectly compatible with the traditional view that we often revise simply by \(\alpha \) rather than by \(\alpha \wedge B\alpha \), and are quite rational in doing so. Not only does this accord better with our pre-theoretical conceptions of things (at least those of the present authors), but it means that this strategy works for reflective agents and unreflective agents alike. Unlike the perspectival strategy, Segerberg’s solution is dependent on the assumption that the agent in question is reflective. Thus, unless an independent motivation is provided for not taking unreflective agents into consideration, the perspectival strategy stands out as the more general solution.
References
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Bonanno, G. (2005). A simple modal logic for belief revision. Synthese, 147, 193–228.
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Hansson, S.-O. (1999). A textbook of belief dynamics: Theory change and database updating. Dordrecht: Kluwer Academic.
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Appendix: Proofs of Main Results
Appendix: Proofs of Main Results
1.1 Proof of Theorem 1
The proof is based on constructing models for \(S_{2D}\) out of models for \(S_{Temp}\), in the following manner:
Definition 7
Given a temporal belief model \(\mathfrak {A}= \langle W,\{B_n\}_{n\in \omega }, \{I_n\}_{n\in \omega }, V \rangle \), we define the two-dimensional revision model
as follows. We set
For all \((u,n),(v,m),(w,k) \in W^*\), we set \((u,n) B_{(v,m)} (w,k)\) iff \(u B_n w\) and \(k = m\). We set \((u,n) R^*_{(v,m)}(X) (w,k)\) iff
-
\(u = w\),
-
\(k = n+1\) and
-
\(Z = I_k(u)\), where \(Z = \{t\in W : (t,m)\in X\}\)
Finally, we set \((u,n)\in V^*(p)\) iff \(u \in V(p)\).
The construction is sound by the following proposition:
Proposition 2
\(\mathfrak {A}_{2D}\) is a two-dimensional revision model, for any temporal belief model \(\mathfrak {A}\).
Proof
We need to check that, for each \(X\subseteq W^*\), if \((u,m) R^*_{(v,n)}(X) (w,k)\) then
-
1.
\(B_{(v,n)}(w,k)\subseteq X\)
-
2.
if \(X\ne \emptyset \) then \(B_{(v,n)}(w,k) \ne \emptyset \)
-
3.
if \(B_{(v,n)}(u,m) \cap X \ne \emptyset \) then \(B_{(v,n)}(w,k) = B_{(v,n)}(u,m)\cap X\)
So suppose \((u,m) R^*_{(v,n)}(X) (w,k)\). Then \(u = w\), \(k = m+1\) and
Now, since
item (1) follows easily by definition of the relation \(B_{(v,n)}\): for if
\((u,m+1) B_{v,n} (w^\prime ,k^\prime )\), then \(k^\prime = n\) and \(u B_{m+1} w^\prime \), so \(w^\prime \in I_{m+1}(u)\), so \((w^\prime ,n) = (w^\prime , k^\prime )\in X\).
For (2), note that \(X \ne \emptyset \) implies \(I_{m+1}(u)\ne \emptyset \), so \(B_{m+1}(u) \ne \emptyset \). Pick \(w^\prime \) such that \(u B_{m+1} w^\prime \). Then \((u,m+1) B_{v,n} (w^\prime ,n)\) so \(B_{(v,n)}(u,m+1)\ne \emptyset \).
Lastly, for (3), suppose \(B_{(v,n)}(u,m) \cap X \ne \emptyset \). Let \((w^\prime ,k^\prime ) \in B_{(v,n)}(u,m) \cap X \ne \emptyset \); then \(k^\prime = n\) and \(u B_{m} w^\prime \). Since \((w^\prime ,n)\in X\), \(w^\prime \in I_{m+1}(u)\). So
and hence
This means that
To see this, suppose \((u,m+1) B_{(v,n)} (s,i)\). Then \(i = n\), and \(u B_{m+1} s\). But then \(u B_m s\) and \(s \in I_{m+1}(u)\). So \((u,m) B_{(v,n)} (s,n)\) and \((s,n) \in X\).
Conversely, suppose \((u,m+1) B_{(v,n)} (s,i)\) and \((s,i) \in X\). By definition of \(B_{v,n}\), \(i = n\). So \((s,n) \in X\) and therefore \( s \in I_{m+1}(u)\). Furthermore, \(u B_{m+1} s\). So \(s \in B_m (u) \cap I_{m+1} (u)\), hence \(s \in B_{m+1}(u)\). By definition this means that \((u,m+1) B_{(v,n)} (s,n)\), i.e. \((u,m+1)B_{(v,n)}(s,i)\) as required.
We now define a mapping \(G\) from pointed temporal belief models to pointed two-dimensional revision models by setting
for each pointed temporal revision model \((\mathfrak {A},u)\). We then have the following result, which gives the key to the soundness result for \(F\):
Lemma 1
For any pointed temporal model \((\mathfrak {A},u)\) and any \(\mathcal {L}_{2D}\)-formula \(\alpha \), we have
Proof
We show, for any formula \(\alpha \), that for each world \(u\) in the universe of \(\mathfrak {A}\), we have both
and
for all \(v\in W\). From \((1)\) and \((2)\) together it follows that
i.e.
as desired.
The proof goes by induction on the length of \(\alpha \). For propositional variables, both clauses are immediate, and the steps for Boolean connectives are easy.
Step for \(B\): Suppose \((\mathfrak {A},u)\vDash \tau _{n,m}(B\alpha )\), i.e. \((\mathfrak {A},u)\vDash B_m \tau _{n,n}(\alpha )\). Let \(v\in W\) and let \((w,k)\) be such that \((u,m) B_{v,n} (w,k)\). Then by definition \(u B_m w\) and \(k = n\), so we must have \((\mathfrak {A},w)\vDash \tau _{n,n} (\alpha )\) and by clause \((1)\) of the IH we get \((\mathfrak {A}_{2D},(v,n),(w,n))\vDash \alpha \). So we must have \((\mathfrak {A}_{2D},(v,n),(u,m))\vDash B\alpha \). This shows that
Suppose that \((\mathfrak {A},u)\nvDash \tau _{n,m}(B\alpha )\), i.e. \((\mathfrak {A},u)\nvDash B_m \tau _{n,n}(\alpha )\). Then there exists \(w\in W\) such that \(u B_m w\) and \((\mathfrak {A},v)\nvDash \tau _{n,n}\alpha \). Let \(v\in W\); then we have \((u,m) B_{(v,n)} (w,n)\) and by clause \((2)\) of IH we have \((\mathfrak {A}_{2D},(v,n),(w,n))\nvDash \alpha \), hence \((\mathfrak {A}_{2D},(v,n),(u,m))\nvDash B \alpha \). We have shown that
as required.
Step for \(*\): Suppose \((\mathfrak {A},u)\vDash \tau _{n,m}([*\alpha ]\beta )\), i.e.
We note that by the IH we have, for each \(v\in W\),
Suppose for \(v\in W\) that \((u,m) R^*_{(v,n)}(\left\| \alpha \right\| _{(v,n)})(w,k)\). Then \(k = m+1\). Furthermore, by definition and by \((\ddagger )\) we get
So \((\mathfrak {A},u)\vDash I_{m+1}(\tau _{n,n}(\alpha ))\). Thus, we get \((\mathfrak {A},u)\vDash \tau _{n,m+1}(\beta )\). By clause \((1)\) of the IH, this gives \((\mathfrak {A}_{2D},(v,n),(w,m+1))\vDash \beta \), i.e. \((\mathfrak {A}_{2D},(v,n),(w,k))\vDash \beta \). So \((\mathfrak {A}_{2D},(v,n),(u,m))\vDash [*\alpha ]\beta \). We have thus shown
Suppose \((\mathfrak {A},u)\nvDash \tau _{n,m}([*\alpha ]\beta )\), i.e. \((\mathfrak {A},u)\vDash I_{m+1}\tau _{n,n}(\alpha )\) but \((\mathfrak {A},u)\nvDash \tau _{n,m+1}(\beta )\). Pick \(v\in W\). Using \((\ddagger )\) we obtain
From this we can conclude that \((u,m) R^*_{(v,n)}(\left\| \alpha \right\| _{(v,n)})(u,m+1)\). Furthermore, by clause \((2)\) of the IH we have \((\mathfrak {A}_{2D},(v,n),(u,m+1)\nvDash \beta )\), so \((\mathfrak {A}_{2D},(v,n),(u,m) \nvDash [*\alpha ]\beta )\). We have thus shown
as required.
Step for \(\dagger \): Given that the IH holds for \(\alpha \), suppose first that we have \((\mathfrak {A},u)\vDash \tau _{n,m}(\dagger \;\alpha )\), i.e. \((\mathfrak {A},u)\vDash \tau _{m,m}(\alpha )\). Then we have by clause \((1)\) of IH: for all \(v\in W\), \((\mathfrak {A}_{2D},(v,m),(u,m))\vDash \alpha \). In particular, \((\mathfrak {A}_{2D},(u,m),(u,m))\vDash \alpha \). This means that, for all \(v\in W\), \((\mathfrak {A}_{2D},(v,n),(u,m))\vDash \dagger \;\alpha \). We have established:
On the other hand, suppose \((\mathfrak {A},u)\nvDash \tau _{n,m}\dagger \,\alpha \), i.e. \((\mathfrak {A},u)\nvDash \tau _{m,m}(\alpha )\). Then we have by clause \((2)\) of IH: for all \(v\in W\), \((\mathfrak {A}_{2D},(v,m),(u,m))\nvDash \alpha \). In particular, \((\mathfrak {A}_{2D},(u,m),(u,m))\nvDash \alpha \). This means that, for all \(v\in W\), \((\mathfrak {A}_{2D},(v,n),(u,m))\nvDash \dagger \;\alpha \). We have established:
This ends the proof.
We now prove Theorem 1 as follows: suppose \(F(\Gamma )\nvDash _{Temp} F(\alpha )\). Then there is a pointed temporal belief model \((\mathfrak {A},u)\) such that \((\mathfrak {A},u)\vDash F(\Gamma )\) but \((\mathfrak {A},u)\nvDash F(\alpha )\). By the previous theorem, \(G(\mathfrak {A},u)\vDash \Gamma \) but \(G(\mathfrak {A},u)\nvDash \alpha \). Hence \(\Gamma \nvDash _{2D} \alpha \). This ends the proof of the theorem.
1.2 Proof of Theorem 2
We use the same strategy as in the previous section:
Definition 8
Given a two-dimensional model \(\mathfrak {A}\) and a world \(v\) in the universe of \(\mathfrak {A}\), we define the conditional belief model
as follows: we set \(W^*= W\) and \(V^*= V\). For each \(u\in W\) and \(X\subseteq W\), we set
It is easily checked that \(\mathfrak {A}_{DEL}[v]\) is a conditional belief model. We define the mapping \(G\) from pointed two-dimensional revision models to pointed conditional belief models by setting \(G(\mathfrak {A},v,u) = (\mathfrak {A}_{DEL}[v],u)\) for a pointed two-dimensional revision model \((\mathfrak {A},v,u)\). We have the following result:
Lemma 2
For any pointed two-dimensional model \((\mathfrak {A},u,v)\) and any static \({\mathcal {L}}_{DEL}\)-formula \(\alpha \) we have
Proof
By induction over the length of static formulas we show that, for all \(v\in W\) we have
The steps for atomic formulas and Boolean connectives are trivial.
Step for \(A\): suppose \((\mathfrak {A},u,v)\vDash \tau (A\alpha )\), i.e. \((\mathfrak {A},u,v)\vDash [*\lnot \tau (\alpha )]B\!\perp \). By seriality of \(R^*_u(\left\| \lnot \tau (\alpha )\right\| ^{\mathfrak {A}}_u)\) there must be some \(w\) such that \( v R^*_u(\left\| \lnot \tau (\alpha )\right\| ^{\mathfrak {A}}_u) w\). Furthermore, clearly we have \(B_u(w) = \emptyset \), and this means that \({\left\| \lnot \tau (\alpha )\right\| ^{\mathfrak {A}}_u} = \emptyset \). Hence \(\left\| \tau (\alpha )\right\| ^\mathfrak {A}_u = W = W^*\). By the IH, \(\left\| \alpha \right\| _{\mathfrak {A}_{DEL}[u]} = W^*\), and so we have \((\mathfrak {A}_{DEL}[u],v)\vDash A\alpha \) as required.
Conversely, suppose \((\mathfrak {A},u,v)\nvDash \tau (A\alpha )\), i.e. \((\mathfrak {A},u,v)\nvDash [*\lnot \tau (\alpha )]B\!\!\perp \). Then there is some \(w\) such that \(v R^*_u (\left\| \lnot \tau (\alpha )\right\| ^\mathfrak {A}_u) w\) and \(B_u(w)\ne \emptyset \). Hence there is some \(s\) such that \(w B_u s\). By the definition of a two-dimensional model, \(s\in \left\| \lnot \tau (\alpha )\right\| \) i.e. \((\mathfrak {A},u,s)\vDash \lnot \tau (\alpha )\). Hence \((\mathfrak {A},u,s)\nvDash \tau (\alpha )\), and by the IH \((\mathfrak {A}_{DEL}[u],s)\nvDash \alpha \). Hence \((\mathfrak {A}_{DEL}[u],v)\nvDash A \alpha \) as required.
Step for \(B\): suppose \((\mathfrak {A},u,v) \vDash \tau (B(\alpha \mid \beta ))\), i.e.
Suppose \(w \in \sigma _v(\left\| \beta \right\| _{\mathfrak {A}_{DEL}[u]})\). By the IH this means that \(w \in \sigma _v (\left\| \tau (\beta )\right\| ^{\mathfrak {A}}_u)\), so there is some \(s\) such that \(v R^*_u(\left\| \alpha \right\| ^\mathfrak {A}_u) s\) and \(s B_u w\). Since \((\mathfrak {A},u,v)\vDash [*\tau (\beta )] B \tau (\alpha )\) we have \((\mathfrak {A},u,s)\vDash B\tau (\alpha )\) so \((\mathfrak {A},u,w)\vDash \tau (\alpha )\). By IH we get \((\mathfrak {A}_{DEL}[u],w)\vDash \alpha \). We have thus shown that \((\mathfrak {A},u,v)\vDash B(\alpha \mid \beta )\) as required.
Conversely, suppose that \((\mathfrak {A},u,v)\nvDash \tau (B(\alpha \mid \beta ))\), i.e.
Then there is some \(s\) such that \(v R^*_u (\left\| \tau (\beta )\right\| ^\mathfrak {A}_u) s\) and \((\mathfrak {A},u,s)\nvDash B\tau (\alpha )\). This means that for some \(w\) we have \(s B_u w\) and \((\mathfrak {A},u,w)\nvDash \tau (\alpha )\). By the IH we have \(v R^*_u (\left\| \beta \right\| _{\mathfrak {A}_{DEL}[u]}) s\), and thus we have \(w\in \sigma _v(\left\| \beta \right\| _{\mathfrak {A}_{DEL}[u]})\). Furthermore, by the IH again, we have \((\mathfrak {A}_{DEL}[u],w)\nvDash \alpha \). Thus \((\mathfrak {A}_{DEL}[u],v)\nvDash B(\alpha \mid \beta )\) as required.
Using the fundamental property of the translation \(\rho \) used in the construction of \(F\), this lemma immediately entails:
Corollary 2
For any pointed two-dimensional model \((\mathfrak {A},u,v)\) and any \({\mathcal {L}}_{DEL}\)-formula \(\alpha \) we have
From this result, we can prove Theorem 2 just like we proved Theorem 1.
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Enqvist, S., Olsson, E. (2014). Segerberg on the Paradoxes of Introspective Belief Change. In: Trypuz, R. (eds) Krister Segerberg on Logic of Actions. Outstanding Contributions to Logic, vol 1. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7046-1_8
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