It has become standard practice in Artificial Intelligence and Game Theory to use possible-worlds models to describe the knowledge and beliefs of a group of agents. In such models, the agents’ knowledge is based on what is true throughout the set of epistemically accessible worlds, the current information range. Following a similar pattern, the agents’ beliefs are based on what is true in the set of most “plausible” worlds. Now, it is often implicitly assumed that the agents arrived at these structures through some process of investigation, but these details are no longer present in the models.
In a number of areas, ranging from epistemology to computer science and decision theory, the need has been recognized for models that keep track of the “reasons”, or the evidence for beliefs and other informational attitudes (cf. [20, 18, 4, 31]). Encoding evidence as the current range of worlds the agent considers possible ignores how the agent arrived at this epistemic state. This also ignores the fine-structure of evidence that allows us to consider or modify just parts of it. One extreme for recording this additional structure are models with complete syntactic details of what the agent has learned so far (including the precise formulation and sources for each piece of evidence) (cf. ). In this paper, we will explore a middle ground in between ranges of possible worlds and syntactic fine-structure, viz. neighborhood models, where the available evidence is recorded as a family of sets of worlds. Our models are not unlike some earlier proposals in the study of conditionals and belief revision (cf. [19, 39, 11, 25]), but we quickly strike out in other directions, and provide a more in-depth logical treatment.
This paper is a continuation of our earlier work in [36, 35] on a new evidence interpretation of neighborhood models.111Neighborhood models have been used to provide a semantics for both normal and non-normal modal logics. See  for an early discussion of neighborhood models and their logics, and [16, 22, 17] for modern motivations and mathematical details. In a neighborhood model, each state is assigned a collection of subsets of the set of states. We view these different collections of subsets as the evidence that the agent has acquired – allowing the agent to have different evidence at different states. Given such an explicit description of the evidence that the agent has acquired, one can explore different notions of beliefs and related cognitive attitudes over neighborhood models. The logical systems that arise on natural model classes of this sort with modalities for evidence and belief have been axiomatized completely in . In this paper, we go one step further, and add some further crucial structure to the neighborhood structures.
In general, there are two ways we can enrich neighborhood structures with descriptions of the agent’s beliefs. The first approach is to add new accessibility relations corresponding to each epistemic or doxastic attitude in the neighborhood structure. We then impose constraints on these new relations to ensure that they are “appropriately grounded” on the available evidence. The second approach is to define the agent’s beliefs and related cognitive attitudes directly using no more than the given evidence structure in the neighborhood structures. The latter “intended models” may be considered as a special case of the former “general models”. This paper offers a careful study of these two approaches.
Our second new contribution is to elaborate on the relationship between our neighborhood models and another general framework for belief change in the modal tradition. Originally used as a semantics for conditionals [[, cf.]]lewis-conditionals, plausibility models are wide-spread in modal logics of belief [32, 33, 5, 13]. The main idea is to endow epistemic ranges with an ordering of relative plausibility on worlds (usually uniform across epistemic equivalence classes): “(according to the agent) world is at least as plausible as ”.222In conditional semantics, such plausibility or ‘similarity’ orders are typically world-dependent. Plausibility orders are typically assumed to be reflexive and transitive, and often even connected, making every two worlds comparable. Connections between evidence structure and plausibility order will occur throughout this paper, and their reflection in logical axioms will be determined.
In all, we shall consider four variants of the logic of evidence-based belief, which depend on some fundamental assumptions one may make about evidence models, in terms of uniformity of evidence across worlds, and total coherence of maximally consistent sets of evidence. For each of the resulting logics, we prove two main results. The first is a characterization theorem for general evidence models as being p-morphic images of intended models. The second result determines a complete deductive calculus for each logic. Here our representation using extended evidence models is crucial, since it permits us to employ familiar techniques from modal logic.
Anyone familiar with Sergei Artemov’s work will have seen a similarity by now. It is very natural to attach to every believed proposition a “justification” for that proposition. This idea was first studied in Artemov’s seminal paper  and applied to epistemic logic in  (see  for an overview and pointers to the relevant literature). In particular, is intended to mean that the agent believes and that is the justification for this belief. Here is a proof term, and sophisticated logical systems have been developed that extend traditional “provability logic” to reason explicitly about the logical structure of these justifications.333By now, these systems have been applied to a variety of issues in epistemology and game theory, and other fields: see  for the relevant references. Moreover, an appealing semantics exists for these systems (cf. ) that combines ideas from possible worlds semantics and the syntax of proof terms. The approach in the current paper is more coarse-grained, since we do not give ourselves proof or evidence terms that can be manipulated in our deductive systems. Nevertheless, we hope to show that even at our chosen level of modeling evidence, an amazing amount of fine-structure exists. A deeper comparison between our approach and justification logic seems a well-worth effort, but it is one step beyond the horizon of this paper.444 For the moment, we can only refer to
For the moment, we can only refer to for a system that merges features of justification logic with plausibility models and syntax dynamics in the style of “dynamic-epistemic logic” , an enterprise close to the spirit of the present paper.
This paper is organized as follows. The next section (Section 2) introduces the logical systems that we will study. Each of these logical systems is defined using sublanguages of a single modal language, which includes both a non-normal modality (the “evidence-for” modality) and normal modalities (two belief modalities and a universal modality). One important theme in this paper is that there are two different types of models that can be used as a semantics for this language. The first type of model is a neighborhood structure where the relations used to interpret the normal modalities are derived from the neighborhood function. The second type of model extends neighborhood models with relations, one for each belief operator. Our first technical contribution is to clarify the relationship between these two types of models (Sections 2.2 and 4). More generally, we compare these models with the more standard plausibility models often used to represent a rational agent’s belief in Section 3. Sections 5 and 6 contain our second main contribution of this paper: proofs that our axiom systems are complete with respect to their intended class of models. Section 7 returns to the original motivation for studying evidence logic from : developing dynamic logics of evidence management. We offer some concluding remarks in Section 8.
2 Logics of evidence and belief
We start by presenting the logical system that we will study in this paper. Given a set of possible worlds or states, one of which represents the “actual” situation, an agent gathers evidence about this situation from a variety of sources. To simplify things, we assume these sources provide binary evidence, i.e., subsets of which (may) contain the actual world. The following modal language can be used to describe what the agent believes given her available evidence (cf. ).
Let be a fixed set of atomic propositions. Let be the smallest set of formulas generated by the following grammar
where . The propositional connectives () are defined as usual and the duals555In other words, , and similarly for other operators. of and are , and , respectively.
A signature is a subset of , written usually as a string rather than a set. Sublanguages will be denoted by , where is a signature, and contains only formulas with modalities in the signature.
The intended interpretation of is “the agent has evidence for ” and says that “the agents believes that is true.” The modality refers to the plausibility order to be introduced below, and it has become standard to read it as “the agent safely believes that is true”, though we will really explore this interpretation only in Section 3. We also include the universal modality (: “ is true in all states”) for technical convenience.666A natural interpretation of in the single-agent context of this paper is as “the agent knows that ”.
Since we do not assume that the sources of the evidence are jointly consistent (or even that a single source is guaranteed to be consistent and provide all the available evidence), the “evidence for” operator () is not a normal modal operator. That is, the agent may have evidence for and evidence for ( without having evidence for their conjunction (). Of course, the two belief and universal operators are normal modal operators. So, the logical systems we study in this paper combine a non-normal modal logic with a normal one.
2.1 Models for and its fragments
Our background assumption is that the agent uses the evidence she has gathered at each state to form her beliefs. There are two ways to make this precise. The first is to ground the agent’s beliefs (defined as relations on the set of states) on the evidence available at each state via a number of technical assumptions. The second way is to define the agent’s beliefs directly in terms of the evidence available at each state. These different options lead us to consider several different signatures. This motivates the following general definition of a model:
Let be a signature containing and not containing777The universal modality will always be interpreted according to its intended meaning and need not be represented explicitly in the models. In addition, we consider evidence the most fundamental notion in our setting and assume it always in our signature. . A -structure is a tuple , where is a non-empty set of worlds and
if then is an evidence relation,
if then is a plausibility order on : is reflexive and transitive.888In the literature is usually assumed to be converse well-founded, but we shall work in a more general setting. Note that the plausibility relation need not be connected.
if then is an (arbitrary) binary relation on , and999For readers familiar with the more standard literature on modal logics of belief, it is natural to wonder why we include a separate relation for our belief modalities (rather than using the plausibility ordering to define belief in the usual way). The reason for working with this more general definition is discussed in Sections 2.2 and 3.
is a valuation function.
We will not distinguish between and its denotation in what follows, and we will write for the set .
A -structure is a -model if it further satisfies the following constraints:
for each , and and
if then for all , if and it follows that and
whenever and , it follows that .
An evidence model is a -model where . We may also write simply model instead of -model when the value of is clear from context.
The following basic assumptions are implicit in the above definition:
Sources may or may not be reliable: a subset recording a piece of evidence need not contain the actual world. Also, agents need not know which evidence is reliable.
The evidence gathered from different sources (or even the same source) may be jointly inconsistent. And so, the intersection of all the gathered evidence may be empty.
Despite the fact that sources may not be reliable or jointly inconsistent, they are all the agent has for forming beliefs.101010Modeling sources and agents’ trust in these is quite feasible in our setting – but we will not pursue this topic here.
The evidential state of the agent is the set of all propositions (i.e., subsets of ) identified by the agent’s sources. In general, this could be any collection of subsets of ; but we do impose some mild conditions, that were stated in Constraint 1 in the above definition of our models:
No evidence set is empty (evidence per se is never contradictory),
The whole universe is an evidence set (agents know their ‘space’).
In addition, one might expect a ‘monotonicity’ assumption:
If the agent has evidence and then the agent has evidence .
To us, however, this is a property of propositions supported by evidence, not of the evidence itself. Therefore, we model this feature differently through the definition of our “evidence for” modality (see Definition 3).
Next, constraint 2 in the definition of our models ensures that the agent’s evidence “coheres” with her opinions about the relative plausibility of the states. The intended interpretation of is that, although the agent does not know which of or is the “actual situation”, she considers at least as plausible as . Thus, if is evidence that the actual state may be (i.e., ), then is also evidence for all states that the agent considers at least as plausible as . The underlying idea is that the agent uses her plausibility ordering to “fill-out” the sets of states identified as evidence. We will also consider the even more constrained situation where the agent defines her plausibility ordering directly from the evidence (this is discussed in Section 2.2).
Finally, constraint 3 on our evidence models generalizes the usual assumption that the agent believes what holds throughout the set of most plausible worlds: Any state the agent believes is possible must be among the most plausible states overall, though we will not require that the converse always holds.
Let us define truth for formulas of in a given model:
Let be an evidence model. Truth of a formula is defined inductively as follows:
iff there exists such that and for all ,
if for all , if , then
iff for all , if , then
iff for all ,
The truth set of is the set . The standard logical notions of satisfiability and validity are defined as usual.
2.2 From general models to intended models
In this section, we consider an alternative way of interpreting our modal language , specializing the above “general models” to a special class where belief is entirely and explicitly evidence-driven. Rather than interpreting each modality as a new relation on the set of states, we will now derive the relevant relations from the evidence sets. As a result, we can interpret the entire language on an evidence model .
Recall that we do not assume that the collection of evidence sets is closed under supersets. Also, even though evidence pieces are non-empty, their combination through the obvious operations of taking intersections need not yield consistent evidence: we allow for disjoint evidence sets, whose combination may lead (and should lead) to trouble. But importantly, even though an agent may not be able to consistently combine all of her evidence, there will be maximal collections of admissible evidence that she can safely put together to form scenarios:
A -scenario is a maximal collection that has the fip (i.e., the finite intersection property: for each finite subfamily we have that ). A collection is called a scenario if it is a -scenario for some state .
Our notion of having evidence for need not imply that the agent believes . In order to believe a proposition , the agent must consider all her evidence for or against . The idea is that each -scenario represents a maximally consistent theory based on (some of) the evidence collected at . 111111Analogous ideas occur in semantics of conditionals [19, 39] and belief revision [11, 25]. Note that the definition of truth of the “evidence for” operator builds in monotonicity. That is, the agent has evidence for at provided there is some evidence available at that implies . This motivates the following definition:
Given an evidence function , we define by if for some -scenario .
In order to derive a plausibility ordering, we borrow a ubiquitous idea, occurring in point-set topology, but also in recent theories of relation merge (cf. [1, 21]): the so-called specialization (pre)-order. Under this ordering, is more plausible than if every set that is evidence for is also evidence for . Formally, we define the following:
Given a evidence function , we define by if whenever are such that , then .
To make this definition a bit more concrete, here is a simple illustration.
Note that the derived relation is uniform throughout the model, even though evidence itself is not. It is indeed possible to define a point-wise variant of the specialization order with as parameter, but we shall not explore this option in this paper.
Given an evidence model , we now define the extended structure
We extend the truth-definitions on to all of by setting . If , we say is an intended model.
We will be interested in the precise relationship between intended models and our earlier general models . This issue will be discussed in detail in the remainder of the paper, but we conclude this section with a brief remark. Notice that, in general, is not necessarily a model according to Definition 2. The problem is that the constraint stating that if then is not necessarily satisfied. A particularly simple example is a uniform model where and . If we take, for example, , then it should be clear that yet .
However, this can never happen over an intended model:
Suppose that is an evidence model, then is a model according to Definition 2.
Most conditions are obvious; the only one that we need to check is that if and then . Since , there is a scenario such that . But since , it follows that also , and since was a -scenario, we have that . ∎
It is important to identify maps between structures which preserve truth-sets. In our setting, the following definition will be very useful:
Given evidence models and , we say a function is a p-morphism if the following conditions hold:
if then there is such that
if then there is such that and
if then there is such that and
if then there is such that
We define p-morphism between structures of smaller signature by restricting the above conditions accordingly.
Then we obtain the following familiar result which we present without its straightforward inductive proof:
If is a p-morphism between -models and and is any formula over the signature , then
If a surjective -morphism exists from to , we will write . In Section 4 we will show that, given a general evidence model , there is an evidence model such that . Thus a general evidence model may always be represented as a -morphic image of an intended evidence model. This intended model , however, will often be much larger.
2.4 Special classes of evidence models
The class of evidence models we have described gives the most general setting such an agent may face. However, there are natural additional assumptions one may consider:
Suppose that is a -model. We say is flat if whenever there is such that .
We say is uniform if and are constant (that is, given , then if and only if , and given , ), and whenever it follows that is -maximal. In this case, we shall treat as a set (of neighborhoods) rather than a function and as a set of points.
Finally, we say is concise if it is flat, uniform and, whenever is -maximal then .
The reason for this terminology will show in the arguments to follow. The definition applies, mutatis mutandis, to models with different signatures (e.g., evidence models). Lemma 4 shows that if is a uniform intended evidence model then every point in is maximal, whereas if is flat as well, then coincides exactly with the set of maximal points. Note, however, that in general the property of being concise is stronger than that of merely being flat and uniform.
Flatness may seem like an odd assumption but it is actually very natural because every finite intended model is flat, though infinite models need not be. Uniformity is also natural because one often wishes to model a situation where an agent eitherhas or does not have evidence right at the beginning, rather than obtaining different evidence at each state.
2.5 The logics
We now turn to logics for reasoning about distinct classes of evidence models. Our first observation is that the language is sensitive to flatness:
If is a flat model, then .
If is an evidence set witnessing (i.e., ), then the singleton can be extended to a -scenario using Zorn’s Lemma, which, in flat structures, has a non-empty intersection. ∎
Meanwhile, the formula is not valid in general. To see this, consider a uniform evidence model with domain and evidence sets for each . The valuation is unimportant, so we may let . Clearly, the only scenario on is all of , but . Hence , i.e., ; yet (this formula is universally valid), and we conclude
From this we get the following corollary:
The logic of evidence models does not have the finite model property, nor does the logic of uniform evidence models.
Every finite model is flat, and hence it validates ; but as we have just shown, this formula of our language is not valid over all uniform evidence models. ∎
With this in mind, we state a list of axioms and rules for evidence logics:
|tautology||all propositional tautologies||all|
no empty evidence
The above table gives a family of axioms which in turn define a family of logics; given a signature , we let denote the logic which uses only those axioms and rules that fall within and are marked “all”. The subscripts denote the addition of the respective axioms; note that the logic includes the additional conciseness axiom which is not present in any other logic. We will denote derivability in the logic by , where is any one of the four combinations that we may form from a given , that is, . If is a (possibly infinite) set of formulas, means that there is a finite such that
These axioms are sound for their respective classes of models:
The logics are sound for the classes of all -models, all uniform -models, all flat -models and all concise -models, respectively.
We assume familiarity with neighborhood semantics and well-known modal logics, so we will restrict ourselves to commenting on the more unusual axioms.
Let be any model; by passing to if necessary, we may assume without loss of generality that .
Let us begin by checking plausible evidence. Suppose that , so that there is such that . Pick and suppose . Then, , so that ; since was arbitrary, and thus , as claimed.
The -monotonicity axiom follows easily from the condition that if and , then .
Finally, the maximality axiom uses the fact that, over concise models, every element of is maximal. For indeed, if holds, there is some and thus some with . Since is maximal we must have that and that is maximal as well, so that and satisfies . But it also satisfies so we have that holds. ∎
The weakest logic will be called general evidence logic, while will be called flat logics and will be called uniform logics. We will write -consistency for consistency over the logic .
3 Belief and plausibility
Now we step aside, and consider a different tradition in modeling beliefs.
A plausibility model is a tuple where is a nonempty set, is a valuation function and is a reflexive, transitive and well-founded order on . We assume the reader is familiar with these well-studied models and the modal languages used to reason about them (see  for details and pointers to the relevant literature). Our evidence models are not intended to replace plausibility models, but rather to complement them. So, what exactly is the relationship between these two frameworks for modeling beliefs? The answer is subtle (see [36, Section 5] for some initial observations).
On plausibility models, it is commonplace to say that an agent believes a formula if it is true in all the -maximal worlds. This makes belief definable in the language containing only the universal modality and the plausibility modality , namely, as .121212This is not quite correct if there is more than one information cell (i.e., equivalence class of the knowledge relation). The formula means that is believed at all states in the model. We are implicitly assuming that there is only one “information cell” for the agent, namely the set of all states . In this case, the universal modality is best interpreted as “knowledge”. This may look like a simple technical ploy, but Baltag and Smets  interpret independently as “a safe belief in ”. Following , they show that this amounts to the beliefs the agent retains under all new true information about the actual world.131313For the same notion in the computer science literature on agency, see . Also, see  for related discussions. Our first observation is that our belief operator is not similarly definable over the class of all evidence models.
The modality is not definable in terms of , even over uniform intended evidence models.
The model from Corollary 1 will do. Observe that is bisimilar to the trivial model where has one point and one evidence set (simply because all propositional variables are false). Yet while the model is finite, and hence it validates . ∎
The situation changes over the class of flat models. Here the following notion will be useful later:
Given a uniform evidence model , write .
This construction suggests an extension to our language that we will briefly discuss before moving on to the main result of this section. While what follows can be skipped without loss of continuity, it does give a more concrete idea of the many natural kinds of evidence structure to be found in our models.
Digression: reliable and unreliable evidence
Suppose that is a uniform evidence model. Then is the set of evidence that is “correct” or reliable at state . Unlike the agent’s full evidential state at , the set of reliable evidence at can always be consistently combined, suggesting a new modality meaning “the agent’s reliable evidence entails ” or, for lack of a better term, “ is reliably believed”. The formal definition is:
There is one technical issue we need to address here, which has interesting consequences. Note that the set of reliable evidence differs from state to state. That is, even if the evidence function is assumed to be constant, the reliable evidence function will not be constant. This means that, unlike plain belief, even in uniform evidence models, reliable belief does not satisfy introspection properties. So, what type of doxastic attitude is ? We do not have the space here for an extensive discussion, but here are a few comments.
First, notice that validates the truth axiom (), so believing something based on reliable evidence implies truth. Of course, it is only the modeler, from a third person perspective, who can actually determine what the agent believes based only on reliable evidence. Since the agent does not have access to the actual world, she herself cannot determine which evidence is reliable and which is not. Second, the restriction to “truthful” evidence suggests that reliable belief might be safe belief on evidence models. However, the derived plausibility ordering is typically not connected, and on such models, safe belief quantifies over all worlds not strictly less plausible than the current world. In particular, if is safely believed at a world in a non-connected plausibility model, then must be true at all worlds that are incomparable with . Now, for a derived plausibility ordering, there are two reasons why a state may be incomparable to the actual state : either there is reliable evidence not containing , or there is evidence containing but not . This suggests yet another modality.
Let be an evidence model and define . This is the unreliable or incorrect evidence at state . The corresponding modality is : “ follows from the unreliable evidence at ”. Of course, the agent cannot necessarily consistently combine this evidence, but in the formal definition we can take the union of these sets:
It is the conjunction of these two operators that corresponds to safe belief on evidence models.
These new operators are not definable in our basic language .
The operators and are not definable in evidence belief language .
Let and be uniform evidence models with the evidence sets:
The dashed line is a -morphisim, so . So, and satisfy the same formulas of . However, we have141414In general, it need not be the case that the agent reliably believes iff the agent unreliably believes . We do not discuss the logic of these operators here, but the reader might note that the set of reliable evidence at a state is a filter while the set of unreliable evidence is an ideal. while . ∎
This ends our digression on reliable and unreliable evidence as an example of richer languages supported by our evidence models. This theme of language extensions will return in Section 7, but for now, we return to the earlier source of variety, the existence of natural subclasses of evidence models.
We are now ready to clarify the relationship between uniform, flat evidence models and plausibility models. It will be helpful to start with the following observation:
If is a uniform evidence model, then every is maximal and every non-empty scenario is of the form for some , so that
If is flat as well as uniform, then is exactly the set of maximal points and every collection of the form with -maximal is a scenario.
First assume that is a uniform evidence model and for some scenario . It follows that (since lies in every element of ), and by maximality of , . But if we had for some , then clearly , which would contradict the maximality of and thus is -maximal.
Furthermore, the elements of are precisely those such that (by definition of and ), but from the maximality of we have that as well.
For the other direction, assume is flat and uniform and is maximal. By Zorn’s lemma, can be extended to a scenario . By flatness, we have that . If , there is some . But then lies in every evidence set containing , yet there is some with and , so by definition , which contradicts the maximality of . We conclude that , and thus . ∎
With this, we may describe the plausibility orders of evidence models.
Let be a plausibility order over . Say is directed if any two elements of have an upper bound in .
A plausibility order satisfies the boundendess condition if every directed set has an upper bound (not necessarily in ).
If an evidence model is flat, then its derived plausibility relation satisfies the boundedness condition.
Assume that is flat and let be any directed set. Consider the family . We claim that has the fip. Indeed, let and . Let be an upper bound for all (it is an easy exercise to show that directed sets have upper bounds for finite subsets). Then, by definition, .
Thus can be extended to a scenario . Since is flat, then is non-empty. But every is an upper bound for . ∎
If is flat and is its derived plausibility relation then for every there is such that is maximal.
If is directed it satisfies the conditions of Zorn’s lemma, and we may use it to find maximal elements above any given . ∎
Now we are ready to show that belief is definable over flat models:
Over the class of uniform evidence models with derived plausibility relation, implies .
Over the class of models that are moreover flat, they are equivalent.
First assume that and let be any scenario. Let . By Lemma 4, is maximal; but since it satisfies , it follows that satisfies .
Now assume that the model is flat and holds and let . We have to show that there is satisfying . Use Corollary 2 to find a maximal . By Lemma 4 we have that lies in for some scenario . Meanwhile, if then we also have , and by the assumption that holds, also satisfies , i.e. satisfies , as desired. ∎
4 The representation theorem
It is generally convenient to work with models that are not necessarily intended, since it is easier to control accessibility relations than scenarios. Fortunately, as we shall show in this section, the two classes of models are equivalent with respect to our logics. For indeed, while it is not the case that every model is intended, we do have that every model is the -morphic image of an intended model.
Our goal is then to define an intended model given an arbitrary model . First, let us define our domain, which depends on the underlying logic. Below, recall that
Let be a model. Define the following sets:
is the set of all tuples where , and .
is the set of all with .
is the set of all such that and .
These will be the domains of our intended models. Note that is defined even if is not a -model, but in this case it will generally not be very useful. We remark that the parameter in the concise case is simply a dummy parameter which we use for uniformity with the other cases.
Let us begin with a simple observation about the size of :
The set is finite if and only if is finite and is flat.
Before we define evidence sets, which are somewhat more elaborate, let us define the accessibility relations.
Given a model and an evidence logic , define a relation on by if and only if , either or , and , where is the upset of .
The function as we have defined it is a -morphism on the restricted structures:
The function is a surjective p-morphism with respect to the signature .
The and conditions are straightforward from the definitions so we focus on . But if and , then and . ∎
We can now define the neighborhoods we shall use.
We define the following sets:
Given , and , define to be the set of all such that and .
Given , and , define to be the set of all such that and .
Our evidence relation will be divided into several parts:
Given and , let be the collection of all sets of the form such that and .
Given , , let be the collection of all sets of the form such that and is not flat for . Note that .
If is uniform, define
Given , to be the collection of all sets of the form such that and for all .
Given , to be the collection of all sets of the form such that and is not flat.
Define to be the collection of all sets of the form such that .
For non-uniform let
We remark that does not depend on but we write it this way to conform to the definition of an evidence relation. It will take some work to check that the belief relation is intended, but less so to check that plausibility is.
If is any model then .
If moreover is uniform then for uniform .
First let us check that . Suppose that and . If there is nothing do do, but otherwise consider three cases:
If with , then we must have , agrees with on and hence does on . It follows that .
If , as before the condition on is satisfied and since we also have .
For the other direction, if , consider now three cases.
If , then define to be identical to except that and define as follows: if there is such that , then ; otherwise, is not flat for and we choose . In either case , agrees with on so yet .
If and at least one of them belongs to (for example, ), find such that . Then, but . The cases where either but or are very similar.
If and , let be such that if . Setting we see that but . If instead , we can set .
We conclude that . The uniform cases are very similar (and simpler). ∎
We can now define our intended models:
Let be a -model.
Define and set
These models are convenient because scenarios are easy to identify. Below, let if is not uniform, .
Given a non-concise , a -model and :
for every set of the form
where , is a -scenario, and ;
if is a -scenario that is not of the above form, then
We shall only consider the non-uniform case; the uniform one is analogous.
If , then either or . It follows that if are such that and , either , , or , where the latter is impossible if is a scenario.
Hence every scenario is either contained in a collection of the form (1) or is of the form (2). Let us first check that collections of the form (1) are indeed scenarios and have the appropriate intersection.
So assume that is of form (1) for some with . First note that if then . For clearly, and every set in is of the form for some which agrees with on , thus on , which means that . It follows that .
Moreover, if , then any element of is of the form . Define a function which is identical to except that . Then, by definition , yet and agrees with on so that and . Since were arbitrary, we conclude that . Thus, , and our claim follows.
Now let us check that if is of form (2), then it has an empty intersection. We begin with three simple observations:
contains at least one element of the form