Dealing With Logical Omniscience:
Expressiveness and Pragmatics
Abstract
We examine four approaches for dealing with the logical omniscience problem and their potential applicability: the syntactic approach, awareness, algorithmic knowledge, and impossible possible worlds. Although in some settings these approaches are equiexpressive and can capture all epistemic states, in other settings of interest (especially with probability in the picture), we show that they are not equiexpressive. We then consider the pragmatics of dealing with logical omniscience—how to choose an approach and construct an appropriate model.
1 Introduction
Logics of knowledge based on possibleworld semantics are useful in many areas of knowledge representation and reasoning, ranging from security to distributed computing to game theory. In these models, an agent is said to know a fact if is true in all the worlds she considers possible. While reasoning about knowledge with this semantics has proved useful, as is well known, it suffers from what is known in the literature as the logical omniscience problem: under possibleworld semantics, agents know all tautologies and know the logical consequences of their knowledge.
While logical omniscience is certainly not always an issue, in many applications it is. For example, in the context of distributed computing, we are interested in polynomialtime algorithms, although in some cases the knowledge needed to perform optimally may require calculations that cannot be performed in polynomial time (unless P=NP) [Moses and TuttleMoses and Tuttle1988]; in the context of security, we may want to reason about computationally bounded adversaries who cannot factor a large composite number, and thus cannot be logically omniscient; in game theory, we may be interested in the impact of computational resources on solution concepts (for example, what will agents do if computing a Nash equilibrium is difficult).
Not surprisingly, many approaches for dealing with the logical omniscience problem have been suggested (see [Fagin, Halpern, Moses, and VardiFagin et al.1995, Chapter 9] and [MorenoMoreno1998]). A far from exhaustive list of approaches includes:

syntactic approaches [EberleEberle1974, Moore and HendrixMoore and Hendrix1979, KonoligeKonolige1986], where an agent’s knowledge is represented by a set of formulas (intuitively, the set of formulas she knows);

awareness [Fagin and HalpernFagin and Halpern1988], where an agent knows if she is aware of and is true in all the worlds she considers possible;

algorithmic knowledge [Halpern, Moses, and VardiHalpern et al.1994] where, roughly speaking, an agent knows if her knowledge algorithm returns “Yes” on a query of ; and

impossible worlds [RantalaRantala1982], where the agent may consider possible worlds that are logically inconsistent (for example, where and may both be true).
Which approach is best to use, of course, depends on the application. One goal of this paper is to elucidate the aspects of the application that make a logic more or less appropriate. We start by considering the expressive power of these approaches. It may seem that there is not much to say with regard to expressiveness, since it has been shown that all these approaches are equiexpressive and, indeed, can capture all epistemic states (see [WansingWansing1990, Fagin, Halpern, Moses, and VardiFagin et al.1995] and Section 2). However, this result holds only if we allow an agent to consider no worlds possible. As we show, this equivalence no longer holds in contexts where agents must consider some worlds possible. This is particularly relevant once we have probability in the picture. But expressive power is only part of the story. We consider here (mainly by example) the pragmatics of dealing with logical omniscience—an issue that has largely been ignored: how to choose an approach and construct an appropriate model.
2 The Four Approaches: A Review
We now review the standard possibleworlds approach and the four approaches to dealing logical omniscience discussed in the introduction. For ease of exposition we focus on the singleagent propositional case. While in many applications it is important to consider more than one agent and to allow firstorder features (indeed, this is true in some of our examples), the issues that arise in dealing with multiple agents and firstorder features are largely orthogonal to those involved in dealing with logical omniscience. Thus, we do not discuss these extensions here.
2.1 The Standard Approach
Starting with a set of propositional formulas, we close off under conjunction, negation, and the operator. Call the resulting language . We give semantics to these formulas using Kripke structures. For simplicity, we focus on approaches that satisfy the K45 axioms (as well as KD45 and S5). In this case, a K45 Kripke structure is a triple , where is a nonempty set of possible worlds (or worlds, for short), is the set of worlds that the agent considers possible, and is an interpretation that associates with each world a truth assignment to the primitive propositions in . Note that the agent need not consider every possible world (that is, each world in ) possible. Then we have

iff if .

iff .

iff and .

iff for all .
This semantics suffers from the logical omniscience problem. In particular, one sound axiom is
which says that an agent’s knowledge is closed under implication. In addition, the knowledge generalization inference rule is sound:
From infer . 
Thus, agents know all tautologies. As is well known, two other axioms are sound in K45 Kripke structures:
and
These are known respectively as the positive and negative introspection axioms. (These properties characterize K45.)
In the structures we consider, we allow to be empty, in which case the agent does not consider any worlds possible. In such structures, the formula is true. A KD45 Kripke structure is a K45 Kripke structure where . Thus, in a KD45 Kripke structure, the agent always considers at least one world possible. In KD45 Kripke structures, the axiom
is sound, which implies that the agent cannot know inconsistent facts. The logic KD45 results when we add this axiom to K45. S5 Kripke structures are KD45 Kripke structures where ; that is, the agent considers all worlds in possible. In S5 Kripke structures, the axiom
which says that the agent can know only true facts, is sound. Adding this axiom to the KD45 axioms gives us the logic S5.
2.2 The Syntactic Approach
The intuition behind the syntactic approach for dealing with logical omniscience is simply to explicitly list, at every possible world , the set of formulas that the agent knows at . A syntactic structure has the form , where is a K45 Kripke structure and associates a set of formulas with every world . The semantics of primitive propositions, conjunction, and negation is just the same as for Kripke structures. For knowledge, we have

iff .
2.3 Awareness
Awareness is based on the intuition that an agent should be aware of a concept before she can know it. The formulas that an agent is aware of are represented syntactically; we associate with every world the set of formulas that the agent is aware of. For an agent to know a formula , not only does have to be true at all the worlds she considers possible, but she has to be aware of as well. A K45 awareness structure is a tuple , where is a K45 Kripke structure and maps worlds to sets of formulas. We now define

iff for all and .^{1}^{1}1In [Fagin and HalpernFagin and Halpern1988], the symbol is reserved for the standard definition of knowledge; the definition we have just given is denoted as , where stands for explicit knowledge. A similar remark applies to the algorithmic knowledge approach below. We use throughout for ease of exposition.
We can define KD45 and S5 awareness structures in the obvious way: is a KD45 awareness structure when is a KD45 structure, and an S5 awareness structure when is an S5 structure.
2.4 Algorithmic Knowledge
In some applications, there is a computational intuition underlying what an agent knows; that is, an agent computes what she knows using an algorithm. Algorithmic knowledge is one way of formalizing this intuition. An algorithmic knowledge structure is a tuple , where is a K45 Kripke structure and is a knowledge algorithm that returns “Yes”, “No”, or “?” given a formula .^{2}^{2}2In [Halpern, Moses, and VardiHalpern et al.1994], the knowledge algorithm is also given an argument that describes the agent’s local state, which, roughly speaking, captures the relevant information that the agent has. However, in our singleagent static setting, there is only one local state, so this argument is unneeded. Intuitively, returns “Yes” if the agent can compute that is true, “No” if the agent can compute that is false, and “?” otherwise. In algorithmic knowledge structures,

iff .
An important class of knowledge algorithms consists of the sound knowledge algorithms. When a sound knowledge algorithm returns “Yes” to a query , then the agent knows (in the standard sense) , and when it returns “No” to a query , then the agent does not know (again, in the standard sense) . Thus, if is a sound knowledge algorithm, then implies for all , and and implies there exists such that . (When , nothing is prescribed.)
2.5 Impossible Worlds
The impossibleworlds approach relies on relaxing the notion of possible world. Take the special case of logical omniscience that says that an agent knows all tautologies. This is a consequence of the fact that a tautology must be true at every possible world. Thus, one way to eliminate this problem is to allow tautologies to be false at some worlds. Clearly, those worlds do not obey the usual laws of logic—they are impossible possible worlds (or impossible worlds, for short).
A K45 (resp., KD45, S5) impossibleworlds structure is a tuple , where is a K45 (resp., KD45, S5) Kripke structure, is the set of worlds that the agent considers possible, and associates with each world in a set of formulas. , the set of worlds the agent considers possible, is not required to be a subset of —the agent may well include impossible worlds in . The worlds in are the impossible worlds. We can also consider a class of impossibleworlds structures intermediate between K45 and KD45 impossibleworlds structures. A KD45 impossibleworlds structure is a K45 impossibleworlds structure where is nonempty. In a KD45 impossibleworlds structure, we do not require that be nonempty.
A formula is true at a world if and only if ; for worlds , the truth assignment is like that in Kripke structures. Thus,

if , then iff ;

if , then iff for all ;

if , then iff .
We remark that when we speak of validity in impossibleworlds structures, we mean truth at all possible worlds in in all impossibleworlds structures .
3 Expressive Power
There is a sense in which all four approaches are equiexpressive, and can capture all states of knowledge.
Theorem 3.1 ():
[WansingWansing1990, Fagin, Halpern, Moses, and VardiFagin et al.1995] For every finite set of formulas and every propositionally consistent set of formulas, there exists a syntactic structure (resp., K45 awareness structure, KD45 impossibleworlds structure, algorithmic knowledge structure) and a world such that if and only if , and for all .^{3}^{3}3This result extends to infinite sets of formulas for syntactic structure, K45 awareness structures, and KD45 impossibleworlds structures. For algorithmic knowledge structures, the result extends to recursive sets of formulas.
Proof.
We review the basic idea of the proof, since it will set the stage for our later results.

For syntactic structures, let , where and is such that for all . (Since is propositionally consistent, there must be a truth assignment that makes all the formulas in true; we can take to be that truth assignment.)

For K45 awareness structure, let , where and makes all the formulas in true.

For KD45 impossibleworlds structure, let , where and makes all the formulas in true.

For algorithmic knowledge, let , where iff and makes all the formulas in true.
∎
Despite the name, the introspective axioms of K45 are not valid in K45 awareness structures or K45 impossibleworlds structures. Indeed, it follows from Theorem 3.1 that no axioms of knowledge are valid in these structures. (Take to be the empty set.) To make this precise, let be the axiom
is a valid formula of propositional logic  (Prop) 
and be the inference rule
From and infer .  (MP) 
Theorem 3.2 ():
is a sound and complete axiomatization of with respect to K45 awareness structures (resp., K45 and KD45 impossibleworlds structures, syntactic structures, algorithmic knowledge structures).
Proof.
Suppose that is consistent with . It suffices to show that is satisfiable in a K45 awareness (resp., K45 and KD45 impossibleworlds structure, syntactic structure, algorithmic knowledge structure). Viewing formulas of the form as primitive propositions, must be propositionally consistent. Thus, there must be a truth assignment to the primitive propositions and formulas of the form that appear in such that evaluates to true under this truth assignment. Let consist of all formulas such that and let consist of all the propositional formulas such that . Let be the structure guaranteed to exist by Theorem 3.1. It is easy to see that . ∎
It follows from Theorem 3.2 that a formula is valid with respect to K45 awareness structures (resp., K45 and KD45 impossibleworlds structures, syntactic structures, algorithmic knowledge structures) if and only if it is propositionally valid, if we treat formulas of the form as primitive propositions. Thus, deciding if a formula is valid is coNP complete, just as it is for propositional logic.
Theorems 3.1 and 3.2 rely on the fact that we are considering K45 awareness structures and KD45 (or K45) impossibleworlds structures. (Whether we consider K45, KD45, or S5 is irrelevant in the case of syntactic structures and algorithmic knowledge structures, since the truth of a formula does not depend on what worlds an agent considers possible.) There are constraints on what can be known if we consider KD45 and S5 awareness structures and impossibleworlds structures. The constraints depend on which structures we consider. To make the constraints precise, we need a few definitions. We say a set of formulas is downward closed if the following conditions hold:

if , then both and are in ;

if , then ;

if , then either or (or both); and

if , then .
We say that is kcompatible with if implies that .
Proposition 3.3 ():
Suppose that is a KD45 awareness structure (resp., KD45 impossibleworlds structure), , and (resp., ). Let and let . Then

is propositionally consistent downwardclosed set of formulas that contains ;

if is a KD45 impossibleworlds structure then is kcompatible with .
Proof.
Suppose that is a KD45 awareness structure. Let , , , and be as in the statement of the theorem. Clearly . Since is a possible world, it is easy to see that satisfies the first three conditions of being downward closed. For the last condition, note that if , then we must have for all worlds , so . Finally, must be propositionally consistent, since is a possible world. The argument is the same if is a KD45 impossibleworlds structure, since in this case. To see that is kcompatible if is a KD45 impossibleworlds structure, suppose that . By the definition of , this means that . It follows that for all . Hence, , so . Note that this argument does not work for awareness structures, since we may not have . ∎
The next result show that the constraints on described in Proposition 3.3 are the only constraints on .
Theorem 3.4 ():
If and are such that is propositionally consistent downwardclosed set of formulas that contains , then there exists a KD45 awareness structure such that iff and for all . If, in addition, is kcompatible with , then there exists a KD45 impossibleworlds structure such that iff and for all . Finally, if , then we can take , so that is an S5 awareness (resp., S5 impossibleworlds) structure.
Proof.
In the case of KD45 awareness structures, let , where makes all the propositional formulas in true, , and . We now prove by induction that if then . This is true by construction in the case of primitive propositions and follows easily from the induction hypothesis in the case of conjunctions. If has the form then, since must be in , it follows from the induction hypothesis that and, by construction, that . Thus, . Finally, if has the form , we consider the possible forms of . If is a primitive proposition it follows from the definition of . If has the form , then , so, by the induction hypothesis, . Hence, . Similarly, the result follows from the definition of downward closure and the induction hypothesis if has the form . Finally, if has the form , then the result follows from the definition on . It is now immediate that iff : if then it follows from the definition of that we must have . Conversely, if , then and (since ), so .
If , then we can take in this argument to get an S5 awareness structure.
In the case of impossibleworlds structures, let , where makes all the propositional formulas in true and . A proof by induction on the structure of formulas much like that above shows that if . To deal with the case that , we use the fact that is kcompatible with to get that , so that . To see that iff , first observe that if then, by construction , and, since , , so . For the converse, if , then , so . ∎
We can characterize these properties axiomatically. Let (for Veridicality) be the standard axiom that says that everything known must be true:
(Ver) 
Let be the axiom system consisting of . The fact that the set of formulas known must be a subset of a downward closed set is characterized by the following axiom:
(DC) 
The key point here is that, as we shall show, a propositionally consistent set of formulas that is downward closed must be consistent with .
The fact that the set of formulas that is known is kcompatible with a downward closed set of formulas is characterized by the following axiom:
(KC)  
Axiom is just the special case of axiom where . It is also easy to see that (and therefore ) follow from .
Let and let .
Theorem 3.5 ():

is a sound and complete axiomatization of with respect to KD45 awareness structures;

is a sound and complete axiomatization of with respect to KD45 impossibleworlds structures;

is a sound and complete axiomatization of with respect to S5 awareness structures and S5 impossibleworlds structures.
Proof.
We first prove soundness. Consider axiom . Suppose that . Let be a KD45 awareness structure. For each world , it easily follows from Proposition 3.3 (taking ) that each instance of axiom holds at , as does each instance of . An easy argument by induction on the length of proof then shows that, if , then . In particular, . It follows that, for each , we must have . Essentially the same argument shows that axiom is sound in KD45 impossibleworlds structures.
A similar argument also shows the soundness of with respect to KD45 impossibleworlds structures. For suppose that is an impossibleworlds structure, , , and . Thus, for all . But since each world in is a model of , if , we must have . Moreover, since , there must be some world . It follows that, for some , . Thus, for all , so . It follows that , as desired.
Finally, as we have already observed, the soundness of in S5 awareness and impossibleworlds structures follows easily from Proposition 3.3.
For completeness, we start with part (a). It suffices to show that, given an consistent formula , there exists a KD45 awareness structure and world such that . So suppose that is consistent. Let be a maximal consistent set containing . Let . We claim that is consistent. If not, then there exists such that . But then by axiom , we have that , contradicting the fact that is consistent. Thus, is consistent with . Let be a maximal consistent set extending . Then it is easy to check that is a propositionallyconsistent downwardclosed set of formulas that contains . Thus, by Theorem 3.4, there is a KD45 awareness structure such that for all . We can assume without loss of generality that and that makes all the primitive propositions in true. (Note that this would not be the case if we were dealing with S5 awareness structures.) An easy induction on the structure of formulas then shows that for all . In particular, .
For part (b), we use much the same argument. Suppose that is consistent. Let be a maximal consistent set containing . Let , and let . We again claim that is consistent. If not, then there exists such that . By axiom , we have that , contradicting the fact that is consistent. Thus, is consistent with . Again, let be a maximal consistent set extending . Then it is easy to check that is a propositionallyconsistent downwardclosed set of formulas that contains ; moreover the construction guarantees that is kcompatible with . Thus, by Theorem 3.4, there is a KD45 impossibleworlds structure such that for all . We can assume without loss of generality that and that makes all the primitive propositions in true. An easy induction on the structure of formulas then shows that for all . In particular, .
Finally, for part (c), let . Suppose that is consistent with AX. Extend to a maximally AXconsistent set of formulas. It suffices to show that is satisfiable in an S5 awareness structure and in an S5 impossibleworlds structure. In the case of awareness structures, consider the structure , where iff and . We now show by induction on the structure of formulas that iff . If is a primitive proposition, then this is immediate from the definition of . If has the form , then the result is immediate from the induction hypothesis. If has the form , this is immediate from the observation that, since is a maximal AXconsistent set and propositional reasoning is sound in AX that iff and . If has the form , note that if then (since ). By the induction hypothesis, . Thus, . For the converse, if , suppose, by way of contradiction, that . Then, by construction, . Thus, , a contradiction.
To show that is satisfiable in an S5 impossibleworlds structure, consider the structure , where iff and . Thus, is the same set of formulas as in the argument for S5 awareness structures. An almost identical argument as in the case of S5 awareness structures now shows that iff . We leave details to the reader. ∎
Corollary 3.6 ():
The satisfiability problem for the language with respect to KD45 awareness structures (resp., KD45 impossibleworlds structures, S5 awareness structures) is NPcomplete.
Proof.
NPhardness follows immediately from the observation that contains propositional logic. The fact that the satisfiability problem with respect to each of these classes of structures is in NP follows from the construction of Theorem 3.5, which shows that if a formula is satisfiable with respect to KD45 awareness structures (resp., KD45 impossibleworlds structures, S5 awareness structures), then it is consistent with respect to (resp. , ), which in turn implies that it is satisfiable in a KD45 awareness structure (resp., KD45 impossibleworlds structure, S5 awareness structure) with two (resp., three, one) world(s). Without loss of generality, we can also assume that, in the case of awareness structures, at each world , is a subset of , the set of subformulas of , and only if is a subformula of ; similarly, in the case of impossibleworlds structures, we can assume that for each impossible world , is a subset of the subformulas of . (If this is not true in , then we can easily modify so that this is true without affecting the truth of or any subformula of in any world.) Thus, we can guess a satisfying structure for and verify that it satisfies in time linear in the length of . ∎
4 Adding Probability
While the differences between K45, KD45, and KD45 impossibleworlds structures may appear minor, they turn out to be important when we add probability to the picture. As pointed out by Cozic \citeyearr:cozic05, standard models for reasoning about probability suffer from the same logical omniscience problem as models for knowledge. In the language considered by Fagin, Halpern, and Megiddo \citeyearFHM (FHM from now on), there are formulas that talk explicitly about probability. A formula such as says that the probability that is prime is . In the FHM semantics, a probability is put on the set of worlds that the agent considers possible. The probability of a formula is then the probability of the set of worlds where is true. Clearly, if and are logically equivalent, then will be true. However, the agent may not recognize that and are equivalent, and so may not recognize that . Problems of logical omniscience with probability can to some extent be reduced to problems of logical omniscience with knowledge in a logic that combines knowledge and probability [Fagin and HalpernFagin and Halpern1994]. For example, the fact that an agent may not recognize when and are equivalent just amounts to saying that if is valid, then we do not necessarily want to hold. However, adding knowledge and awareness does not prevent from holding. This is not really a problem if we interpret as the objective probability of ; if and are equivalent, it is an objective fact about the world that their probabilities are equal, so should hold. On the other hand, if represents the agent’s subjective view of the probability of , then we do not want to require to hold. This cannot be captured in all approaches.
To make this precise, we first clarify the logic we have in mind. Let be extended with linear inequality formulas involving probability (called likelihood formulas), in the style of FHM. A likelihood formula is of the form , where and are integers. (For ease of exposition, we restrict to be propositional formulas in likelihood formulas; however, the techniques presented here can be extended to deal with formulas that allow arbitrary nesting of and ). We give semantics to these formulas by extending Kripke structures with a probability distribution over the worlds that the agent considers possible. A probabilistic KD45 (resp., S5) Kripke structure is a tuple , where is KD45 (resp., S5) Kripke structure, and is a probability distribution over . To interpret likelihood formulas, we first define , for a propositional formula . We then extend the semantics of with the following rule for interpreting likelihood formulas:

iff .
Note that the truth of a likelihood formula at a world does not depend on that world; if a likelihood formula is true at a world of a structure , then it is true at every world of .
FHM give an axiomatization for likelihood formulas in probabilistic structures. Aside from propositional reasoning axioms, one axiom captures reasoning with linear inequalities. A basic inequality formula is a formula of the form , where are (not necessarily distinct) variables. A linear inequality formula is a Boolean combination of basic linear inequality formulas. A linear inequality formula is valid if the resulting inequality holds under every possible assignment of real numbers to variables. For example, the formula is a valid linear inequality formula. To get an instance of , we replace each variable that occurs in a valid formula about linear inequalities by a likelihood term of the form (naturally, each occurrence of the variable must be replaced by the same primitive expectation term ). (We can replace by a sound and complete axiomatization for Boolean combinations of linear inequalities; one such axiomatization is given in FHM.)
The other axioms of FHM are specific to probabilistic reasoning, and capture the defining properties of probability distributions:
It is straightforward to extend all the approaches in Section 2 to the probabilistic setting. In this section, we only consider probabilistic awareness structures and probabilistic impossibleworlds structures, because the interpretation of both algorithmic knowledge and knowledge in syntactic structures does not depend on the set of worlds or any probability distribution over the set of worlds.
A KD45 (resp., S5) probabilistic awareness structure is a tuple where is a KD45 (resp., S5) awareness structure and is a probability distribution over the worlds in . Similarly, a KD45 (resp., KD45, S5) probabilistic impossibleworlds structure is a tuple where is a KD45 (resp., KD45, S5) impossibleworlds structure and is a probability distribution over the worlds in . Since the set of worlds that are assigned probability must be nonempty, when dealing with probability, we must restrict to KD45 awareness structures and KD45 impossibleworlds structures, extended with a probability distribution over the set of worlds the agent considers possible. As we now show, adding probability to the language allows finer distinctions between awareness structures and impossibleworlds structures.
In probabilistic awareness structures, the axioms of probability described by FHM are all valid. For example, is valid in probabilistic awareness structures if and are equivalent formulas. Using arguments similar to those in Theorem 3.4, we can show that is valid in probabilistic awareness structures. Similarly, since is valid in probability structures, is valid in probabilistic awareness structures.
We can characterize properties of knowledge and likelihood in probabilistic awareness structures axiomatically. Let denote a substitution instance of a valid formula in probabilistic logic (using the FHM axiomatization). By the observation above, is sound in probabilistic awareness structures. Our reasoning has to take this into account. There is also an axiom that connects knowledge and likelihood:
(KL) 
Let denote the axiom system consisting of . Let be the following strengthening of , somewhat in the spirit of :
()  
Finally, even though is not sound in KD45 probabilistic awareness structures, a weaker version, restricted to likelihood formulas, is sound, since there is a single probability distribution in probabilistic awareness structures. Let be the following axiom:
(WVer) 
Let be the axiom system obtained by replacing in by and adding , , and .
Theorem 4.1 ():

is a sound and complete axiomatization of with respect to KD45 probabilistic awareness structures.

is a sound and complete axiomatization of with respect to S5 probabilistic awareness structures.
Proof.
We first prove soundness. We have already argued that is sound in KD45 probabilistic awareness structures. It is easy to see that is sound: let be a KD45 probabilistic awareness structure, and let be a world in such that . This means that is true at every world , and therefore, , that is, . Similarly, is sound: let be a KD45 probabilistic awareness structure, and let be a world in such that , with a likelihood formula. This means that is true at every world , and because is a likelihood formula, the truth of does not depend on the world. Thus, if is true at some world, it is true at every world; in particular, it is true at , so that , as required. Finally, we show soundness of , using an argument similar to that in the proof of Theorem 3.5. Suppose that is a KD45 probabilistic awareness structure, , , for likelihood formulas , and . Thus, for all . But since each world in is a model of , if , we must have . Since , let be an element of . For some , we must have . Because is a likelihood formula, and therefore its truth does not depend on the world, if is true at some world, then is true at every world. In particular, , and it follows that , as desired.
The soundness of in S5 probabilistic awareness structures follows easily by induction on the structure of in , using the fact that —the special case of when is a likelihood formula—is sound in probabilistic awareness structures, and the argument for the soundness of in S5 awareness structures.
For completeness, first consider part (a). Completeness follows from combining techniques from the FHM completeness proof with those of Theorem 3.5. We briefly sketch the main ideas here. Define to be the least set containing , closed under subformulas, and containing if it contains a propositional formula . It is easy to see that . Suppose that is consistent with . Let be a maximal consistent subset of that includes . Let consist of all truth assignments to primitive propositions. Using techniques of FHM, we can show that there must be a probability measure on that makes all the likelihood formulas in true. We remark for future reference that the FHM proof shows that we can take the set of truth assignments which get positive probability to be polynomial in the size of , and we can assume that the probability is rational, with a denominator whose size is polynomial in .
Let . Arguments almost identical to those in Theorem 3.5 show that must be consistent. Hence there is a maximal consistent subset of that contains . We now construct a KD45 awareness structure as follows. There is a world in corresponding to each truth assignment such that and a world corresponding to ; we define on so that and . Define so that , iff and iff . Finally, define so that , and . Now the same ideas as in the proof of Theorem 3.5 show that, for each formula we have that iff and iff . Thus, .
The proof of completeness for part (b) is similar in spirit; the modifications required are exactly those needed to prove Theorem 3.5(c). We leave details to the reader. ∎
Things change significantly when we move to probabilistic impossibleworlds structures. In particular, is no longer sound. For example, even if is valid, is not valid, because we can have an impossible possible world with positive probability where both and are true. Similarly, is not valid. Indeed, both and are both satisfiable in impossibleworlds structures: the former requires that there be an impossible possible world that gets positive probability where both and are true, while the latter requires an impossible possible world with positive probability where neither is true. As a consequence, it is not hard to show that both and are satisfiable in such impossibleworlds structures.^{4}^{4}4We remark that Cozic \citeyearr:cozic05, who considers the logical omniscience problem in the context of probabilistic reasoning, makes somewhat similar points. Although he does not formalize things quite the way we do, he observes that, in his setting, impossibleworlds structures seem more expressive than awareness structures. In fact, the only constraint on probability in probabilistic impossibleworlds structures is that it must be between 0 and 1. This constraint is expressed by the following axiom :
(Bound) 
We can characterize properties of knowledge and likelihood in probabilistic impossibleworlds structures axiomatically. Let . We can think of as being the core of probabilistic reasoning in impossibleworlds structures.
Let denote the axiom system consisting of . Let denote the following extension of :
()  
Here again, is a special case of . Let obtained by replacing in by and adding , and .
Theorem 4.2 ():

is a sound and complete axiomatization of with respect to KD45 probabilistic impossibleworlds structures.

is a sound and complete axiomatization of with respect to KD45 probabilistic impossibleworlds structures.

is a sound and complete axiomatization of with respect to S5 probabilistic impossibleworlds structures with probabilities.
Proof.
We first prove soundness. The argument is similar to the argument for soundness in Theorem 4.1. That and are sound in probabilistic impossibleworlds structures follows from the same argument as in Theorem 4.1. To show that is sound, note that for any probabilistic impossibleworlds structure , , so that . Because this is independent of the actual world, holds.
We show soundness of with respect to KD45 probabilistic impossibleworlds structures. For suppose that is a KD45 probabilistic impossibleworlds structure, , , where each either a likelihood formula or of the form , and . Thus, for all . But since each world in is a model of , if , we must have . Moreover, since , there must be some world . It follows that, for some , . There are two cases. If is a likelihood formula, then its truth does not depend on the world, so that if is true at some world, then is true at every world. In particular, , and it follows that , as desired. If is a formula of the form , then for all , so , that is, . It follows that , as desired.
Finally, as in the proof of Theorem 4.1, the soundness of in S5 probabilistic impossibleworlds structures follows by induction on the structure of in .
For completeness, we prove part (a). Given a formula consistent with , let be a maximal consistent subset of that includes . Consider the basic likelihood formulas in . ¿From these, we can get a system of linear inequalities by replacing each term by a variable . We add an inequality for each formula . Using the arguments of FHM, we can show that this set of inequalities must be satisfiable (otherwise would not be consistent.) Take a solution. Without loss of generality, we have subformulas listed so that