Logic and Topology for Knowledge, Knowability, and Belief
aa r X i v : . [ c s . L O ] M a r Logic and Topology for Knowledge, Knowability, and Belief
Adam Bjorndahl and Ayb¨uke ¨Ozg¨un
Abstract
In recent work, Stalnaker proposes a logical framework in which belief is realized as aweakened form of knowledge [30]. Building on Stalnaker’s core insights, and using frameworksdeveloped in [11] and [4], we employ topological tools to refine and, we argue, improve on thisanalysis. The structure of topological subset spaces allows for a natural distinction betweenwhat is known and (roughly speaking) what is knowable ; we argue that the foundational axiomsof Stalnaker’s system rely intuitively on both of these notions. More precisely, we argue thatthe plausibility of the principles Stalnaker proposes relating knowledge and belief relies ona subtle equivocation between an “evidence-in-hand” conception of knowledge and a weaker“evidence-out-there” notion of what could come to be known . Our analysis leads to a trimodallogic of knowledge, knowability, and belief interpreted in topological subset spaces in whichbelief is definable in terms of knowledge and knowability . We provide a sound and completeaxiomatization for this logic as well as its uni-modal belief fragment. We then consider weakerlogics that preserve suitable translations of Stalnaker’s postulates, yet do not allow for anyreduction of belief. We propose novel topological semantics for these irreducible notions ofbelief, generalizing our previous semantics, and provide sound and complete axiomatizationsfor the corresponding logics.
Epistemology has long been concerned with the relationship between knowledge and belief. There isa long tradition of attempting to strengthen the latter to attain a satisfactory notion of the former:belief might be improved to true belief, to “justified” true belief, to “correctly justified” true belief[14], to “undefeated justified” true belief [25, 26, 23, 24], and so on (see, e.g, [22, 29] for a survey).There has also been some interest in reversing this project—deriving belief from knowledge—or, atleast, putting “knowledge first” [40]. In this spirit, Stalnaker has proposed a framework in whichbelief is realized as a weakened form of knowledge [30]. More precisely, beginning with a logicalsystem in which both belief and knowledge are represented as primitives, Stalnaker formalizes somenatural-seeming relationships between the two, and proves on the basis of these relationships thatbelief can be defined out of knowledge.This project is of both conceptual and technical interest. Philosophically speaking, it providesa new perspective from which to investigate knowledge, belief, and their interplay. Mathematically,it offers a potential route by which to represent belief in formal systems that are designed to handleonly knowledge. Both these themes underlie the present work. Building on Stalnaker’s core insights,we employ topological tools to refine and, we argue, improve on Stalnaker’s original system.Our work brings together two distinct lines of research. Stalnaker’s epistemic-doxastic axiomshave motivated and inspired several prior topological proposals for the semantics of belief [28, 2,1, 4], including most recently and most notably a proposal by Baltag et al. [4] that is essentiallyrecapitulated in our strongest logic for belief (Section 3). Our development of this logic, however(as well as the new, weaker logics we study in Section 4), relies crucially on a semantic frameworkdefined in recent work by Bjorndahl [11] that distinguishes what is known from (roughly speaking)what is knowable .We argue that the foundational axioms of Stalnaker’s system rely intuitively on both of thesenotions at various points. More precisely, we argue that the plausibility of the principles Stalnakerproposes relating knowledge and belief relies on a subtle equivocation between an “evidence-in-hand” conception of knowledge and a weaker “evidence-out-there” notion of what could come tobe known . As such, we find it quite natural to study Stalnaker’s principles in the richer semanticsetting developed in [11], which is based on topological subset spaces , a class of epistemic models ofgrowing interest in recent years [27, 15, 11, 34, 35]. These models support a careful reworking ofStalnaker’s system in a manner that respects the distinction described above, yielding a trimodallogic of knowledge, knowability, and belief that is our main object of study.Subset spaces have been employed in the representation of a variety of epistemic notions, includ-ing knowledge, learning, and public announcement (see, e.g., [27, 19, 9, 8, 1, 39, 38, 20]), but to thebest of our knowledge this paper contains the first formalization of belief in subset space semantics.Stalnaker’s original system is an extension of the basic logic of knowledge S4 ; belief emerges as astandard KD45 modality, as it is often assumed to be, while knowledge turns out to satisfy thestronger but somewhat less common S4 . axioms. Our system, by contrast, is an extension of thebasic bimodal logic of knowledge-and-knowability introduced in [11]; belief is similarly KD45 , whileknowledge is S5 and knowability is S4 ; thus, our approach preserves what are arguably the desirableproperties of belief while cleanly dividing “knowledge” into two conceptually distinct and familiarlogical constructs.In Stalnaker’s system, belief can be defined in terms of knowledge; in our system, we provethat belief can be defined in terms of knowledge and knowability (Proposition 3). This yields apurely topological interpretation of belief that coincides with that previously proposed by Baltaget al. [4]: roughly speaking, while knowledge is interpreted (as usual) as “truth in all possiblealternatives”, belief becomes “truth in most possible alternatives”, with the meaning of “most”cashed out topologically. The conceptual underpinning of this interpretation of belief as developedby Baltag et al., and its connection to the present work, is discussed further in Section 5.In this richer topological setting, the translation of Stalnaker’s postulates do not in themselvesentail that belief is reducible to knowledge (or even knowledge-and-knowability): our character-ization of belief in these terms relies on two additional principles we call “weak factivity” and“confident belief”. This motivates the study of weaker logical systems obtained by rejecting oneor both of these principles. We initiate the investigation of these systems by proposing noveltopological semantics that aim to capture the corresponding, irreducible notions of belief.This rest of the paper is organized as follows. In Section 2 we present Stalnaker’s originalsystem, motivate our objections to it, and introduce the formal logical framework that supportsour revision. In Section 3 we present our revised system, explore its relationship to Stalnaker’ssystem, and prove an analogue to Stalnaker’s characterization result: belief can be defined out ofknowledge and knowability . We also establish that our system is sound and complete with respectto the class of topological subset models, and that the pure logic of belief it embeds is axiomatizedby the standard KD45 system. In Section 4 we investigate weaker logics as discussed above anddevelop the semantic tools needed to interpret belief in this more general context; we also providesoundness and completeness results for each of these logics. Section 5 concludes. Due to length2estrictions, several of the longer proofs are omitted from the main body; we include them forreference in Appendix A.
Given unary modalities ⋆ , . . . , ⋆ k , let L ⋆ ,...,⋆ k denote the propositional language recursively gen-erated by ϕ ::= p | ¬ ϕ | ϕ ∧ ψ | ⋆ i ϕ, where p ∈ prop , the (countable) set of primitive propositions , and 1 ≤ i ≤ k . Our focus in thispaper is the trimodal language L K, ✷ ,B and various fragments thereof, where we read Kϕ as “theagent knows ϕ ”, ✷ ϕ as “ ϕ is knowable” or “the agent could come to know ϕ ”, and Bϕ as “theagent believes ϕ ”. The Boolean connectives ∨ , → and ↔ are defined as usual, and ⊥ is defined asan abbreviation for p ∧ ¬ p . We also employ ˆ K as an abbreviation for ¬ K ¬ , ✸ for ¬ ✷ ¬ , and ˆ B for ¬ B ¬ .(K ⋆ ) ⊢ ⋆ ( ϕ → ψ ) → ( ⋆ϕ → ⋆ψ ) Distribution(D ⋆ ) ⊢ ⋆ϕ → ¬ ⋆ ¬ ϕ Consistency(T ⋆ ) ⊢ ⋆ϕ → ϕ Factivity(4 ⋆ ) ⊢ ⋆ϕ → ⋆ ⋆ ϕ Positive introspection(.2 ⋆ ) ⊢ ¬ ⋆ ¬ ⋆ ϕ → ⋆ ¬ ⋆ ¬ ϕ Directedness(5 ⋆ ) ⊢ ¬ ⋆ ϕ → ⋆ ¬ ⋆ ϕ Negative introspection(Nec ⋆ ) from ⊢ ϕ infer ⊢ ⋆ϕ NecessitationTable 1: Some axiom schemes and a rule of inference for ⋆ Let
CPL denote an axiomatization of classical propositional logic. Then, following standardnaming conventions, we define the following logical systems: K ⋆ = CPL + (K ⋆ ) + (Nec ⋆ ) S4 ⋆ = K ⋆ + (T ⋆ ) + (4 ⋆ ) S4 . ⋆ = S4 ⋆ + (.2 ⋆ ) S5 ⋆ = S4 ⋆ + (5 ⋆ ) KD45 ⋆ = K ⋆ + (D ⋆ ) + (4 ⋆ ) + (5 ⋆ ) . Stalnaker [30] works with the language L K,B , augmenting the logic S4 K with the additional axiomsschemes presented in Table 2. Let Stal denote this combined logic. Stalnaker proves that this systemyields the pure belief logic
KD45 B ; moreover, he shows that Stal proves the following equivalence: Bϕ ↔ ˆ KKϕ . Thus, belief in this system is reducible to knowledge; every formula of L K,B can betranslated into a provably equivalent formula in L K . Stalnaker also shows that although only the S4 K system is assumed for knowledge, Stal actually derives the stronger system S4 . K .What justifies the assumption of these particular properties of knowledge and belief? It is, ofcourse, possible to object to any of them (including the features of knowledge picked out by thesystem S4 K ); however, in this paper we focus on the relationships expressed in (KB) and (FB).3D B ) ⊢ Bϕ → ¬ B ¬ ϕ Consistency of belief(sPI) ⊢ Bϕ → KBϕ
Strong positive introspection(sNI) ⊢ ¬ Bϕ → K ¬ Bϕ Strong negative introspection(KB) ⊢ Kϕ → Bϕ Knowledge implies belief(FB) ⊢ Bϕ → BKϕ
Full beliefTable 2: Stalnaker’s additional axiom schemesThat knowing implies believing is widely taken for granted—loosely speaking, it corresponds to aconception of knowledge as a special kind of belief. Full belief, on the other hand, may seem morecontentious; this is because it is keyed to a rather strong notion of belief. The English verb “tobelieve” has a variety of uses that vary quite a bit in the nature of the attitude ascribed to thesubject. For example, the sentence, “I believe Mary is in her office, but I’m not sure” makes aclearly possibilistic claim, whereas, “I believe that nothing can travel faster than the speed of light”might naturally be interpreted as expressing a kind of certainty. It is this latter sense of beliefthat Stalnaker seeks to capture: belief as subjective certainty . On this reading, (FB) essentiallystipulates that being certain is not subjectively distinguishable from knowing: an agent who feelscertain that ϕ is true also feels certain that she knows that ϕ is true.At a high level, then, each of (KB) and (FB) have a certain plausibility. Crucially, however,we contend that their joint plausibility is predicated on an abstract conception of knowledge thatpermits a kind of equivocation. In particular, tension between the two emerges when knowledge isinterpreted more concretely in terms of what is justified by a body of evidence.Consider the following informal account of knowledge: an agent knows something just in case itis entailed by the available evidence. To be sure, this is still vague since we have not yet specifiedwhat “evidence” is or what “available” means (we return to formalize these notions in Section 2.2).But it is motivated by a very commonsense interpretation of knowledge, as for example in a cardgame when one player is said to know their opponent is not holding two aces on the basis of thefact that they are themselves holding three aces.Even at this informal level, one can see that something like this conception of knowledge liesat the root of the standard possible worlds semantics for epistemic logic. Roughly speaking, suchsemantics work as follows: each world w is associated with a set of accessible worlds R ( w ), and theagent is said to know ϕ at w just in case ϕ is true at all worlds in R ( w ). A standard intuition forthis interpretation of knowledge is given in terms of evidence: the worlds in R ( w ) are exactly thosecompatible with the agent’s evidence at w , and so the agent knows ϕ just in case the evidence rulesout all not- ϕ possibilities. Suppose, for instance, that you have measured your height and obtaineda reading of 5 feet and 10 inches ± know that you are less than 6 feet tall, having ruled out the possibility that you are taller.Call this the evidence-in-hand conception of knowledge. Observe that it fits well with the (KB)principle: evidence-in-hand that entails ϕ should surely also cause you to believe ϕ . On the otherhand, it does not sit comfortably with (FB): presumably you can be (subjectively) certain of ϕ without simultaneously being certain that you currently have evidence-in-hand that guarantees ϕ ,lest we lose the distinction between belief and knowledge. However, the intuition for (FB) can be Stalnaker calls this property “strong belief” but we, following [2, 3], adopt the term “full belief” instead. This assumes, roughly speaking, that evidence-in-hand is “transparent” in the sense that the agent cannot be thereis evidence entailing ϕ —even if you don’t happen to personally have it in hand at the moment.This corresponds to a transition from the known to the knowable. On this account, (FB) is recastas “If you are certain of ϕ , then you are certain that there is evidence entailing ϕ ”, a sort ofdictum of responsible belief: do not believe anything unless you think you could come to knowit. Returning to (KB), on the other hand, we see that it is not supported by this weaker sense ofevidence-availability: the fact that you could, in principle, discover evidence entailing ϕ should notin itself imply that you believe ϕ .This way of reconciling Stalnaker’s proposed axioms with an evidence-based account of knowledge—namely, by carefully distinguishing between knowledge and knowability—is the focus of the remain-der of this paper. We begin by defining a class of models rich enough to interpret both of thesemodalities at once. A subset space is a pair ( X, S ) where X is a nonempty set of worlds and S ⊆ X is a collectionof subsets of X . A subset model X = ( X, S , v ) is a subset space ( X, S ) together with a function v : prop → X specifying, for each primitive proposition p ∈ prop , its extension v ( p ).Subset space semantics interpret formulas not at worlds x but at epistemic scenarios of theform ( x, U ), where x ∈ U ∈ S . Let ES ( X ) denote the collection of all such pairs in X . Given anepistemic scenario ( x, U ) ∈ ES ( X ), the set U is called its epistemic range ; intuitively, it representsthe agent’s current information as determined, for example, by the measurements she has taken.We interpret L K in X as follows:( X , x, U ) | = p iff x ∈ v ( p )( X , x, U ) | = ¬ ϕ iff ( X , x, U ) = ϕ ( X , x, U ) | = ϕ ∧ ψ iff ( X , x, U ) | = ϕ and ( X , x, U ) | = ψ ( X , x, U ) | = Kϕ iff ( ∀ y ∈ U )(( X , y, U ) | = ϕ ) . Thus, knowledge is cashed out as truth in all epistemically possible worlds, analogously to thestandard semantics for knowledge in relational models. A formula ϕ is said to be satisfiable in X if there is some ( x, U ) ∈ ES ( X ) such that ( X , x, U ) | = ϕ , and valid in X if for all ( x, U ) ∈ ES ( X )we have ( X , x, U ) | = ϕ . The set [[ ϕ ]] U X = { x ∈ U : ( X , x, U ) | = ϕ } is called the extension of ϕ under U . We sometimes drop mention of the subset model X whenit is clear from context.Subset space models are well-equipped to give an account of evidence-based knowledge and its dynamics . Elements of S can be thought of as potential pieces of evidence, while the epistemicrange U of an epistemic scenario ( x, U ) corresponds to the “evidence-in-hand” by means of whichthe agent’s knowledge is evaluated. This is made precise in the semantic clause for Kϕ , whichstipulates that the agent knows ϕ just in case ϕ is entailed by the evidence-in-hand.In this framework, stronger evidence corresponds to a smaller epistemic range, and whether agiven proposition can come to be known corresponds (roughly speaking) to whether there exists a mistaken about what evidence she has or what it entails. A model rich enough to represent this kind of uncertaintyabout evidence might therefore be of interest; we leave the development of such a model to other work. topologically .A topological space is a pair ( X, T ) where X is a nonempty set and T ⊆ X is a collectionof subsets of X that covers X and is closed under finite intersections and arbitrary unions. Thecollection T is called a topology on X and elements of T are called open sets. In what follows weassume familiarity with basic topological notions; for a general introduction to topology we referthe reader to [16, 17].A topological subset model is a subset model X = ( X, T , v ) in which T is a topology on X . Clearly every topological space is a subset space. But topological spaces possess additionalstructure that enables us to study the kinds of epistemic dynamics we are interested in. Moreprecisely, we can capture a notion of knowability via the following definition: for A ⊆ X , say that x lies in the interior of A if there is some U ∈ T such that x ∈ U ⊆ A . The set of all points in theinterior of A is denoted int ( A ); it is easy to see that int ( A ) is the largest open set contained in A .Given an epistemic scenario ( x, U ) and a primitive proposition p , we have x ∈ int ([[ p ]] U ) preciselywhen there is some evidence V ∈ T that is true at x and that entails p . We therefore interpret theextended language L K, ✷ that includes the “knowable” modality in X via the additional recursiveclause ( X , x, U ) | = ✷ ϕ iff x ∈ int ([[ ϕ ]] U ) . The formula ✷ ϕ thus represents knowability as a restricted existential claim over the set T ofavailable pieces of evidence. The dual modality correspondingly satisfies( X , x, U ) | = ✸ ϕ iff x ∈ cl ([[ ϕ ]] U ) , where cl denotes the topological closure operator. Since the formula ✷ ¬ ϕ reads as “the agent couldcome to know that ϕ is false”, one intuitive reading of its negation, ✸ ϕ , is “ ϕ is unfalsifiable”.It is worth noting that the intuition behind reading ✷ ϕ as “ ϕ is knowable” can falter when ϕ is itself an epistemic formula. In particular, if ϕ is the Moore sentence p ∧ ¬ Kp , then Kϕ is notsatisfiable in any subset model, so in this sense ϕ can never be known; nonetheless, ✷ ϕ is satisfiable.Loosely speaking, this is because our language abstracts away from the implicit temporal dimensionof knowability; ✷ ϕ might be more accurately glossed as “one could come to know what ϕ used toexpress (before you came to know it)”. Since primitive propositions do not change their truthvalue based on the agent’s epistemic state, this subtlety is irrelevant for propositional knowledgeand knowability. For the purposes of this paper, we opt for the simplified “knowability” gloss ofthe ✷ modality, and leave further investigation of this subtlety to future work. Like Stalnaker, we augment a basic logic of knowledge with some additional axiom schemes thatspeak to the relationship between belief and knowledge. Unlike Stalnaker, however, we work with It is not hard to see that [[ ✷ ϕ ]] U = int ([[ ϕ ]] U ) as one might expect; however, since the closure of [[ ϕ ]] U need notbe a subset of U , we have [[ ✸ ϕ ]] U = cl ([[ ϕ ]] U ) ∩ U . This reading suggests a strong link to conditional belief modalities , which are meant to capture an agent’s revisedbeliefs about how the world was before learning the new information. More precisely, a conditional belief formula B ϕ ψ is read as “if the agent would learn ϕ , then she would come to believe that ψ was the case (before the learning)”[7, p. 14]. Borrowing this interpretation, we might say that ✷ ϕ represents hypothetical, conditional knowledge of ϕ where the condition consists in having some piece of evidence V entailing ϕ as evidence-in-hand: “if the agent wereto have V as evidence-in-hand, she would know ϕ was the case (before having had the evidence)”. L K, ✷ ,B and take as our “basic logic of knowledge” the system EL K, ✷ = S5 K + S4 ✷ + (KI) , where (KI) denotes the axiom scheme Kϕ → ✷ ϕ . As noted in Section 2.1, the evidence-in-handconception of knowledge captured by the semantics for K is based on the premise that evidence-in-hand is completely transparent to the agent. That is, the agent is aware that she has the evidenceshe does and of what it entails and does not entail. In this sense, the agent is fully introspectivewith regard to the evidence-in-hand, and as such, K naturally emerges as an S5 -type modality.The system EL K, ✷ was defined by Bjorndahl [11] and shown to be exactly the logic of topologicalsubset spaces. Theorem 1 ([11]) . EL K, ✷ is a sound and complete axiomatization of L K, ✷ with respect to the classof all topological subset spaces: for every ϕ ∈ L K, ✷ , ϕ is provable in EL K, ✷ if and only if ϕ is validin all topological subset models. We strengthen EL K, ✷ with the additional axiom schemes given in Table 3. Let SEL K, ✷ ,B denote(K B ) ⊢ B ( ϕ → ψ ) → ( Bϕ → Bψ ) Distribution of belief(sPI) ⊢ Bϕ → KBϕ
Strong positive introspection(KB) ⊢ Kϕ → Bϕ Knowledge implies belief(RB) ⊢ Bϕ → B ✷ ϕ Responsible belief(wF) ⊢ Bϕ → ✸ ϕ Weak factivity(CB) ⊢ B ( ✷ ϕ ∨ ✷ ¬ ✷ ϕ ) Confident beliefTable 3: Additional axioms schemes for SEL K, ✷ ,B the resulting logic. (sPI) and (KB) occur here just as they do in Stalnaker’s original system (Table2), and though (K B ) is not an axiom of Stal , it is derivable in that system. The remaining axiomsinvolve the ✷ modality and thus cannot themselves be part of Stalnaker’s system; however, if weforget the distinction between ✷ and K (and between ✸ and ˆ K ), all of them do hold in Stal , asmade precise in Proposition 2.
Proposition 2.
Let t : L K, ✷ ,B → L K,B be the map that replaces each instance of ✷ with K . Thenfor every ϕ that is an instance of an axiom scheme from Table 3, we have ⊢ Stal t ( ϕ ) .Proof. This is trivial for (sPI), (KB), and (RB). (K B ) follows immediately from the fact that Stal validates
KD45 B . After applying t , (wF) becomes Bϕ → ˆ Kϕ , which follows easily from the factthat ⊢ Stal Bϕ ↔ ˆ KKϕ . Finally, under t , (CB) becomes B ( Kϕ ∨ K ¬ Kϕ ), which follows directlyfrom the aforementioned equivalence together with the fact that ⊢ S4 K ˆ KK ( Kϕ ∨ K ¬ Kϕ ).Thus, modulo the distinction between knowledge and knowablity, we make no assumptions be-yond what follows from Stalnaker’s own stipulations. Of course, the distinction between knowledgeand knowability is crucial for us. Responsible belief differs from full belief in that K is replacedby ✷ , exactly as motivated in Section 2.1; it says that if you are sure of ϕ , then you must alsobe sure that there is some evidence that entails ϕ . Weak factivity and confident belief do notdirectly correspond to any of Stalnaker’s axioms, but they are necessary to establish an analogue ofStalnaker’s reduction of belief to knowledge (Proposition 3). Of course, one need not adopt these7rinciples; indeed, rejecting them allows one to assent to the spirit of Stalnaker’s premises withoutcommitting oneself to his conclusion that belief can be defined out of knowledge (or knowability).We return in Section 4 to consider weaker logics that omit one or both of (wF) and (CB).Weak factivity can be understood, given (KI), as a strengthening of the formula Bϕ → ˆ Kϕ (which is provable in Stal ). Intuitively, it says that if you are certain of ϕ , then ϕ must be compatiblewith all the available evidence (in hand or not). Thus, while belief is not required to be factive—youcan believe false things—(wF) does impose a weaker kind of connection to the world—you cannotbelieve knowably false things.Confident belief expresses a kind of faith in the justificatory power of evidence. Consider thedisjunction ✷ ϕ ∨ ✷ ¬ ✷ ϕ , which says that ϕ is either knowable or, if not, that you could come to knowthat it is unknowable. This is equivalent to the negative introspection axiom for the ✷ modality,and does not hold in general; topologically speaking, it fails at the boundary points of the extensionof ✷ ϕ —where no measurement can entail ϕ yet every measurement leaves open the possibility thatsome further measurement will. What (CB) stipulates is that the agent is sure that they are not insuch a “boundary case”—that every formula ϕ is either knowable or, if not, knowably unknowable.Stalnaker’s reduction of belief to knowledge has an analogue in this setting: every formula in L K, ✷ ,B is provably equivalent to a formula in L K, ✷ via the following equivalence. Proposition 3.
The formula Bϕ ↔ K ✸✷ ϕ is provable in SEL K, ✷ ,B .Proof. We present an abridged derivation:1. Bϕ → ✸✷ ϕ (RB), (wF)2. KBϕ → K ✸✷ ϕ (Nec K ), (K K )3. Bϕ → KBϕ (sPI)4. Bϕ → K ✸✷ ϕ CPL : 2, 35. B ( ✷ ϕ ∨ ✷ ¬ ✷ ϕ ) (CB)6. ( ✷ ϕ ∨ ✷ ¬ ✷ ϕ ) → ( ✸✷ ϕ → ϕ ) (T ✷ ), CPL B ( ✷ ϕ ∨ ✷ ¬ ✷ ϕ ) → B ( ✸✷ ϕ → ϕ ) (Nec K ), (KB), (K B )8. B ( ✸✷ ϕ → ϕ ) CPL : 5, 79. B ✸✷ ϕ → Bϕ (K B )10. K ✸✷ ϕ → B ✸✷ ϕ (KB)11. K ✸✷ ϕ → Bϕ CPL : 9, 1012. Bϕ ↔ K ✸✷ ϕ CPL : 4, 11.Thus, rather than being identified with the “epistemic possibility of knowledge” [30] as in Stal-naker’s framework, to believe ϕ in this framework is to know that the knowability of ϕ is unfalsifiable.This is a bit of a mouthful, so consider for a moment the meaning of the subformula ✸✷ ϕ : in theinformal language of evidence, this says that every piece of evidence is compatible not only with thetruth of ϕ , but with the knowability of ϕ . In other words: no possible measurement can rule outthe prospect that some further measurement will definitively establish ϕ . To believe ϕ , accordingto Proposition 3, is to know this.This equivalence also tells us exactly how to extend topological subset space semantics to thelanguage L K, ✷ ,B : ( X , x, U ) | = Bϕ iff ( X , x, U ) | = K ✸✷ ϕ iff ( ∀ y ∈ U )( y ∈ cl ( int ([[ ϕ ]] U )))iff U ⊆ cl ( int ([[ ϕ ]] U )) . ϕ at ( x, U ) just in case the interior of [[ ϕ ]] U is dense in U . The collectionof sets that have dense interiors on U forms a filter, making it a good mathematical notion of largeness : sets with dense interior can be thought of as taking up “most” of the space. Thisprovides an appealing intuition for the semantics of belief that runs parallel to that for knowledge:the agent knows ϕ at ( x, U ) iff ϕ is true at all points in U , whereas the agent believes ϕ at ( x, U )iff ϕ is true at most points in U .As mentioned in the introduction, this interpretation of belief as “truth at most points” (in agiven domain) was first studied by Baltag et al. as a topologically natural, evidence-based notion ofbelief [4]. Though their motivation and conceptual underpinning differ from ours, the semantics forbelief we have derived in this section essentially coincide with those given in [4]. We discuss thisconnection further in Section 5. Let (EQ) denote the scheme Bϕ ↔ K ✸✷ ϕ . It turns out that this equivalence is not only provablein SEL K, ✷ ,B , but in fact it characterizes SEL K, ✷ ,B as an extension of EL K, ✷ . To make this precise,let EL + K, ✷ = EL K, ✷ + (EQ) . We then have:
Proposition 4. EL + K, ✷ and SEL K, ✷ ,B prove the same theorems. From this it is not hard to establish soundness and completeness of
SEL K, ✷ ,B : Theorem 5.
SEL K, ✷ ,B is a sound and complete axiomatization of L K, ✷ ,B with respect to the classof topological subset models: for every ϕ ∈ L K, ✷ ,B , ϕ is valid in all topological subset models if andonly if ϕ is provable in SEL K, ✷ ,B . Much work in belief representation takes the logical principles of
KD45 B for granted (see, e.g.,[21, 6, 36]). An important feature of SEL K, ✷ ,B is that it derives these principles: Proposition 6.
For every ϕ ∈ L B , if ⊢ KD45 B ϕ , then ⊢ SEL K, ✷ ,B ϕ . In fact,
KD45 B is not merely derivable in our logic—it completely characterizes belief as in-terpreted in topological models. That is, KD45 B proves exactly the validities expressible in thelanguage L B : Theorem 7.
KD45 B is a sound and complete axiomatization of L B with respect to the class of alltopological subset spaces: for every ϕ ∈ L B , ϕ is provable in KD45 B if and only if ϕ is valid in alltopological subset models. Soundness follows easily from the above. The proof of completeness is more involved; we includeit as an appendix. A nonempty collection of subsets forms a filter if it is closed under taking supersets and finite intersections. Weaker Notions of Belief
In Section 3, we motivated the axioms of our system
SEL K, ✷ ,B in part by the fact that they al-lowed us to achieve a reduction of belief to knowledge-and-knowability in the spirit of Stalnaker’sresult. SEL K, ✷ ,B includes several of Stalnaker original axioms (or modifications thereof), but alsotwo new schemes: weak factivity (wF) and confident belief (CB). As noted, if we forget the dis-tinction between knowledge and knowability, each of these schemes holds in Stal (Proposition 2).Nonetheless, in our tri-modal logic these two principles do not follow from the others: one canadopt (our translations of) Stalnaker’s original principles while rejecting one or both of (wF) and(CB). In particular, this allows one to essentially accept all of Stalnaker’s premises without beingforced to the conclusion that belief is reducible to knowledge (or even knowledge-and-knowability).We are therefore motivated to generalize our earlier semantics in order to study weaker logics inwhich the belief modality is not definable and so requires its own semantic machinery.In this section we do just this: we augment EL K, ✷ with the axiom schemes given in Table 4 toform the logic EL K, ✷ ,B , and prove that this system is sound and complete with respect to the newsemantics defined below. We then consider logics intermediate in strength between EL K, ✷ ,B and SEL K, ✷ ,B —specifically, those obtained by augmenting EL K, ✷ ,B with the axioms (D B ) (consistencyof belief), (wF), or (CB)—and establish soundness and completeness results for these logics as well.(K B ) ⊢ B ( ϕ → ψ ) → ( Bϕ → Bψ ) Distribution of belief(sPI) ⊢ Bϕ → KBϕ
Strong positive introspection(KB) ⊢ Kϕ → Bϕ Knowledge implies belief(RB) ⊢ Bϕ → B ✷ ϕ Responsible beliefTable 4: Additional axiom schemes for EL K, ✷ ,B As before, we rely on topological subset models X = ( X, T , v ) for the requisite semantic structure(see Section 2.2); however, we define the evaluation of formulas with respect to epistemic-doxastic(e-d) scenarios , which are tuples of the form ( x, U, V ) where ( x, U ) is an epistemic scenario, V ∈ T ,and V ⊆ U . We call V the doxastic range . The semantic evaluation for the primitive propositions and the Boolean connectives is definedas usual; for the modal operators, we make use of the following semantic clauses:( X , x, U, V ) | = Kϕ iff U = [[ ϕ ]] U,V ( X , x, U, V ) | = ✷ ϕ iff x ∈ int ([[ ϕ ]] U,V )( X , x, U, V ) | = Bϕ iff V ⊆ [[ ϕ ]] U,V , where [[ ϕ ]] U,V = { x ∈ U : ( X , x, U, V ) | = ϕ } . Thus, the modalities K and ✷ are interpreted essentially as they were before, while the modality B is rendered as universal quantification over the doxastic range. Intuitively, we might think of V as the agent’s “conjecture” about the way the world is, typically stronger than what is guaranteedby her evidence-in-hand U . On this view, states in V might be conceptualized as “more plausible” If we want to insist on consistent beliefs, we should add the axiom (D B ): Bϕ → ˆ Bϕ (or, equivalently, ˆ B ⊤ ) andrequire that V = ∅ . We begin with the more general case, without these assumptions. U V from the agent’s perspective, with belief being interpreted as truth in all thesemore plausible states (see, e.g., [18, 31, 7, 33, 37] for more details on plausibility models for belief).Note that we do not require that x ∈ V ; this corresponds to the intuition that the agent may havefalse beliefs. Note also that none of the modalities alter either the epistemic or the doxastic range;this is essentially what guarantees the validity of the strong introspection axioms. In order to distinguish these semantics from those previous, we refer to them as epistemic-doxastic (e-d) semantics for topological subset spaces.
Theorem 8. EL K, ✷ ,B is a sound and complete axiomatization of L K, ✷ ,B with respect to the classof all topological subset spaces under e-d semantics. Call an e-d scenario ( x, U, V ) consistent if V = ∅ , and call it dense if V is dense in U (i.e., if U ⊆ cl ( V )). Theorem 9. EL K, ✷ ,B + (D B ) is a sound and complete axiomatization of L K, ✷ ,B with respect to theclass of all topological subset spaces under e-d semantics for consistent e-d scenarios. EL K, ✷ ,B +(wF) is a sound and complete axiomatization of L K, ✷ ,B with respect to the class of all topological subsetspaces under e-d semantics for dense e-d scenarios. It turns out that the strong semantics for the belief modality presented in Section 3, namely( X , x, U ) | = Bϕ iff U ⊆ cl ( int ([[ ϕ ]] U )) , does not arise as a special case of our new e-d semantics: there is no condition (e.g., density) onecan put on the doxastic range V so that these two interpretations of Bϕ agree in general. Roughlyspeaking, this is because the formulas of the form ✷ ϕ ∨ ✷ ¬ ✷ ϕ correspond to the open and densesets, but in general one cannot find a (nonempty) open set V that is simultaneously contained inevery open, dense set. As such, one cannot hope to validate (CB) in the e-d semantics presentedabove without also validating B ⊥ .However, we can validate (CB) on topological subset spaces by altering the semantic interpreta-tion of the belief modality so that, intuitively, it “ignores” nowhere dense sets . Loosely speaking,this works because nowhere dense sets are exactly the complements of sets with dense interiors.More precisely, we work with the same notion of e-d scenarios as before, but use the followingsemantics clauses:( X , x, U, V ) |≈ p iff x ∈ v ( p )( X , x, U, V ) |≈ ¬ ϕ iff ( X , x, U, V ) |6≈ ϕ ( X , x, U, V ) |≈ ϕ ∧ ψ iff ( X , x, U, V ) |≈ ϕ and ( X , x, U, V ) |≈ ψ ( X , x, U, V ) |≈ Kϕ iff U = [( ϕ )] U,V ( X , x, U, V ) |≈ ✷ ϕ iff x ∈ int ([( ϕ )] U,V )( X , x, U, V ) |≈ Bϕ iff V ⊆ ∗ [( ϕ )] U,V , where [( ϕ )] U,V = { x ∈ U : ( X , x, U, V ) |≈ ϕ } , We could, of course, consider even more general semantics that do not validate these axioms, but as our goalhere is to understand the role of weak factivity and confident belief in the context of Stalnaker’s reduction of beliefto knowledge, we leave such investigations to future work. A subset S of a topological space is called nowhere dense if its closure has empty interior: int ( cl ( S )) = ∅ . A ⊆ ∗ B iff A B is nowhere dense. In other words, we interpret everything as beforewith the exception of the belief modality, which now effectively quantifies over almost all worlds inthe doxastic range V rather than over all worlds. Theorem 10. EL K, ✷ ,B + (CB) is a sound and complete axiomatization of L K, ✷ ,B with respect tothe class of all topological subset spaces under e-d semantics using the semantics given above: for allformulas ϕ ∈ L K, ✷ ,B , if |≈ ϕ , then ⊢ EL K, ✷ ,B +(CB) ϕ . Moreover, SEL K, ✷ ,B is sound and completewith respect to these semantics for e-d scenarios where V = U . When we think of knowledge as what is entailed by the “available evidence”, a tension between twofoundational principles proposed by Stalnaker emerges. First, that whatever the available evidenceentails is believed ( Kϕ → Bϕ ), and second, that what is believed is believed to be entailed bythe available evidence ( Bϕ → BKϕ ). In the former case, it is natural to interpret “available”as, roughly speaking, “currently in hand”, whereas in the latter, intuition better accords with abroader interpretation of availability as referring to any evidence one could potentially access.Being careful about this distinction leads to a natural division between what we might call“knowledge” and “knowability”; the space of logical relationships between knowledge, knowability,and belief turns out to be subtle and interesting. We have examined several logics meant to capturesome of these relationstips, making essential use of topological structure, which is ideally suitedto the representation of evidence and the epistemic/doxastic attitudes it informs. In this refinedsetting, belief can also be defined in terms of knowledge and knowability, provided we take on twoadditional principles, “weak factivity” (wF) and “confident belief” (CB); in this case, the semanticsfor belief have a particularly appealing topological character: roughly speaking, a proposition isbelieved just in case it is true in most possible alternatives, where “most” is interpreted topologicallyas “everywhere except on a nowhere dense set”.This interpretation of belief first appeared in the topological belief semantics presented in [4]:Baltag et al. take the believed propositions to be the sets with dense interiors in a given evidential topology. Interestingly, however, although these semantics essentially coincide with those we presentin Section 3, the motivations and intuitions behind the two proposals are quite different. Baltaget al. start with a subbase model in which the (subbasic) open sets represent pieces of evidencethat the agent has obtained directly via some observation or measurement. They do not distinguishbetween evidence-in-hand and evidence-out-there as we do; moreover, the notion of belief theyseek to capture is that of justified belief , where “justification”, roughly speaking, involves havingevidence that cannot be defeated by any other available evidence. (They also consider a weaker,defeasible type of knowledge, correctly justified belief , and obtain topological semantics for it underwhich Stalnaker’s original system
Stal is sound and complete.) The fact that two rather differentconceptions of belief correspond to essentially the same topological interpretation is, we feel, quitestriking, and deserves a closer look.Despite the elegance of this topological characterization of belief, our investigation of the inter-play between knowledge, knowability, and belief naturally leads to consideration of weaker logics inwhich belief is not interpreted in this way. In particular, we focus on the principles (wF) and (CB)and what is lost by their omission. Again we rely on topological subset models to interpret these Given a subset A of a topological space X , we say that a property P holds for almost all points in A just in case A ⊆ ∗ { x : P ( x ) } . doxastic range and, perhaps more dramatically, a modification to the semantic satisfactionrelation ( |≈ ) that builds the topological notion of “almost everywhere” quantification directly intothe foundations of the semantics. We believe this approach is an interesting area for future research,and in this regard our soundness and completeness results may be taken as proof-of-concept. References [1] Balbiani, P., van Ditmarsch, H., and Kudinov, A. (2013) Subset space logic with arbitraryannouncements.
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Proofs
A.1 Soundness and completeness of
SEL K, ✷ ,B Let e : L K, ✷ ,B → L K, ✷ be the map that replaces each instance of B with K ✸✷ . Lemma 11.
For all ϕ ∈ L K, ✷ ,B , we have ⊢ EL + K, ✷ ϕ ↔ e ( ϕ ) .Proof. This is a straightforward induction on the structure of ϕ using (EQ). Proposition 12. EL + K, ✷ and SEL K, ✷ ,B prove the same theorems.Proof. In light of Proposition 3, it suffices to show that EL + K, ✷ proves everything in Table 3. ByLemma 11, then, it suffices to show that for every ϕ that is an instance of an axiom scheme fromTable 3, we have ⊢ EL K, ✷ e ( ϕ ). And for this, by Theorem 1, we need only show that each such e ( ϕ )is valid in all topological subset models.Let X = ( X, T , v ) be a topological subset model and ( x, U ) ∈ ES ( X ).(K B ) Suppose ( x, U ) | = K ✸✷ ( ϕ → ψ ) and ( x, U ) | = K ✸✷ ϕ . Then U ⊆ cl ( int ([[ ϕ → ψ ]] U )) ∩ cl ( int ([[ ϕ ]] U )). Let y ∈ U and let V be an open set containing y . Then we must have V ∩ int ([[ ϕ → ψ ]] U ) = ∅ and so, since this set is also open, V ∩ int ([[ ϕ → ψ ]] U ) ∩ int ([[ ϕ ]] U ) = ∅ ∴ V ∩ int ([[ ϕ → ψ ]] U ∩ [[ ϕ ]] U ) = ∅ ∴ V ∩ int ([[ ψ ]] U ) = ∅ , which establishes that y ∈ cl ( int ([[ ψ ]] U )). This shows that U ⊆ cl ( int ([[ ψ ]] U )), and therefore( x, U ) | = K ✸✷ ψ .(sPI) Suppose ( x, U ) | = K ✸✷ ϕ . Then U = [[ ✸✷ ϕ ]] U , and so for all y ∈ U we have ( y, U ) | = K ✸✷ ϕ .This implies that U = [[ K ✸✷ ϕ ]] U , hence ( x, U ) | = KK ✸✷ ϕ .(KB) Suppose ( x, U ) | = Kϕ . Then U = [[ ϕ ]] U , and so (since U is open), U ⊆ cl ( int ([[ ϕ ]] U )), whichimplies ( x, U ) | = K ✸✷ ϕ .(RB) Suppose ( x, U ) | = K ✸✷ ϕ . Then U ⊆ cl ( int ([[ ϕ ]] U )), so U ⊆ cl ( int ( int ([[ ϕ ]] U ))), hence U =[[ ✸✷✷ ϕ ]] U , which implies that ( x, U ) | = K ✸✷✷ ϕ .(wF) Suppose ( x, U ) | = K ✸✷ ϕ . Then x ∈ U ⊆ cl ( int ([[ ϕ ]] U )) ⊆ cl ([[ ϕ ]] U ), which implies that( x, U ) | = ✸ ϕ .(CB) Observe that [[ ✷ ϕ ∨ ✷ ¬ ✷ ϕ ]] U = int ([[ ϕ ]] U ) ∪ int ( X int ([[ ϕ ]] U ))is an open set. Moreover, it is dense in U ; to see this, let y ∈ U and let V be an openneighbourhood of y . Then either V ∩ int ([[ ϕ ]] U ) = ∅ or, if not, V ⊆ X int ([[ ϕ ]] U ), hence V ⊆ int ( X int ([[ ϕ ]] U )). We therefore have U ⊆ cl ( int ([[ ✷ ϕ ∨ ✷ ¬ ✷ ϕ ]] U )) , whence ( x, U ) | = K ✸✷ ( ✷ ϕ ∨ ✷ ¬ ✷ ϕ ). 16 roposition 13. EL + K, ✷ is a sound axiomatization of L K, ✷ ,B with respect to the class of topologicalsubset models: for every ϕ ∈ L K, ✷ ,B , if ϕ is provable in EL + K, ✷ then ϕ is valid in all topologicalsubset models.Proof. This follows from the soundness of EL K, ✷ (Theorem 1) together with the fact that thesemantics for the B modality ensures that (EQ) is valid is all topological subset models. Corollary 14.
SEL K, ✷ ,B is a sound axiomatization of L K, ✷ ,B with respect to the class of topologicalsubset models.Proof. Immediate from Propositions 12 and 13.
Theorem 15.
SEL K, ✷ ,B is a complete axiomatization of L K, ✷ ,B with respect to the class of topo-logical subset models: for every ϕ ∈ L K, ✷ ,B , if ϕ is valid in all topological subset models then ϕ isprovable in SEL K, ✷ ,B .Proof. We show the contrapositive. Let ϕ ∈ L K, ✷ ,B be such that SEL K, ✷ ,B ϕ . By Lemma 11 andProposition 12 we have ⊢ SEL K, ✷ ,B ϕ ↔ e ( ϕ ), and so also SEL K, ✷ ,B e ( ϕ ). Since e ( ϕ ) ∈ L K, ✷ and SEL K, ✷ ,B is an extension of EL K, ✷ , we know that EL K, ✷ e ( ϕ ). Thus, by Theorem 1, there exists atopological subset model X and ( x, U ) ∈ ES ( X ) such that ( X , x, U ) = e ( ϕ ) and so, by the soundnessof SEL K, ✷ ,B , we obtain ( X , x, U ) = ϕ . A.2
KD45 B and the doxastic fragment L B Proposition 16.
For every ϕ ∈ L B , if ⊢ KD45 B ϕ , then ⊢ SEL K, ✷ ,B ϕ .Proof. It suffices to show that
SEL K, ✷ ,B derives all the axioms and the rule of inference of KD45 B .(K B ) is itself an axiom of SEL K, ✷ ,B . It is not hard to see, using (wF) and S4 ✷ , that ⊢ SEL K, ✷ ,B ¬ B ⊥ ;given this, (D B ) follows from (K B ) with ψ replaced by ⊥ . (4 B ) follows easily from (sPI) and(KB). To derive (5 B ), first observe that by (5 K ) we have ⊢ SEL K, ✷ ,B ¬ K ✸✷ ϕ → K ¬ K ✸✷ ϕ ; fromProposition 3 it then follows that ⊢ SEL K, ✷ ,B ¬ Bϕ → K ¬ Bϕ , and so from (KB) we can deduce (5 B ).Lastly, (Nec B ) follows directly from (Nec K ) together with (KB). Theorem 17.
KD45 B is a sound and complete axiomatization of L B with respect to the class ofall topological subset spaces: for every ϕ ∈ L B , ϕ is provable in KD45 B if and only if ϕ is valid inall topological subset models. Soundness follows immediately from Proposition 16 together with the soundness of
SEL K, ✷ ,B (Corollary 14). The remainder of this section is devoted to developing the tools needed to provecompleteness. Our proof relies crucially on the standard Kripke-style interpretation of L B in re-lational models and the completeness results pertaining thereto. We therefore begin with a briefreview of these notions (for a more comprehensive overview, we direct the reader to [12, 13]).A relational frame is a pair ( X, R ) where X is a nonempty set and R is a binary relation on X .A relational model is a relational frame ( X, R ) equipped with a valuation function v : prop → X .The language L B is interpreted in a relational model M = ( X, R, v ) by extending the valuationfunction via the standard recursive clauses for the Boolean connectives together with the following:(
M, x ) | = Bϕ iff ( ∀ y ∈ X )( xRy implies ( M, y ) | = ϕ ) . k ϕ k M = { x ∈ X : ( M, x ) | = ϕ } . A belief frame is a frame ( X, R ) where R is serial, transitive,and Euclidean. Theorem 18.
KD45 B is a sound and complete axiomatization of L B with respect to the class ofbelief frames.Proof. See, e.g., [13, Chapter 5] or [12, Chapters 2, 4].A frame (
X, R ) is called a brush if there exists a nonempty subset C ⊆ X such that R = X × C .If such a C exists, clearly it is unique; call it the final cluster of the brush. A brush is called a pin if | X C | = 1. It is not hard to see that every brush is a belief frame. Conversely, the followingLemma shows that every belief frame ( X, R ) is a disjoint union of brushes. Lemma 19.
Let ( X, R ) be a belief frame, and define x ∼ y iff ( ∃ z ∈ X )( xRz and yRz ) . Then ∼ is an equivalence relation extending R . Moreover, if [ x ] denotes the equivalence class of x under ∼ , then ([ x ] , R | [ x ] ) is a brush, and ( X, R ) is the disjoint union of all such brushes.Proof. Reflexivity of ∼ follows from seriality of R , and symmetry is immediate. To see that ∼ istransitive, suppose x ∼ x ′ and x ′ ∼ x ′′ . Then there exist y, z ∈ X such that xRy , x ′ Ry , x ′ Rz and x ′′ Rz . Because R is Euclidean, it follows that yRz ; because R is transitive, we can deduce that xRz ; it follows that x ∼ x ′′ . To see that ∼ extends R , suppose xRy . Then because R is Euclidean,we have yRy , which implies x ∼ y .The fact that ∼ is an equivalence relation tells us that the sets [ x ] partition X ; furthermore,since xRy implies [ x ] = [ y ], we also know that the sets R | [ x ] partition R . Thus ( X, R ) is the disjointunion of the frames ([ x ] , R | [ x ] ).Finally we show that each such frame ([ x ] , R | [ x ] ) is a brush. Set C x = { y ∈ [ x ] : yRy } ; that C x = ∅ follows easily from R being serial and Euclidean. Let y ∈ C x . Then for all x ′ ∈ [ x ] we have x ′ ∼ y , so there is some z ∈ X with x ′ Rz and yRz ; now because R is Euclidean, we can deduce that zRy , so by transitivity x ′ Ry . It follows that [ x ] × { y } ⊆ R , hence [ x ] × C x ⊆ R . On the other hand,if y / ∈ C x , then for every x ′ ∈ [ x ] we have ¬ ( x ′ Ry ), or else the Euclidean property would imply yRy , a contradiction. Thus, R | [ x ] = [ x ] × C x , so ([ x ] , R | [ x ] ) is a brush with final cluster C x . Corollary 20.
KD45 B is a sound and complete axiomatization of L B with respect to the class ofbrushes and with respect to the class of pins. There is a close connection between the relational semantics for L B presented above and ourtopological semantics for this language. For any frame ( X, R ), let R + denote the reflexive closure of R : R + = R ∪ { ( x, x ) : x ∈ X } . Given a transitive frame (
X, R ), the set B R + = { R + ( x ) : x ∈ X } constitutes a topological basis on X ; denote by T R + the topology generated by B R + (see, e.g., [10, 32] for a more detailed discussionof this construction). It is well-known that ( X, T R + ) is an Alexandroff space and, for every x ∈ X ,the set R + ( x ) is the smallest open neighborhood of x . A relation is serial if ( ∀ x )( ∃ y )( xRy ); it is transitive if ( ∀ x, y, z )(( xRy & yRz ) ⇒ xRz ); it is Euclidean if( ∀ x, y, z )(( xRy & xRz ) ⇒ yRz ). A frame (
X, R ) is said to be a disjoint union of frames ( X i , R i ) provided the X i partition X and the R i partition R . emma 21. Let ( X, R ) be a belief frame. For each x ∈ X , let C x denote the final cluster of thebrush ([ x ] , R | [ x ] ) as in Lemma 19, and let int and cl denote the interior and closure operators,respectively, in the topological space ( X, T R + ) . Then for all x ∈ X and every A ⊆ X :1. [ x ] ∈ T R + , and so ( x, [ x ]) ∈ ES ( X M ) ;2. R ( x ) = C x ∈ T R + ;3. int ( A ) ∩ C x = ∅ if and only if A ⊇ C x ;4. cl ( A ) ⊇ [ x ] if and only if A ∩ C x = ∅ .Proof.
1. This follows from the fact that y ∈ [ x ] implies R + ( y ) ⊆ [ x ], which in turn follows from thefact that ∼ extends R (Lemma 19).2. That R ( x ) = C x follows from the fact that R | [ x ] = [ x ] × C x (Lemma 19). To see that C x isopen, observe that if y ∈ C x , then R + ( y ) = R ( y ) = C y = C x .3. Since C x is open, it follows immediately that if A ⊇ C x then int ( A ) ⊇ C x , so in particular int ( A ) ∩ C x = ∅ . Conversely, if y ∈ int ( A ) ∩ C x then R + ( y ) ⊆ A , since R + ( y ) is the smallestopen neigbourhood of y ; therefore, since R + ( y ) = R ( y ) = C x , we have A ⊇ C x .4. First suppose that y ∈ A ∩ C x and let z ∈ [ x ]. By part 2, R + ( z ) ⊇ R ( z ) = C x , and so since R + ( z ) is the smallest open neighbourhood of z and y ∈ C x , it follows that z ∈ cl ( { y } ) ⊆ cl ( A ),hence [ x ] ⊆ cl ( A ). Conversely, suppose that A ∩ C x = ∅ . Then since C x is open it followsthat C x ∩ cl ( A ) = ∅ , which shows that [ x ] cl ( A ).Given a transitive model M = ( X, R, v ), let X M denote the topological subset model constructedfrom M , namely ( X, T R + , v ). Lemma 22.
Let M = ( X, R, v ) be a belief frame. Then for every formula ϕ ∈ L B , for every x ∈ X we have ( M, x ) | = ϕ iff ( X M , x, [ x ]) | = ϕ. Proof.
The proof follows by induction on the structure of ϕ ; cases for the primitive propositions andthe Boolean connectives are elementary. So assume inductively that the result holds for ϕ ; we mustshow that it holds also for Bϕ . Note that the inductive hypothesis implies that [[ ϕ ]] [ x ] = k ϕ k M ∩ [ x ],since by Lemma 19, y ∈ [ x ] implies [ y ] = [ x ].( M, x ) | = Bϕ iff R ( x ) ⊆ k ϕ k M iff C x ⊆ k ϕ k M (Lemma 21.2)iff C x ⊆ k ϕ k M ∩ [ x ] (since C x ⊆ [ x ])iff C x ⊆ [[ ϕ ]] [ x ] (inductive hypothesis)iff int ([[ ϕ ]] [ x ] ) ∩ C x = ∅ (Lemma 21.3)iff cl ( int ([[ ϕ ]] [ x ] )) ⊇ [ x ] (Lemma 21.4)iff ( X M , x, [ x ]) | = Bϕ.
Completeness is an easy consequence of this lemma: if ϕ ∈ L B is such that KD45 B ϕ , then byTheorem 18 there is a belief frame M that refutes ϕ at some point x . Then, by Lemma 22, ϕ isalso refuted in X M at the epistemic scenario ( x, [ x ]). This completes the proof of Theorem 17.19 .3 Soundness and completeness of EL K, ✷ ,B Theorem 23. EL K, ✷ ,B is a sound axiomatization of L K, ✷ ,B with respect to the class of all topo-logical subset spaces under e-d semantics.Proof. The validity of the axioms without the modality B follows as in Theorem 1, since the onlydifference here lies in the semantic clause for B . Let X = ( X, T , v ) be a topological subset model,( x, U, V ) an e-d scenario, and ϕ, ψ ∈ L K, ✷ ,B .(K B ) Suppose ( x, U, V ) | = B ( ϕ → ψ ) and ( x, U, V ) | = Bϕ . This means V ⊆ [[ ϕ → ψ ]] U,V =( U \ [[ ϕ ]] U,V ) ∪ [[ ψ ]] U,V and V ⊆ [[ ϕ ]] U,V , from which we obtain V ⊆ [[ ψ ]] U,V , i.e., ( x, U, V ) | = Bψ .(sPI) Suppose ( x, U, V ) | = Bϕ . This means V ⊆ [[ ϕ ]] U,V . As such, for every y ∈ U we have( y, U, V ) | = Bϕ , which implies that [[ Bϕ ]] U,V = U , so ( x, U, V ) | = KBϕ .(KB) Suppose ( x, U, V ) | = Kϕ . This means [[ ϕ ]] U,V = U . As V ⊆ U (by definition of ( x, U, V )), weobtain ( x, U, V ) | = Bϕ .(RB) Suppose ( x, U, V ) | = Bϕ . This means V ⊆ [[ ϕ ]] U,V . Thus, since V is open, we obtain V ⊆ int ([[ ϕ ]] U,V ). As int ([[ ϕ ]] U,V ) = [[ ✷ ϕ ]] U,V , we have V ⊆ [[ ✷ ϕ ]] U,V , i.e., ( x, U, V ) | = B ✷ ϕ .Completeness follows from a fairly straightforward canonical model construction, similar to thecompleteness proof of EL K, ✷ in [11]. Roughly speaking, we extend the canonical model in [11] inorder to be able to prove the truth lemma for the belief modality B .Let X c be the set of all maximal EL K, ✷ ,B -consistent sets of formulas. Define binary relations ∼ and R on X c by x ∼ y iff ( ∀ ϕ ∈ L K, ✷ ,B )( Kϕ ∈ x ⇔ Kϕ ∈ y ) and xRy iff ( ∀ ϕ ∈ L K, ✷ ,B )( Bϕ ∈ x ⇒ ϕ ∈ y ) . It is not hard to see that ∼ is an equivalence relation, hence, it induces equivalence classes on X c .Let [ x ] denote the equivalence class of x induced by the relation ∼ and let R ( x ) = { y ∈ X c | xRy } .Define b ϕ = { y ∈ X c | ϕ ∈ y } , so x ∈ b ϕ iff ϕ ∈ x .The axioms of EL K, ✷ ,B that relate K and B induce the following important links between ∼ and R : Lemma 24.
For any x, y ∈ X c , the following holds:1. if x ∼ y then ( ∀ ϕ ∈ L K, ✷ ,B )( Bϕ ∈ x iff Bϕ ∈ y ) ;2. if x ∼ y then R ( x ) = R ( y ) ;3. R ( x ) ⊆ [ x ] ;4. either R ( x ) ∩ R ( y ) = ∅ or R ( x ) = R ( y ) .Proof. Let x, y ∈ X c .1. Suppose x ∼ y and let ϕ ∈ L K, ✷ ,B such that Bϕ ∈ x . By (sPI), we have KBϕ ∈ x . As x ∼ y , we have KBϕ ∈ y . Thus, by (T K ), we conclude Bϕ ∈ y . The other direction followsanalogously. In fact, this is equivalent to ( ∀ ϕ ∈ L K, ✷ ,B )( Kϕ ∈ x ⇒ ϕ ∈ y ), since K is an S5 modality.
20. Suppose x ∼ y and take z ∈ R ( x ); let ϕ ∈ L K, ✷ ,B be such that Bϕ ∈ y . Since x ∼ y , byLemma 24.1, we have Bϕ ∈ x . Therefore, z ∈ R ( x ) implies that ϕ ∈ z . This shows that z ∈ R ( y ), hence R ( x ) ⊆ R ( y ). The reverse inclusion follows similarly.3. Let z ∈ R ( x ) and ϕ ∈ L K, ✷ ,B ; we will show that Kϕ ∈ x iff Kϕ ∈ z . Suppose Kϕ ∈ x . Then,by (4 K ), we have KKϕ ∈ x . This implies, by (KB), that BKϕ ∈ x . Hence, since z ∈ R ( x ),we obtain Kϕ ∈ z . For the converse, suppose Kϕ x , i.e., ¬ Kϕ ∈ x . Then, by (5 K ), wehave K ¬ Kϕ ∈ x . Again by (KB), we obtain B ¬ Kϕ ∈ x . Thus, since z ∈ R ( x ), we obtain ¬ Kϕ ∈ z , i.e., Kϕ z . We therefore conclude that z ∈ [ x ], hence R ( x ) ⊆ [ x ].4. Suppose R ( x ) ∩ R ( y ) = ∅ . This means there is z ∈ X c such that z ∈ R ( x ) and z ∈ R ( y ). Then,by Lemma 24.3, we have x ∼ z and y ∼ z . Thus, by Lemma 24.2, R ( x ) = R ( z ) = R ( y ).Let T c be the topology on X c generated by the collection B = { [ x ] ∩ c ✷ ϕ | x ∈ X c , ϕ ∈ L K, ✷ ,B } ∪ { R ( x ) ∩ c ✷ ϕ | x ∈ X c , ϕ ∈ L K, ✷ ,B } . It is not hard to prove that B is in fact a basis for T c . Define the canonical model X c to be thetuple ( X c , T c , v c ), where v c ( p ) = b p . Observe that since d ✷ ⊤ = X c , we have [ x ] , R ( x ) ∈ T c for all x ∈ X c ; therefore, by Lemma 24.3, for each x ∈ X c the tuple ( x, [ x ] , R ( x )) is an e-d scenario. Lemma 25 (Truth Lemma) . For every ϕ ∈ L K, ✷ ,B and for each x ∈ X c , ϕ ∈ x iff ( X c , x, [ x ] , R ( x )) | = ϕ. Proof.
The proof proceeds as usual by induction on the structure of ϕ ; cases for the primitivepropositions and the Boolean connectives are elementary and the case for K is presented in [11,Theorem 1, p. 16]. So assume inductively that the result holds for ϕ ; we must show that it holdsalso for ✷ ϕ and Bϕ .Case for ✷ ϕ :( ⇒ ) Let ✷ ϕ ∈ x . Then, observe that x ∈ c ✷ ϕ ∩ [ x ] ⊆ { y ∈ [ x ] | ϕ ∈ y } (by x ∈ [ x ] and (T ✷ )).Since c ✷ ϕ ∩ [ x ] is open, it follows that x ∈ int { y ∈ [ x ] | ϕ ∈ y } (1)By (IH), we also have { y ∈ [ x ] | ϕ ∈ y } = { y ∈ [ x ] | ( y, [ y ] , R ( y )) | = ϕ } = { y ∈ [ x ] | ( y, [ x ] , R ( x )) | = ϕ } (Lemma 24)= [[ ϕ ]] [ x ] ,R ( x ) Therefore, by (1), we conclude that x ∈ int ([[ ϕ ]] [ x ] ,R ( x ) ), i.e., ( x, [ x ] , R ( x )) | = ✷ ϕ .( ⇐ ) Now suppose that ( x, [ x ] , R ( x )) | = ✷ ϕ . This means, by the semantics, that x ∈ int ([[ ϕ ]] [ x ] ,R ( x ) ).As above, this is equivalent to x ∈ int { y ∈ [ x ] | ϕ ∈ y } . It then follows that there exists U ∈ B such that x ∈ U ⊆ { y ∈ [ x ] | ϕ ∈ y } . By definition of B , the basic open neighbourhood U can be of the following forms:21. U = [ z ] ∩ c ✷ ψ , for some z ∈ X c and ψ ∈ L K, ✷ ,B ;2. U = R ( z ) ∩ c ✷ ψ , for some z ∈ X c and ψ ∈ L K, ✷ ,B .However, since x ∈ U , we can simply replace the above cases by:1. U = [ x ] ∩ c ✷ ψ , for some ψ ∈ L K, ✷ ,B ;2. U = R ( x ) ∩ c ✷ ψ , for some ψ ∈ L K, ✷ ,B , respectively.The case for U = [ x ] ∩ c ✷ ψ follows similarly as in [11, Theorem 1, p. 16]. We here onlyprove the case for U = R ( x ) ∩ c ✷ ψ . We therefore have x ∈ R ( x ) ∩ c ✷ ψ ⊆ { y ∈ [ x ] | ϕ ∈ y } (2)This means that for every y ∈ R ( x ), if ✷ ψ ∈ y then ϕ ∈ y . Thus, we obtain that { χ | Bχ ∈ x } ∪ {¬ ( ✷ ψ → ϕ ) } is an inconsistent set. Otherwise, it could be extended toa maximally consistent set y such that y ∈ R ( x ), ✷ ψ ∈ y and ϕ y , contradicting (2).Thus, there exists a finite subset Γ ⊆ { χ | Bχ ∈ x } such that ⊢ ^ χ ∈ Γ χ → ( ✷ ψ → ϕ ) , which implies by S4 ✷ that ⊢ ^ χ ∈ Γ ✷ χ → ✷ ( ✷ ψ → ϕ ) . Observe that, since x ∈ R ( x ), we have { χ | Bχ ∈ x } ⊆ x . Moreover, by (RB), we alsoobtain that { ✷ χ | Bχ ∈ x } ⊆ { χ | Bχ ∈ x } ⊆ x . We therefore obtain that V χ ∈ Γ ✷ χ ∈ x ,thus, that ✷ ( ✷ ψ → ϕ ) ∈ x . Then, by S4 ✷ , we have ✷ ψ → ✷ ϕ ∈ x . As x ∈ c ✷ ψ , weconclude ✷ ϕ ∈ x .Case for Bϕ :( ⇒ ) Let Bϕ ∈ x . Then, by defn. of R , we have ϕ ∈ y for all y ∈ R ( x ). Then, by (IH), weobtain ( ∀ y ∈ R ( x ))( y, [ y ] , R ( y )) | = ϕ . By Lemma 24.3, y ∈ R ( x ) implies x ∼ y . Thus,as [ y ] = [ x ] and R ( x ) = R ( y ) (Lemma 24.2), we obtain, ( ∀ y ∈ R ( x ))( y, [ x ] , R ( x )) | = ϕ .This means, R ( x ) ⊆ [[ ϕ ]] [ x ] ,R ( x ) , thus, ( x, [ x ] , R ( x )) | = Bϕ .( ⇐ ) Let Bϕ x . This implies, { ψ | Bψ ∈ x } ∪ {¬ ϕ } is consistent. Otherwise, there exists afinite subset Γ ⊆ { ψ | Bψ ∈ x } such that ⊢ ^ χ ∈ Γ χ → ϕ. Then, by normality of B , ⊢ ^ χ ∈ Γ Bχ → Bϕ.
Since Bχ ∈ x for all χ ∈ Γ, we have Bϕ ∈ x , contradicting the fact that x is a consistentset. 22hen, by Lindenbaum’s Lemma, { ψ | Bψ ∈ x } ∪ {¬ ϕ } can be extended to a maximallyconsistent set y . ¬ ϕ ∈ y means that ϕ y . Thus, by IH, ( y, [ y ] , R ( y )) = ϕ . Since { ψ | Bψ ∈ x } ⊆ y , we have y ∈ R ( x ). This means, by Lemma 24.3 and Lemma24.2, [ y ] = [ x ] and R ( x ) = R ( y ). Therefore, as [ y ] = [ x ] and R ( x ) = R ( y ), we have( y, [ x ] , R ( x )) = ϕ . Thus, y ∈ R ( x ) but y [[ ϕ ]] [ x ] ,R ( x ) implying that ( x, [ x ] , R ( x )) = Bϕ .Moreover, Lemma 24.3 guarantees that the evaluation tuple ( x, [ x ] , R ( x )) is of desired kind(more precisely, the construction of the canonical model guarantees that R ( x ) ∈ T c , for all x ∈ X c and the aforementioned lemma makes sure that R ( x ) ⊆ [ x ].) Corollary 26. EL K, ✷ ,B is a complete axiomatization of L K, ✷ ,B with respect to the class of alltopological subset spaces under e-d semantics.Proof. Let ϕ ∈ L K, ✷ ,B such that EL K, ✷ ,B ϕ . Then, {¬ ϕ } is consistent and can be extended to amaximally consistent set x ∈ X c . Then, by Lemma 25, we obtain that ( X c , x, [ x ] , R ( x )) = ϕ . A.4 Consistent belief and weak factivity
Proposition 27. EL K, ✷ ,B + (D B ) is a sound axiomatization of L K, ✷ ,B with respect to the class ofall topological subset spaces under e-d semantics for consistent e-d scenarios.Proof. The validity of the axioms of EL K, ✷ ,B follows as in Theorem 23, we only need to provethe validity of (D B ) for consistent e-d scenarios. Let X = ( X, T , v ) be a topological subset model,( x, U, V ) a consistent e-d scenario, and ϕ ∈ L K, ✷ ,B .(D B ) Suppose ( x, U, V ) | = Bϕ . This means V ⊆ [[ ϕ ]] U,V . Then, since V = ∅ , we have V U \ [[ ϕ ]] U,V ,therefore, ( x, U, V ) | = ¬ B ¬ ϕ .The completeness proof follows similarly to the completeness proof of EL K, ✷ ,B and the onlydifference lies in the requirement of a consistent e-d scenario in the corresponding Truth Lemma.We therefore only need to prove that the canonical epistemic scenario ( x, [ x ] , R ( x )) of the system EL K, ✷ ,B + (D B ) is consistent, i.e., we need to show that R ( x ) = ∅ for any maximally consistent setsof EL K, ✷ ,B + (D B ). The canonical model for the system EL K, ✷ ,B + (D B ) is constructed as usual,exactly the same way as the one for EL K, ✷ ,B . Lemma 28.
The relation R of the canonical model X c = ( X c , T c , ν c ) for the system EL K, ✷ ,B + (D B )is serial.Proof. For any x ∈ X c , the set { ψ | Bψ ∈ x } is consistent. Otherwise, there is a finite subsetΓ ⊆ { ψ | Bψ ∈ x } and ϕ ∈ { ψ | Bψ ∈ x } such that ⊢ ^ χ ∈ Γ χ → ¬ ϕ. Then, by normality of B , ⊢ ^ χ ∈ Γ Bχ → B ¬ ϕ. Since Bχ ∈ x for all χ ∈ Γ, we have B ¬ ϕ ∈ x . On the other hand, since Bϕ ∈ x and ⊢ Bϕ → ¬ B ¬ ϕ ((D B )-axiom), we obtain ¬ B ¬ ϕ ∈ x , contradicting the fact that x a maximallyconsistent set. Therefore, { ψ | Bψ ∈ x } can be extended to a maximally consistent set y and, since { ψ | Bψ ∈ x } ⊆ y , we have xRy . 23 orollary 29. Let X c = ( X c , T c , ν c ) be the canonical model of the system EL K, ✷ ,B + (D B ). Then,for all x ∈ X c , we have R ( x ) = ∅ . Proposition 30. EL K, ✷ ,B + (D B ) is a complete axiomatization of L K, ✷ ,B with respect to the classof all topological subset spaces under e-d semantics for consistent e-d scenarios.Proof. Follows from Corollary 29 similarly to the proof of Corollary 26.
Proposition 31. EL K, ✷ ,B + (wF) is a sound axiomatization of L K, ✷ ,B with respect to the class ofall topological subset spaces under e-d semantics for dense e-d scenarios.Proof. The validity of the axioms of EL K, ✷ ,B follows as in Theorem 23, we only need to prove thevalidity of (wF) for dense e-d scenarios. Let X = ( X, T , v ) be a topological subset model, ( x, U, V )a dense e-d scenario, and ϕ ∈ L K, ✷ ,B .(wF) Suppose ( x, U, V ) | = Bϕ . This means V ⊆ [[ ϕ ]] U,V . Then, since x ∈ U ⊆ cl ( V ), we obtain x ∈ U ⊆ cl ([[ ϕ ]] U,V ), meaning that ( x, U, V ) | = ✸ ϕ .The completeness result for EL K, ✷ ,B +(wF) follows similarly to the above case: the only key stepwe need to show is that the canonical epistemic scenario ( x, [ x ] , R ( x )) of the system EL K, ✷ ,B + (D B )is dense. Lemma 32.
Let X c = ( X c , T c , ν c ) be the canonical model of the system EL K, ✷ ,B + (wF). Then,for all x ∈ X c , we have that R ( x ) is dense in [ x ] , i.e., that [ x ] ⊆ cl ( R ( x )) .Proof. Let x ∈ X c and y ∈ [ x ]. We want to show that y ∈ cl ( R ( x )), i.e., for all U ∈ B with y ∈ U ,we should show that U ∩ R ( x ) = ∅ holds. Let U ∈ B such that y ∈ U . By definition of B , the basicopen neighbourhood U can be of the following forms:1. U = R ( z ) ∩ c ✷ ϕ , for some z ∈ X c and ϕ ∈ L K, ✷ ,B ;2. U = [ z ] ∩ c ✷ ϕ , for some z ∈ X c and ϕ ∈ L K, ✷ ,B .However, since y ∈ [ x ] and y ∈ U , we can simply replace the above cases by:1. U = R ( x ) ∩ c ✷ ϕ , for some ϕ ∈ L K, ✷ ,B ;2. U = [ x ] ∩ c ✷ ϕ , for some ϕ ∈ L K, ✷ ,B , respectively.If (1) is the case, the result follows trivially since y ∈ U = R ( x ) ∩ c ✷ ϕ = U ∩ R ( x ).If (2) is the case, U ∩ R ( x ) = ([ x ] ∩ c ✷ ϕ ) ∩ R ( x ) = c ✷ ϕ ∩ R ( x ) (by Lemma 24.3). Therefore, weneed to show that R ( x ) ∩ c ✷ ϕ = ∅ :Consider the set { ψ | Bψ ∈ y } ∪ { ✷ ϕ } . This set is consistent, otherwise, there exists a finitesubset Γ ⊆ { ψ | Bψ ∈ y } such that ⊢ ^ χ ∈ Γ χ → ✸ ¬ ϕ. Then, by normality of B , ⊢ ^ χ ∈ Γ Bχ → B ✸ ¬ ϕ. We also have 24. ⊢ B ✸ ¬ ϕ → ✸✸ ¬ ϕ (wF)2. ⊢ ✸✸ ¬ ϕ → ✸ ¬ ϕ (4 ✷ )3. ⊢ B ✸ ¬ ϕ → ✸ ¬ ϕ CPL: 1, 2Hence, ⊢ ^ χ ∈ Γ Bχ → ✸ ¬ ϕ. Therefore, since Bχ ∈ y for all χ ∈ Γ, we have ✸ ¬ ϕ ∈ y . But we know that ✷ ϕ (:= ¬ ✸ ¬ ϕ ) ∈ y (since y ∈ U = [ x ] ∩ c ✷ ϕ ), contradicting the maximal consistency of y . Therefore, { ψ | Bψ ∈ y } ∪ { ✷ ϕ } is consistent. Moreover, by Lindenbaum’s Lemma, it can be extended to a maximallyconsistent set z . Therefore, as { ψ | Bψ ∈ y } ⊆ z , we have z ∈ R ( y ) = R ( x ) (since y ∈ [ x ], wehave R ( x ) = R ( y ) (by Lemma 24.2)). Moreover, ✷ ϕ ∈ z , i.e., z ∈ c ✷ ϕ . We therefore conclude that z ∈ c ✷ ϕ ∩ R ( x ) = ∅ . Corollary 33.
Let X c = ( X c , T c , ν c ) be the canonical model of the system EL K, ✷ ,B + (wF). Then,for all x ∈ X c , the e-d scenario ( x, [ x ] , R ( x )) is dense. Proposition 34. EL K, ✷ ,B + (wF) is a complete axiomatization of L K, ✷ ,B with respect to the classof all topological subset spaces under e-d semantics for dense e-d scenarios.Proof. Follows from Corollary 33 similarly to the proof of Corollary 26.
A.5 Confident belief
Lemma 35.
Let X be a topological space and A an open subset of X . Then for any B ⊆ X , wehave A ⊆ ∗ B iff A ⊆ cl ( int ( B )) .Proof. First suppose that A cl ( int ( B )). Then there is some x ∈ A and some open set U with x ∈ U and U ∩ int ( B ) = ∅ . Since A is open, so is U ∩ A . In fact, U ∩ A ⊆ cl ( A B ); to see this, takeany y ∈ U ∩ A and any open V containing y and observe that if V ∩ ( A B ) = ∅ , then it followsthat V ∩ A ⊆ B , and therefore y ∈ V ∩ U ∩ A ⊆ int ( B ), so y ∈ U ∩ int ( B ), a contradiction. Wehave therefore shown that int ( cl ( A B )) = ∅ , so A ∗ B .Conversely, suppose that A ∗ B . Then there is some nonempty open set U with U ⊆ cl ( A B ).Note that this implies that U ∩ ( A B ) = ∅ , so in particular there is some x ∈ U ∩ A . Observe that( A ∩ int ( B )) ∩ ( A B ) = ∅ ; as such, ( A ∩ int ( B )) ∩ cl ( A B ) = ∅ , so we must have U ∩ A ∩ int ( B ) = ∅ .It then follows that ( U ∩ A ) ∩ cl ( int ( B )) = ∅ ; this shows that x / ∈ cl ( int ( B )), so since x ∈ A , wehave A cl ( int ( B )).Let α : L K, ✷ ,B → L K, ✷ ,B be the map that replaces every occurence of B with B ✸✷ . Lemma 36.
For all topological subset models X and every e-d scenario ( x, U, V ) therein, we have ( X , x, U, V ) |≈ ϕ iff ( X , x, U, V ) | = α ( ϕ ) . Proof.
This follows from Lemma 35 using structural induction on ϕ . Lemma 37.
For all ϕ ∈ L K, ✷ ,B , if ⊢ EL K, ✷ ,B α ( ϕ ) , then ⊢ EL K, ✷ ,B +(CB) ϕ . roof. This follows by structural induction on ϕ using the easy fact that ⊢ EL K, ✷ ,B + (CB) Bϕ ↔ B ✸✷ ϕ . Theorem 38. EL K, ✷ ,B + (CB) is a complete axiomatization of L K, ✷ ,B with respect to the class ofall topological subset spaces under e-d semantics using the semantics given above: for all formulas ϕ ∈ L K, ✷ ,B , if |≈ ϕ , then ⊢ EL K, ✷ ,B +(CB) ϕ .Proof. Suppose that |≈ ϕ . Then by Lemma 36 we know that | = α ( ϕ ). By Corollary 26, then, wecan deduce that ⊢ EL K, ✷ ,B α ( ϕ ), and so by Lemma 37 we obtain ⊢ EL K, ✷ ,B +(CB) ϕϕ