TY - JOUR
AU - Zikos,, Yorgos
AB - Abstract The actions of an agent operating in a complex environment are based on her knowledge and beliefs. Epistemic states are typically incomplete, thus the agent has often to estimate whether a fact is true or false before proceeding to actions. We introduce a modal framework for reasoning about Knowledge, Belief and Estimation, three attitudes involved in an agent’s decision-making process. In this modal account, Knowledge and Belief are captured by |$\mathbf{S4.2}$| which has been advocated as a ‘correct’ logic of knowledge by W. Lenzen and R. Stalnaker. Estimation is a non-normal modal operator interpreted as a ‘majority’ quantifier; the approach employs the ‘weak filters’, a general notion of ‘big’ subsets introduced within KR by K. Schlechta and V. Jauregui. Its axiomatization reveals that this ‘majority’ operator has been introduced by J. Burgess (1969) and used also by A. Herzig (2003). We provide a full account of the logic |$\mathbf{KBE}$| in which estimation is complete: exactly one of |$\varphi$| and |$\neg\varphi$| is estimated to be true. We prove soundness and completeness with respect to a class of frames combining relational Kripke frames with Scott-Montague semantics in which neighbourhoods are ‘weak ultrafilters’ and present a tableaux proof procedure. It comes out that believing |$\varphi$| can be equivalently defined in |$\mathbf{KBE}$| as ‘estimating that |$\varphi$| is known’, an interesting fact and an indication of the intuitive correctness of the introduced estimation operator. The assumption on complete estimation is rather strong; yet, it is not hard to define |$\mathbf{KBiE}$|⁠, a relaxed variant in which estimation is still consistent but not complete. We provide the technical details for soundness and completeness with respect to the larger class of frames where neighbourhoods are weak filters. Finally, it is proved that |$\mathbf{KBiE}$| invalidates a rule introduced in W. Lenzen’s probabilistic analysis of weak belief and thus the weak filter semantics is essentially different than its probabilistic counterpart. 1 Introduction The various logics of Knowledge and Belief, mainly based on relational or Scott-Montague possible-worlds semantics, have found very important applications in Knowledge Representation, Distributed Computing, Security and Cryptography, Game Theory and Economics. Within Knowledge Representation (KR) the importance of these two notions can be hardly overstated: ‘An agent’s choice of actions at any point in time can, however, be based only on its local knowledge and beliefs’ declares the opening statement of Y. Shoham’s article in the Handbook of KR [37]. This very phrase, implies already the distinction between the two notions and creates the space for introducing the questions on their relation and interaction, which have always been very important in AI and Philosophy. On the other hand, it is natural to ask whether knowledge and belief suffice to guide the decision-making process of an agent acting in a complex environment. Given the fact that an agent typically reasons in terms of incomplete information, it is natural to consider that its epistemic state is incomplete; the same for its belief set. In the absence of knowledge, the agent can proceed to an estimation on the truth (or falsity) of a certain fact, in view of the available evidence and in several situations where a decision has to be taken at any rate, this estimation should necessarily be accomplished. As it is the case with most of the propositional attitudes (as philosophers prefer to call these notions), there exist many variants of ‘estimation’ encountered in the decision-making processes of human reasoners. This is also true in the restricted case of scientific experimentations in Robotics and Control. Yet the interaction of Knowledge, Belief and Estimation crops all over, implicitly or explicitly: The CEO of a large e-company, in addressing the shareholders, estimates that the new futuristic delivery-by-drone service, will be available in less than two years. He knows that this is a strong argument in favour of the company, given that a success in this project will firmly establish the company’s position. He declares his belief that it will cause a sales rise by 40%. Talos, the small indoor robot, knows that objects of size less than X, can be safely avoided in normal circumstances. It estimates that the motion and velocity of the identified obstacle allows for an avoidance-plan generation and it believes that the size of the object will be verified to be at most X. Stella, a graduate student, has just taken a difficult exam. She managed to answer only 5 (out of 10) questions and she knows that it is the last but one course in order to fulfill the requirements for the degree. She believes that there will be a ‘loose’ grading policy due to the difficulty of the exam and she estimates that she has been successful. That is, she can proceed to prepare for PhD and funding applications. As these examples suggest, there exists a fine interaction between knowledge, belief and estimation. In this article, we introduce |$\mathbf{KBE}$|⁠, a multi-modal logic intended to capture their relation and interaction and |$\mathbf{KBiE}$| which is a more relaxed variant of this logic, accounting for not necessarily complete estimation. We work in the frame of |$\mathbf{S4.2}$| which has been advocated by W. Lenzen as the ‘correct’ logic of knowledge; the reader is referred to Section 2 on the epistemic importance of |$\mathbf{S4.2}$|. Thus, in |$\mathbf{KBE}$|⁠, knowledge is a normal |$\mathbf{S4}$| operator, (consistent) belief is actually a |$\mathbf{KD45}$| operator, estimation is a non-normal operator and the interaction among them is described by a set of ‘bridge’ axioms. Estimation is intended to capture the intuition that the agent can estimate that |$\varphi$| is true (in the sense that she ‘bets’ on its truth rather than its falsity) in the case |$\varphi$| is true in ‘many’ alternative situations to the one the agent is situated in. The axiomatization of |$\mathbf{KBE}$| comprises four ‘bridge’ axioms that pin down the relationship of estimation to knowledge and belief, as suggested by our intuition on what ‘estimation’ actually means. In |$\mathbf{KBE}$| we assume that estimation is consistent and complete having in mind applications in which the decision-making process requires that an ‘estimation’ is necessarily reached. We prove some ‘introspective’ properties of estimation and it turns out that, in |$\mathbf{KBE}$|⁠, belief can be equivalently defined in terms of estimation and knowledge: believing that |$\varphi$| is true amounts exactly to estimating that the agent knows |$\varphi$|⁠. This is clearly close to our intuition on ‘estimation’ as a weak version of belief, which traditionally comes in many facets and many variants. Regarding the model theory of |$\mathbf{KBE}$|⁠, we work with a class of frames which combine a subclass of |$\mathbf{S4.2}$| relational frames (those with a final cluster), endowed with Scott-Montague semantics; more on this in Section 3.2. Given the intuition that estimating |$\varphi$| means that |$\varphi$| holds in ‘many’ (a ‘large’ set of) epistemic alternatives, we have opted for employing weak ultrafilters, the completed version of the ‘weak filters’ introduced independently by K. Schlechta [34] and V. Jauregui [24] as an embodiment of a collection of ‘large’ sets. We provide soundness and completeness results for |$\mathbf{KBE}$|⁠. Given then the very strong assumption on the completeness of estimation, we proceed to define a relaxed version, the logic |$\mathbf{KBiE}$|⁠, by dropping the completeness assumption. The model-theoretic vehicle now is the class of weak filters whose choice simplifies a lot the technical constructions used in |$\mathbf{KBE}$|⁠. The study of the relationship between knowledge and belief has its origins in Plato’s Meno and Theaetetus. In general, knowledge is considered to be different from belief. Following Hintikka’s seminal work [22] epistemic logic has blossomed. Important contributions on the relationship between knowledge and belief include W. Lenzen’s [28] and R. Stalnaker’s work [38], several papers by J. Halpern, including [19, 20] and the Logic of Objective Knowledge and Rational Belief of F. Voorbraak [43]. In contrast to the discussions in Epistemology and Philosophy on the ‘appropriate’ logic for knowledge and belief, logic-based KR is interested in identifying the relation between the various notions and putting them to use [4]. Succinctly stated: ‘a rich repertoire of epistemic and doxastic attitudes have been identified and analyzed in the epistemic logic literature. The challenge for a logician is not to argue that one particular account of belief or knowledge is primary, but, rather, to explore the logical space of definitions and identify interesting relationships between the different notions.’ ([32], see also [4]). In this line of research, our logic attempts to elucidate the relationship between classical epistemic reasoning (in the form of |$\mathbf{S4.2}$|⁠) and a natural notion of ‘estimation’ which is useful in Knowledge Representation. The article is organized as follows: in Section 2, we gather the background material for the many notions involved in the logic we introduce. Section 3 comprises the axiomatization and the possible-worlds semantics of the logic |$\mathbf{KBE}$|⁠, followed by soundness and completeness theorems, along with a discussion on some |$\mathbf{KBE}$| non-theorems. In Section 4 we define |$\mathbf{KBiE}$|⁠, a variant of |$\mathbf{KBE}$| in which estimation is not complete; we sketch the necessary modifications for a proof of soundness and completeness with respect to a simpler semantics based on weak filters. In subsection 4.2 we examine the relation of |$\mathbf{KBiE}$| to Lenzen’s probabilistic semantics for weak belief [29]. Then, in Section 5 we provide a prefix tableaux decision procedure for |$\mathbf{KBE}$| based on the structure of the logic’s model theory. Finally, we conclude in Section 6 with a review of related work and some interesting questions for future research. 2 Background material We assume that the reader has a working knowledge of Modal Logic, as presented in the books [8, 15]. The reader is referred to [3, 14, 38] for a tour in epistemic logic. The seminal work of W. Lenzen in [27, 28] had a significant influence in the field, in particular with respect to the acceptability of introspective properties, the relationship between knowledge and belief and the epistemic importance of |$\mathbf{S4.2}$|⁠. The language |$\mathcal{L}_{\mathsf{K}}$| of propositional epistemic logic — built over a set |$\Phi$| of propositional variables (atomic propositions, atoms) — is endowed with an epistemic operator |${\mathsf{K}}$|⁠; if the attitude considered is belief, it is usually written as |${\mathsf{B}}$|⁠. The formula |${\mathsf{K}}\varphi$| reads as ‘it is known that |$\varphi$|’ and |${\mathsf{B}}\varphi$| reads as ‘it is believed that |$\varphi$|’. Normal Epistemic Logics are defined as sets of |$\mathcal{L}_{\mathsf{K}}$|-formulas that contain classical propositional logic, all the instances of schema |$\mathbf{K}.\; {\mathsf{K}}\varphi\land{\mathsf{K}}(\varphi\supset\psi)\supset{\mathsf{K}}\psi$|⁠, and are closed under the rules Uniform Substitution as well as |$\displaystyle \mathbf{MP}.\,\frac{\varphi,\varphi\supset\psi}{\psi}$| and |$\displaystyle\mathbf{RN}_{{\mathsf{K}}}.\,\frac{\varphi}{{\mathsf{K}}\varphi}$|⁠. We denote by |$\mathbf{KA_1\ldots A_n}$|⁠, the smallest normal modal logic containing the axiom schemata |$\mathbf{A_1}$| to |$\mathbf{A_n}$|⁠. Unlike modal logic texts, we will occasionally follow the convention of subscripting modal languages, modal axioms, rules and logics by the modal operator(s) of the language employed, in an attempt to improve readability. For instance, |$\mathbf{K}_{{\mathsf{K}}}$| denotes the version of axiom |$\mathbf{K}$| for the epistemic modality, |$\mathbf{S4}_{{\mathsf{K}}}$| denotes the well-known logic |$\mathbf{S4}$| in the language |$\mathcal{L}_{{\mathsf{K}}}$| of epistemic logic, while |$\mathbf{KD45}_{{\mathsf{B}}}$| denotes the well-known |$\mathbf{KD45}$| in the language |$\mathcal{L}_{{\mathsf{B}}}$| of doxastic logic. This convention appears to be handy when logics that combine knowledge, belief and other attitudes come into play. A ‘basic’ epistemic logic is |$\mathbf{S4}_{{\mathsf{K}}}$|⁠, alias |$\mathbf{KT4}$|⁠. It is axiomatized by |$\mathbf{K_{{\mathsf{K}}}.}\,{\mathsf{K}}\varphi\land{\mathsf{K}}(\varphi\supset\psi)\supset{\mathsf{K}}\psi$| (The consequences of knowledge constitute knowledge), |$\mathbf{T_{{\mathsf{K}}}.}\,{\mathsf{K}}\varphi\supset\varphi$| (Only true facts are known) and |$\mathbf{4_{{\mathsf{K}}}.} \, {\mathsf{K}}\varphi\supset{\mathsf{K}}{\mathsf{K}}\varphi$| (Positive introspection with respect to knowledge). Another important logic is doxastic |$\mathbf{KD45_{\mathsf{B}}}$|⁠, axiomatized by |$\mathbf{K_{\mathsf{B}}}$|⁠, |$\mathbf{4_{\mathsf{B}}}$| and the axioms |$\mathbf{D_{\mathsf{B}}.}\,{\mathsf{B}}\varphi\supset\neg{\mathsf{B}}\neg\varphi$| (Belief is consistent, called |$\mathbf{CB}$| by R. Stalnaker [38]) and |$\mathbf{5_{\mathsf{B}}.}\, \neg{\mathsf{B}}\varphi\supset{\mathsf{B}}\neg{\mathsf{B}}\varphi$| (Negative introspection with respect to belief). Logics of Knowledge and Belief. The logics combining knowledge and belief employ a bimodal language |$\mathcal{L}_{{\mathsf{K}}{\mathsf{B}}}$| and require the bridge axioms or interaction axioms that attempt to capture the interaction between the two attitudes (see [4] for a fine analysis of interaction axioms). There exist several bridge axioms in the literature. Following is a list of the ones we will use, with the name R. Stalnaker uses in [38]; inside the parenthesis is the name used by W. Lenzen for the same axiom. |$\mathbf{KB.}\, {\mathsf{K}}\varphi\supset{\mathsf{B}}\varphi$| — knowledge implies belief (⁠|$\mathbf{B1}$|⁠, entailment property in [20]), |$\mathbf{B2.3}\ {\mathsf{B}}\varphi\supset\neg{\mathsf{B}}\neg{\mathsf{K}}\varphi$| — assuming that something is believed to be true, it cannot be the case that it is believed not to be known1, |$\mathbf{PIB.}\ {\mathsf{B}}\varphi\supset{\mathsf{K}}{\mathsf{B}}\varphi$| — positive introspection regarding belief (⁠|$\mathbf{B2.4}$|⁠), |$\mathbf{NIB.}\ \neg{\mathsf{B}}\varphi\supset{\mathsf{K}}\neg{\mathsf{B}}\varphi$| — negative introspection regarding belief, |$\mathbf{SB.}\ {\mathsf{B}}\varphi\supset{\mathsf{B}}{\mathsf{K}}\varphi$| —‘strong belief’, ‘subjective certainty’ (⁠|$\mathbf{B2.1}$|⁠). The epistemic importance of |$\mathbf{S4.2}$|. Consider the axiom |$\mathbf{G.}\ \neg{\mathsf{K}}\neg{\mathsf{K}}\varphi\supset{\mathsf{K}}\neg{\mathsf{K}}\neg\varphi$| and the axiom |$\mathbf{DB.}\ {\mathsf{B}}\varphi\equiv\neg{\mathsf{K}}\neg{\mathsf{K}}\varphi$| which essentially defines belief in terms of knowledge. We will use the logic(s) |$\mathbf{S4.2} = \mathbf{S4}+\mathbf{G}$| (in the language |$\mathcal{L}_{\mathsf{K}}$|⁠) and |$ \mathbf{S4.2_{{\mathsf{K}}{\mathsf{B}}}}$| (in the language |$\mathcal{L}_{{\mathsf{K}}{\mathsf{B}}}$|⁠). It is the same modal logic of course, but we reserve the original name |$\mathbf{S4.2}$| for the monomodal epistemic logic in which |${\mathsf{B}}\varphi$| is just an abbreviation for |$\neg{\mathsf{K}}\neg{\mathsf{K}}\varphi$|⁠, and |$\mathbf{S4.2_{{\mathsf{K}}{\mathsf{B}}}}$| for the bimodal epistemic logic, in which belief is axiomatized by |$\mathbf{DB.}$| It is merely a notational convenience, whose purpose is explained by the following Proposition, which gathers two important results on the epistemic significance of |$\mathbf{S4.2}$|⁠. Proposition 2.1 S4+KD45B+B1+B2.3+B2.4=S4.2KB[28,Lenzen]S4+CB+KB+SB+PIB+NIB=S4.2KB[38,Stalnaker] We further assume that the reader is well aware of the basic model theory of modal (and epistemic) logic. Normal modal logics are interpreted over Kripke (or relational) possible-worlds models: a Kripke model|${\mathfrak{M}}={\langle{W}{{\mathcal{R}}}{V}\rangle}$| consists of a non-empty set |$W$| of possible worlds (states, situations), a binary accessibility relation between them |${\mathcal{R}}\subseteq W\times W$| and a valuation |$V: \Phi \rightarrow 2^{W}$| assigning to each propositional variable from |$\Phi$| a set of worlds. The pair |${\mathfrak{F}}={\langle{W}{{\mathcal{R}}}\rangle}$| is called the frame underlying |${\mathfrak{M}}$|⁠. We denote by |${\mathcal{R}}(w)$|the set of epistemically alternative states to|$w$|⁠: |${\mathcal{R}}(w)=\{v\in W\ |\ w{\mathcal{R}} v\}$|⁠. Logics with the axiom |$\mathbf{4}$| are called transitive logics. A fine analysis of the structure of transitive frames (and their logics) is carried out through the notion of a cluster: a maximal subset of the frame’s states, within which the accessibility relation is universal (or the degenerate case of a single, irreflexive world). The reader should consult [35] and [15, Chapter 8] for details. To obtain logics weaker than normal, one has to resort to the Scott-Montague semantics, also called neighbourhood semantics [10, 31, 35]. A neighbourhood model is a triple |${\mathfrak{N}}={\langle{W}{{\mathcal{N}}}{V}\rangle}$|⁠, where |$W$| is a set of possible worlds (states), |${\mathcal{N}}: W\to\mathcal{P}(\mathcal{P}(W))$| is a neighbourhood function assigning to a state its ‘neighbourhood’ and |$V$| is a valuation. Inside a state, formulas of the form |${\mathsf{K}}\varphi$| become true at |$w$| iff the set of states |$\overline{V}(\varphi)$| where |$\varphi$| holds (called the truth set of |$\varphi$|⁠, |$\overline{V}(\varphi)=\{v\in W\ |\ {\mathfrak{N}},v\Vdash\varphi\}$|⁠), belongs to the neighbourhood of |$w$|⁠: |$\overline{V}(\varphi)\in{\mathcal{N}}(w)$|⁠. The pair |$\mathfrak{F}={\langle{W}{{\mathcal{N}}}\rangle}$| is called a Scott-Montague (neighbourhood) frame. A logic |$\Lambda$| is determined by a class of frames iff it is sound and complete with respect to this class. Now, about the frames of |$\mathbf{S4.2}$|⁠: the axiom |$\mathbf{G}$| canonically corresponds to the class of frames with the weak directedness or Church-Rosser property: (∀w,v,u∈W)(wRv & wRu⟹(∃s∈W)(vRs & uRs)) and it is well-known that this logic is determined by the class of reflexive, transitive and weakly-directed frames. It turns out [15, p. 30] that in |$\mathbf{S4.2}$|⁠, weak directedness can be substituted for directedness: |$(\forall v,u\in W)(\exists s\in W)(v{\mathcal{R}} s\ \&\ u{\mathcal{R}} s)$| and it is also known that |$\mathbf{S4.2}$| is also determined by the class of finite |$\mathbf{S4.2}$|-frames, in which every cluster is non-degenerate and there exists a final cluster. Note that, a final cluster need not exist in every |$\mathbf{S4.2}$|-frame: take for instance the frame |$\langle\omega,\leq\rangle$|⁠, in which each cluster is simple and there is no final cluster. In [26] it is also proved that |$\mathbf{S4.2}$| is determined by the class of reflexive and transitive frames with a final cluster (which are always directed) and it is this characterization which will be very useful to us. Notation. Given a consistent logic |$\Lambda$|⁠, |$\Lambda$|-maximal consistent sets (⁠|$mc\Lambda$|-sets) do exist, by a standard Lindenbaum argument. We denote by |$[\varphi]_\Lambda$| (or by |$[\varphi]$|⁠, when |$\Lambda$| is understood) the set of all |$mc\Lambda$|-sets, which contain |$\varphi$|⁠. We use small-caps Greek capital characters (⁠|${{\mathsf{\scriptstyle \Gamma}}}, {{ \mathsf{\scriptstyle \Delta}}}$| etc.) for denoting maximal consistent sets. We use the notion of strong completeness: given a consistent logic |$\Lambda$| and a class of frames |$\mathsf{S}$|⁠, we say that |$\Lambda$| is strongly complete with respect to |$\mathsf{S}$| iff for every theory |$I$| consistent with |$\Lambda$| and formula |$\varphi$|⁠, if |$I$| semantically entails |$\varphi$|⁠, then |$\varphi$| is deducible in |$\Lambda$| from |$I$|⁠. To prove strong completeness with respect to |$\mathsf{S}$|⁠, it suffices to show that every |$\Lambda$|-consistent theory |$I$| is satisfiable in a frame from |$\mathsf{S}$| (see [8]). Large Sets of worlds. To cope with ‘large’ sets of worlds, we will use the notion introduced by V. Jauregui [24]: a non-empty collection |$F$| of subsets of |$W$| is a collection of large subsets iff (i)|$X \in F$| and |$X \subseteq Y \subseteq W$| implies |$Y \in F$| (|$F$| is upwards closed), (ii)|$X \notin F\ $| or |$\ (W \setminus X) \notin F$| (it cannot be the case that both a set and its complement are large). In [33] a different, but provably equivalent, notion of large sets had been given by K. Schlechta: it replaces the second condition with the requirement that, |$X \in F$| and |$Y \in F$| implies that |$X \cap Y \neq\emptyset$| (There cannot be disjoint large sets). Throughout this article we will be freely switching between the two definitions. K. Schlechta’s definition makes clear why this can be considered as a notion of ‘weak filter’, a loose version of the ‘filters’ used in model theory (a filter requires closure under intersections, not only pairwise disjointness). In defining the models of |$\mathbf{KBE}$| we need to extend this by a requirement of completeness, proceeding to a notion of ‘weak ultrafilter’. This, strengthens Jauregui’s definition which requires that it cannot be the case that both a set and its complement are ‘large’ (but, conceivably none), by requesting that exactly one of them should be ‘large’. This introduces a notion of a ‘complete’ collection of large subsets in a very interesting way. Weak Ultrafilters are a proper superclass of classical ultrafilters and possess many interesting set-theoretic properties. We are going to prove that for every weak filter |$F$| over a set |$W$|⁠, there exists a weak ultrafilter |$U$| over |$W$| which extends |$F$|⁠. The proof uses the Teichmüller–Tukey lemma, an equivalent of the Axiom of Choice. Definition 2.2 [1, 24, 33] Consider sets |$W\neq\emptyset$|⁠, |$Z\subseteq W$|⁠, |$F\subseteq\mathcal{P}(W)$| and the following properties: |$\mathbf{(wf1)}$||$\ \ W\in F$| |$\mathbf{(wf2)}$||$\ \ (\forall X\in F)(\forall Y\subseteq W)(X\subseteq Y\Longrightarrow Y\in F)$| |$\mathbf{(wf3)}$||$\ \ (\forall X\subseteq W)(X\in F\Longrightarrow W\setminus X\notin F)$| |$\mathbf{(wuf)}$||$\ \ (\forall X\subseteq W)(X\notin F\Longrightarrow W\setminus X\in F)$| |$\mathbf{(inZ)}$||$\ \ (\forall A,D\in F)\;\; (A\cap D\cap Z\neq\emptyset)$| If (wf1) to (wf3) hold for |$F$|⁠, then it is called a weak filter over |$W$|. If (wuf) holds for a weak filter |$F$|⁠, then it is called a weak ultrafilter over |$W$|. If (inZ) holds for a weak filter |$F$| then it is called a weak filter over |$W$| with intersections in |$Z$|. A non-empty set |$C$| is said to be of finite character iff [|$S\in C\Longleftrightarrow$| each finite subset of |$S$| is in |$C$|]. The following lemma is equivalent to the Axiom of Choice. Lemma 2.3 (Teichmüller-Tukey Lemma) Every set of finite character has an inclusion maximal element. Proposition 2.4 Consider sets |$W\neq\emptyset$|⁠, |$Z\subseteq W$| and a weak filter |$F$| over |$W$| with intersections in |$Z$|⁠. Then, assuming the Axiom of Choice, there exists a weak ultrafilter |$U$| over |$W$| with intersections in |$Z$|⁠, extending |$F$|⁠. Proof. We define |$C=\{S\in\mathcal{P}({{\mathcal{P}}}(W))\mid\forall A,D\in S\cup F\;\; A\cap D\cap Z\neq\emptyset\}$|⁠. Observe that |$C\neq\emptyset$| (because |$F\in C$|⁠) and that it is of finite character (any two subsets |$A\mbox{ and }D$| of |$W$| either have the property or not). Let us call |$U$| the maximal member provided by the Teichmüller-Tukey lemma. We will prove that |$U$| extends |$F$| and has all properties (wf1),(wf2),(wf3),(wuf),(inZ). |$\mathbf{U\supseteq F}$|: If a member of |$F$| was not in |$U$| we could add it (otherwise |$U$| would not have the property |$\forall A,D\in U\cup F\;\; A\cap D\cap Z\neq\emptyset$|⁠) and find a bigger set. (wf1):|$W\in F$| and |$U\supseteq F$| (wf2): If |$A,D\in U\mbox{ and }B\supseteq A$|⁠, we have (wf3): If |$A,\overline{A}\in U$| then |$A\cap\overline{A}\cap Z=\emptyset$|⁠. A contradiction. (wuf): Let |$A\subseteq W$| and suppose |$U$| includes neither a |$A$| nor |$\overline{A}$|⁠. Then it must be the case that |$\exists D,D'\in U$| s.t. |$A\cap D\cap Z=\emptyset$| and |$\overline{A}\cap D'\cap Z=\emptyset$|⁠. In other words |$w\in D\cap D'\Rightarrow(w\notin A\mbox{ or }w\notin Z)\;\&\;(w\in A\mbox{ or }w\notin Z)$|⁠. If |$w\notin Z$| we have a contradiction (because |$D,D'\in U$| so |$D\cap D'\cap Z\neq\emptyset$|⁠). If |$w\notin A$| and |$w\in A$| the contradiction is obvious. (inZ): It suffices to observe that |$U\cup F=U$|⁠. ■ An easy corollary follows, given the observation that a weak ultrafilter extending |$F$| includes either |$S$| or |$W\backslash S$|⁠. Corollary 2.5 Consider sets |$W\neq\emptyset,\;\;Z,S\subseteq W$| and a weak filter |$F$| over |$W$| with intersections in |$Z$|⁠. Then, assuming the axiom of choice, there exists a weak ultrafilter |$U$| over |$W$| with intersections in |$Z$|⁠, extending |$F\cup\{S\}$| or |$F\cup\{W\backslash S\}$|⁠. 3 The logic $\mathbf{KBE}$ The logic |$\mathbf{S4.2}$| has been advocated by W. Lenzen as the ‘correct’ logic of knowledge, as, by Proposition 2.1 it contains practically every one of the ‘plausible’ principles governing knowledge, belief, and their interaction. Let us note again that this is more clear with bimodal |$\mathbf{S4}+\mathbf{KD45_{\mathsf{B}}}+\mathbf{B1}+\mathbf{B2.3}+\mathbf{B2.4}$|⁠, but this essentially coincides with |$\mathbf{S4.2_{{\mathsf{K}}{\mathsf{B}}}}=\mathbf{S4.2}+\mathbf{DB}$|⁠. The axiom |$\mathbf{SB.}\ {\mathsf{B}}\varphi\supset{\mathsf{B}}{\mathsf{K}}\varphi$| belongs to |$\mathbf{S4.2_{{\mathsf{K}}{\mathsf{B}}}}$|⁠, and thus, the notion of belief embedded in |$\mathbf{S4.2_{{\mathsf{K}}{\mathsf{B}}}}$| is quite strong, hence the name ‘strong belief’ by R. Stalnaker and W. Lenzen. Given the discussion in the introductory section, it seems appropriate to enrich |$\mathbf{S4.2+DB}$| with a modal ‘estimation’ operator. Let us write down a few intuitive requirements, that this operator viewed as a particular form of weak belief should fulfill. (i) a ‘strong belief’ phenomenon should be avoided: |${\mathsf{E}}\varphi\supset{\mathsf{E}}{\mathsf{K}}\varphi$| is hardly acceptable. (ii) estimation should correspond to a ‘degree of certainty’, somewhat weaker than belief: |${\mathsf{B}}\varphi\supset{\mathsf{E}}\varphi$| is desirable. (iii) estimation should be able to be distributed through knowledge (see axiom |$\mathbf{EK}$| below). (iv) introspection of estimation with respect to knowledge is naturally acceptable (see axiom |$\mathbf{PIE}$| below). 3.1 Axiomatization of $KBE$ |$\mathbf{S4.2}+\mathbf{DB}$| is a normal, monomodal |${\mathsf{K}}$|-logic, in which, belief |${\mathsf{B}}$| is an abbreviation. |$\mathbf{KBE}$| is based on it; since |$\mathbf{DB}$| is an abbreviation, we will simply refer to |$\mathbf{S4.2}$|⁠. We consider the propositional bimodal language |$\mathcal{L}_{KBE}$| with the propositional variables |$\Phi=\{p_0,p_1,\ldots\}$|⁠, the falsum|$\bot$|⁠, the implication connective |$\supset$| (all the other propositional connectives can be defined) and the modal operators |${\mathsf{K}}$| and |${\mathsf{E}}$|⁠. The intended interpretation is that |${\mathsf{K}}\varphi$| reads as ‘the agent knows|$\varphi$|’, |${\mathsf{B}}\varphi$| (we keep in mind that this is an abbreviation) reads as ‘the agent believes|$\varphi$|’, |${\mathsf{E}}\varphi$| reads as ‘the agent estimates that |$\varphi$| is true’. We proceed now to list the axioms of |$\mathbf{KBE}$|⁠, including the abbreviation for belief. Abbreviation $\mathbf{DB.}$$\ {\mathsf{B}}\varphi\equiv\neg{\mathsf{K}}\neg{\mathsf{K}}\varphi$ Belief definition. Axioms $\mathbf{K.}$$\ {\mathsf{K}}\varphi\land{\mathsf{K}}(\varphi\supset\psi)\supset{\mathsf{K}}\psi$ Knowledge is closed under logical consequence. $\mathbf{T.}$$\ {\mathsf{K}}\varphi\supset\varphi$ Only true things are known. $\mathbf{4.}$$\ {\mathsf{K}}\varphi\supset{\mathsf{K}}{\mathsf{K}}\varphi$ Positive introspection, with respect to knowledge. $\mathbf{CB.}$$\ {\mathsf{B}}\varphi\supset\neg{\mathsf{B}}\neg\varphi$ Belief is consistent. $\mathbf{BE.}$$\ {\mathsf{B}}\varphi\supset{\mathsf{E}}\varphi$ Beliefs are estimations. $\mathbf{CCE.}$$\ {\mathsf{E}}\varphi\equiv\neg{\mathsf{E}}\neg\varphi$ Estimation is consistent and complete. $\mathbf{EK.}$$\ {\mathsf{E}}\varphi\land{\mathsf{K}}(\varphi\supset\psi)\supset{\mathsf{E}}\psi$ Estimation can be safely inferred through knowledge. $\mathbf{PIE.}$$\ {\mathsf{E}}\varphi\supset{\mathsf{K}}{\mathsf{E}}\varphi$ Introspection with respect to estimation. Definition 3.1 |$\mathbf{KBE}$| is the propositional bimodal logic axiomatized by |$\mathbf{K}$|⁠, |$\mathbf{T}$|⁠, |$\mathbf{4}$|⁠, |$\mathbf{CB}$|⁠, |$\mathbf{BE}$|⁠, |$\mathbf{CCE}$|⁠, |$\mathbf{EK}$|⁠, |$\mathbf{PIE}$| and closed under the rule |$\displaystyle\mathbf{RN}_{{\mathsf{K}}}.\,\frac{\varphi}{{\mathsf{K}}\varphi}$|⁠. |$\mathbf{KBE}$| is normal with respect to |${\mathsf{K}}$| and this readily implies (cf. [10]) Fact 3.2 |$\mathbf{KBE}$| is closed under the rule |$\displaystyle\mathbf{RM}_{{\mathsf{K}}}.\,\frac{\varphi\supset\psi}{{\mathsf{K}}\varphi\supset{\mathsf{K}}\psi}$|⁠. On the other hand, we will later show that |$\mathbf{KBE}$| is not normal with respect to |${\mathsf{E}}$|⁠. Yet, the following closure results wrt |${\mathsf{E}}$| do hold. Fact 3.3 |$\mathbf{KBE}$| is closed under the rules Proof. About |$\mathbf{RN}_{{\mathsf{E}}}$|⁠, note that |$\mathbf{KB}$| is a theorem of |$\mathbf{S4.2}+\mathbf{DB}$| (Proposition 2.1) and thus of |$\mathbf{KBE}$|⁠. The result follows immediately from |$\mathbf{RN}_{{\mathsf{K}}}$| and axiom |$\mathbf{BE}$|⁠. |$\mathbf{RM}_{{\mathsf{E}}}$|⁠, is certified by the next derivation: 1.φ⊃ψhypothesis2.K(φ⊃ψ)1,RNK3.Eφ⊃Eφ∧K(φ⊃ψ)2,PC,MP4.Eφ∧K(φ⊃ψ)⊃EψEK5.Eφ⊃Eψ3,4,PC,MP ■ The axiom |$\mathbf{PIE}$| describes an intuitively acceptable property of estimation: as soon as the agent is able to make an estimation, she should be able to know it. It is clearly worth asking what happens with introspection on non-estimations: if the agent does not estimate something, should she be certain that she does not estimate it? In addition, it is worth checking how does |$\mathbf{KBE}$| behave with respect to the property of ‘negative certainty’ [20], corresponding to the axiom |$\neg{\mathsf{B}}\varphi\supset{\mathsf{B}}\neg{\mathsf{K}}\varphi$|⁠, which in turn, belongs to Voorbraak’s ‘logic of objective knowledge and rational belief’ (⁠|$\mathbf{OK\&ORIB}$|⁠, [43]). Proposition 3.4 (i) Positive ‘Introspection’ wrt estimation can be proven for all three epistemic ‘degrees’: |${\mathsf{E}}\varphi\supset{\mathsf{K}}{\mathsf{E}}\varphi,\ {\mathsf{E}}\varphi\supset{\mathsf{B}}{\mathsf{E}}\varphi,\ {\mathsf{E}}\varphi\supset{\mathsf{E}}{\mathsf{E}}\varphi\ \in\mathbf{KBE}$| (ii) So can negative ‘Introspection’ wrt estimation: |$\neg{\mathsf{E}}\varphi\supset{\mathsf{K}}\neg{\mathsf{E}}\varphi,\ \neg{\mathsf{E}}\varphi\supset{\mathsf{B}}\neg{\mathsf{E}}\varphi,\ \neg{\mathsf{E}}\varphi\supset{\mathsf{E}}\neg{\mathsf{E}}\varphi\ \in\mathbf{KBE}$| (iii) Non-estimation implies introspection wrt ignorance and ‘lack of certainty’: |$\neg{\mathsf{E}}\varphi\supset{\mathsf{K}}\neg{\mathsf{K}}\varphi,\ \neg{\mathsf{E}}\varphi\supset{\mathsf{B}}\neg{\mathsf{K}}\varphi,\ \neg{\mathsf{E}}\varphi\supset{\mathsf{E}}\neg{\mathsf{K}}\varphi\ \in\mathbf{KBE}$||$\neg{\mathsf{E}}\varphi\supset{\mathsf{K}}\neg{\mathsf{B}}\varphi,\ \neg{\mathsf{E}}\varphi\supset{\mathsf{B}}\neg{\mathsf{B}}\varphi,\ \neg{\mathsf{E}}\varphi\supset{\mathsf{E}}\neg{\mathsf{B}}\varphi\ \in\mathbf{KBE}$| (iv) |${\mathsf{K}}{\mathsf{E}}\varphi\equiv{\mathsf{E}}\varphi\ \in\mathbf{KBE}$| (v) |${\mathsf{E}}{\mathsf{K}}\varphi\equiv{\mathsf{B}}\varphi\ \in\mathbf{KBE}$| (vi) Estimation can be also inferred through belief - the analog of |$\mathbf{EK}$| holds for belief: |$\ {\mathsf{E}}\varphi\land{\mathsf{B}}(\varphi\supset\psi)\supset{\mathsf{E}}\psi \ \in\mathbf{KBE}$| Proof. (i) Immediate from |$\mathbf{PIE}$|⁠, |$\mathbf{KB}$| and |$\mathbf{BE}$|⁠. (ii) First, note the following derivation: 1.¬Eφ⊃E¬φCCE,PC,MP2.E¬φ⊃KE¬φPIE3.E¬φ⊃¬EφCCE,PC,MP4.KE¬φ⊃K¬Eφ3,RMK5.¬Eφ⊃K¬Eφ1,2,4,PC,MP The rest,readily follow by |$\mathbf{KB}$| and |$\mathbf{BE}$|⁠. (iii). The second triple of theorems can be found below, in the lines 4, 6 and 8 of the derivation: 1.Bφ⊃EφBE2.¬Eφ⊃¬Bφ1,PC,MP3.¬Bφ⊃K¬BφNIB (theorem of S4.2+DB,Proposition 2.1)4.¬Eφ⊃K¬Bφ2,3,PC,MP5.K¬Bφ⊃B¬BφKB6.¬Eφ⊃B¬Bφ4,5,PC,MP7.B¬Bφ⊃E¬BφBE8.¬Eφ⊃E¬Bφ6,7,PC,MP By |$\mathbf{KB}$| and the rules |$\mathbf{RM}_{\mathsf{K}}$|⁠, |$\mathbf{RM}_{\mathsf{B}}$| and |$\mathbf{RM}_{\mathsf{E}}$| we can obtain the three remaining theorems. (iv) Immediate by |$\mathbf{T}$| and |$\mathbf{PIE}$|⁠. (v) Check the following derivation: 1.¬Bφ⊃K¬BφNIB2.Kφ⊃BφKB3.¬Bφ⊃¬Kφ2,PC,MP4.K¬Bφ⊃K¬Kφ3,RMK5.K¬Kφ⊃B¬KφKB6.B¬Kφ⊃E¬KφBE7.E¬Kφ⊃¬EKφCCE8.¬Bφ⊃¬EKφ1,4,5,6,7,PC,MP9.EKφ⊃Bφ8,PC,MP10.Bφ⊃BKφSB (theorem of S4.2+DB)11.BKφ⊃EKφBE12.Bφ⊃EKφ10,11,PC,MP13.EKφ≡Bφ9,12,PC,MP (vi) Check the following derivation: 1.Eφ⊃KEφPIE2.B(φ⊃ψ)⊃EK(φ⊃ψ)Proposition 3.4.(v)3.Eφ∧K(φ⊃ψ)⊃EψEK4.KEφ⊃K(K(φ⊃ψ)⊃Eψ)3,PC,RMK5.EK(φ⊃ψ)∧K(K(φ⊃ψ)⊃Eψ)⊃EEψEK6.¬Eψ⊃E¬EψProposition 3.4.(ii)7.¬E¬Eψ⊃Eψ6,PC,MP8.EEψ⊃¬E¬EψCCE9.Eφ∧B(φ⊃ψ)⊃Eψ1,4,2,5,8,7,MP ■ Remark 3.5 Note that by Proposition 3.4.(v) above, belief can be equivalently defined as ‘estimation that the agent knows’. Defining knowledge in terms of belief and vice versa, is a very interesting topic in epistemic logic (see [4, 23]). In that respect, it is interesting that belief can be equivalently defined in an |$\mathbf{S4.2}$| framework, in a rather intuitive way, through an ‘estimation’ operator. In the same fashion, item (iv) says that knowledge about estimation amounts exactly to estimation itself.} 3.2 The possible-worlds models of $\mathbf{KBE}$ In this section, we define the frames and models of |$\mathbf{KBE}$|⁠. These structures properly mix the subclass of |$\mathbf{S4.2}$|-frames in which we are interested (the reflexive, transitive, directed frames, with a final cluster FC), with general Scott-Montague semantics, in which each neighbourhood is a complete collection of large sets on the epistemic alternatives of the world at hand. In the following definition, the properties |$\mathbf{(cce)}$| and |$\mathbf{(ek)}$| are essentially |$\mathbf{(wf2)}$|⁠, |$\mathbf{(wf3)}$| and |$\mathbf{(wuf)}$| of Definition 2.2 of weak ultrafilters, properly stated, as we define weak ultrafilters on |${\mathcal{R}}(w)$|⁠. Definition 3.6 Consider the triple |${\mathfrak{F}}={\langle{W}{{\mathcal{R}}}{{\mathcal{N}}}\rangle}$|⁠, where |$W$| is a non-empty set, |${\mathcal{R}}\subseteq W\times W$|⁠, |${\mathcal{N}}:W\to\mathcal{P}(\mathcal{P}(W))$| and |${\mathcal{R}}$| is reflexive, transitive and has a nonempty final cluster |$FC=$||$\{v\in W\ |\ (\forall w\in W)\ w{\mathcal{R}} v\}$|⁠. |${\mathcal{N}}$| is such, that |$\forall w\in W$| |$\mathbf{(nr)}$||$\ \ {\mathcal{N}}(w)\subseteq\mathcal{P}({\mathcal{R}}(w))$| |$\mathbf{(be)}$||$\ \ FC\in {\mathcal{N}}(w)$| |$\mathbf{(pie)}$||$\ \ \forall X\subseteq{\mathcal{R}}(w)\;\forall u\in W\;\big(X\in{\mathcal{N}}(w)\;\&\; w{\mathcal{R}} u\Longrightarrow X\cap{\mathcal{R}}(u)\in{\mathcal{N}}(u)\big)$| |$\mathbf{(cce)}$||$\ \ \forall X\subseteq{\mathcal{R}}(w)\;\big(X\in{\mathcal{N}}(w)\Longleftrightarrow{\mathcal{R}}(w)\setminus X\notin{\mathcal{N}}(w)\big)$| |$\mathbf{(ek)}$||$\ \ \forall X,Y\subseteq{\mathcal{R}}(w)\;\big(X\in{\mathcal{N}}(w)\;\&\; Y\supseteq X\Longrightarrow Y\in{\mathcal{N}}(w)\big)$| |${\mathfrak{F}}$| is called a kbe-frame. |${\mathfrak{M}}={\langle{{\mathfrak{F}}}{V}\rangle}$| is called a kbe-model, if it is based on a kbe-frame and |$V:\Phi\to \mathcal{P}(W)$| is a valuation. Fact 3.7 The class of all kbe-frames is non-empty. Proof. Consider the frame |${\mathfrak{F}}_1={\langle{W}{{\mathcal{R}}}{{\mathcal{N}}}\rangle}$| where |$W=\{w,u_1,u_2,u_3\}$| and |${\mathcal{R}}$| is the relation shown in Figure 1. Furthermore, assume |${\mathcal{N}}(w)=\big\{X\subseteq W\ |\ |X|\geq 3\big\}\cup\big\{\{u_1,u_2\},\{u_1,u_3\},\{u_2,u_3\}\big\}$|⁠, |${\mathcal{N}}(u_1)={\mathcal{N}}(u_2)={\mathcal{N}}(u_3)=\big\{\{u_1,u_2\},\{u_1,u_3\},\{u_2,u_3\},FC\big\}$|⁠, where |$FC=\{u_1,u_2,u_3\}$| is the final cluster of the structure |${\langle{W}{{\mathcal{R}}}\rangle}$|⁠. With a little effort it can be checked that all properties of Definition 3.6 do hold, hence, |${\mathfrak{F}}_1$| is a kbe-frame. ■ Figure 1. Open in new tabDownload slide |$W$| and |${\mathcal{R}}$| of frame |${\mathfrak{F}}_1$|⁠. Figure 1. Open in new tabDownload slide |$W$| and |${\mathcal{R}}$| of frame |${\mathfrak{F}}_1$|⁠. Now, given a model |${\mathfrak{M}}={\langle{W}{{\mathcal{R}}}{{\mathcal{N}},V}\rangle}$| for the language |$\mathcal{L}_{KBE}$|⁠, the valuation |$V:\Phi\to \mathcal{P}(W)$| can be extended to all formulae of |$\mathcal{L}_{KBE}$| in a straightforward way: Definition 3.8 Consider the model |${\mathfrak{M}}={\langle{W}{{\mathcal{R}}}{{\mathcal{N}},V}\rangle}$| for the language |$\mathcal{L}_{KBE}$|⁠. The function |$\overline{V}:\mathcal{L}_{KBE}\to\mathcal{P}(W)$| is defined recursively as follows: |$\overline{V}(p)=V(p),\quad (\forall p\in\Phi)$|⁠, |$\overline{V}(\bot)=\emptyset$|⁠,and |$(\forall\varphi,\psi\in\mathcal{L}_{KBE})$|⁠: |$\overline{V}(\varphi\supset\psi)=(W\setminus\overline{V}(\varphi))\cup\overline{V}(\psi)$|⁠, |$\overline{V}({\mathsf{K}}\varphi)=\{w\in W\ |\ {\mathcal{R}}(w)\subseteq\overline{V}(\varphi)\}$| and |$\overline{V}({\mathsf{E}}\varphi)=\{w\in W\ |\ {\mathcal{R}}(w)\cap\overline{V}(\varphi)\in{\mathcal{N}}(w)\}$|⁠. As usual, we will write |${\mathfrak{M}},w\Vdash\varphi$| instead of |$w\in\overline{V}(\varphi)$|⁠. 3.3 Soundness of ${KBE}$ Theorem 3.9 (Soundness) |$\mathbf{KBE}$| is sound w.r.t. the class of all kbe-frames. Proof. It suffices to show that all axioms of |$\mathbf{KBE}$| are valid in every kbe-frame. So, consider any kbe-frame |${\mathfrak{F}}={\langle{W}{{\mathcal{R}}}{{\mathcal{N}}}\rangle}$|⁠. First, since |${\mathfrak{F}}$| is a Kripke-frame w.r.t. |${\mathsf{K}}$| and the corresponding relation |${\mathcal{R}}$| is reflexive and transitive, we know (e.g., see [8], [15]) that the axioms |$\mathbf{K}$|⁠, |$\mathbf{T}$| and |$\mathbf{4}$| are valid in |${\mathfrak{F}}$|⁠. Next, consider any kbe-model |${\mathfrak{M}}={\langle{{\mathfrak{F}}}{V}\rangle}$| and any world |$w\in W$|⁠. CB. Assume that |${\mathfrak{M}},w\Vdash{\mathsf{B}}\varphi$|⁠, i.e. by |$\mathbf{DB}$|⁠, |${\mathfrak{M}},w\Vdash\neg{\mathsf{K}}\neg{\mathsf{K}}\varphi$|⁠. Then, there must be a |$v\in W$| s.t. |$w{\mathcal{R}} v$| and |${\mathfrak{M}},v\Vdash{\mathsf{K}}\varphi\quad(1)$| Now, let |$u$| be any world s.t. |$w{\mathcal{R}} u$|⁠. Since |$FC\neq\emptyset$|⁠, there is an |$s\in FC$|⁠, and, by the definition of the final cluster |$FC$|⁠, |$u{\mathcal{R}} s$| and |$v{\mathcal{R}} s$|⁠, therefore, by |$(1)$|⁠, |${\mathfrak{M}},s\Vdash\varphi$|⁠, hence, since |$u{\mathcal{R}} s$|⁠, |${\mathfrak{M}},u\Vdash\neg{\mathsf{K}}\neg\varphi$|⁠, so, since |$u$| is an arbitrary world s.t. |$w{\mathcal{R}} u$|⁠, |${\mathfrak{M}},w\Vdash{\mathsf{K}}\neg{\mathsf{K}}\neg\varphi$|⁠, i.e. by |$\mathbf{DB}$|⁠, |${\mathfrak{M}},w\Vdash\neg{\mathsf{B}}\neg\varphi$|⁠. BE. Assume again that |${\mathfrak{M}},w\Vdash{\mathsf{B}}\varphi$|⁠, i.e. |${\mathfrak{M}},w\Vdash\neg{\mathsf{K}}\neg{\mathsf{K}}\varphi$|⁠. Then, there is a |$v\in W$| s.t. |${\mathfrak{M}},v\Vdash{\mathsf{K}}\varphi$|⁠, hence, for every |$u\in FC$|⁠, since |$v{\mathcal{R}} u$|⁠, |${\mathfrak{M}},u\Vdash\varphi$|⁠, so, |${\mathfrak{M}},FC\Vdash\varphi$|⁠, i.e. |$FC\subseteq\overline{V}(\varphi)$|⁠. But, by the definition of FC, |$FC\subseteq{\mathcal{R}}(w)$|⁠, therefore, |$FC\subseteq{\mathcal{R}}(w)\cap\overline{V}(\varphi)$|⁠, hence by |$\mathbf{(be)}$| and |$\mathbf{(ek)}$| of Definition 3.6, |${\mathcal{R}}(w)\cap\overline{V}(\varphi)\in{\mathcal{N}}(w)$|⁠, i.e. by Definition 3.8, |${\mathfrak{M}},w\Vdash{\mathsf{E}}\varphi$|⁠. CCE.|${\mathfrak{M}},w\Vdash{\mathsf{E}}\varphi\stackrel{\textrm{3.8}}{\iff}{\mathcal{R}}(w)\cap\overline{V}(\varphi)\in{\mathcal{N}}(w)\stackrel{\textrm{3.6(cce)}}{\iff}$| |${\mathcal{R}}(w)\cap(W\setminus\overline{V}(\varphi))\notin{\mathcal{N}}(w)\iff$||${\mathcal{R}}(w)\cap\overline{V}(\neg\varphi))\notin{\mathcal{N}}(w)\stackrel{\textrm{3.8}}{\iff}{\mathfrak{M}},w\Vdash\neg{\mathsf{E}}\neg\varphi$|⁠. EK. Suppose that |${\mathfrak{M}},w\Vdash{\mathsf{E}}\varphi\land{\mathsf{K}}(\varphi\supset\psi)$|⁠. Then, by Definition 3.8, |${\mathcal{R}}(w)\cap\overline{V}(\varphi)\in{\mathcal{N}}(w)\quad(1)\ $| and |$\ {\mathcal{R}}(w)\subseteq\overline{V}(\varphi\supset\psi)\quad(2)$| Consider now any |$v\in{\mathcal{R}}(w)\cap\overline{V}(\varphi)$|⁠. Then, |$v\notin\overline{V}(\neg\varphi)$|⁠, but by |$(2)$|⁠, |$v\in\overline{V}(\neg\varphi)$| or |$v\in\overline{V}(\psi)$|⁠, hence |$v\in\overline{V}(\psi)$|⁠. Therefore, |${\mathcal{R}}(w)\cap\overline{V}(\varphi)\subseteq{\mathcal{R}}(w)\cap\overline{V}(\psi)\qquad(3)$| Now, by |$(1)$|⁠, |$(3)$|⁠, and |$\mathbf{(ek)}$| of Definition 3.6, |${\mathcal{R}}(w)\cap\overline{V}(\psi)\in{\mathcal{N}}(w)$|⁠, i.e. by Definition 3.8, |${\mathfrak{M}},w\Vdash{\mathsf{E}}\psi$|⁠. PIE. Assume that |${\mathfrak{M}},w\Vdash{\mathsf{E}}\varphi$|⁠. Then, by Definition 3.8, |${\mathcal{R}}(w)\cap\overline{V}(\varphi)\in{\mathcal{N}}(w)$|⁠, therefore, by |$\mathbf{(pie)}$| of Definition 3.6, |$\forall u\in W, w{\mathcal{R}} u\Rightarrow \overline{V}(\varphi)\cap{\mathcal{R}}(w)\cap{\mathcal{R}}(u)=\overline{V}(\varphi)\cap{\mathcal{R}}(u)\in{\mathcal{N}}(u)$|⁠, hence, by using Definition 3.8 twice, |${\mathcal{R}}(w)\subseteq\overline{V}({\mathsf{E}}\varphi)$| and |${\mathfrak{M}},w\Vdash{\mathsf{K}}{\mathsf{E}}\varphi$|⁠. ■ An important result follows now trivially. Corollary 3.10 |$\mathbf{KBE}$| is consistent. Proof. If |$\mathbf{KBE}$| were inconsistent, then |$\bot\in\mathbf{KBE}$|⁠, hence, by Theorem 3.9 and Fact 3.7, |${\mathfrak{F}}_1\Vdash\bot$|⁠, which is absurd. ■ Using the soundness of |$\mathbf{KBE}$| wrt kbe-frames, we can proceed to show that various ‘introspective’ principles are not |$\mathbf{KBE}$|-axioms. Fact 3.11 (i) |${\mathsf{E}}\varphi\supset{\mathsf{B}}\varphi\notin\mathbf{KBE}$| (ii) |$\neg{\mathsf{B}}\neg\varphi\supset{\mathsf{B}}\varphi\notin\mathbf{KBE}$| (iii) |${\mathsf{E}}\varphi\supset{\mathsf{K}}{\mathsf{K}}\varphi,\ {\mathsf{E}}\varphi\supset{\mathsf{K}}{\mathsf{B}}\varphi\notin\mathbf{KBE}$| |${\mathsf{E}}\varphi\supset{\mathsf{B}}{\mathsf{K}}\varphi,\ {\mathsf{E}}\varphi\supset{\mathsf{B}}{\mathsf{B}}\varphi\notin\mathbf{KBE}$| |${\mathsf{E}}\varphi\supset{\mathsf{E}}{\mathsf{K}}\varphi,\ {\mathsf{E}}\varphi\supset{\mathsf{E}}{\mathsf{B}}\varphi\notin\mathbf{KBE}$| (iv) |${\mathsf{E}}{\mathsf{K}}\varphi\supset{\mathsf{K}}\varphi\notin\mathbf{KBE}$| (v) |${\mathsf{E}}\varphi\land{\mathsf{E}}(\varphi\supset\psi)\supset{\mathsf{E}}\psi\notin\mathbf{KBE}$| Proof. Assume the model |${\mathfrak{M}}_1={\langle{{\mathfrak{F}}_1}{V}\rangle}$| based on the frame |${\mathfrak{F}}_1={\langle{W}{{\mathcal{R}}}{{\mathcal{N}}}\rangle}$| of Fact 3.7, with the valuation |$V$|⁠, shown in Figure 2. Applying Theorem 3.9, it suffices to find a world of |${\mathfrak{M}}_1$| that falsifies the formulae at hand. Firstly, notice that |${\mathfrak{M}}_1,w\Vdash{\mathsf{E}} p_0\land\neg{\mathsf{B}} p_0\land\neg{\mathsf{B}}\neg p_0\land{\mathsf{E}}{\mathsf{K}} p_2\land\neg{\mathsf{K}} p_2\land{\mathsf{E}}(p_0\supset p_1)\land\neg{\mathsf{E}} p_1$|⁠, so, items (i), (ii), (iv), and (v) are immediately falsified. Then, we observe that |${\mathfrak{M}}_1,w\Vdash{\mathsf{E}} p_0\land\neg{\mathsf{E}}{\mathsf{B}} p_0 $|⁠, and thus schema |${\mathsf{E}}\varphi\supset{\mathsf{E}}{\mathsf{B}}\varphi$| is falsified. Because of axioms |$\mathbf{KB}$| and |$\mathbf{BE}$| and rules |$\mathbf{RM}_{\mathsf{K}}$|⁠, |$\mathbf{RM}_{\mathsf{B}}$|2, and |$\mathbf{RM}_{\mathsf{E}}$| (Fact. 3.3), if any of the remaining formulae of case (iii) was a |$\mathbf{KBE}$|-theorem, then we would have |$\vdash_\mathbf{KBE}{\mathsf{E}}\varphi\supset{\mathsf{E}}{\mathsf{B}}\varphi$| — a contradiction. ■ Figure 2. Open in new tabDownload slide |$W$|⁠, |${\mathcal{R}}$| and |$V$| of model |${\mathfrak{M}}_1$|⁠. Figure 2. Open in new tabDownload slide |$W$|⁠, |${\mathcal{R}}$| and |$V$| of model |${\mathfrak{M}}_1$|⁠. Remark 3.12 Having in mind that ‘estimation’ is conceived as a weak form of a belief-like attitude, the fact that |${\mathsf{E}}\varphi\supset{\mathsf{B}}\varphi$| is not a |$\mathbf{KBE}$|-theorem is consistent with our intuition. From the formulae of Fact 3.11 (iii), |${\mathsf{E}}\varphi\supset{\mathsf{E}}{\mathsf{K}}\varphi$| deserves a comment. The fact that it is not a theorem of |$\mathbf{KBE}$| is welcomed; otherwise, given that |${\mathsf{E}}{\mathsf{K}}\varphi\supset{\mathsf{B}}\varphi\in\mathbf{KBE}$| (Proposition 3.4 (v)) and |$\mathbf{BE}$|⁠, estimation would collapse to belief (⁠|${\mathsf{E}}\varphi\equiv{\mathsf{B}}\varphi$| would be a theorem of |$\mathbf{KBE}$|⁠) and this would immediately invalidate our attempt to define estimation as a weak form of belief. In the same fashion, it is really good news that |${\mathsf{E}}{\mathsf{K}}\varphi\supset{\mathsf{K}}\varphi$| is not a |$\mathbf{KBE}$|-theorem. This formula introduces a strong form of the ‘infallibility argument’ or else the ‘paradox of the perfect believer’ (see [14]): in view of axiom |$\mathbf{T}$|⁠, it finally requires that something is true whenever our agent estimates that she knows it. Finally, upon inspection of the properties |$\mathbf{(cce)}$| and |$\mathbf{(ek)}$| as compared to the defining properties of weak ultrafilters in Definition 2.2, one could observe that the property |$\mathbf{(pie)}$|⁠, which corresponds to the axiom |$\mathbf{PIE}$|⁠, is not directly involved in the definition. It is tempting to ask whether |$\mathbf{PIE}$| can be proved form the rest of the |$\mathbf{KBE}$| axioms and the following Corollary clarifies that this is not the case. Corollary 3.13 The axiom |$\mathbf{PIE}$| is independent from the rest of the axioms of |$\mathbf{KBE}$|⁠. Proof. Let |$\mathbf{KBE^-}$| the logic defined as |$\mathbf{KBE}$| in Definition 4.1, but without axiom |$\mathbf{PIE}$|⁠. We will prove that |$\mathbf{PIE}\notin\mathbf{KBE^-}$|⁠. Let us call kbe|$^-$|-frames, the frames that satisfy all the properties of Definition 3.6 except |$\mathbf{(pie)}$|⁠. It is easy to check that |$\mathbf{KBE^-}$| is sound wrt the kbe|$^-$|-frames. It suffices to find a kbe|$^-$|-frame, in which |$\mathbf{PIE}$| can be falsified. Consider model |${\mathfrak{M}}^-={\langle{W}{{\mathcal{R}}}{{\mathcal{N}},V}\rangle}$| such that |$W=\{w,v,u_1,u_2,u_3\}$|⁠, where |${\mathcal{R}}$| and |$V$| are shown in Figure 3. Let |${\mathcal{N}}(w)=\{X\subseteq W\ |\ |X|\geq 3\}$| |${\mathcal{N}}(v)=\big\{X\subseteq \{v,u_1,u_2,u_3\}\ |\ |X|\geq 3\big\}\cup\big\{\{u_1,u_2\},\{u_1,u_3\},\{u_1,v\}\big\}$| |${\mathcal{N}}(u_1)={\mathcal{N}}(u_2)={\mathcal{N}}(u_3)=\{X\subseteq FC\ |\ |X|\geq 2\}$|⁠, where |$FC=\{u_1,u_2,u_3\}$| is the final cluster of the |$\mathbf{S4.2}$|-model |${\langle{W}{{\mathcal{R}}}\rangle}$|⁠. It is not hard to verify that this is a kbe|$^-$|-frame. Moreover it can easily be verified that |${\mathfrak{M}}^-,w\Vdash{\mathsf{E}} p_0\land\neg{\mathsf{K}}{\mathsf{E}} p_0$|⁠. ■ Figure 3. Open in new tabDownload slide |$W$|⁠, |${\mathcal{R}}$| and |$V$| of model |${\mathfrak{M}}^-$|⁠. Figure 3. Open in new tabDownload slide |$W$|⁠, |${\mathcal{R}}$| and |$V$| of model |${\mathfrak{M}}^-$|⁠. 3.4 Completeness of ${KBE}$ Now, we will prove that |$\mathbf{KBE}$| is strongly complete w.r.t. the class of all kbe-frames. To simplify the notation we assume that |$\Lambda$| denotes the logic |$\mathbf{KBE}$| and that c|$\Lambda$| and mc|$\Lambda$| mean |$\Lambda$|-consistent and maximal |$\Lambda$|-consistent theories respectively. We provide next, a definition and a series of lemmata, which will be used in the proof of completeness. This proof will be based on a variation of a point generated submodel of the canonical model of |$\Lambda$|⁠. Definition 3.14 Consider the relation |$P$| on the set of all mc|$\Lambda$| theories, which is defined as follows (∀Γ,Δ:mΛc)(ΓPΔ⟺(∀φ∈LKBE)(Kφ∈Γ⇒φ∈Δ)) and assume that |${{\mathsf{\scriptstyle \Xi}}}$| is any mc|$\Lambda$| theory. Furthermore, consider the theory |$B=\{\varphi\supset{\mathsf{K}}\neg{\mathsf{K}}\neg\varphi\ |\ \varphi\in\mathcal{L}_{KBE}\}$|⁠. Then, the canonical model |${\mathfrak{M}}$| of |$\Lambda$| generated by |${{\mathsf{\scriptstyle \Xi}}}$| (we write |${\mathfrak{M}}$| instead of |${\mathfrak{M}}^\Lambda_{{\mathsf{\scriptstyle \Xi}}}$|⁠) is the quadruple |${\langle{W}{{\mathcal{R}},{\mathcal{N}}}{V}\rangle}$| where (i) |$\ \ W=\{{{\mathsf{\scriptstyle \Gamma}}}\subseteq\mathcal{L}_{KBE}\ |\ {{\mathsf{\scriptstyle \Gamma}}}: m\Lambda c\ \&\ {{\mathsf{\scriptstyle \Xi}}} P^*{{\mathsf{\scriptstyle \Gamma}}}\}\quad$| (⁠|$P^*$| is the transitive closure of |$P$|⁠) (ii) |$\ \ {\mathcal{R}}$| is the restriction of |$P$| in |$W\times W$|⁠, i.e. |$(\forall{{\mathsf{\scriptstyle \Gamma}}},{{ \mathsf{\scriptstyle \Delta}}}\in W)({{\mathsf{\scriptstyle \Gamma}}}{\mathcal{R}}{{ \mathsf{\scriptstyle \Delta}}}\iff{{\mathsf{\scriptstyle \Gamma}}} P{{ \mathsf{\scriptstyle \Delta}}})$| (iii) |$\ \ {\mathcal{N}}^{-}({{\mathsf{\scriptstyle \Xi}}})=\{S\subseteq{\mathcal{R}}({{\mathsf{\scriptstyle \Xi}}})\mid S\supseteq\Sigma\mbox{ or }\exists\varphi\in\mathcal{L}_{KBE}\;\exists{{\mathsf{\scriptstyle \Gamma}}}\in W\mbox{ s.t. }S\supseteq[\varphi]\;\&\;{\mathsf{E}}\varphi\in{{\mathsf{\scriptstyle \Gamma}}}\}$| where We take |${\mathcal{N}}({{\mathsf{\scriptstyle \Xi}}})$| to be a weak ultrafilter with intersections in |$\Sigma$| extending |${\mathcal{N}}^{-}({{\mathsf{\scriptstyle \Xi}}})$| (see Proposition 2.4). For |${{\mathsf{\scriptstyle \Gamma}}}\neq{{\mathsf{\scriptstyle \Xi}}}$| we define |${\mathcal{N}}({{\mathsf{\scriptstyle \Gamma}}})=\{S\cap{\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}})\mid S\in{\mathcal{N}}({{\mathsf{\scriptstyle \Xi}}})\}$|⁠. (iv) |$\ \ (\forall p\in\Phi)\ V(p)=[p]$| The frame |${\mathfrak{F}}$| underlying |${\mathfrak{M}}$| is called the canonical frame of |$\Lambda$| generated by |${{\mathsf{\scriptstyle \Xi}}}$|. The following two lemmata are classic. Lemma 3.15 (Lindenbaum) Every c|$\Lambda$| theory can be extended to an mc|$\Lambda$| theory. Lemma 3.16 |$(\forall{{\mathsf{\scriptstyle \Gamma}}}:m\Lambda c)(\forall\varphi,\psi\in\mathcal{L}_{KBE})$| (i) |$\ \ \textrm{If }\ \varphi,\ \varphi\supset\psi\in{{\mathsf{\scriptstyle \Gamma}}},\ \textrm{then }\ \psi\in{{\mathsf{\scriptstyle \Gamma}}}$| (ii) |$\ \ \varphi\in{{\mathsf{\scriptstyle \Gamma}}}\ \textrm{ or }\ \neg\varphi\in{{\mathsf{\scriptstyle \Gamma}}}$| (iii) |$\ \ \textrm{If }\ \varphi\lor\psi\in{{\mathsf{\scriptstyle \Gamma}}},\ \textrm{then }\ \varphi\in{{\mathsf{\scriptstyle \Gamma}}}\ \textrm{ or }\ \psi\in{{\mathsf{\scriptstyle \Gamma}}}$| (iv) |$\ \ \Lambda\subseteq{{\mathsf{\scriptstyle \Gamma}}}$| Fact 3.17 |$W\neq\emptyset$| Proof. Since |$\Lambda$| is consistent (Corollary 3.10), there do exist mc|$\Lambda$| theories, therefore, it is meaningful to point out one of them (theory |${{\mathsf{\scriptstyle \Xi}}}$|⁠). But axiom |$\mathbf{T}$| belongs to |$\Lambda$|⁠, therefore, by Lemma 3.16 and Definition 3.14, |${{\mathsf{\scriptstyle \Xi}}} P{{\mathsf{\scriptstyle \Xi}}}$|⁠, i.e. |${{\mathsf{\scriptstyle \Xi}}}\in W$|⁠. ■ The following fact is a useful reformulation of the definition of |$P$| (and |${\mathcal{R}}$|⁠), which follows immediately by Lemma 3.16 (ii). Fact 3.18 |$(\forall{{\mathsf{\scriptstyle \Gamma}}},{{ \mathsf{\scriptstyle \Delta}}}\in W)\big({{\mathsf{\scriptstyle \Gamma}}}{\mathcal{R}}{{ \mathsf{\scriptstyle \Delta}}}\iff(\forall\varphi\in\mathcal{L}_{KBE})(\varphi\in{{ \mathsf{\scriptstyle \Delta}}}\Rightarrow\neg{\mathsf{K}}\neg\varphi\in{{\mathsf{\scriptstyle \Gamma}}})\big)$| Now, another classic Lemma can be proved. Lemma 3.19 (Existence Lemma) |$(\forall{{\mathsf{\scriptstyle \Gamma}}}\in W)(\forall\varphi\in\mathcal{L}_{KBE})$| ¬K¬φ∈Γ⟺(∃Δ∈W)(Δ∈R(Γ) & φ∈Δ) and equivalently Kφ∈Γ⟺(∀Δ∈W)(Δ∈R(Γ)⇒φ∈Δ) The following results are specific for |$\mathbf{KBE}$|⁠, therefore, we will provide the corresponding proofs. Lemma 3.20 |$(\forall{{\mathsf{\scriptstyle \Gamma}}}\in W)(\forall\varphi,\psi\in\mathcal{L}_{KBE})$| (R(Γ)∩[φ]⊆R(Γ)∩[ψ]  & Eφ∈Γ) ⟹ Eψ∈Γ Proof. Assuming that |${\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}})\cap[\varphi]\subseteq{\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}})\cap[\psi]$| and that |${\mathsf{E}}\varphi\in{{\mathsf{\scriptstyle \Gamma}}}$|⁠, we have (∀Δ∈W)((Δ∈R(Γ) & φ∈Δ)⇒ψ∈Δ) hence, by Lemma 3.16, (∀Δ∈W)(Δ∈R(Γ)⇒φ⊃ψ∈Δ) therefore, by Lemma 3.19, |${\mathsf{K}}(\varphi\supset\psi)\in{{\mathsf{\scriptstyle \Gamma}}}$|⁠, and since |${\mathsf{E}}\varphi\in{{\mathsf{\scriptstyle \Gamma}}}$| and |$\mathbf{EK}\in\Lambda\subseteq{{\mathsf{\scriptstyle \Gamma}}}$|⁠, by Lemma 3.16 (i), |${\mathsf{E}}\psi\in{{\mathsf{\scriptstyle \Gamma}}}$|⁠. ■ Lemma 3.21 Consider the set |$B$| of Definition 3.14. Then (∀Γ,Δ∈W)((B⊆Γ &ΓRΔ)⇒ΔRΓ) Proof. Consider any |$\varphi\in{{\mathsf{\scriptstyle \Gamma}}}$|⁠. Since |${\mathsf{K}}\neg{\mathsf{K}}\neg\varphi\in B\subseteq{{\mathsf{\scriptstyle \Gamma}}}$| and |${{\mathsf{\scriptstyle \Gamma}}}{\mathcal{R}}{{ \mathsf{\scriptstyle \Delta}}}$|⁠, by Definition 3.14 (ii), |$\neg{\mathsf{K}}\neg\varphi\in{{ \mathsf{\scriptstyle \Delta}}}$|⁠, therefore, by Fact 3.18, |${{ \mathsf{\scriptstyle \Delta}}}{\mathcal{R}}{{\mathsf{\scriptstyle \Gamma}}}$|⁠. ■ Lemma 3.22 (i) |$\quad\vdash_\Lambda{\mathsf{B}}(\varphi\supset{\mathsf{K}}\neg{\mathsf{K}}\neg\varphi)\quad(\forall\varphi\in\mathcal{L}_{KBE})$| (ii) |$\quad\vdash_\Lambda{\mathsf{B}}\varphi_0\land\ldots\land{\mathsf{B}}\varphi_n\supset{\mathsf{B}}(\varphi_0\land\ldots\land\varphi_n)\quad(\forall n\in\mathbb{N})(\forall\varphi_0,\ldots,\varphi_n\in\mathcal{L}_{KBE})$| Proof. (i) 1.¬B¬φ∨B¬φPC2.K¬B¬φ∨B¬φ1,NIB,PC,MP3.B¬B¬φ∨B¬φ2,KB,PC,MP4.B(¬B¬φ∨¬φ)3,DB,RMK,PC,MP5.¬B¬φ∨¬φ⊃(φ⊃K¬K¬φ)4,DB,PC,MP6.B(¬B¬φ∨¬φ)⊃B(φ⊃K¬K¬φ)5,RMB7.B(φ⊃K¬K¬φ)4,6,MP (ii) By induction. The case |$n=0$| is trivial. 1.φ0∧…∧φn⊃(φn+1⊃φ0∧…∧φn+1)PC2.B(φ0∧…∧φn)⊃B(φn+1⊃φ0∧…∧φn+1)1,RMB3.B(φn+1⊃φ0∧…∧φn+1)⊃(Bφn+1⊃B(φ0∧…∧φn+1))theorem of Λ [28, p.40]4.B(φ0∧…∧φn)⊃(Bφn+1⊃B(φ0∧…∧φn+1))2,3,PC,MP5.Bφ0∧…∧Bφn⊃B(φ0∧…∧φn)Ind. Hyp.6.Bφ0∧…∧Bφn+1⊃B(φ0∧…∧φn+1)5,4,PC,MP ■ Lemma 3.23 Consider the final cluster |$FC$| of |${\mathfrak{M}}$|⁠, i.e. the set |$FC=\{{{ \mathsf{\scriptstyle \Delta}}}\in W\ |$||$(\forall{{\mathsf{\scriptstyle \Gamma}}}\in W)\ {{\mathsf{\scriptstyle \Gamma}}}{\mathcal{R}}{{ \mathsf{\scriptstyle \Delta}}}\}$|⁠, and the set |$\Sigma$| of Definition 3.14 (iii). Then (i) |$\ \ \Sigma\neq\emptyset$| (ii) |$\ \ \Sigma=FC$| (iii) |$\ \ (\forall{{ \mathsf{\scriptstyle \Delta}}}\in\Sigma)(\forall{{\mathsf{\scriptstyle Z}}}\in W)({{ \mathsf{\scriptstyle \Delta}}}{\mathcal{R}}{{\mathsf{\scriptstyle Z}}}\Rightarrow{{\mathsf{\scriptstyle Z}}}\in \Sigma)$| (iv) |$\ \ (\forall{{\mathsf{\scriptstyle \Gamma}}}\in W)(\forall\varphi\in\mathcal{L}_{KBE})(\Sigma\subseteq{\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}})\cap[\varphi]\Rightarrow{\mathsf{E}}\varphi\in{{\mathsf{\scriptstyle \Gamma}}})$| Proof. (i) Consider the theory |$T=B\cup\{\varphi\in\mathcal{L}_{KBE}\ |\ {\mathsf{K}}\varphi\in{{\mathsf{\scriptstyle \Xi}}}\}$|⁠, and suppose, for the sake of contradiction, that |$T$| was inconsistent with |$\Lambda$|⁠, i.e. that |$T\vdash_\Lambda\bot$|⁠. Then, since |$\Lambda$| is consistent, there would be formulae |$\varphi_1,\ldots,\varphi_n,\psi_1,\ldots,\psi_m$| s.t. ⊢Λ(φ1⊃K¬K¬φ1)∧…∧(φn⊃K¬K¬φn)∧ψ1∧…∧ψm⊃⊥ where |${\mathsf{K}}\psi_1,\ldots,{\mathsf{K}}\psi_m\in{{\mathsf{\scriptstyle \Xi}}}$|⁠, i.e. since, by axioms |$\mathbf{DB}$| and |$\mathbf{T}$|⁠, |$\vdash_\Lambda{\mathsf{K}}\varphi\supset{\mathsf{B}}\varphi$|⁠, by Lemma 3.16, |${\mathsf{B}}\psi_1,\ldots,{\mathsf{B}}\psi_m\in{{\mathsf{\scriptstyle \Xi}}}\quad(2)$| But then, by |$\mathbf{RM}_{\mathsf{B}}$| and Lemma 3.22 (ii) ⊢ΛB(φ1⊃K¬K¬φ1)∧…∧B(φn⊃K¬K¬φn)∧Bψ1∧…∧Bψm⊃B⊥ Therefore, using repeatedly Lemma 3.16, by |$(2)$| and Lemma 3.22 (i), |${\mathsf{B}}\bot\in{{\mathsf{\scriptstyle \Xi}}}$|⁠, so, by |$\mathbf{CB}$|⁠, |$\neg{\mathsf{B}}\top\in{{\mathsf{\scriptstyle \Xi}}}$|⁠, therefore, by |$\mathbf{DB}$| and |$\mathbf{T}$|⁠, |$\neg{\mathsf{K}}\top\in{{\mathsf{\scriptstyle \Xi}}}$|⁠. But obviously, |$\vdash_\Lambda\top$|⁠, hence, by |$\mathbf{RN}_{\mathsf{K}}$|⁠, |$\vdash_\Lambda{\mathsf{K}}\top$|⁠, so, |${\mathsf{K}}\top\in{{\mathsf{\scriptstyle \Xi}}}$|⁠, therefore, |${{\mathsf{\scriptstyle \Xi}}}$| would be inconsistent, which is not true. We conclude that |$T$| is consistent with |$\Lambda$|⁠, hence, by Lindenbaum’s Lemma, there is a mc|$\Lambda$| theory |${{ \mathsf{\scriptstyle \Delta}}}$| containing |$T$|⁠, so, if |${\mathsf{K}}\varphi\in{{\mathsf{\scriptstyle \Xi}}}$|⁠, then |$\varphi\in T\subseteq{{ \mathsf{\scriptstyle \Delta}}}$|⁠, hence, by Definition 3.14, |${{\mathsf{\scriptstyle \Xi}}} P{{ \mathsf{\scriptstyle \Delta}}}$|⁠, therefore, |${{ \mathsf{\scriptstyle \Delta}}}\in W$|⁠, and since |$B\subseteq T\subseteq{{ \mathsf{\scriptstyle \Delta}}}$|⁠, |${{ \mathsf{\scriptstyle \Delta}}}\in\Sigma$|⁠. (ii) (⁠|$\subseteq$|⁠) Consider any |${{ \mathsf{\scriptstyle \Delta}}}\in\Sigma$| and |${{\mathsf{\scriptstyle Z}}}\in W$|⁠. Then, by Definition 3.14, |${{\mathsf{\scriptstyle \Xi}}} P^*{{\mathsf{\scriptstyle Z}}}$|⁠, so, using the transitivity of |$P$| (which follows from the fact that |$\mathbf{4}\in\Lambda$|⁠), it can be shown by a trivial induction, that |${{\mathsf{\scriptstyle \Xi}}} P{{\mathsf{\scriptstyle Z}}}$|⁠. But, since |$\mathbf{DB}, \mathbf{CB}\in\Lambda$|⁠, |$\mathbf{G}.\neg{\mathsf{K}}\neg{\mathsf{K}}\varphi\supset{\mathsf{K}}\neg{\mathsf{K}}\neg\varphi\in\Lambda$|⁠, and since we know that this axiom is canonical for the property of weak directedness, |$P$| is weakly directed, therefore, there must be a mc|$\Lambda$| theory |${{\mathsf{\scriptstyle H}}}$| s.t. |${{\mathsf{\scriptstyle Z}}} P{{\mathsf{\scriptstyle H}}}$| and |${{ \mathsf{\scriptstyle \Delta}}} P{{\mathsf{\scriptstyle H}}}$|⁠. Furthermore, |${{ \mathsf{\scriptstyle \Delta}}}\in W$| i.e. |${{\mathsf{\scriptstyle \Xi}}} P^*{{ \mathsf{\scriptstyle \Delta}}}$|⁠, hence, |${{\mathsf{\scriptstyle \Xi}}} P^*{{\mathsf{\scriptstyle H}}}$|⁠, so, |${{\mathsf{\scriptstyle H}}}\in W$| and, by Definition 3.14 (ii), |${{ \mathsf{\scriptstyle \Delta}}}{\mathcal{R}}{{\mathsf{\scriptstyle H}}}$|⁠. Therefore, since |$B\subseteq{{ \mathsf{\scriptstyle \Delta}}}$|⁠, by Lemma 3.21, |${{\mathsf{\scriptstyle H}}}{\mathcal{R}}{{ \mathsf{\scriptstyle \Delta}}}$| i.e. |${{\mathsf{\scriptstyle H}}} P{{ \mathsf{\scriptstyle \Delta}}}$|⁠, and since |$P$| is transitive, |${{\mathsf{\scriptstyle Z}}} P{{ \mathsf{\scriptstyle \Delta}}}$|⁠, hence, since |${{\mathsf{\scriptstyle Z}}},{{ \mathsf{\scriptstyle \Delta}}}\in W$|⁠, |${{\mathsf{\scriptstyle Z}}}{\mathcal{R}}{{ \mathsf{\scriptstyle \Delta}}}$|⁠. It has been proved that |${{ \mathsf{\scriptstyle \Delta}}}\in FC$|⁠. (⁠|$\supseteq$|⁠) Assume that |${{ \mathsf{\scriptstyle \Delta}}}\in FC$| and that |$\varphi\in\mathcal{L}_{KBE}$|⁠. If |$\neg\varphi\in{{ \mathsf{\scriptstyle \Delta}}}$|⁠, then, by Lemma 3.16, |$\varphi\supset{\mathsf{K}}\neg{\mathsf{K}}\neg\varphi\in{{ \mathsf{\scriptstyle \Delta}}}$|⁠. If |$\neg\varphi\notin{{ \mathsf{\scriptstyle \Delta}}}$|⁠, then, by Lemma 3.16 (ii), |$\varphi\in{{ \mathsf{\scriptstyle \Delta}}}$|⁠. Consider now any |${{\mathsf{\scriptstyle Z}}}\in W$| s.t. |${{ \mathsf{\scriptstyle \Delta}}}{\mathcal{R}}{{\mathsf{\scriptstyle Z}}}$|⁠. Since |${{ \mathsf{\scriptstyle \Delta}}}\in FC$|⁠, |${{\mathsf{\scriptstyle Z}}}{\mathcal{R}}{{ \mathsf{\scriptstyle \Delta}}}$|⁠, and since |$\varphi\in{{ \mathsf{\scriptstyle \Delta}}}$|⁠, by Fact 3.18, |$\neg{\mathsf{K}}\neg\varphi\in{{\mathsf{\scriptstyle Z}}}$|⁠, therefore, by Lemma 3.19, |${\mathsf{K}}\neg{\mathsf{K}}\neg\varphi\in{{ \mathsf{\scriptstyle \Delta}}}$|⁠, so again, by Lemma 3.16, |$\varphi\supset{\mathsf{K}}\neg{\mathsf{K}}\neg\varphi\in{{ \mathsf{\scriptstyle \Delta}}}$|⁠. Hence, |$B\subseteq{{ \mathsf{\scriptstyle \Delta}}}$| i.e. |${{ \mathsf{\scriptstyle \Delta}}}\in\Sigma$|⁠. (iii) Consider any |${{ \mathsf{\scriptstyle \Delta}}}\in\Sigma$| and |${{\mathsf{\scriptstyle Z}}}\in W$| s.t. |${{ \mathsf{\scriptstyle \Delta}}}{\mathcal{R}}{{\mathsf{\scriptstyle Z}}}$|⁠, and any |${{\mathsf{\scriptstyle \Gamma}}}\in W$|⁠. Since |${{ \mathsf{\scriptstyle \Delta}}}\in\Sigma$|⁠, by (ii), |${{ \mathsf{\scriptstyle \Delta}}}\in FC$|⁠, therefore, |${{\mathsf{\scriptstyle \Gamma}}}{\mathcal{R}}{{ \mathsf{\scriptstyle \Delta}}}$|⁠, and since |${\mathcal{R}}$| is transitive, |${{\mathsf{\scriptstyle \Gamma}}}{\mathcal{R}}{{\mathsf{\scriptstyle Z}}}$|⁠. Hence, |${{\mathsf{\scriptstyle Z}}}\in FC$| i.e. by (ii), |${{\mathsf{\scriptstyle Z}}}\in\Sigma$|⁠. (iv) Suppose that |${{\mathsf{\scriptstyle \Gamma}}}\in W$| and |$\varphi\in\mathcal{L}_{KBE}$| s.t. |$\Sigma\subseteq{\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}})\cap[\varphi]$|⁠. By (i) and (ii), there exists a |${{ \mathsf{\scriptstyle \Delta}}}\in FC$|⁠, so, |${{\mathsf{\scriptstyle \Gamma}}}{\mathcal{R}}{{ \mathsf{\scriptstyle \Delta}}}$|⁠. Now, consider any |${{\mathsf{\scriptstyle Z}}}\in W$| s.t. |${{ \mathsf{\scriptstyle \Delta}}}{\mathcal{R}}{{\mathsf{\scriptstyle Z}}}$|⁠. Then, by (iii), |${{\mathsf{\scriptstyle Z}}}\in\Sigma$|⁠, therefore, since |$\Sigma\subseteq{\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}})\cap[\varphi]$|⁠, |${{\mathsf{\scriptstyle Z}}}\in[\varphi]$| i.e. |$\varphi\in{{\mathsf{\scriptstyle Z}}}$|⁠. Hence, by Lemma 3.19, |${\mathsf{K}}\varphi\in{{ \mathsf{\scriptstyle \Delta}}}$|⁠, which means that we have found a |${{ \mathsf{\scriptstyle \Delta}}}\in W$| s.t. |${{\mathsf{\scriptstyle \Gamma}}}{\mathcal{R}}{{ \mathsf{\scriptstyle \Delta}}}$| and |${\mathsf{K}}\varphi\in{{ \mathsf{\scriptstyle \Delta}}}$|⁠. Therefore, again by Lemma 3.19, |$\neg{\mathsf{K}}\neg{\mathsf{K}}\varphi\in{{\mathsf{\scriptstyle \Gamma}}}$|⁠, hence, using Lemma 3.16 twice, by axiom |$\mathbf{DB}$|⁠, |${\mathsf{B}}\varphi\in{{\mathsf{\scriptstyle \Gamma}}}$|⁠, and finally, by axiom |$\mathbf{BE}$|⁠, |${\mathsf{E}}\varphi\in{{\mathsf{\scriptstyle \Gamma}}}$|⁠. ■ Lemma 3.24 |$\forall {{\mathsf{\scriptstyle \Gamma}}}\in W\;\;{\mathcal{N}}({{\mathsf{\scriptstyle \Gamma}}})$| is a weak ultrafilter. Proof. For |${{\mathsf{\scriptstyle \Gamma}}}={{\mathsf{\scriptstyle \Xi}}}$|⁠: In fact we show that |${\mathcal{N}}({{\mathsf{\scriptstyle \Gamma}}})$| is a weak ultrafilter with intersections in |$FC$|⁠. Having in mind Proposition 2.4 we only have to show that |${\mathcal{N}}^{-}({{\mathsf{\scriptstyle \Xi}}})$| is a weak filter with intersections in |$FC$|⁠. For the latter we need to show that |$\forall X,Y\in{\mathcal{N}}^{-}({{\mathsf{\scriptstyle \Xi}}})\;X\cap Y \cap FC\neq \emptyset$|⁠. We discern the following cases: |$X\supseteq [\varphi],Y\supseteq [\psi]$| for some formulas |$\varphi,\psi$| and worlds |${{\mathsf{\scriptstyle \Gamma}}},{{ \mathsf{\scriptstyle \Delta}}}$| such that |${\mathsf{E}}\varphi\in{{\mathsf{\scriptstyle \Gamma}}},{\mathsf{E}}\psi\in{{ \mathsf{\scriptstyle \Delta}}}$|⁠. Let |${{\mathsf{\scriptstyle Z}}}\in FC$|⁠. Due to axiom |$\mathbf{PIE}$| and Lemma 3.19, |${\mathsf{E}}\varphi,{\mathsf{E}}\psi\in{{\mathsf{\scriptstyle Z}}}$|⁠. For the sake of contradiction assume |$X\cap Y \cap FC = \emptyset$|⁠. Then |$[\varphi]\cap[\psi]\cap FC = \emptyset$| i.e. no world in |$FC$| includes both |$\varphi,\psi$|⁠. By Lemmas 3.16 and 3.19, |${\mathsf{K}} (\varphi\supset\neg\psi)\in{{\mathsf{\scriptstyle Z}}}$|⁠. By axiom |$\mathbf{EK}$|⁠, |${\mathsf{E}}\neg\psi\in{{\mathsf{\scriptstyle Z}}}$| and by axiom |$\mathbf{CCE}$||$\neg{\mathsf{E}}\psi\in{{\mathsf{\scriptstyle Z}}}$|⁠. But |${{\mathsf{\scriptstyle Z}}}$| is a consistent set, hence the contradiction. |$X\supseteq [\varphi], Y\supseteq FC$|⁠. In this case |$X\cap Y \cap FC = X \cap FC$|⁠. Let |${{\mathsf{\scriptstyle Z}}}\in FC$|⁠. Again by |$\mathbf{PIE}$| we have |${\mathsf{E}}\varphi\in{{\mathsf{\scriptstyle Z}}}$|⁠. Also, since |$\top$| is valid in FC, by Lemmas 3.16 and 3.19 we have |${\mathsf{K}} (\varphi\supset\top)\in{{\mathsf{\scriptstyle Z}}}$|⁠. For the sake of contradiction, let |$X \cap FC = \emptyset$|⁠. Then again by the same Lemmas, |${\mathsf{K}} (\varphi\supset\bot)\in{{\mathsf{\scriptstyle Z}}}$|⁠. By axiom |$\mathbf{EK}$|⁠, |${\mathsf{E}}\top,{\mathsf{E}}\bot\in{{\mathsf{\scriptstyle Z}}}$|⁠, which by axiom |$\mathbf{CCE}$| leads to another contradiction. |$X,Y\supseteq FC$|⁠. The property is obvious. A fortiori, |${\mathcal{N}}^{-}({{\mathsf{\scriptstyle \Xi}}})$| does not include disjoint sets. It also closed under supersets by definition, and so it also includes |$W\supseteq FC$|⁠. Thus it is a weak filter with intersections in |$FC$|⁠. For |${{\mathsf{\scriptstyle \Gamma}}}\neq{{\mathsf{\scriptstyle \Xi}}}$|⁠: We show that the required properties hold. (wf1): |$W\in {\mathcal{N}}({{\mathsf{\scriptstyle \Xi}}})\Rightarrow W\cap{\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}})={\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}})\in {\mathcal{N}}({{\mathsf{\scriptstyle \Gamma}}})$|⁠. (wf2): Suppose there are sets |$X,Y\subseteq{\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}})$| such that |$X\in{\mathcal{N}}({{\mathsf{\scriptstyle \Gamma}}})$| and |$Y\supseteq X$|⁠. There must exist a set |$Z\in{\mathcal{N}}({{\mathsf{\scriptstyle \Xi}}})$| such that |$Z\cap{\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}})=X$|⁠. Since |${\mathcal{N}}({{\mathsf{\scriptstyle \Xi}}})$| is closed under supersets |$Z\cup Y\in{\mathcal{N}}({{\mathsf{\scriptstyle \Xi}}})$|⁠. By Definition 3.14 |$(Z\cup Y)\cap{\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}})=X\cup Y=Y\in{\mathcal{N}}({{\mathsf{\scriptstyle \Gamma}}})$|⁠. (wf3): For the sake of contradiction let |$X,{\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}})\setminus X \in {\mathcal{N}}({{\mathsf{\scriptstyle \Gamma}}})$|⁠. There must exist sets |$Y,Z\in{\mathcal{N}}({{\mathsf{\scriptstyle \Xi}}})$| such that |$Y\cap{\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}})=X$| and |$Z\cap{\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}}) = {\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}})\setminus X$|⁠. Since |${\mathcal{N}}({{\mathsf{\scriptstyle \Gamma}}})$| has intersections in |$FC$|⁠, and |$FC\subseteq{\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}})$| we have |$Y\cap Z \cap FC = Y\cap X \cap {\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}}) \cap FC = X \cap {\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}})\setminus X \cap FC \neq \emptyset$|⁠. Hence the contradiction. (wuf): For the sake of contradiction suppose there is a set |$X\subseteq{\mathcal{R}}(w)$| such that |$X,{\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}}) \setminus X \notin {\mathcal{N}}({{\mathsf{\scriptstyle \Gamma}}})$|⁠. Then it must be the case that |$X,{\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}}) \setminus X \notin {\mathcal{N}}({{\mathsf{\scriptstyle \Xi}}})$|⁠. Since |${\mathcal{N}}({{\mathsf{\scriptstyle \Xi}}})$| is a weak ultrafilter this is absurd. ■ Corollary 3.25 |$(\forall{{\mathsf{\scriptstyle \Gamma}}}\in W)(\forall\varphi\in\mathcal{L}_{KBE})({\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}})\cap[\varphi]\in{\mathcal{N}}({{\mathsf{\scriptstyle \Gamma}}})\iff{\mathsf{E}}\varphi\in{{\mathsf{\scriptstyle \Gamma}}})$| Proof. First, assume that |${\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}})\cap[\varphi]\in{\mathcal{N}}({{\mathsf{\scriptstyle \Gamma}}})$| and |${\mathsf{E}}\varphi\notin{{\mathsf{\scriptstyle \Gamma}}}$|⁠. Since |${{\mathsf{\scriptstyle \Gamma}}}$| is maximal |$\neg{\mathsf{E}}\varphi\in{{\mathsf{\scriptstyle \Gamma}}}$| and by axiom |$\mathbf{CCE}$|⁠, |${\mathsf{E}}\neg\varphi\in{{\mathsf{\scriptstyle \Gamma}}}$|⁠. Consequently by Definition 3.14 (iii), |$[\neg\varphi]\in{\mathcal{N}}({{\mathsf{\scriptstyle \Xi}}})$| so |${\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}})\cap[\neg\varphi]\in{\mathcal{N}}({{\mathsf{\scriptstyle \Gamma}}})$|⁠. Therefore |${\mathcal{N}}({{\mathsf{\scriptstyle \Gamma}}})$| is a weak ultrafilter which includes complementary subsets. We derive a contradiction. Conversely, if |${\mathsf{E}}\varphi\in{{\mathsf{\scriptstyle \Gamma}}}$| then by Definition 3.14 (iii) |$[\varphi]\in{\mathcal{N}}({{\mathsf{\scriptstyle \Xi}}})\Rightarrow{\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}})\cap[\varphi]\in{\mathcal{N}}({{\mathsf{\scriptstyle \Gamma}}})$|⁠. ■ Lemma 3.26 (Truth Lemma) |$(\forall{{\mathsf{\scriptstyle \Gamma}}}\in W)(\forall\varphi\in\mathcal{L}_{KBE})({\mathfrak{M}},{{\mathsf{\scriptstyle \Gamma}}}\Vdash\varphi\iff\varphi\in{{\mathsf{\scriptstyle \Gamma}}})$| Proof. By induction on |$\varphi$|⁠. Since the ind. base follows immediately from Definition 3.14 (v) and from Lemma 3.16 (for |$\bot$|⁠), and since the case |$\varphi\supset\psi$| of the ind. step is an immediate consequence of the ind. hypothesis, we focus on the cases |${\mathsf{K}}\varphi$| and |${\mathsf{E}}\varphi$|⁠. M,Γ⊩Kφ⟺3.8(∀Δ∈W)(Δ∈R(Γ)⇒M,Δ⊩φ)⟺Ind.Hyp. (∀Δ∈W)(Δ∈R(Γ)⇒φ∈Δ)⟺3.19Kφ∈Γ Now, for any |${{ \mathsf{\scriptstyle \Delta}}}\in W$|⁠, |${{ \mathsf{\scriptstyle \Delta}}}\in\overline{V}(\varphi)\stackrel{\textrm{3.8}}{\iff}{\mathfrak{M}},{{ \mathsf{\scriptstyle \Delta}}}\Vdash\varphi\stackrel{\textrm{Ind.Hyp.}}{\iff}\varphi\in{{ \mathsf{\scriptstyle \Delta}}}\stackrel{\textrm{3.14 (ii)}}{\iff}{{ \mathsf{\scriptstyle \Delta}}}\in[\varphi]$|⁠, hence, |$\overline{V}(\varphi)=[\varphi]\qquad(*)$| So, |${\mathfrak{M}},{{\mathsf{\scriptstyle \Gamma}}}\Vdash{\mathsf{E}}\varphi\stackrel{\textrm{3.8}}{\iff} {\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}})\cap\overline{V}(\varphi)\in{\mathcal{N}}({{\mathsf{\scriptstyle \Gamma}}})\stackrel{(*)}{\iff} {\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}})\cap[\varphi]\in{\mathcal{N}}({{\mathsf{\scriptstyle \Gamma}}})\stackrel{\textrm{3.25}}{\iff} {\mathsf{E}}\varphi\in{{\mathsf{\scriptstyle \Gamma}}}$| ■ Theorem 3.27 (Completeness) |$\mathbf{KBE}$| is strongly complete w.r.t. the class of all kbe-frames. Proof. It suffices to show (see [8, p. 194]) that every c|$\Lambda$| theory |$T$| is satisfiable in a kbe-model. By Lindenbaum’s Lemma, |$T$| is contained in a mc|$\Lambda$| theory |${{\mathsf{\scriptstyle \Xi}}}$|⁠, hence, by the Truth Lemma (3.26), |${\mathfrak{M}}^\Lambda_{{\mathsf{\scriptstyle \Xi}}},{{\mathsf{\scriptstyle \Xi}}}\Vdash T$|⁠. Therefore, it remains to check that |${\mathfrak{M}}^\Lambda_{{\mathsf{\scriptstyle \Xi}}}$| is a kbe-model (we simply write again |${\mathfrak{M}}$| instead of |${\mathfrak{M}}^\Lambda_{{\mathsf{\scriptstyle \Xi}}}$|⁠, which was defined in Definition 3.14). By Fact 3.17, |$W\neq\emptyset$|⁠. Since axioms |$\mathbf{T}$| and |$\mathbf{4}$| are canonical for the properties of reflexivity and transitivity respectively, |$P$| (as a relation on the set of all mc|$\Lambda$| theories) is reflexive and transitive, so is |${\mathcal{R}}$|⁠, by Definition 3.14 (ii). Furthermore, |$FC\neq\emptyset$|⁠, by Lemma 3.23 (i)–(ii). Consider now any |${{\mathsf{\scriptstyle \Gamma}}}\in W$|⁠. Properties |$\mathbf{(nr)}$| and |$\mathbf{(be)}$| hold by Definition 3.14 and Lemma 3.23 (ii). Properties |$\mathbf{(cce)}$| and |$\mathbf{(ek)}$| hold by Lemma 3.24. Regarding property |$\mathbf{(pie)}$|⁠: Let |${{\mathsf{\scriptstyle \Gamma}}},{{ \mathsf{\scriptstyle \Delta}}}\in W$|⁠, |${{\mathsf{\scriptstyle \Gamma}}} {\mathcal{R}} {{ \mathsf{\scriptstyle \Delta}}}$|⁠, and |$X\in{\mathcal{N}}({{\mathsf{\scriptstyle \Gamma}}})$|⁠. For |${{\mathsf{\scriptstyle \Gamma}}}={{\mathsf{\scriptstyle \Xi}}}$| the property holds by Definition 3.14. For |${{\mathsf{\scriptstyle \Gamma}}}\neq{{\mathsf{\scriptstyle \Xi}}}$|⁠: there must exist a set |$Y\in{\mathcal{N}}({{\mathsf{\scriptstyle \Xi}}})$| such that |$Y\cap{\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}})=X$|⁠. But then, again by definition and since |${\mathcal{R}}({{ \mathsf{\scriptstyle \Delta}}})\subseteq{\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}})$|⁠, |$X\cap{\mathcal{R}}({{ \mathsf{\scriptstyle \Delta}}})=Y\cap{\mathcal{R}}({{ \mathsf{\scriptstyle \Delta}}})\in{\mathcal{N}}({{ \mathsf{\scriptstyle \Delta}}})$|⁠. ■ 4 A relaxed variant: the logic $\mathbf{KBiE}$ The reader has certainly noticed that in |$\mathbf{KBE}$| we have deliberately opted for a ‘complete’ estimation operator, including thus the axiom |$\mathbf{CCE}$| in the axiomatization and working with complete weak filters (weak ultrafilters) in the model theory of the logic. Obviously, this is a very strong assumption and is related to our intention to model a rational agent whose knowledge (and belief) need not be complete but (the weak form of belief we call) estimation should be complete and it is based on ‘majority’ considerations. Nevertheless, a relaxed version of this modal account, the logic |$\mathbf{KBiE}$| (Knowledge, Belief, incomplete Estimation) can be easily obtained. The details follow. 4.1 $\mathbf{KBiE}$: estimation is consistent but not complete |$\mathbf{KBiE}$| is obtained by replacing |$\mathbf{CCE}$| for its ‘half’, the axiom |$\mathbf{D}$| for estimation. Definition 4.1 |$\mathbf{KBiE}$| is the propositional bimodal logic axiomatized by |$\mathbf{K}$|⁠, |$\mathbf{T}$|⁠, |$\mathbf{4}$|⁠, |$\mathbf{CB}$|⁠, |$\mathbf{BE}$|⁠, |$\mathbf{CE}$|⁠, |$\mathbf{EK}$|⁠, |$\mathbf{PIE}$| and closed under the rule |$\mathbf{RN}_{{\mathsf{K}}}$|⁠, where |$\mathbf{CE}$| is the axiom |$ {\mathsf{E}}\varphi \rightarrow \neg{\mathsf{E}}\neg\varphi$| (Estimation is consistent). Apparently, the semantics of |$\mathbf{KBiE}$| will be quite similar: the only difference lies in the fact that now the neighbourhoods need only be weak filters and not necessarily (weak) ultrafilters. The following definition highlights this. Definition 4.2 A kbie frame is a triple |${\mathfrak{F}}={\langle{W}{{\mathcal{R}}}{{\mathcal{N}}}\rangle}$| satisfying the items of Definition 3.6, with the exception that |$\mathbf(cce)$| is replaced by |$\mathbf{(ce)}$||$\ \ \forall X\subseteq{\mathcal{R}}(w)\;\big(X\in{\mathcal{N}}(w)\Longrightarrow{\mathcal{R}}(w)\setminus X\notin{\mathcal{N}}(w)\big)$| |${\mathfrak{M}}={\langle{{\mathfrak{F}}}{V}\rangle}$| is called a kbie-model, if it is based on a kbie-frame and |$V:\Phi\to \mathcal{P}(W)$| is a valuation. We will sketch below the proof of soundness and completeness for |$\mathbf{KBiE}$| with respect to the class of kbie-frames, emphasizing the items that should be modified in the relevant proofs for |$\mathbf{KBE}$|⁠, while providing some technical insight on the necessary modifications. The proof of soundness imitates the proof for |$\mathbf{KBE}$|⁠. For completeness, again we employ a completeness-via-canonicity proof. Most intermediate results can be repeated with only trivial changes. However, the definition of the canonical model and in particular the part that defines the neighbourhoods has to be stated. Definition 4.3 The canonical model |${\mathfrak{M}}$| of |$\mathbf{KBiE}$| generated by |${{\mathsf{\scriptstyle \Xi}}}$| is defined as in Definition 3.14 except for the following item (iii) |${\mathcal{N}}({{\mathsf{\scriptstyle \Gamma}}})=\{S\subseteq{\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}})\mid S\supseteq\Sigma\mbox{ or }\exists\varphi\in\mathcal{L}_{KBE}\;\mbox{ s.t. }S\supseteq {\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}}) \cap [\varphi]\;\&\;{\mathsf{E}}\varphi\in{{\mathsf{\scriptstyle \Gamma}}}\}$| A technical comment is in order: in |$\mathbf{KBE}$|⁠, to construct the neighbourhood |${\mathcal{N}}^{-}({{\mathsf{\scriptstyle \Xi}}})$| of the ‘root’ world |${{\mathsf{\scriptstyle \Xi}}}$| of the model, one has to take into account all formulae of the form |${\mathsf{E}} \varphi $| in the worlds accessible. The reason is that the canonical model there involves the expansion of the neighbourhood to a weak ultrafilter. Due to property |$\mathbf{(pie)}$| (that the canonical model should satisfy to be a kbe-model) this expansion cannot be arbitrary; it has to be compatible with the aforementioned ‘estimations’ found in the successor worlds. On the other hand, neighbourhoods in |$\mathbf{KBiE}$| need only be weak filters, and so the definition is simplified respectively. All results, and their proofs, up to Lemma 3.24, can be repeated without changes. The following lemma calls for our attention. Lemma 4.4 |$\forall {{\mathsf{\scriptstyle \Gamma}}}\in W\;\;{\mathcal{N}}({{\mathsf{\scriptstyle \Gamma}}})$| is a weak filter. Proof. We show that the required properties hold. (wf1): |${\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}}) \supseteq FC$| and so |${\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}}) \in {\mathcal{N}}({{\mathsf{\scriptstyle \Gamma}}})$|⁠. (wf2): |${\mathcal{N}}({{\mathsf{\scriptstyle \Gamma}}})$| is upwards closed by definition. (wf3): We will equivalently prove that |$\forall X,Y \in {\mathcal{N}}({{\mathsf{\scriptstyle \Gamma}}})$|⁠, |$X \cap Y \neq \emptyset$|⁠. We discern the following cases. Let |$X\supseteq [\varphi],Y\supseteq [\psi]$| for some formulas |$\varphi,\psi$| such that |${\mathsf{E}}\varphi, {\mathsf{E}}\psi\in{{\mathsf{\scriptstyle \Gamma}}}$|⁠. For the sake of contradiction assume |$X \cap Y = \emptyset$| i.e. no world includes both |$\varphi,\psi$|⁠. By Lemmas 3.16 and 3.19, |${\mathsf{K}} (\varphi\supset\neg\psi)\in {{\mathsf{\scriptstyle \Gamma}}}$|⁠. By axiom |$\mathbf{EK}$|⁠, |${\mathsf{E}}\neg\psi\in{{\mathsf{\scriptstyle \Gamma}}}$| and by axiom |$\mathbf{CE}$||$\neg{\mathsf{E}}\psi\in{{\mathsf{\scriptstyle \Gamma}}}$|⁠. But |${{\mathsf{\scriptstyle \Gamma}}}$| is a consistent set, hence the contradiction. Let |$X\supseteq [\varphi], Y\supseteq FC$|⁠. It suffices to prove that |$X \cap FC \neq \emptyset$|⁠. Let |${{\mathsf{\scriptstyle Z}}}\in FC$|⁠. By axiom |$\mathbf{PIE}$| and Lemma 3.19 we have |${\mathsf{E}}\varphi\in{{\mathsf{\scriptstyle Z}}}$|⁠. Also, since |$\top$| is valid, by Lemmas 3.16 and 3.19 we have |${\mathsf{K}} (\varphi\supset\top)\in{{\mathsf{\scriptstyle Z}}}$|⁠. For the sake of contradiction, let |$X \cap FC = \emptyset$|⁠. Then again by the same Lemmas, |${\mathsf{K}} (\varphi\supset\bot)\in{{\mathsf{\scriptstyle Z}}}$|⁠. By axiom |$\mathbf{EK}$|⁠, |${\mathsf{E}}\top,{\mathsf{E}}\bot\in{{\mathsf{\scriptstyle Z}}}$|⁠, which by axiom |$\mathbf{CE}$| leads to another contradiction. |$X,Y\supseteq FC$|⁠. The property is obvious. The proof is complete. ■ Corollary 4.5 |$(\forall{{\mathsf{\scriptstyle \Gamma}}}\in W)(\forall\varphi\in\mathcal{L}_{KBE})({\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}})\cap[\varphi]\in{\mathcal{N}}({{\mathsf{\scriptstyle \Gamma}}})\iff{\mathsf{E}}\varphi\in{{\mathsf{\scriptstyle \Gamma}}})$| Proof. First, assume that |${\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}})\cap[\varphi]\in{\mathcal{N}}({{\mathsf{\scriptstyle \Gamma}}})$|⁠. If |${\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}})\cap[\varphi] \supseteq \Sigma$| then by Lemma 3.23 (iv) |${\mathsf{E}}\varphi \in {{\mathsf{\scriptstyle \Gamma}}}$|⁠. If |${\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}})\cap[\varphi] \supseteq {\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}})\cap[\psi]$| for some |${\mathsf{E}} \psi \in {{\mathsf{\scriptstyle \Gamma}}}$|⁠, then by Lemmas 3.16 and 3.19, |${\mathsf{K}} (\psi \supset \varphi)\in {{\mathsf{\scriptstyle \Gamma}}}$|⁠. By axiom |$\mathbf{EK}$|⁠, |${\mathsf{E}} \varphi \in {{\mathsf{\scriptstyle \Gamma}}}$|⁠. Conversely, if |${\mathsf{E}}\varphi\in{{\mathsf{\scriptstyle \Gamma}}}$| then by Definition 4.3 (iii) |${\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}})\cap[\varphi]\in{\mathcal{N}}({{\mathsf{\scriptstyle \Gamma}}})$|⁠. ■ Lemma 3.26 can also be imitated with only trivial changes. Finally we have: Theorem 4.6 (Completeness) |$\mathbf{KBiE}$| is strongly complete w.r.t. the class of all kbie-frames. Proof. It suffices to show that every c|$\Lambda$| theory |$T$| is satisfiable in a kbie-model. By Lindenbaum’s Lemma, |$T$| is contained in a mc|$\Lambda$| theory |${{\mathsf{\scriptstyle \Xi}}}$|⁠, hence, by the Truth Lemma, |${\mathfrak{M}}^\Lambda_{{\mathsf{\scriptstyle \Xi}}},{{\mathsf{\scriptstyle \Xi}}}\Vdash T$|⁠. Therefore, it remains to check that |${\mathfrak{M}}^\Lambda_{{\mathsf{\scriptstyle \Xi}}}$| is a kbie-model (we simply write again |${\mathfrak{M}}$| instead of |${\mathfrak{M}}^\Lambda_{{\mathsf{\scriptstyle \Xi}}}$|⁠, which was defined in Definition 4.3). By Fact 3.17, |$W\neq\emptyset$|⁠. Since axioms |$\mathbf{T}$| and |$\mathbf{4}$| are canonical for the properties of reflexivity and transitivity respectively, |$P$| (as a relation on the set of all mc|$\Lambda$| theories) is reflexive and transitive, so is |${\mathcal{R}}$|⁠, by Definition 3.14 (ii). Furthermore, |$FC\neq\emptyset$|⁠, by Lemma 3.23 (i)–(ii). Consider now any |${{\mathsf{\scriptstyle \Gamma}}}\in W$|⁠. Properties |$\mathbf{(nr)}$| and |$\mathbf{(be)}$| hold by Definition 4.3 and Lemma 3.23 (ii). Properties |$\mathbf{(ce)}$| and |$\mathbf{(ek)}$| hold by Lemma 4.4. Regarding property |$\mathbf{(pie)}$|⁠: Let |${{\mathsf{\scriptstyle \Gamma}}},{{ \mathsf{\scriptstyle \Delta}}}\in W$|⁠, |${{\mathsf{\scriptstyle \Gamma}}} {\mathcal{R}} {{ \mathsf{\scriptstyle \Delta}}}$|⁠, and |$X\in{\mathcal{N}}({{\mathsf{\scriptstyle \Gamma}}})$|⁠. Let |$X \supseteq FC$|⁠. We have |${\mathcal{R}}({{ \mathsf{\scriptstyle \Delta}}}) \supseteq FC$| and so by Definition 4.3 |$X \cap {\mathcal{R}}({{ \mathsf{\scriptstyle \Delta}}}) \in {\mathcal{N}}({{ \mathsf{\scriptstyle \Delta}}})$|⁠. Otherwise, |$X \supseteq {\mathcal{R}}({{\mathsf{\scriptstyle \Gamma}}}) \cap [\varphi]$| and |${\mathsf{E}} \varphi \in {{\mathsf{\scriptstyle \Gamma}}}$|⁠. By axiom |$\mathbf{PIE}$| and Lemma 3.19, |${\mathsf{E}} \varphi \in {{ \mathsf{\scriptstyle \Delta}}}$|⁠, and because |$X \cap {\mathcal{R}}({{ \mathsf{\scriptstyle \Delta}}}) \supseteq {\mathcal{R}}({{ \mathsf{\scriptstyle \Delta}}}) \cap [\varphi]$|⁠, we have |$X \cap {\mathcal{R}}({{ \mathsf{\scriptstyle \Delta}}}) \in {\mathcal{N}}({{ \mathsf{\scriptstyle \Delta}}})$|⁠. The proof is complete. ■ 4.2 Relation to probabilistic semantics for weak belief Having defined |$\mathbf{KBiE}$|⁠, it is natural to ask about the relation of its weak filter based semantics to the probabilistic semantics for weak belief, as introduced for instance by W. Lenzen in [29]. It is worth noting here that similar considerations appear in A. Herzig’s work [21], where it is proved that the relevant logic is not complete with respect to the probabilistic semantics [21, Theorem 4.4]. We will also prove here that |$\mathbf{KBiE}$| cannot be given such a semantics, using a different technique. Weak belief is defined by Lenzen with the following probabilistic semantics: given a probability distribution over a set of worlds we have (using |${\mathsf{E}}$| as the modality for weak belief) M,w⊨Eφ iff ∑{P(w)∣M,w⊨φ}>0.5. It is not hard to see that in this approach, only one direction of axiom |$\mathbf{CCE}$| is acceptable. However, |$\mathbf{KBiE}$| could conceivably be given such a semantics. Yet, according to Lenzen [29, p. 26] the following rule of inference should be valid: {φ1,…,φn}⇒{ψ1,…,ψn}⊢Eφ1∧¬E¬φ2∧⋯∧¬E¬φn⊃Eψ1∨⋯∨Eψn, where, the relation |$ \{ \varphi_1, \dots, \varphi_n \} \Rightarrow \{ \psi_1, \dots, \psi_n \} $| (attributed to K. Segerberg [36]) holds in a world |$w$| iff |$w$| satisfies at least as many formulas in |$\{ \psi_1, \dots, \psi_n \}$| as it does in |$\{\varphi_1, \dots, \varphi_n \}$|⁠. Note that both sets have the same (finite) cardinality. We will exhibit a |$\mathbf{KBiE}$| model and corresponding formulas |$\varphi_i$|⁠, |$\psi_j$| such that {φ1,…,φn}⇒{ψ1,…,ψn} is valid in that model, but |${\mathsf{E}} \varphi_1 \wedge \neg {\mathsf{E}} \neg \varphi_2 \wedge \dots \wedge \neg {\mathsf{E}} \neg \varphi_n \supset {\mathsf{E}} \psi_1 \vee \dots \vee {\mathsf{E}} \psi_n$| is not. Consider the following formulas, where |$p, q, r, s$| are propositional variables (atoms). |$\varphi_{1}\equiv$| |$(p\wedge q\wedge r\wedge s)\vee(p\wedge q\wedge r\wedge\neg s)\vee(p\wedge q\wedge\neg r\wedge s)\vee(p\wedge q\wedge\neg r\wedge\neg s)$| |$\varphi_{2}\equiv$| |$(p\wedge q\wedge r\wedge s)\vee(p\wedge\neg q\wedge r\wedge s)\vee(p\wedge\neg q\wedge r\wedge\neg s)\vee(p\wedge\neg q\wedge\neg r\wedge s)$| |$\varphi_{3}\equiv$| |$(p\wedge q\wedge r\wedge\neg s)\vee(p\wedge\neg q\wedge r\wedge s)\vee(p\wedge\neg q\wedge\neg r\wedge\neg s)\vee(\neg p\wedge q\wedge r\wedge s)$| |$\varphi_{4}\equiv$| |$(p\wedge q\wedge\neg r\wedge s)\vee(p\wedge\neg q\wedge r\wedge\neg s)\vee(p\wedge\neg q\wedge\neg r\wedge\neg s)\vee(\neg p\wedge q\wedge r\wedge\neg s)$| |$\varphi_{5}\equiv$| |$(p\wedge q\wedge\neg r\wedge\neg s)\vee(p\wedge\neg q\wedge\neg r\wedge s)\vee(\neg p\wedge q\wedge r\wedge s)\vee(\neg p\wedge q\wedge r\wedge\neg s)$| |$\varphi_{1}\equiv$| |$(p\wedge q\wedge r\wedge s)\vee(p\wedge q\wedge r\wedge\neg s)\vee(p\wedge q\wedge\neg r\wedge s)\vee(p\wedge q\wedge\neg r\wedge\neg s)$| |$\varphi_{2}\equiv$| |$(p\wedge q\wedge r\wedge s)\vee(p\wedge\neg q\wedge r\wedge s)\vee(p\wedge\neg q\wedge r\wedge\neg s)\vee(p\wedge\neg q\wedge\neg r\wedge s)$| |$\varphi_{3}\equiv$| |$(p\wedge q\wedge r\wedge\neg s)\vee(p\wedge\neg q\wedge r\wedge s)\vee(p\wedge\neg q\wedge\neg r\wedge\neg s)\vee(\neg p\wedge q\wedge r\wedge s)$| |$\varphi_{4}\equiv$| |$(p\wedge q\wedge\neg r\wedge s)\vee(p\wedge\neg q\wedge r\wedge\neg s)\vee(p\wedge\neg q\wedge\neg r\wedge\neg s)\vee(\neg p\wedge q\wedge r\wedge\neg s)$| |$\varphi_{5}\equiv$| |$(p\wedge q\wedge\neg r\wedge\neg s)\vee(p\wedge\neg q\wedge\neg r\wedge s)\vee(\neg p\wedge q\wedge r\wedge s)\vee(\neg p\wedge q\wedge r\wedge\neg s)$| |$\varphi_{1}\equiv$| |$(p\wedge q\wedge r\wedge s)\vee(p\wedge q\wedge r\wedge\neg s)\vee(p\wedge q\wedge\neg r\wedge s)\vee(p\wedge q\wedge\neg r\wedge\neg s)$| |$\varphi_{2}\equiv$| |$(p\wedge q\wedge r\wedge s)\vee(p\wedge\neg q\wedge r\wedge s)\vee(p\wedge\neg q\wedge r\wedge\neg s)\vee(p\wedge\neg q\wedge\neg r\wedge s)$| |$\varphi_{3}\equiv$| |$(p\wedge q\wedge r\wedge\neg s)\vee(p\wedge\neg q\wedge r\wedge s)\vee(p\wedge\neg q\wedge\neg r\wedge\neg s)\vee(\neg p\wedge q\wedge r\wedge s)$| |$\varphi_{4}\equiv$| |$(p\wedge q\wedge\neg r\wedge s)\vee(p\wedge\neg q\wedge r\wedge\neg s)\vee(p\wedge\neg q\wedge\neg r\wedge\neg s)\vee(\neg p\wedge q\wedge r\wedge\neg s)$| |$\varphi_{5}\equiv$| |$(p\wedge q\wedge\neg r\wedge\neg s)\vee(p\wedge\neg q\wedge\neg r\wedge s)\vee(\neg p\wedge q\wedge r\wedge s)\vee(\neg p\wedge q\wedge r\wedge\neg s)$| |$\varphi_{1}\equiv$| |$(p\wedge q\wedge r\wedge s)\vee(p\wedge q\wedge r\wedge\neg s)\vee(p\wedge q\wedge\neg r\wedge s)\vee(p\wedge q\wedge\neg r\wedge\neg s)$| |$\varphi_{2}\equiv$| |$(p\wedge q\wedge r\wedge s)\vee(p\wedge\neg q\wedge r\wedge s)\vee(p\wedge\neg q\wedge r\wedge\neg s)\vee(p\wedge\neg q\wedge\neg r\wedge s)$| |$\varphi_{3}\equiv$| |$(p\wedge q\wedge r\wedge\neg s)\vee(p\wedge\neg q\wedge r\wedge s)\vee(p\wedge\neg q\wedge\neg r\wedge\neg s)\vee(\neg p\wedge q\wedge r\wedge s)$| |$\varphi_{4}\equiv$| |$(p\wedge q\wedge\neg r\wedge s)\vee(p\wedge\neg q\wedge r\wedge\neg s)\vee(p\wedge\neg q\wedge\neg r\wedge\neg s)\vee(\neg p\wedge q\wedge r\wedge\neg s)$| |$\varphi_{5}\equiv$| |$(p\wedge q\wedge\neg r\wedge\neg s)\vee(p\wedge\neg q\wedge\neg r\wedge s)\vee(\neg p\wedge q\wedge r\wedge s)\vee(\neg p\wedge q\wedge r\wedge\neg s)$| Consider also the following propositional valuations. The following table succintly summarizes the situation: Now, let |$\psi_i = \neg \varphi_i$| for |$i \in \{1,\dots,5\}$|⁠. Consider the kbie-model |${\mathfrak{M}}$| where |$W = FC$| is the set of all |$2^{4}$| valuations for the variables |$p,q,r,s$|⁠. For every |$w$| let |${\mathcal{N}} (w)$| be the weak filter that contains all |$\overline{V}(\varphi_i)$|⁠, for |$i \in \{1,\dots,5\}$|⁠, and their supersets. To see that this is indeed a weak filter note that it is upwards closed (by definition) and for each pair |$\varphi_i, \varphi_j$| (⁠|$i\neq j$|⁠) there exists at least (and in fact exactly) one common valuation that satisfies both formulae. Consequently their truth sets have non-empty intersections. It can be checked that |$ \{ \varphi_1, \dots, \varphi_n \} \Rightarrow \{ \psi_1, \dots, \psi_n \}$| is valid in |${\mathfrak{M}}$|⁠: any world |$w$| satisfies at most two of |$\varphi_1, \dots, \varphi_5 $| (and thus does not satisfy ‘at least’ three) so it satisfies at least three from |$\psi_1, \dots, \psi_5$|⁠. On the other hand, there exists a world |$w$| (in fact all of them), such that |${\mathsf{E}} \varphi_1 \wedge \dots \wedge {\mathsf{E}} \varphi_5$| is true, and consequently |${\mathsf{E}} \varphi_1 \wedge \neg {\mathsf{E}} \neg \varphi_2 \wedge \dots \wedge \neg {\mathsf{E}} \neg \varphi_n$|⁠. But no |${\mathsf{E}} \psi_i$| is true; as |$\overline{V}(\psi_i) = W \setminus \overline{V}(\varphi_i) \notin {\mathcal{N}}(w).$| It follows that |$\mathbf{KBiE}$| invalidates Lenzen’s rule for the probabilistic semantics. 5 Tableaux for $\mathbf{KBE}$ In this section we present a tableau system for |$\mathbf{KBE}$| using prefixed formulas. We prove soundness and completeness with respect to the class of kbe-models, and then modify the systematic procedure used to prove completeness, to show that |$\mathbf{KBE}$| is decidable and has the finite model property. A reminder on terminology is in order: a prefix is a finite sequence of natural numbers, separated by periods. A prefixed formula is an expression of the form |$\sigma\;\varphi$|⁠, where |$\sigma$| is a prefix and |$\varphi$| is a formula. A tableau branch is closed if it contains both |$\sigma\;\varphi$| and |$\sigma\;\neg\varphi$| for some prefix |$\sigma$| and formula |$\varphi$|⁠. A tableau is closed if all of its branches are closed. A tableau or branch is open if it is not closed. The terminology and most of the techniques we use, draw from [12, 13]. The intention for the prefixes is that they name worlds in a model, and the world named by |$\sigma.n$| is accessible from the world named by |$\sigma$|⁠. The worlds of a kbe-model either belong to its final cluster or not, so we will be using two kinds of prefixed formulas; of the form |$0.n,n\in\mathbb{N}$| to represent the first, and of the form |$1.\sigma$| to represent the latter. Prefixes of the form |$0.n$| do not allow tracking of some accessibility relation, but are sufficient for the final cluster, exactly because it is a cluster, i.e. relation |${\mathcal{R}}$| is universal in it. 5.1 Tableaux rules Before presenting the rules themselves we need a notion of accessibility between prefixes, proper for kbe-models. Definition 5.1 A prefix |$\sigma'$| is accessible from a prefix |$\sigma$| if and only if |$\sigma$| is an initial segment of |$\sigma'$| (proper or otherwise), or |$\sigma'$| is of the form |$0.n,n\in\mathbb{N}$|⁠. For the alphabet of our tableaux, we assume |${\mathsf{B}}\varphi,\left\langle {\mathsf{K}}\right\rangle \varphi,\left\langle {\mathsf{E}}\right\rangle \varphi,\varphi\supset\psi,\varphi\equiv\psi$| are abbreviations for |$\neg{\mathsf{K}}\neg{\mathsf{K}}\varphi,\neg{\mathsf{K}}\neg\varphi,\neg{\mathsf{E}}\neg\varphi,\neg\varphi\vee\psi,(\varphi\supset\psi)\wedge(\psi\supset\varphi)$| respectively, thus no corresponding rules have to be specified. Definition 5.2 A kbe-tableau for a formula |$\varphi$| is a tableau that starts with the prefixed formulas |$1\;\neg\varphi$| and |$0.1\;\top$| and is extended using any of the rules below. For prefixes |$\sigma$| of the form |$1.\sigma'$| : For prefixes |$0.n$| we have, in essence, the same rules, but the exact notation for rules introducing a new world is: The double negation, conjunctive and disjunctive rules, are standard for the propositional part of any modal logic. Moreover, we need rules to reflect the meaning of box and diamond for each modal operator. In our case, |$\mathbf{KBE}$| is normal with respect to |${\mathsf{K}}$|⁠, and our rules for |${\mathsf{K}}$| (and |$\neg{\mathsf{K}}$|⁠) state what they should, regarding semantics. What makes the rules appropriate for an |$\mathbf{S4.2}$|-frame, is that we introduce at least a prefix for the final cluster with |$0.1\;\top$|⁠, and reflexivity, transitivity and that the final cluster is both ‘final’ and a cluster, is integrated into our notion of prefix accessibility. Regarding [|$\mathbf{CCE}$|-rule] and [|$\mathbf{PIE}$|-rule], as their name suggests, they exist to tend to axioms |$\mathbf{CCE}$| and |$\mathbf{PIE}$|⁠. Axiom |$\mathbf{CCE}$| is in fact an equivalence, but we decide to transform all |$\left\langle {\mathsf{E}}\right\rangle $| into |${\mathsf{E}}$| and have no use for the other direction. Axiom |$\mathbf{PIE}$| is also not exactly what our rule implies, but for the sake of shortening proofs, one can observe the only applicable rule to a formula |${\mathsf{K}}{\mathsf{E}}\varphi$| is [|${\mathsf{K}}\nu$|-rule]; we do it outright. Finally, regarding modality |${\mathsf{E}}$|⁠, |$\mathbf{KBE}$| is monotonic with respect to it. The proper rule, is that for any pair |$\left\langle {\mathsf{E}}\right\rangle \varphi$|⁠, |${\mathsf{E}}\psi$| there is a world such that |$\varphi,\psi$| hold (see [12] regarding the Logic |$\mathbf{U}$|⁠, and specifically Chapter 6.13 for a tableau for |$\mathbf{U}$|⁠). In our case, |$\left\langle {\mathsf{E}}\right\rangle $| has turned into |${\mathsf{E}}$|⁠, and not just any world will do, but one accessible with respect to |${\mathcal{R}}$|⁠; [|$\mathbf{E}$|-rule] is created accordingly. Also note that |$\varphi$| can be the same as |$\psi$|⁠. Definition 5.3 A closed kbe-tableau for a formula |$\varphi$| is a kbe-tableau proof for |$\varphi$|⁠. We give tableaux proofs for formulas of Proposition 3.4 Example 1 |${\mathsf{E}}{\mathsf{K}} p\equiv\neg{\mathsf{K}}\neg{\mathsf{K}}p$| 1 |$\neg((\neg{\mathsf{E}}{\mathsf{K}} p\vee\neg{\mathsf{K}}\neg{\mathsf{K}} p)\wedge(\neg\neg{\mathsf{K}}\neg{\mathsf{K}} p\vee{\mathsf{E}}{\mathsf{K}} p))$| 1. 0.1 |$\top$| 2. 1 |$\neg(\neg{\mathsf{E}}{\mathsf{K}} p\vee\neg{\mathsf{K}}\neg{\mathsf{K}} p)$| 3. |$\qquad{}$| 1 |$\neg(\neg\neg{\mathsf{K}}\neg{\mathsf{K}} p\vee{\mathsf{E}}{\mathsf{K}} p)$|10. 1 |$\neg\neg{\mathsf{E}}{\mathsf{K}} p$| 4. 1 |$\neg\neg\neg{\mathsf{K}}\neg{\mathsf{K}} p$| 11. 1 |$\neg\neg{\mathsf{K}}\neg{\mathsf{K}} p$| 5. 1 |$\neg{\mathsf{E}}{\mathsf{K}} p$| 12. 1 |${\mathsf{E}}{\mathsf{K}} p$| 6. 1 |$\neg{\mathsf{K}}\neg{\mathsf{K}} p$| 13. 1 |${\mathsf{K}}\neg{\mathsf{K}} p$| 7. 1 |${\mathsf{E}}\neg{\mathsf{K}} p$| 14. 1.1 |${\mathsf{K}} p$| 8. 1.1 |$\neg\neg{\mathsf{K}} p$| 15. 1.1 |$\neg{\mathsf{K}} p$| 9. 1.1 |${\mathsf{K}} p$| 16. 0.1 |${\mathsf{E}}\neg{\mathsf{K}} p$| 17. 0.2 |$\neg{\mathsf{K}} p$| 18. 0.3 |$\neg p$| 19. 0.3 |$p$| 20. Example 1 |${\mathsf{E}}{\mathsf{K}} p\equiv\neg{\mathsf{K}}\neg{\mathsf{K}}p$| 1 |$\neg((\neg{\mathsf{E}}{\mathsf{K}} p\vee\neg{\mathsf{K}}\neg{\mathsf{K}} p)\wedge(\neg\neg{\mathsf{K}}\neg{\mathsf{K}} p\vee{\mathsf{E}}{\mathsf{K}} p))$| 1. 0.1 |$\top$| 2. 1 |$\neg(\neg{\mathsf{E}}{\mathsf{K}} p\vee\neg{\mathsf{K}}\neg{\mathsf{K}} p)$| 3. |$\qquad{}$| 1 |$\neg(\neg\neg{\mathsf{K}}\neg{\mathsf{K}} p\vee{\mathsf{E}}{\mathsf{K}} p)$|10. 1 |$\neg\neg{\mathsf{E}}{\mathsf{K}} p$| 4. 1 |$\neg\neg\neg{\mathsf{K}}\neg{\mathsf{K}} p$| 11. 1 |$\neg\neg{\mathsf{K}}\neg{\mathsf{K}} p$| 5. 1 |$\neg{\mathsf{E}}{\mathsf{K}} p$| 12. 1 |${\mathsf{E}}{\mathsf{K}} p$| 6. 1 |$\neg{\mathsf{K}}\neg{\mathsf{K}} p$| 13. 1 |${\mathsf{K}}\neg{\mathsf{K}} p$| 7. 1 |${\mathsf{E}}\neg{\mathsf{K}} p$| 14. 1.1 |${\mathsf{K}} p$| 8. 1.1 |$\neg\neg{\mathsf{K}} p$| 15. 1.1 |$\neg{\mathsf{K}} p$| 9. 1.1 |${\mathsf{K}} p$| 16. 0.1 |${\mathsf{E}}\neg{\mathsf{K}} p$| 17. 0.2 |$\neg{\mathsf{K}} p$| 18. 0.3 |$\neg p$| 19. 0.3 |$p$| 20. Open in new tab Example 1 |${\mathsf{E}}{\mathsf{K}} p\equiv\neg{\mathsf{K}}\neg{\mathsf{K}}p$| 1 |$\neg((\neg{\mathsf{E}}{\mathsf{K}} p\vee\neg{\mathsf{K}}\neg{\mathsf{K}} p)\wedge(\neg\neg{\mathsf{K}}\neg{\mathsf{K}} p\vee{\mathsf{E}}{\mathsf{K}} p))$| 1. 0.1 |$\top$| 2. 1 |$\neg(\neg{\mathsf{E}}{\mathsf{K}} p\vee\neg{\mathsf{K}}\neg{\mathsf{K}} p)$| 3. |$\qquad{}$| 1 |$\neg(\neg\neg{\mathsf{K}}\neg{\mathsf{K}} p\vee{\mathsf{E}}{\mathsf{K}} p)$|10. 1 |$\neg\neg{\mathsf{E}}{\mathsf{K}} p$| 4. 1 |$\neg\neg\neg{\mathsf{K}}\neg{\mathsf{K}} p$| 11. 1 |$\neg\neg{\mathsf{K}}\neg{\mathsf{K}} p$| 5. 1 |$\neg{\mathsf{E}}{\mathsf{K}} p$| 12. 1 |${\mathsf{E}}{\mathsf{K}} p$| 6. 1 |$\neg{\mathsf{K}}\neg{\mathsf{K}} p$| 13. 1 |${\mathsf{K}}\neg{\mathsf{K}} p$| 7. 1 |${\mathsf{E}}\neg{\mathsf{K}} p$| 14. 1.1 |${\mathsf{K}} p$| 8. 1.1 |$\neg\neg{\mathsf{K}} p$| 15. 1.1 |$\neg{\mathsf{K}} p$| 9. 1.1 |${\mathsf{K}} p$| 16. 0.1 |${\mathsf{E}}\neg{\mathsf{K}} p$| 17. 0.2 |$\neg{\mathsf{K}} p$| 18. 0.3 |$\neg p$| 19. 0.3 |$p$| 20. Example 1 |${\mathsf{E}}{\mathsf{K}} p\equiv\neg{\mathsf{K}}\neg{\mathsf{K}}p$| 1 |$\neg((\neg{\mathsf{E}}{\mathsf{K}} p\vee\neg{\mathsf{K}}\neg{\mathsf{K}} p)\wedge(\neg\neg{\mathsf{K}}\neg{\mathsf{K}} p\vee{\mathsf{E}}{\mathsf{K}} p))$| 1. 0.1 |$\top$| 2. 1 |$\neg(\neg{\mathsf{E}}{\mathsf{K}} p\vee\neg{\mathsf{K}}\neg{\mathsf{K}} p)$| 3. |$\qquad{}$| 1 |$\neg(\neg\neg{\mathsf{K}}\neg{\mathsf{K}} p\vee{\mathsf{E}}{\mathsf{K}} p)$|10. 1 |$\neg\neg{\mathsf{E}}{\mathsf{K}} p$| 4. 1 |$\neg\neg\neg{\mathsf{K}}\neg{\mathsf{K}} p$| 11. 1 |$\neg\neg{\mathsf{K}}\neg{\mathsf{K}} p$| 5. 1 |$\neg{\mathsf{E}}{\mathsf{K}} p$| 12. 1 |${\mathsf{E}}{\mathsf{K}} p$| 6. 1 |$\neg{\mathsf{K}}\neg{\mathsf{K}} p$| 13. 1 |${\mathsf{K}}\neg{\mathsf{K}} p$| 7. 1 |${\mathsf{E}}\neg{\mathsf{K}} p$| 14. 1.1 |${\mathsf{K}} p$| 8. 1.1 |$\neg\neg{\mathsf{K}} p$| 15. 1.1 |$\neg{\mathsf{K}} p$| 9. 1.1 |${\mathsf{K}} p$| 16. 0.1 |${\mathsf{E}}\neg{\mathsf{K}} p$| 17. 0.2 |$\neg{\mathsf{K}} p$| 18. 0.3 |$\neg p$| 19. 0.3 |$p$| 20. Open in new tab Item 1 is the negation of the formula we want to prove expressed in the tableaux language and item 2 is standard. Items 3 and 10 are from 1 by [Disjunctive Rule]. Items 4 and 5 are from 3 by a [Conjunctive Rule]. Items 6 and 7 are from 4 and 5 respectively by [Double negation rule]. Item 8 is from 6 by [|${\mathsf{E}}$|-rule]. Item 9 is from 7 by [|${\mathsf{K}}\nu$|-rule]. Item 11 and 12 are from 10 by a [Conjunctive rule]. Item 13 is from 11 by [Double negation rule]. Item 14 is from 12 by [|$\mathbf{CCE}$|-rule]. Item 15 is from 13 by [|${\mathsf{K}}\pi$|-rule]. Item 16 is from 15 by [Double negation rule]. Item 17 is from 14 by [|$\mathbf{PIE}$|-rule]. Item 18 is from 17 by [|${\mathsf{E}}$|-rule]. Item 19 is from 18 by [|${\mathsf{K}}\pi$|-rule] and item 20 is from 16 by [|${\mathsf{K}}\nu$|-rule]. 5.2 Soundness Definition 5.4 (Satisfiable) Suppose |$S$| is a set of prefixed formulas. We say |$S$| is kbe-satisfiable if there is a kbe-model |$\left\langle W,{\mathcal{R}},{\mathcal{N}},V\right\rangle $| and a function |$\theta:prefixes\rightarrow W$| such that: (i) for any prefixes |$\sigma',\sigma$| that occur in |$S$|⁠, if |$\sigma'$| is accessible from |$\sigma$| then |$\theta(\sigma){\mathcal{R}}\theta(\sigma')$|⁠. (ii) for any |$\sigma\;\varphi\in S$|⁠, it holds that |$\theta(\sigma)\Vdash\varphi$| We say a tableau is kbe-satisfiable if some branch of it is kbe-satisfiable. A branch is kbe-satisfiable if the set of prefixed formulas on it is kbe-satisfiable. Proposition 5.5 A closed tableau is not kbe-satisfiable Proof. Suppose a tableau was closed and satisfiable. This means that for some formula |$\varphi$| and prefix |$\sigma$|⁠, both |$\sigma\;\varphi$| and |$\sigma\;\neg\varphi$| occur on a tableau’s branch, and there is a model |$\left\langle W,{\mathcal{R}},{\mathcal{N}},V\right\rangle $| and function |$\theta$| such that (Definition 5.4 (ii)) |$\theta(\sigma)\Vdash\varphi$| and |$\theta(\sigma)\Vdash\neg\varphi$|⁠. We derive a contradiction. ■ Proposition 5.6 Applying any of the rules to a kbe-satisfiable tableau, gives another kbe-satisfiable tableau. Proof. Suppose we have a satisfiable tableau and let |$\mathcal{B}$| be the branch we apply any of the tableaux rules. If there is another branch |$\mathcal{B}'$| that is satisfiable, then the resulting tableau is trivially also satisfiable. So we assume the only satisfiable branch is |$\mathcal{B}$|⁠. [Conjunctive rules]: Suppose |$\sigma\;\varphi\wedge\psi$| occurs on |$\mathcal{B}$|⁠. By applying the corresponding rule we add |$\sigma\;\varphi$| and |$\sigma\;\psi$| to the end of |$\mathcal{B}$|⁠. Since |$\mathcal{B}$| is satisfiable there are the aforementioned model |$\left\langle W,{\mathcal{R}},{\mathcal{N}},V\right\rangle$| and function |$\theta$| such that |$\theta(\sigma)\Vdash\varphi\wedge\psi$|⁠. Consequently |$\theta(\sigma)\Vdash\varphi$| and |$\theta(\sigma)\Vdash\psi$|⁠, so the extended branch is also satisfiable, using the same model and function |$\theta$|⁠. The other conjunctive rule, as well as the double negation rule, are treated similarly. [Disjunctive rules]: Suppose |$\sigma\;\varphi\vee\psi$| occurs on |$\mathcal{B}$|⁠. By applying the corresponding rule we split the end of |$\mathcal{B}$|⁠, adding |$\sigma\;\varphi$| to the left fork and |$\sigma\;\psi$| to the right. Since |$\mathcal{B}$| is satisfiable there is a model |$\left\langle W,{\mathcal{R}},{\mathcal{N}},V\right\rangle $| and a function |$\theta$| such that |$\theta(\sigma)\Vdash\varphi\vee\psi$|⁠. Consequently |$\theta(\sigma)\Vdash\varphi$| or |$\theta(\sigma)\Vdash\psi$|⁠, and so at least one extension of |$\mathcal{B}$| is satisfiable. The other disjunctive rule is treated similarly. [|${\mathsf{K}}\nu$|-rule]: Suppose |$\sigma\;{\mathsf{K}}\varphi$| occurs on |$\mathcal{B}$|⁠. By applying the corresponding rule we add |$\sigma'\;\varphi$| to the end of |$\mathcal{B}$|⁠, for some prefix |$\sigma'$| accessible from |$\sigma$|⁠. Since |$\mathcal{B}$| is satisfiable there is a model |$\left\langle W,{\mathcal{R}},{\mathcal{N}},V\right\rangle $| and a function |$\theta$| such that |$\theta(\sigma)\Vdash{\mathsf{K}}\varphi$| and |$\theta(\sigma){\mathcal{R}}\theta(\sigma')$|⁠. Consequently |$\theta(\sigma')\Vdash\varphi$|⁠, so the extended branch is also satisfiable. [|${\mathsf{K}}\pi$|-rule]: Suppose |$\sigma\;\neg{\mathsf{K}}\varphi$| occurs on |$\mathcal{B}$|⁠. By applying the corresponding rule we add |$\sigma.n\;\varphi$| to the end of |$\mathcal{B}$|⁠, where |$\sigma.n$| is new to |$\mathcal{B}$|⁠. Since |$\mathcal{B}$| is satisfiable there is a model |$\left\langle W,{\mathcal{R}},{\mathcal{N}},V\right\rangle $| and a function |$\theta$| such that |$\theta(\sigma)\Vdash\neg{\mathsf{K}}\varphi$|⁠. Consequently there is a world |$w$| such that |$w\Vdash\varphi$| and |$\theta(\sigma){\mathcal{R}} w$|⁠. We need only define |$\theta(\sigma.n)=w$|⁠, and the extended branch is shown to be satisfiable. We used the notation for prefixes of the form |$1.\sigma$|⁠; prefixes of the form |$0.n$| are treated similarly. [|$\mathbf{CCE}$|-rule]: Suppose |$\sigma\;\neg{\mathsf{E}}\varphi$| occurs on |$\mathcal{B}$|⁠. By applying the corresponding rule we add |$\sigma\;{\mathsf{E}}\neg\varphi$| to the end of |$\mathcal{B}$|⁠. Since |$\mathcal{B}$| is satisfiable there is a model |$\left\langle W,{\mathcal{R}},{\mathcal{N}},V\right\rangle $| and a function |$\theta$| such that |$\theta(\sigma)\Vdash\neg{\mathsf{E}}\varphi$|⁠. We already know the axiomatization includes axiom |$\mathbf{CCE}$| so it is valid in any kbe-model. Consequently |$\theta(\sigma)\Vdash{\mathsf{E}}\neg\varphi$|⁠. [|$\mathbf{PIE}$|-rule]: Suppose |$\sigma\;{\mathsf{E}}\varphi$| occurs on |$\mathcal{B}$|⁠, and by applying the corresponding rule we add |$\sigma'\;{\mathsf{E}}\varphi$| to the end of |$\mathcal{B}$|⁠, for some prefix |$\sigma'$| accessible from |$\sigma$|⁠. Since |$\mathcal{B}$| is satisfiable there is a model |$\left\langle W,{\mathcal{R}},{\mathcal{N}},V\right\rangle $| and a function |$\theta$| such that |$\theta(\sigma)\Vdash{\mathsf{E}}\varphi$| and |$\theta(\sigma){\mathcal{R}}\theta(\sigma')$|⁠. We already know the axiomatization includes axiom |$\mathbf{PIE}$| so it is valid in any kbe-model. Consequently |$\theta(\sigma)\Vdash{\mathsf{K}}{\mathsf{E}}\varphi$| and since |$\theta(\sigma){\mathcal{R}}\theta(\sigma')$| we have |$\theta(\sigma')\Vdash{\mathsf{E}}\varphi$|⁠. [|${\mathsf{E}}$|-rule]: Suppose |$\sigma\;{\mathsf{E}}\varphi$| and |$\sigma\;{\mathsf{E}}\psi$| occur on |$\mathcal{B}$|⁠. By applying the corresponding rule we add |$\sigma.n\;\varphi$| and |$\sigma.n\;\psi$| to the end of |$\mathcal{B}$|⁠, where |$\sigma.n$| is new to |$\mathcal{B}$|⁠. Since |$\mathcal{B}$| is satisfiable there is a model |$\left\langle W,{\mathcal{R}},{\mathcal{N}},V\right\rangle $| and a function |$\theta$| such that |$\theta(\sigma)\Vdash{\mathsf{E}}\varphi$| and |$\theta(\sigma)\Vdash{\mathsf{E}}\psi$|⁠. That is |${\mathcal{R}}(\theta(\sigma))\cap|\varphi|,{\mathcal{R}}(\theta(\sigma))\cap|\psi|\in N(w)$|⁠. By definition of large sets |$({\mathcal{R}}(\theta(\sigma))\cap|\varphi|)\cap({\mathcal{R}}(\theta(\sigma))\cap|\psi|)={\mathcal{R}}(\theta(\sigma))\cap|\varphi|\cap|\psi|\neq\emptyset.$| So there is a world |$w$| such that |$w\Vdash\varphi,w\Vdash\psi$| and |$\theta(\sigma){\mathcal{R}} w$|⁠. We need only define |$\theta(\sigma.n)=w$| and the extended branch is satisfiable. Again, prefixes of the form |$0.n$| are treated similarly. ■ Theorem 5.7 (Soundness) If |$\varphi$| is not valid in kbe-frames (kbe-valid), then there is no kbe-tableau proof for |$\varphi$|⁠. Proof. If |$\varphi$| is not kbe-valid, there is a kbe-model |$\mathfrak{M}$| and a world |$w$| of |$\mathfrak{M}$|⁠, such that |$\mathfrak{M},w\Vdash\neg\varphi$|⁠. So the tableau |$\mathcal{T}$| with the two prefixed formulas |$1\;\neg\varphi$| and |$0.1\;\top$| is satisfiable, using the model |$\mathfrak{M}$| and defining |$\theta(1)=\theta(0.1)=w$|⁠. Now for the sake of contradiction suppose there is a kbe-tableau proof for |$\varphi$|⁠, so by applying tableaux rules to |$\mathcal{T}$| we get a closed tableau. But due to Proposition 5.6, the resulting tableau will also be satisfiable, hence the contradiction due to Proposition 5.5. ■ 5.3 Completeness To achieve completeness is to show that, if by no means of the selected proof system, in our case the kbe-tableaux, can a formula |$\varphi$| be proven, then it must be the case that |$\varphi$| is not kbe-valid. The technique which also we adopt, is to provide a systematic procedure of applying the tableaux rules, making sure everything that can be derived actually is. And if the systematic procedure fails to produce a proof, then construct, or show the existence, of a kbe-model satisfying |$\neg\varphi.$|⁠, namely a counter-model. 5.3.1 Systematic procedure Notation.|${\mathsf{K}}\sigma$| and |${\mathsf{E}}\sigma$| for some prefix |$\sigma$| are sets (intended to serve as registries so as to remember |${\mathsf{K}}$| and |${\mathsf{E}}$| formulas that were found on a branch). Stage 1: Put down |$1\;\neg\varphi$| and |$0.1\;\top$|⁠. Also |${\mathsf{K}}\sigma={\mathsf{E}}\sigma=\emptyset$| for all prefixes. After stage |$n$| we stop when tableau is closed or all occurrences of prefixed formulas are finished (see below). Otherwise we proceed with stage |$n+1.$| Stage |$n+1:$| Reading the formulas starting with the leftmost branch and from top to bottom, we encounter the first unfinished occurrence of prefixed formula |$F$|⁠. (1) If $F$ is $\sigma\;\neg\neg\varphi,\sigma\;\varphi\wedge\psi,\sigma\;\neg(\varphi\vee\psi),\sigma\;\varphi\vee\psi,\sigma\;\neg(\varphi\wedge\psi),\sigma\;\neg{\mathsf{E}}\varphi$ use the appropriate rule, for each open branch including $F$⁠. That is, for the disjunctive case we split the end of each branch and for the rest cases we just add the appropriate formulas at the end of the branch provided they do not already occur. (2) If $F$ is $\sigma\;\neg{\mathsf{K}}\varphi$⁠, for each open branch that includes $F$ and that for no $n$ includes $\sigma.n\;\neg\varphi$⁠, we add $\sigma.k\;\neg\varphi$ and $\sigma.k\;\psi$ for all $\psi\in{\mathsf{K}}\sigma'$ to the end of the branch, where $k$ is the smallest positive integer such that $\sigma.k$ is new to the branch and $\sigma$ is accessible from $\sigma'$⁠. If $F$ is $0.n\;\neg{\mathsf{K}}\varphi$ we use the respective notation. (3) If $F$ is $\sigma\;{\mathsf{K}}\varphi$⁠, for each open branch including $F$ we add $\sigma'\;\varphi$ to the end of the branch, for each prefix $\sigma'$accessible from $\sigma$ and already present on the branch, if $\sigma'\;\varphi$ is not already present. We also add $\varphi$ to ${\mathsf{K}}\sigma$⁠. (4) If $F$ is $\sigma\; {\mathsf{E}}\varphi$⁠, for each open branch including $F$ (i) for each prefix $\sigma'$ accessible from $\sigma$ and already present on the branch, we add $\sigma'\;{\mathsf{E}}\varphi$ to the end of the branch, if $\sigma'\;{\mathsf{E}}\varphi$ is not already present. We also add $\varphi$ to ${\mathsf{E}}\sigma$ and ${\mathsf{E}}\varphi$ to ${\mathsf{K}}\sigma$⁠. (ii) for each formula $\psi\in{\mathsf{E}}\sigma$⁠, if for no integer $n$ does the branch include $\sigma.n\;\varphi$ and $\sigma.n\;\psi$⁠, we add $\sigma.k\;\varphi$⁠, $\sigma.k\;\psi$⁠, $\sigma.k\;\theta$ for all $\theta\in{\mathsf{K}}\sigma'$ to the end of the branch, where $k$ is the smallest positive integer such that $\sigma.k$ is new to the branch and $\sigma$ is accessible from $\sigma'$⁠. If $F$ is $0.n\;\neg{\mathsf{K}}\varphi$ we use the respective notation. |$F$| might not fall into one of the above cases (e.g. |$\sigma\; P$|⁠, |$P$| atomic) but then we just skip it. After the above we declare that occurrence of |$F$|finished. 5.3.2 Existence of a counter-model Definition 5.8 Given a branch of a tableau we define |$[\varphi]=\{\sigma\mid\sigma\;\varphi\mbox{ is on the branch}\}$|⁠. Given a model, |$|\varphi|=\overline{V}(\varphi)$| is the truth set of |$\varphi$|⁠. Let |$\mathcal{T}$| be an open tableau generated by the systematic procedure, |$\mathcal{B}$| an open branch of |$\mathcal{T}$|⁠. We define a model |$\mathfrak{M}=\left\langle W,{\mathcal{R}},{\mathcal{N}},V\right\rangle $| as follows: - |$W$| is the set of prefixes on |$\mathcal{B}$|⁠. -We define |${\mathcal{R}}$| according to our concept of accessibility, that is |$\sigma{\mathcal{R}}\sigma'$| if and only if |$\sigma'$|accessible from |$\sigma$|⁠. - If |$\sigma\; P$|⁠, |$P$| atomic, occurs on the branch then |$\sigma\Vdash P$|⁠. Otherwise |$\sigma\Vdash\neg P$|⁠. - We define |${\mathcal{N}}^{-}(1)=\{S\subseteq{\mathcal{R}}(1)\mid(S\supseteq FC)\mbox{ or }(S\supseteq[\varphi]\;\&\;\sigma\;{\mathsf{E}}\varphi\mbox{ occurs for any}\sigma)\}$.| Then we take |${\mathcal{N}}(1)$| to be a weak ultrafilter with intersections in |$FC$| (see Proposition 2.4) extending |${\mathcal{N}}^{-}(1)$|⁠. For |$\sigma\neq1$| we define |${\mathcal{N}}(\sigma)=\{S\cap{\mathcal{R}}(\sigma)\mid S\in{\mathcal{N}}(1)\}$|⁠. Proposition 5.9 |$\mathfrak{M}$| is a kbe-model. Proof. |$W$| is of course non-empty. |${\mathcal{R}}$| is reflexive, transitive and the final cluster |$FC$| is non-empty. Also (nr) and (be) are satisfied by definition. (pie) For |$w=1$| the property is guaranteed by definition. For |$w\neq1$| there must exist a set |$X\in{\mathcal{N}}(1)$| such that |$X\cap{\mathcal{R}}(w)=Y$|⁠. But then, again by definition and since |${\mathcal{R}}(u)\subseteq{\mathcal{R}}(w)$|⁠, |$X\cap{\mathcal{R}}(u)=Y\cap{\mathcal{R}}(u)\in{\mathcal{N}}(u)$|⁠. (cce) |$(\Rightarrow)$| For |$w=1$|⁠. |${\mathcal{N}}(1)$| is a weak ultrafilter (with intersections in |$FC$|⁠) if |${\mathcal{N}}^{-}(1)$| is a weak filter with intersections in |$FC$|⁠. Since |${\mathcal{N}}^{-}(1)$| is closed under supersets by definition, it suffices to show that for any |$X,Y\in{\mathcal{N}}^{-}(1)$|⁠, $X\cap Y\cap FC\neq\emptyset$. We discern three cases. - |$X\supseteq[\varphi],Y\supseteq[\psi]$| for some formulas |$\varphi,\psi$| such that |$\sigma\;{\mathsf{E}}\varphi,\sigma'\;{\mathsf{E}}\psi$| occur somewhere on the branch. Again due to [|$\mathbf{PIE}$|-rule], |${\mathsf{E}}\varphi\mbox{ and },{\mathsf{E}}\psi$| occur for some prefix in |$FC$|⁠, and due to [|${\mathsf{E}}$|-rule], so do |$\varphi,\psi$|⁠. Therefore |$[\varphi]\cap[\psi]\cap FC \neq\emptyset\Rightarrow X\cap Y\cap FC\neq\emptyset$|⁠. - |$X\supseteq[\varphi],Y\supseteq FC$| for some formula |$\varphi$| such that |$\sigma\;{\mathsf{E}}\varphi$| occurs somewhere on the branch. Due to having applied [|$\mathbf{PIE}$|-rule] during the systematic procedure, |${\mathsf{E}}\varphi$| occurs for some prefix in |$FC$|⁠, and due to having applied [|${\mathsf{E}}$|-rule] so does |$\varphi$|⁠. That is |$[\varphi]\cap FC\neq\emptyset\Rightarrow X\cap Y\cap FC \neq\emptyset$|⁠. - |$X,Y\supseteq FC$|⁠. Obviously |$X\cap Y\cap FC\neq\emptyset$|⁠. For |$w\neq1$|⁠. For the sake of contradiction let |$X,{\mathcal{R}}(w)\setminus X \in {\mathcal{N}}(w)$|⁠. There must exist sets |$Y,Z\in{\mathcal{N}}(1)$| such that |$Y\cap{\mathcal{R}}(w)=X$| and |$Z\cap{\mathcal{R}}(w) = {\mathcal{R}}(w)\setminus X$|⁠. Since |${\mathcal{N}}(1)$| has intersections in |$FC$|⁠, and |$FC\subseteq{\mathcal{R}}(w)$| we have |$Y\cap Z \cap FC = Y\cap Z \cap {\mathcal{R}}(w) \cap FC = X \cap {\mathcal{R}}(w)\setminus X \cap FC \neq \emptyset$|⁠. Hence the contradiction. |$(\Leftarrow)$| For |$w=1$| it holds by definition; we proved above that |${\mathcal{N}}(1)$| is a weak ultrafilter. For |$w\neq1$|⁠: for the sake of contradiction suppose there is a set |$X\subseteq{\mathcal{R}}(w)$| such that |$X,{\mathcal{R}}(w)\setminus X\notin{\mathcal{N}}(w)$|⁠. Then it must be the case that |$X,{\mathcal{R}}(1)\setminus X\notin{\mathcal{N}}(1)$|⁠. Hence the contradiction. (ek) For |$w=1$|⁠, as stated above, it is obvious by definition. For |$w\neq1$|⁠: suppose there are sets |$Y,Z\subseteq{\mathcal{R}}(w)$| such that |$Y\in{\mathcal{N}}(w)$| and |$Z\supseteq Y$|⁠. There must exist a set |$X\in{\mathcal{N}}(1)$| such that |$X\cap{\mathcal{R}}(w)=Y$|⁠. Since |${\mathcal{N}}(1)$| is closed under supersets |$X\cup Z\in{\mathcal{N}}(1)$|⁠. By definition |$(X\cup Z)\cap{\mathcal{R}}(w)=Y\cup Z=Z\in{\mathcal{N}}(w)$|⁠. ■ Proposition 5.10 (Key Fact) Let |$\mathfrak{M}$| be a model as in Definition 5.8. For any prefix |$\sigma$| and formula |$\varphi$|⁠: (i) if |$\sigma\;\varphi$| occurs on |$\mathcal{B}$| then |$\mathfrak{M},\sigma\Vdash\varphi$| (ii) if |$\sigma\;\neg\varphi$| occurs on |$\mathcal{B}$| then |$\mathfrak{M},\sigma\Vdash\neg\varphi$| Proof. The proof is by induction on the complexity of |$\varphi$|⁠. Base case: |$\varphi$| is atomic. If |$\sigma\; P$| occurs on |$\mathcal{B}$| then |$\sigma\Vdash P$| by definition. If |$\sigma\;\neg P$| occurs on |$\mathcal{B}$| then |$\sigma\; P$| does not occur, because |$\mathcal{B}$| is open, and again by definition |$\sigma\Vdash\neg P$|⁠. Induction step: - |$\varphi$| is |$\neg\psi$|⁠. If |$\sigma\;\neg\psi$| occurs on |$\mathcal{B}$|⁠, due to the induction hypothesis |$\sigma\Vdash\neg\psi$|⁠. If |$\sigma\;\neg\neg\psi$| occurs, having followed the systematic procedure, |$\sigma\;\psi$| also occurs. Due to the induction hypothesis |$\sigma\Vdash\psi$|⁠. - |$\varphi$| is |$\psi\wedge\chi$|⁠. If |$\sigma\;\psi\wedge\chi$| occurs on |$\mathcal{B}$|⁠, having followed the systematic procedure, |$\sigma\;\psi$| and |$\sigma\;\chi$| also occur. Due to the induction hypothesis |$\sigma\Vdash\psi$|⁠,|$\sigma\Vdash\chi$| so |$\sigma\Vdash\psi\wedge\chi$|⁠. If |$\sigma\;\neg(\psi\wedge\chi)$| occurs on |$\mathcal{B}$|⁠, having followed the systematic procedure, |$\sigma\;\neg\psi$| or |$\sigma\;\neg\chi$| occurs. Due to the induction hypothesis |$\sigma\Vdash\neg\psi$| or|$\sigma\Vdash\neg\chi$| so |$\sigma\Vdash\neg(\psi\wedge\chi)$|⁠. The disjunctive case is treated similarly. - |$\varphi$| is |${\mathsf{K}}\psi$|⁠. If |$\sigma\;{\mathsf{K}}\psi$| occurs on |$\mathcal{B}$|⁠, having followed the systematic procedure, |$\sigma'\;\psi$| occurs for all |$\sigma'$| accessible from |$\sigma$|⁠, that is for all |$\sigma'$| such that |$\sigma{\mathcal{R}}\sigma'$|⁠. Due to the induction hypothesis |$\sigma'\Vdash\psi$| for all |$\sigma'$| such that |$\sigma{\mathcal{R}}\sigma'$| so |$\sigma\Vdash{\mathsf{K}}\psi$|⁠. If |$\sigma\;\neg{\mathsf{K}}\psi$| occurs on |$\mathcal{B}$|⁠, having followed the systematic procedure, |$\sigma.n\;\neg\psi$| also occurs. Due to the induction hypothesis |$\sigma.n\Vdash\neg\psi$| and since |$\sigma{\mathcal{R}}\sigma.n$| we have |$\sigma\Vdash\neg{\mathsf{K}}\psi$|⁠. The possibility case |$\neg{\mathsf{K}}\psi$| is treated similarly. - |$\varphi$| is |${\mathsf{E}}\psi$|⁠. Due to the induction hypothesis |$[\psi]\subseteq|\psi|$| and |$[\neg\psi]\subseteq|\neg\psi|$|⁠. If |$\sigma\;{\mathsf{E}}\psi$| occurs on |$\mathcal{B}$| then by definition |$|\psi|\in{\mathcal{N}}(1)$|⁠. Consequently |$1\Vdash{\mathsf{E}}\psi$| and then due to the fact that axiom |$\mathbf{PIE}$| is valid in |$\mathfrak{M}$| (⁠|$\mathfrak{M}$| is a |$\mathbf{KBE}$|-model), |$\sigma\Vdash{\mathsf{E}}\psi$|⁠. If |$\sigma\;\neg{\mathsf{E}}\psi$| occurs then so does |$\sigma\;{\mathsf{E}}\neg\psi$|⁠. By definition |$|\neg\psi|\in{\mathcal{N}}(1)$|⁠. Consequently |$1\Vdash{\mathsf{E}}\neg\psi$| and again due to axiom |$\mathbf{PIE}$| we have |$\sigma\Vdash{\mathsf{E}}\neg\psi$|⁠. Since |$\mathfrak{M}$| is a KBE-model, axiom |$\mathbf{CCE}$| is also valid, and so we have |$\sigma\Vdash\neg{\mathsf{E}}\psi$|⁠. The possibility case |$\neg{\mathsf{E}}\psi$| is treated similarly. ■ Theorem 5.11 (Completeness) If |$\varphi$| has no kbe-tableau proof, |$\varphi$| is not kbe-valid. Proof. Since |$\varphi$| has no kbe-tableau proof, the tableau generated by following the systematic procedure has an open branch from which we construct the counter model. |$1\;\neg\varphi$| occurs on the branch, and by the Key Fact (Proposition 5.10) |$1\Vdash\neg\varphi$|⁠. So |$\neg\varphi$| is kbe-satisfiable i.e. |$\varphi$| is not kbe-valid. ■ 5.4 Decidability One may notice that the systematic procedure we described may be infinite. In fact, that can easily be the case, e.g. if |$1.\sigma\;{\mathsf{E}}\varphi$| or |$1.\sigma\;{\mathsf{K}}\neg{\mathsf{K}}\varphi$| occurs on a tableau branch. We will be modifying the systematic procedure so that it always terminates and the counter-model constructed is finite. In other words, we show that |$\mathbf{KBE}$| is decidable and has the finite model property. Notice that up until now we talked about ‘existence’ and not ‘construction’ of a counter-model. The reason is the existence of a counter-model was based on Proposition 2.4 which in turn is based on an equivalent of the Axiom of Choice. On that note, we deliberately avoided up until now to provide examples of failed tableau proofs, since the modified systematic procedure is probably more aesthetically pleasant, not to mention computationally feasible. Definition 5.12 Fact 5.13 |$S(\varphi)$| is finite Proof. Obvious, given that for any formula |$\varphi$| the number of its subformulas is finite. ■ Definition 5.14 Given a tableau branch |$\mathcal{B}$| and a prefix |$\sigma$|⁠, we define |$\sigma_{\mathcal{B}}=\{\varphi\in\mathcal{L}_{KBE}\mid\sigma\;\varphi\mbox{ occurs on }\mathcal{B}\}$| Fact 5.15 For any tableau attempt to prove |$\varphi$|⁠, it holds that |$\sigma_{\mathcal{B}}\subseteq S(\varphi)$|⁠. Proof. This can easily be seen by inspection of the tableaux rules. The formulas introduced are either a subformula, a subformula preceded by negation, or due to [|$\mathbf{CCE}$|-rule], a subformula preceded by a negation and the modal operator |${\mathsf{E}}$|⁠. ■ Fact 5.16 If a tableau is infinite, it must have an infinite branch. Proof. A special case of the proof of König’s Lemma (each existing branch, if split, gives two branches). ■ Fact 5.17 Any tableau resulting from the systematic procedure, has a finite number of |$0.n$| prefixes. Proof. The only way to introduce new prefixes is by applying [|${\mathsf{K}}\pi$|-rule] or [|${\mathsf{E}}$|-rule]. So the number of prefixes is at most the number of pairs of formulas that can appear, which is finite. ■ Fact 5.18 Any tableau resulting from the systematic procedure, has a finite number of |$1.\sigma.n$| prefixes, for any |$\sigma$|⁠. Proof. In other words, a prefix |$1.\sigma$| cannot have infinite simple extensions. Again the only way to introduce new prefixes is by applying [|${\mathsf{K}}\pi$|-rule] or [|${\mathsf{E}}$|-rule], similarly with Fact 5.17, the number of ‘children’ must be finite. ■ Putting these together, suppose the systematic procedure for some tableau is infinite. By Fact 5.16 it must have an infinite branch. By Facts 5.15, 5.17, 5.18 that infinite branch contains infinite chains of prefixes (consider it also as a special case of the proof of König’s Lemma, this time branching being model theoretic and not due to disjunction), each prefix a simple extension of the preceding. Also by Fact 5.15 there must be prefixes |$\sigma,\sigma'$| on each chain such that |$\sigma_{\mathcal{B}}=\sigma'_{\mathcal{B}}$|⁠. So we modify the systematic procedure to prove a formula |$\varphi$| by following some additional rules In order to shorten and tidy up the procedure: We add prefixed formulas starting with |$0.n$| prefixes, only after we have finished with all |$1.\sigma$| prefixes. We apply |${\mathsf{E}}$|-rules first, and |${\mathsf{K}}\pi$|-rules later. In order to make sure the procedure terminates: We write everything we can about a prefix and update its registries before moving to a new prefix. Let |$m=depth(\varphi)$| (the modal depth of formula |$\varphi$|⁠). For prefixes of length at least |$m+2$|⁠, when we have written all we can about a prefix |$\sigma$|⁠, we check if |$\sigma_{\mathcal{B}}=\sigma'_{\mathcal{B}}$| for some proper initial segment |$\sigma'$| of |$\sigma$|⁠. If that is the case, we declare all occurrences of prefixed formulas of |$\sigma$|finished. We call |${{\sigma}}$| an endpoint. The reason for checking only prefixes of length at least |$m+2$| is that this way, any formulas that are written for an immediate ‘ancestor’ prefix must be due to the application of a |${\mathsf{K}}\nu$| or |$\mathbf{PIE}$| rule; only by these rules is information allowed to be carried over further than to a simple extension and this is a fact we use in the proof of Proposition 5.21. This way we make sure all chains of prefixes are finite. Following our reasoning backwards, no infinite chains of prefixes means no infinite branch, and consequently a finite number of prefixes and a finite systematic procedure. 5.5 Finite model property Definition 5.19 Let |$\mathcal{T}$| be an open tableau generated by the modified systematic procedure, |$\mathcal{B}$| the shortest open branch of |$\mathcal{T}$|⁠. We define a model |$\mathfrak{M'}=\left\langle W,{\mathcal{R}},{\mathcal{N}},V\right\rangle $| as in Definition 5.8 except that - |$W$| is the set of prefixes on |$\mathcal{B}$| minus any endpoints. - |${\mathcal{N}}(1)$| is constructed algorithmically as follows: (1) |${\mathcal{N}}(1)={\mathcal{N}}^{-}(1)$|⁠. (2) Search any |$S\subseteq{\mathcal{R}}(1)$|⁠, such that |$S$|⁠,|$S\backslash{\mathcal{R}}(1)\notin{\mathcal{N}}(1)$|⁠. If there is none then stop. Otherwise add either |$S$| or |$S\backslash{\mathcal{R}}(1)$| (see Cor. 2.5) to |${\mathcal{N}}(1)$|⁠. (3) Repeat step 2. Proposition 5.20 |$\mathfrak{M}'$| is a |$\mathbf{KBE}$|-model. Proof. Proof is identical to that of Proposition 5.9. ■ Proposition 2.4 Let |$\mathfrak{M'}$| be a model as in Definition 5.19. For any prefix |$\sigma$| and formula |$\varphi$| (i) if |$\sigma\;\varphi$| occurs on |$\mathcal{B}$| then |$\mathfrak{M'},\sigma\Vdash\varphi$|⁠. (ii) if |$\sigma\;\neg\varphi$| occurs on |$\mathcal{B}$| then |$\mathfrak{M'},\sigma\Vdash\neg\varphi$|⁠. Proof. The proof is by induction on the complexity of |$\varphi$|⁠. Base case: |$\varphi$| is atomic. Proof identical to that of Proposition 5.10. Induction step: - For propositional cases proof is also identical to that of Proposition 5.10. - |$\varphi$| is |${\mathsf{K}}\psi$|⁠. If |$\sigma\;{\mathsf{K}}\psi$| occurs on |$\mathcal{B}$|⁠, proof is identical to that of Proposition 5.10. If |$\sigma\;\neg{\mathsf{K}}\psi$| occurs on |$\mathcal{B}$|⁠, having followed the systematic procedure, if some endpoint is not a simple extension of |$\sigma$| then |$\sigma.n\;\neg\psi$| also occurs and the proof continues normally. If some endpoint is a simple extension of |$\sigma$|⁠, then |$\sigma$| has length at least |$m+1$|⁠. Then the occurrence of |$\sigma\;\neg{\mathsf{K}}\psi$| is (i) due to the existence of a formula |$\sigma'\;{\mathsf{K}}\chi$|⁠, for some initial segment |$\sigma'$| of |$\sigma$|⁠, and the application of [|${\mathsf{K}}\nu$|-rule] or (ii) due to the existence of a formula |$\sigma'\;{\mathsf{E}}\chi$|⁠, for some initial segment |$\sigma'$| of |$\sigma$|⁠, and the application of [|$\mathbf{PIE}$|-rule]. In any case, due to the systematic procedure, the same rule is applied for prefix |$0.1$|⁠. So we can expect the occurrence of |$0.n\;\neg{\mathsf{K}}\psi$| and consequently |$0.m\;\neg\psi$| for some |$n,m\in\mathbb{N}$|⁠. Since |$\sigma{\mathcal{R}}0.1$| and the induction hypothesis holds for |$\psi$| we have that |$\sigma\Vdash\neg{\mathsf{K}}\psi$|⁠. The possibility case |$\neg{\mathsf{K}}\psi$| is treated similarly. - |$\varphi$| is |${\mathsf{E}}\psi$|⁠. Proof is again the same as in Proposition 5.10. ■ We now show, using the methods described, that |${\mathsf{E}} p\supset{\mathsf{B}} p\notin\mathbf{KBE}$|⁠. In order to simplify this example we will actually be making an exception and allow endpoints of length less than |$m+2$|⁠, since, as will soon become obvious, all information carried over from prefix 1, is due to a preceding application of [|${\mathsf{K}}\nu$|-rule] or [|$\mathbf{PIE}$|-rule]. Finally, all registry sets |${\mathsf{K}}\sigma,{\mathsf{E}}\sigma$| end up being the same but we write them down anyway. That is all we can do for prefix 1, we can now proceed with new ones. Applying [|${\mathsf{E}}$|-rule] we create prefix |$1.1$| and [|${\mathsf{K}}\pi$|-rule] to create prefix 1.2: Nothing else to do for these prefixes other than continue with the [|${\mathsf{E}}$|-rule] and the [|${\mathsf{K}}\pi$|-rule]. We now observe |$1.1.1_{\mathcal{B}}=1.1_{\mathcal{B}}$| and |$1.2.2_{\mathcal{B}}=1.2_{\mathcal{B}}$| so we declare all associated prefixed formulas finished. However, we have to continue with prefixes |$1.1.2$| and |$1.2.1$|⁠. This time all chains of prefixes are stopping; |$1.1.2.1_{\mathcal{B}}=1.1_{\mathcal{B}},1.1.2.2_{\mathcal{B}}=1.1.2_{\mathcal{B}},1.2.1.1_{\mathcal{B}}=1.2.1_{\mathcal{B}},1.2.1.2_{\mathcal{B}}=1.2_{\mathcal{B}}$|⁠. We proceed with the final cluster and the systematic procedure ends. This is what the derived finite model looks like. Regarding relation |${\mathcal{R}}$|⁠, all worlds see |$FC$| and of course |${\mathcal{R}}$| includes the reflexive and transitive closure of the relation depicted. |$V(p)$| is also shown. The only |${\mathsf{E}}$|-formula written on our tableau is |${\mathsf{E}} p$| and |$[p]=\{1.1,1.2.1,0.2\}$| so |${\mathcal{N}}^{-}(1)=\{S\supseteq FC\}\cup\{S\supseteq\{1.1,1.2.1,0.2\}\}$|⁠. Now given that there are 8 worlds in our model, and |${\mathcal{N}}(1)$| must include a set of worlds or its complement, it must be of size $\dfrac{1}{2}\underset{k=1}{\overset{8}{\sum}}\left(\begin{array}{c} 8\\ k \end{array}\right)$ and we will refrain from writing it explicitly. However in our case, it can be easily described. The set |$\{0.2\}$| has a non-empty intersection with both |$FC$| and |$\{1.1,1.2.1,0.2\}$| so we can add all the supersets of |$\{0.2\}$|⁠. |${\mathcal{N}}(1)=\{S\supseteq\{0.2\}\}$| is a weak filter generated by a singleton, and thus a weak ultrafilter. For |$w\neq 1$|⁠, we have |${\mathcal{N}}(w)=\{S\cap{\mathcal{R}}(w)\mid S\in{\mathcal{N}}(1)\}$| e.g. N(1.1.2) ={{0.2},{0.2,1.1.2},{0.2,0.1},{0.2,0.3},{0.2,0.1,0.3},{0.2,0.1,1.1.2}, {0.2,1.1.2,0.3},{0.2,1.1.2,0.1,0.3}}. 6 Related work and future research In this article, we have introduced (two versions of) a logic for reasoning about knowledge, belief and (a qualitative form of) estimation. The logic |$\mathbf{KBE}$| resembles the approach of J. Burgess in [9], where a ‘probably’ operator is added to |$\mathbf{S5}$|⁠. In [21], Andreas Herzig employs the same operator, providing an axiomatization which is very close to the ‘most’ modality underlying our estimation operator. However, our approach is the first to combine such a generalized quantifier with an epistemic modal logic (J. Burgess added this modality to |$\mathbf{S5}$| without any particular application in mind), providing a full completeness proof both for the Hilbert-style axiomatization and the tableaux proof procedure introduced. There exist similar approaches, involving the notion of certainty. The relation of knowledge, belief and certainty has been investigated by Halpern [19], Lenzen [27] and other authors. Certainty is also called ‘robust belief’ by some authors, as opposed to ‘strong belief’; belief is a delicate interesting notion with a lot of useful variants. In Halpern’s work [19], certainty is identified with belief, interpreted probabilistically: an agent is said to be certain of |$\varphi$| if |$\varphi$| holds with probability 1. It is shown that certainty is a |$\mathbf{KD45}$| notion. Also in [30], Lamarre and Shoham define a logic of knowledge, (conditional) belief and (conditional) certainty, extending Lenzen’s original work, which introduces axioms for the same set of epistemic and doxastic operators. The work of Lamarre and Shoham introduces a notion of evidence, an important notion also in the work of J. van Benthem and E. Pacuit [40] which develops Scott-Montague models for evidence-based belief, investigating the logics these models support. To the best of our knowledge (belief and estimation) our work is the first to provide a modal treatment of (qualitative) estimation, with respect to its interaction with knowledge and belief. In the literature of ‘reasoning about uncertainty’, various papers can be found on belief (and belief-like) attitudes, treated probabilistically or with fuzzy techniques. Our approach on the epistemic aspects of ‘estimation’ is purely qualitative, and avoids the complications of probabilistic treatment. The analysis of |$\mathbf{KBE}$| and |$\mathbf{KBiE}$| is in line with the tradition of possible-worlds analysis in epistemic logic and sheds light on the nature of belief as ‘estimation that |$\varphi$| is known’. It seems very challenging to try to embed a similar modal ‘estimation’ operator in first-order modal epistemic logic. This is bound to raise several technical and philosophical issues, but it seems a very challenging and interesting problem. In a series of papers [5-7] A. Baltag et al. have introduced a topological semantics for doxastic logics and logics of relevant attitudes including knowledge, evidence and evidence-based justification. This series of results has as a starting point the ‘evidence models’ of J. van Benthem and E. Pacuit [40] and the topology semantics built by the same authors for Stalnaker’s epistemic/doxastic axioms from [38]. Given that |$\mathbf{S4}$| is the logic of topological spaces (a result that goes back to the work of McKinsey and Tarski from 1944, see [16]) and the fact that |$\mathbf{S4.2}$| (the ‘basis’ of |$\mathbf{KBE}$|⁠) is characterized by the extremally disconnected topological spaces (the topological spaces in which the closure of every open subset is also open) [39, p. 253] it seems challenging to pursue the formal connection between the topological semantics and the semantics we have introduced in this article. This is a very interesting avenue for future research; we expect that the topology of weak ultrafilters should be investigated and this points to the very beautiful classical results in the topology of spaces of ultrafilters from Set Theory [11 § 14]. Acknowledgements An extended abstract concerning |$\mathbf{KBE}$| appeared in JELIA 2014 [25]. We wish to thank the anonymous JLC referee for his/her constructive comments which led to a significantly improved article. 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TI - A modal logic of knowledge, belief and estimation
JF - Journal of Logic and Computation
DO - 10.1093/logcom/exx016
DA - 2017-12-01
UR - https://www.deepdyve.com/lp/oxford-university-press/a-modal-logic-of-knowledge-belief-and-estimation-VlzMSppOSA
SP - 2303
VL - 27
IS - 8
DP - DeepDyve
ER -