The Fundamental Problem of Logical Omniscience

Abstract

We propose a solution to the problem of logical omniscience in what we take to be its fundamental version: as concerning arbitrary agents and the knowledge attitude per se. Our logic of knowledge is a spin-off from a general theory of thick content, whereby the content of a sentence has two components: (i) an intension, taking care of truth conditions; and (ii) a topic, taking care of subject matter. We present a list of plausible logical validities and invalidities for the logic of knowledge per se for arbitrary agents, and isolate three explanatory factors for them: (1) the topic-sensitivity of content; (2) the fragmentation of knowledge states; (3) the defeasibility of knowledge acquisition. We then present a novel dynamic epistemic logic that yields precisely the desired validities and invalidities, for which we provide expressivity and completeness results. We contrast this with related systems and address possible objections.

Appendices

Appendix A: Proof of Theorem 2

We call a sentence ψ of $\mathcal {EL}$Boolean if it does not include occurrences of $i, \Box , K_{i},$ or [i : φ]. The following auxiliary lemma will be useful in the proof of Theorem 2:

Lemma 4

For all epistemic update models${\mathcal{M}}=\langle W, @, \mathbb {T}, \oplus , \mu , v, t \rangle$, all$\varphi \in \mathcal {EL}$, $i\in \mathcal {F}$, and all Boolean sentencesψ, we have$|\psi |_{{\mathcal{M}}}=|\psi |_{{\mathcal{M}}^{i}_{\varphi }}$.

Proof

Follows via an easy induction on the structure of ψ. □

Proof of Theorem 2:

Invalidities: :

In figures of counterexamples, white nodes represent possible worlds, black nodes represent possible topics. Valuation and topic assignment are given by labelling each node with atomic formulae. We omit labelling when a node is assigned every element in $\mathcal {A}\mathcal {T}$.

Counterexample for (9)–(11) and (15)–(16): These schemas are invalid due to topicality. Without loss of generality (w.l.o.g.), let $\mathcal {F}=\{i\}$ and consider the model ${\mathcal{M}}_{1}=\langle W, @, \mathbb {T}, \oplus , \mu , v, t \rangle$ such that W = {@}, $\mathbb {T}=\{a, b, c\}$ with b≠c, b < a, and c < a, v(i) = v(p) = v(q) = {@}, and t(p) = t(i) = b and t(q) = c (Fig. 3). (We do not need to specify μ since it is irrelevant for these schemas.) Then, (9) is invalid since ${\mathcal{M}}_{1}, @\not \vDash \Box (p\supset q)\supset (Kp\supset Kq)$: ${\mathcal{M}}_{1}, @\vDash \Box (p\supset q)$ (since |p| = |q|) and ${\mathcal{M}}_{1}, @\vDash K_{i} p$ (since v(i) = |p| and t(p) ≤ t(i)). However, ${\mathcal{M}}_{1}, @\not \vDash K_{i} q$ since t(q)≦̸t(i). As $\mathcal {F}=\{i\}$, we obtain that ${\mathcal{M}}_{1}, @\not \vDash \Box (p\supset q) \supset (Kp\supset Kq)$. Similarly, (10) is invalid since ${\mathcal{M}}_{1}, @\not \vDash \Box q\supset K q$, and (11) is invalid since (q ∨¬q) is valid but ${\mathcal{M}}_{1}, @\not \vDash K(q\vee \neg q)$ for t(q)≦̸t(i). Moreover, (15) and (16) are invalid since ${\mathcal{M}}_{1}, @ \not \vDash K\neg \neg p\supset K\neg (\neg p\wedge q)$ and ${\mathcal{M}}_{1}, @ \not \vDash Kp\supset K\neg (\neg p\wedge q)$, respectively: it is easy to verify that ${\mathcal{M}}_{1}, @ \vDash K\neg \neg p$ and ${\mathcal{M}}_{1}, @ \vDash K p$, but ${\mathcal{M}}_{1}, @\not \vDash K\neg (\neg p\wedge q)$ as t(¬(¬p ∧ q)) = t(p) ⊕ t(q) = a≦̸t(i).

Fig. 3

Counterexample ${\mathcal{M}}_{1}$

Counterexample for (17)–(19): These schemas are invalid due to fragmentation. W.l.o.g., let $\mathcal {F}=\{i, j\}$ and consider the model ${\mathcal{M}}_{2}=\langle W, @, \mathbb {T}, \oplus , \mu , v, t \rangle$ such that W = {@,w₁,w₂}, v(i) = {@,w₁}, v(j) = v(p) = {@,w₂}, and v(q) = {@}, $\mathbb {T}=\{a\}$, and t(φ) = a for all $\varphi \in \mathcal {EL}$. (Since every $\varphi \in \mathcal {EL}$ is mapped to the same topic, the topicality constraints will be trivially satisfied.) Then, (17) is invalid since ${\mathcal{M}}_{2}, @\not \vDash K(p\supset q) \supset (Kp\supset Kq)$: ${\mathcal{M}}_{2}, @\vDash K_{i} (p\supset q)$ (since $v(i)= |p\supset q|$) and ${\mathcal{M}}_{2}, @\vDash K_{j} p$ (since v(j) = |p|), thus, ${\mathcal{M}}_{2}, @\vDash K (p\supset q)$ and ${\mathcal{M}}_{2}, @\vDash K p$. However, ${\mathcal{M}}_{2}, @\not \vDash Kq$ since ${\mathcal{M}}_{2}, @ \not \vDash K_{i} q$ and ${\mathcal{M}}_{2}, @ \not \vDash K_{j} q$ (as $v(i)\not \subseteq |q|$ and $v(j)\not \subseteq |q|$, respectively). For (18) and (19), consider ${\mathcal{M}}_{2}$ with μ(P,Q) = P ∩ Q for all $P, Q\in \mathbb {I}_{@}$. Then, (18) is invalid since ${\mathcal{M}}_{2}, @\vDash \neg [j: p ][i: p\supset q] Kq$: ${\mathcal{M}}_{2}, @\vDash p$ and $({\mathcal{M}}_{2})^{j}_{p}, @\vDash p\supset q$, but, $({({\mathcal{M}}_{2})^{j}_{p})}^{i}_{p \supset q}, @\not \vDash K_{i} q \vee K_{j} q$ (since ${({v_{p}^{j}})}^{i}_{p\supset q}(j)= \{@, w_{2}\}$ and ${({v_{p}^{j}})}^{i}_{p\supset q}(i)= \{@, w_{1}\}$, and $({({\mathcal{M}}_{2})^{j}_{p})}^{i}_{p \supset q}, w_{1}\not \vDash q$ and $({({\mathcal{M}}_{2})^{j}_{p})}^{i}_{p \supset q}, w_{2}\not \!\vDash \! q$). Similarly, (19) is invalid since ${\mathcal{M}}_{2}, @\!\vDash \! \neg [i{}:{} p\!\supset \! q][j{}:{} p ] Kq$.

Counterexample for (20) and (21): These schemas are invalid due to non-monotonicty of knowledge update. The counterexample ${\mathcal{M}}_{2}$ in Fig. 4 with $\mu ^{\prime }(P, Q)$ = Q for all $P, Q\in \mathbb {I}_{@}$ invalidates (20) and (21): ${\mathcal{M}}_{2}, @\vDash K_{i}(p\supset q)\wedge K_{j} p$, but ${\mathcal{M}}_{2}, @\not \vDash [i:j]Kq$ and ${\mathcal{M}}_{2}, @\not \vDash [j:i]Kq$. For ${\mathcal{M}}_{2}, @\not \vDash [i:j]Kq$, observe that ${v_{j}^{i}}(i)= {v_{j}^{i}}(j)= \{@, w_{2}\}\not \subseteq |q|$, thus, $({\mathcal{M}}_{2})^{i}_{j}\not \vDash K_{i}q\vee K_{j} q$. For ${\mathcal{M}}_{2}, @\not \vDash [j:i]Kq$, observe that ${v_{i}^{j}}(i)= {v_{i}^{j}}(j)= \{@, w_{1}\}\not \subseteq |q|$, thus, $({\mathcal{M}}_{2})^{j}_{i}\not \vDash K_{i}q\vee K_{j} q$.

Fig. 4

Counterexample ${\mathcal{M}}_{2}$

Counterexample for (22)–(27): These principles are invalid due to non-monocity of knowledge update. It is easy to see that counterexamples invalidating (26) and (27) are also counterexamples for (24) and (25), respectively. Moreover, (24) and (25) are special cases of (22) and (23), respectively. Consider the model ${\mathcal{M}}_{3}=\langle W, @, \mathbb {T}, \oplus , \mu , v, t \rangle$ such that W = {@,w₁}, $\mathbb {T}=\{a\}$, v(i) = v(p) = {@} for all $i\in \mathcal {F}$, and v(q) = W, t(φ) = a for all $\varphi \in \mathcal {EL}$, and μ(P,Q) = Q for all $P, Q\in \mathbb {I}_{@}$ (Fig. 5). As every frame of mind is mapped to the same set of possible wolds, we do not need to consider fragmentation. Similarly since every $\varphi \in \mathcal {EL}$ is mapped to the same topic, the topicality constraints will be trivially satisfied. Then, (26) is invalid since ${\mathcal{M}}_{3}, @\not \vDash [i: p][i: q] K_{i} p$: ${\mathcal{M}}_{3}, @\vDash p$ and $({\mathcal{M}}_{3})^{i}_{p}, @\vDash q$ but ${(({\mathcal{M}}_{3})^{i}_{p})}^{i}_{q}, @\not \vDash K_{i}p$ since ${({v_{p}^{i}})}_{q}^{i}(i)=W$ and ${(({\mathcal{M}}_{3})^{i}_{p})}^{i}_{q} , w_{1}\not \vDash p$. Similarly, (27) is invalid since ${\mathcal{M}}_{3}, @\not \vDash K_{i}p\supset [i: q]K_{i} p$: ${\mathcal{M}}_{3}, @ \vDash K_{i}p$ and ${\mathcal{M}}_{3}, @ \vDash q$, but $({\mathcal{M}}_{3})^{i}_{q}, @\not \vDash K_{i}p$ since ${v_{q}^{i}}(i)=W$ and $({\mathcal{M}}_{3})^{i}_{q}, w_{1}\not \vDash p$.

Fig. 5

Counterexample ${\mathcal{M}}_{3}$

Validities: :

Let ${\mathcal{M}} = \langle W,\! @,\! \mathbb {\!T}, \!\oplus ,\! \mu ,\! v,\! t \rangle$ be an epistemic update model and w ∈ W:

(12) $\vDash \Box (\varphi \supset \psi )\supset (\Box \varphi \supset \Box \psi )$: Suppose ${\mathcal{M}}, w\vDash \Box (\varphi \supset \psi )$ and ${\mathcal{M}}, w\vDash \Box \varphi$. While the former means that $|\varphi |\subseteq |\psi |$, the latter means |φ| = W. Therefore, |ψ| = W, i.e., ${\mathcal{M}}, w\vDash \Box \psi$.

(13) $\vDash K(\varphi \wedge \psi )\supset K\varphi$: Suppose ${\mathcal{M}}, w\vDash K(\varphi \wedge \psi )$. This means, by the definition of K, that there is an $i\in \mathcal {F}$ such that ${\mathcal{M}}, w\vDash K_{i}(\varphi \wedge \psi )$, i.e., that $v(i)\subseteq |\varphi \wedge \psi |$ and t(φ ∧ ψ) ≤ t(i). Since $v(i)\subseteq |\varphi \wedge \psi |\subseteq |\varphi |$ and t(φ) ≤ t(φ) ⊕ t(ψ) = t(φ ∧ ψ) ≤ t(i), we obtain ${\mathcal{M}}, w\vDash K_{i}\varphi$. Thus, ${\mathcal{M}}, w\vDash K\varphi$. The validity proofs of (14), and (30)–(32) follow similarly.

(28) ⊧ [p]Kp: Let $i\in \mathcal {F}$ and suppose ${\mathcal{M}}, @\vDash p$. We then have that ${v^{i}_{p}}(i)=\mu (v(i), |p|)\subseteq |p|= |p|_{{\mathcal{M}}^{i}_{p}}$ (by Lemma 4). Moreover, ${t^{i}_{p}}(i)=t(i)\oplus t(p)\geq t(p)={t^{i}_{p}}(p)$. Therefore, ${\mathcal{M}}^{i}_{p}, w\vDash K_{i}p$, implying that ${\mathcal{M}}^{i}_{p}, w\vDash Kp$. Thus, ${\mathcal{M}}, w\vDash [i: p] Kp$. Since i has been chosen arbitrarily from $\mathcal {F}$, we obtain that ${\mathcal{M}}, w\vDash [p] Kp$.

(29) $\vDash Kp\supset [q]\neg K\neg p$: Suppose ${\mathcal{M}}, w\vDash Kp$ and let $i\in \mathcal {F}$ such that ${\mathcal{M}}, @\vDash q$. The former implies that ${\mathcal{M}}, w\vDash K_{j}p$ for some $j\in \mathcal {F}$. In particular, since @ ∈ v(k) for all $k\in \mathcal {F}$, we have ${\mathcal{M}}, @\vDash p$. Then, by Lemma 4, we obtain that ${\mathcal{M}}^{i}_{q}, @ \vDash p$. By the definition of μ, we also have that $@\in {v^{i}_{q}}(k)$ for all $k\in \mathcal {F}$. Therefore, ${\mathcal{M}}^{i}_{q}, w\vDash \neg K_{k}\neg p$ for all $k\in \mathcal {F}$, meaning that ${\mathcal{M}}^{i}_{q}, w\vDash \neg K\neg p$. Hence, ${\mathcal{M}}, w\vDash [i:q]\neg K\neg p$. Since i has been chosen arbitrarily from $\mathcal {F}$, we obtain that ${\mathcal{M}}, w\vDash [q]\neg K\neg p$, thus, ${\mathcal{M}}, w\vDash Kp \supset [q]\neg K\neg p$. □

Appendix B: Soundness and Completeness for $\mathcal {EL}^{+}_{-}$, $\mathcal {EL}^{*}_{-}$, and $\mathcal {EL}^{*}$

We provide soundness and completeness results for $\mathcal {EL}^{+}_{-}$, $\mathcal {EL}^{*}_{-}$, and $\mathcal {EL}^{*}$. The following syntactic abbreviations will matter: for any $\varphi \in \mathcal {EL}^{*}$, $\mathcal {A}\mathcal {T}(\varphi )$ denotes the set of atomic formulae occurring in φ and we will use ‘$\overline {\varphi }$’ to denote the tautology $\bigwedge _{x\in \mathcal {A}\mathcal {T}(\varphi })(x\vee \neg x)$, following a similar idea in [26].

All our soundness and completeness results are proven with respect to the class of epistemic update models whose topic assignment function is defined in the following way:

• $t: \mathcal {A}\mathcal {T} \to \mathbb {T}$ is a topic function assigning a topic to each element in $\mathcal {A}\mathcal {T}$. t extends to the whole $\mathcal {EL}^{*}$ by taking the topic of a sentence φ as the fusion of the elements in $\mathcal {A}\mathcal {T}(\varphi )$:

  $$ t(\varphi)=\oplus \mathcal{A}\mathcal{T}(\varphi)= t(x_{1})\oplus {\dots} \oplus t(x_{k}) $$

  where $\mathcal {A}\mathcal {T}(\varphi )=\{x_{1}, \dots , x_{k}\}$.

≤ again denotes topic inclusion as defined in Section 4.1. The topic of a complex sentence φ, defined from its primitive components in $\mathcal {A}\mathcal {T}(\varphi )$, makes all the logical connectives and modal operators in $\mathcal {EL}^{*}$ topic-transparent, that is,

• $t(\Box \varphi ) = t(A\varphi ) = t(\neg \varphi ) = t(\varphi )$

• $t(K_{\psi }\varphi ) = t(\varphi \wedge \psi ) = t(\varphi \vee \psi ) =t(\varphi \supset \psi ) =t(\varphi \equiv \psi )= t(\varphi )\oplus t(\psi )$

• t([i : φ]ψ) = (t(i) ⊕ t(φ)) ⊕ t(ψ)

Topic-transparency of the truth-functional connectives has already been argued for in Section 1 and formally captured in Definition 1. The only further constraint we impose here is the topic-transparency of the modal operators K_ψφ, $\Box \varphi$, Aφ, and [i : φ]ψ. This is admittedly an idealization, but for technical purposes, in this paper we can live with this idealization, as Theorem 2 still holds with respect to epistemic update models with the above constraint. In the remainder of this appendix, all models are implicitly assumed to obey this constraint on t.

We first provide a sound and complete axiomatization for $\mathcal {EL}^{*}_{-}$ (Appendix B.1). The completeness result for $\mathcal {EL}^{+}_{-}$ follows similarly, so we omit many details and only point out the differences (Appendix B.1.1). The completeness for $\mathcal {EL}^{*}$ will follow from the completeness of $\mathcal {EL}^{*}_{-}$ via a set of sound reduction axioms (Appendix B.2).

B.1 The (static) Logic of Knowledge Over $\mathcal {EL}^{*}_{-}$

Since $\mathcal {EL}^{*}_{-}$ does not have the dynamic operator, the update function μ does not play any role in its interpretation in epistemic update models. We therefore opt for simplicity and interpret $\mathcal {EL}^{*}_{-}$ in what we call epistemic models, ${\mathcal{M}}=\langle W, @, \mathbb {T}, \oplus , v, t \rangle$, obtained by removing μ from epistemic update models.

To recap, given an epistemic model ${\mathcal{M}}=\langle W, @, \mathbb {T}, \oplus , v, t \rangle$ and w ∈ W, we define the satisfaction relation$\vDash$ for the atomic formulae, Booleans, and $\Box \varphi$ as in Definition 2; for K_ψφ and Aφ we have:

$$ \begin{array}{lcl} \mathcal{M}, w \vDash K_{\varphi} \psi & \text{ iff } & t(\psi) \leq t(\varphi) \text{ and } (\forall u\in W)(\text{ if }\mathcal{M}, u\vDash\varphi \text{ then } \mathcal{M}, u \vDash \psi)\\ \mathcal{M}, w \vDash A\varphi & \text{ iff } & \mathcal{M}, @ \vDash \varphi. \end{array} $$

Truth in a model and validity are defined as before (see Section 4.1). Soundness and completeness are defined in a standard way with respect to the global notion of validity denoted by $\vDash$.

Lemma 5

Let$\langle W, @, \mathbb {T}, \oplus , v, t \rangle$be an epistemic model. Then, for any$\varphi \in \mathcal {EL}^{*}_{-}$, w ∈ W, and update functionμ (as described in Definition 1), we have

$$\langle W, @, \mathbb{T}, \oplus, v, t \rangle\vDash \varphi \text{ iff } \langle W, @, \mathbb{T}, \oplus, \mu, v, t\rangle, w\vDash \varphi.$$

Proof

Follows via an easy induction on the structure of φ as μ does not play any role in the interpretation of the sentences in $\mathcal {EL}^{*}_{-}$. □

Observation 6

• (1) For all $\varphi \in \mathcal {EL}_{-}^{*}$, $\vDash \overline {\varphi }$.

• (2) $\vDash K_{\varphi }\psi \equiv (K_{\varphi }\overline {\psi } \wedge \Box (\varphi \supset \psi ))$.¹

Table 1 presents a sound and complete axiomatization EL^∗ of epistemic logic over $\mathcal {EL}_{-}^{*}$. The notion of derivation, denoted by $\vdash _{\mathsf {EL}^{*}}$, in EL^∗ is defined as usual. Thus, $\vdash _{\mathsf {EL}^{*}}\varphi$ means φ is a theorem of EL^∗. For any set of formulas ${\Gamma }\subseteq \mathcal {EL}_{-}^{*}$ and any $\varphi \in \mathcal {EL}_{-}^{*}$, we write ${\Gamma } \vdash _{\mathsf {EL}^{*}} \varphi$ if there exists finitely many formulas $\varphi _{1}, \dots , \varphi _{n}\in {\Gamma }$ such that $\vdash _{\mathsf {EL}^{*}}(\varphi _{1}\wedge {\dots } \wedge \varphi _{n})\supset \varphi$.

Table 1 Axiomatization EL^∗ (over $\mathcal {EL}_{-}^{*}$)

Theorem 7

The following are derivable fromEL^∗:

• (1) from$\vdash _{\mathsf {EL}^{*}} \varphi \equiv \psi , \vdash _{\mathsf {EL}^{*}} K_{\varphi }\overline {\psi }, and \vdash _{\mathsf {EL}^{*}} K_{\psi }\overline {\varphi }$, infer$\vdash _{\mathsf {EL}^{*}} K_{\chi } \varphi \equiv K_{\chi } \psi$and$\vdash _{\mathsf {EL}^{*}}K_{\varphi }\chi \equiv K_{\psi }\chi$,

• (2) (Nec_A) from$\vdash _{\mathsf {EL}^{*}}\varphi$infer$\vdash _{\mathsf {EL}^{*}}A\varphi$,

• (3) $\vdash _{\mathsf {EL}^{*}}A(\varphi \wedge \psi ) \equiv A\varphi \wedge A\psi$,

Proof

(1) follows from Ax4, Ax6, and $\mathsf {S5}_{\Box }$. Item (2) follows from necessitation for $\Box$ and Ax1, and (3) is derived from K_A and (Nec_A) in a standard way.

□

Soundness of EL^∗ is a matter of routine validity check, so we skip its proof. The rest of this section is devoted to the completeness proof of EL^∗, which is presented in full detail.

We say that Γ is EL^∗-consistent if ${\Gamma } \not \vdash _{\mathsf {EL}^{*}}\bot$, and EL^∗-inconsistent otherwise. We omit the tag EL^∗ and say (in)consistent when the logic is contextually clear. A sentence φ is consistent with Γ if Γ ∪{φ} is consistent (or, equivalently, if ${\Gamma }\not \vdash _{\mathsf {EL}^{*}}\neg \varphi$). Finally, a set of formulas Γ is a maximally consistent set (or, in short, mcs) if it is consistent and any set of formulas properly containing Γ is inconsistent [8]. ²

Lemma 8

For every mcs Γ ofEL^∗and$\varphi , \psi \in \mathcal {EL}_{-}^{*}$, the following hold:

• (1) ${\Gamma }\vdash _{\mathsf {EL}^{*}} \varphi$iffφ ∈Γ,

• (2) if φ ∈Γ and$\varphi \supset \psi \in {\Gamma }$, thenψ ∈Γ,

• (3) if$\vdash _{\mathsf {EL}^{*}} \varphi$thenφ ∈Γ,

• (4) φ ∈Γ andψ ∈Γ iffφ ∧ ψ ∈Γ,

• (5) φ ∈Γ iff ¬φ∉Γ.

Proof

Standard. □

Lemma 9 (Lindenbaum’s Lemma)

EveryEL^∗-consistent set can be extended to a maximally consistent one.

Proof

Standard. □

Let $\mathcal {X}$ be the set of all maximally consistent sets of EL^∗. Define $\sim _{\Box }$ on $\mathcal {X}$ as

$${\Gamma} \sim_{\Box} {\Delta} \text{ iff } \forall\varphi\in \mathcal{EL}_{-}^{*}(\Box\varphi\in {\Gamma} \text{ implies } \varphi\in {\Delta}).$$

It is standard to prove that $\sim _{\Box }$ is an equivalence relation, as $\Box$ is an S5 operator. To define the canonical model, we need some auxiliary definitions and lemmas.

Lemma 10

For all${\Gamma }\in \mathcal {X}$, $@_{\Gamma } =\{\varphi \in \mathcal {EL}_{-}^{*} : A\varphi \in {\Gamma }\}$is a maximally consistent set such that${\Gamma } \sim _{\Box } @_{\Gamma }$and$1\wedge \dots \wedge n\in @_{\Gamma }$.

Proof

Suppose that $@_{\Gamma } =\{\varphi \in \mathcal {EL}_{-}^{*} : A\varphi \in {\Gamma }\}$ is not consistent. This means that there are finitely many $\varphi _{1}, \dots , \varphi _{k}\in @_{\Gamma }$ such that $\vdash _{\mathsf {EL}^*} (\varphi _1\wedge {\dots } \wedge \varphi _k)\supset \bot$. This implies, in particular, that $\vdash _{\mathsf {EL}^*} (\varphi _1\wedge {\dots } \wedge \varphi _{k-1})\supset \neg \varphi _k$. Then, by K_A, Theorem 7.(2) and 7.(3), we have $\vdash _{\mathsf {EL}^{*}}(A\varphi _{1} \wedge {\dots } \wedge A\varphi _{k-1})\supset A\neg \varphi _{k}$. As $A\varphi _{1}, \dots , A\varphi _{k-1}\in {\Gamma }$, we have A¬φ_k ∈Γ. This means, by Dua_A, that ¬Aφ_k ∈Γ, contradicting consistency of Γ. Therefore, @_Γ is consistent. Now suppose that @_Γ is not maximal. This means that there is a $\psi \in \mathcal {EL}_{-}^{*}$ such that ψ∉@_Γ and @_Γ ∪{ψ} is consistent. But, as ψ∉@_Γ, we have Aψ∉Γ, thus, by Lemma 8.(5) and Dua_A, A¬ψ ∈Γ. This means that ¬ψ ∈ @_Γ, which makes @_Γ ∪{ψ} inconsistent. Therefore, @_Γ is maximal too. By Ax3 and Lemma 8.(3), we have $A(1\wedge {\dots } \wedge n)\in {\Gamma }$, therefore, $1\wedge {\dots } \wedge n\in @_{\Gamma }$. Now take an arbitrary $\varphi \in \mathcal {EL}_{-}^{*}$ and suppose that $\Box \varphi \in {\Gamma }$. Then, by Ax1 and Lemmas 8.(2) and 8.(3), Aφ ∈Γ. This means that φ ∈ @_Γ. Therefore, we also obtain that ${\Gamma } \sim _{\Box } @_{\Gamma }$. □

For ${\Gamma } \in \mathcal {X}$, let $\approx _{\Gamma }\subseteq \mathcal {EL}_{-}^{*} \times \mathcal {EL}_{-}^{*}$ such that

$$\varphi\approx_{\Gamma} \psi \text{ iff } K_{\varphi}\overline{\psi}, K_{\psi}\overline{\varphi}\in {\Gamma}.$$

In the following proofs, we make repeated use of Lemma 8 in a standard way as in the proof of Lemma 10 and omit mention of it.

Lemma 11

For all${\Gamma } \in \mathcal {X}$, ≈_Γis an equivalence relation. Moreover, for all${\Gamma }, {\Delta }\in \mathcal {X}$such that${\Gamma } \sim _{\Box } {\Delta }$, we have ≈_Γ =≈_Δ.

Proof

Let ${\Gamma } \in \mathcal {X}$ and $\varphi , \psi , \chi \in \mathcal {EL}_{-}^{*}$.

• reflexivity: By Ax8, we have $\vdash _{\mathsf {EL}^{*}} K_{\varphi }\overline {\varphi }$, thus, φ ≈_Γφ.

• symmetry: Suppose φ ≈_Γψ. This means, by the defn. of ≈_Γ, that $K_{\varphi }\overline {\psi }, K_{\psi }\overline {\varphi }\in {\Gamma }$. Therefore, $K_{\psi }\overline {\varphi }, K_{\varphi }\overline {\psi }\in {\Gamma }$, i.e., ψ ≈_Γφ.

• transitivity: Suppose φ ≈_Γψ and ψ ≈_Γχ. This means that (a) $K_{\psi }\overline {\varphi }\in {\Gamma }$, (b) $K_{\varphi }\overline {\psi }\in {\Gamma }$, (c) $K_{\psi }\overline {\chi }\in {\Gamma }$, and (d) $K_{\chi }\overline {\psi }\in {\Gamma }$. Then, by Ax6, (b), and (c), $K_{\varphi }\overline {\chi }\in {\Gamma }$. Similarly, by Ax6, (a), and (d), $K_{\chi }\overline {\varphi }\in {\Gamma }$. Therefore, φ ≈_Γχ.

For the last part, let ${\Gamma }, {\Delta }\in \mathcal {X}$ such that ${\Gamma } \sim _{\Box } {\Delta }$. Suppose φ ≈_Γψ. This means that $K_{\varphi }\overline {\psi }, K_{\psi }\overline {\varphi }\in {\Gamma }$. Then, by Ax5, we obtain that $\Box K_{\varphi }\overline {\psi }, \Box K_{\psi }\overline {\varphi }\in {\Gamma }$. As ${\Gamma } \sim _{\Box } {\Delta }$, we conclude that $K_{\varphi }\overline {\psi }, K_{\psi }\overline {\varphi }\in {\Delta }$, i.e., φ ≈_Δψ. For the other direction, use the symmetry of $\sim _{\Box }$. □

Let $[\varphi ]_{\Gamma }=\{ \psi \in \mathcal {EL}_{-}^{*} : \varphi \approx _{\Gamma } \psi \}$, i.e., [φ]_Γ is the equivalence class of φ with respect to ≈_Γ.

Definition 3 (Canonical Model for Γ₀)

Let Γ₀ be a mcs of EL^∗. The canonical model for Γ₀ is the tuple ${\mathcal{M}}^{c}= \langle W^{c}, @_{{\Gamma }_{0}}, \mathbb {T}^{c}, \oplus ^{c}, v^{c}, t^{c} \rangle$, where

• $W^{c}=\{\Gamma \in \mathcal {X} : {\Gamma }_{0}\sim _{\Box } {\Gamma }\}$,

• $@_{{\Gamma }_{0}}=\{\varphi \in \mathcal {EL}_{-}^{*} : A\varphi \in {\Gamma }_{0}\}$,

• $\mathbb {T}^{c}=\{[\varphi ]_{{\Gamma }_{0}} : \varphi \in \mathcal {EL}_{-}^{*}\}$ (we omit the subscript Γ₀ when it is clear from the context),

• $\oplus ^{c}: \mathbb {T}^{c} \times \mathbb {T}^{c} \to \mathbb {T}^{c}$ such that $[\varphi ]\oplus ^{c} [\psi ]=[\varphi \wedge \psi ]$,

• $t^{c}: \mathcal {EL}_{-}^{*} \to \mathbb {T}^{c}$ such that, for all $x\in \mathcal {A}\mathcal {T}$, t^c(x) = [x] and $t^{c}(\varphi )=\oplus ^{c}\mathcal {A}\mathcal {T}(\varphi )$,

• $v^{c}: \mathcal {A}\mathcal {T}\to \mathcal {P}(W^{c})$ such that $v^{c}(x)=\{\Gamma \in W^{c} : x\in {\Gamma }\}$.

The topic inclusion relation ≤^c on the canonical model is defined in the usual way.

Lemma 12

Given a mcs Γ₀ofEL^∗and the canonical model${\mathcal{M}}^{c}= \langle W^{c}, @_{{\Gamma }_{0}}, \mathbb {T}^{c}, \oplus ^{c}, v^{c}, t^{c} \rangle$ for Γ₀as described above, we have

• (1) $@_{{\Gamma }_{0}}\in W^{c}$,

• (2) $i\in @_{{\Gamma }_{0}}$for all$i\in \mathcal {F}$,

• (3) for all$\varphi \in \mathcal {EL}_{-}^{*}$and Γ ∈ W^c, if$\varphi \in @_{{\Gamma }_{0}}$thenAφ ∈Γ,

• (4) for all Γ ∈ W^c, $@_{\Gamma }=@_{{\Gamma }_{0}}$.

Proof

• (1) By Lemma 10.

• (2) By Lemmas 10 and 8.(4).

• (3) Let $\varphi \in \mathcal {EL}_{-}^{*}$ such that $\varphi \in @_{{\Gamma }_{0}}$, i.e., Aφ ∈Γ₀. Then, by Ax2, we have $\Box A\varphi \in {\Gamma }_{0}$. Hence, by the definition of W^c, Aφ ∈Γ f or all ${\Gamma }\in W^{c}$.

• (4) $@_{{\Gamma }_{0}}\subseteq @_{\Gamma }$ follows directly from item (3). For $@_{\Gamma }\subseteq @_{{\Gamma }_{0}}$, suppose φ ∈ @_Γ, i.e., that Aφ ∈Γ. Then, by Ax2, we have $\Box A\varphi \in {\Gamma }$. As Γ ∈ W^c, i.e., ${\Gamma }\sim _{\Box } {\Gamma }_{0}$, we obtain that Aφ ∈Γ₀, i.e., $\varphi \in @_{{\Gamma }_{0}}$.

□

Lemma 13

The canonical model${\mathcal{M}}^{c}= \langle W^{c}, @_{{\Gamma }_{0}}, \mathbb {T}^{c}, \oplus ^{c}, v^{c}, t^{c} \rangle$is an epistemic model.

Proof

• $@_{{\Gamma }_{0}}\in W^{c}$ and $@_{{\Gamma }_{0}}\in v^{c}(i)$ for all $i\in \mathcal {F}$: The former is shown in Lemma 12.(1) and the latter is a consequence of Lemma 12.(2) and the definition of v^c.

• ⊕^c is idempotent, associative, and commutative: All these properties of ⊕^c follow from its definition and Ax8.

□

In order to prove an appropriate Truth Lemma, we need the following auxiliary lemmas.

Lemma 14

For all$\varphi \in \mathcal {EL}_{-}^{*}$, t^c(φ) = [φ].

Proof

Let $\varphi \in \mathcal {EL}_{-}^{*}$.

$$ \begin{array}{@{}rcl@{}} t^{c}(\varphi) & =&\oplus^{c}\mathcal{A}\mathcal{T}(\varphi) \hspace{148pt}(\text{by the definition of}\ t^{c}(\varphi))\\ &=& t^{c}(x_{1})\oplus^{c} {\dots} \oplus^{c}t^{c}(x_{k}) \hspace{90pt}(\text{for}\ \mathcal{A}\mathcal{T}(\varphi)=\{x_{1}, \dots, x_{k}\})\\ &=&[x_{1}]\oplus^{c} {\dots} \oplus^{c}[x_{k}] \hspace{118pt}(\text{by the definition of}\ t^{c}(x))\\ &=&[x_{1}\wedge {\dots} \wedge x_{k}] \hspace{144pt}(\text{by the definition of}\ \oplus^{c})\\ &=&[\varphi] \hspace{266.5pt}(\mathsf{Ax}8) \end{array} $$

□

Lemma 15

For all Γ ∈ W^cand$\varphi , \psi \in \mathcal {EL}_{-}^{*}$, $t^{c}(\psi )\leq ^{c} t^{c}(\varphi )$iff$K_{\varphi }\overline {\psi }\in {\Gamma }$.

Proof

Let ${\Gamma }\in W^{c}$, and $\varphi , \psi \in \mathcal {EL}_{-}^{*}$, and observe that

$$ \begin{array}{@{}rcl@{}} t^{c}(\psi)\leq^{c} t^{c}(\varphi) &&\text{ iff } t^{c}(\psi)\oplus^{c} t^{c}(\varphi)=t^{c}(\varphi) \hspace{61pt}(\text{by the definition of} \leq^{c})\\ && \text{ iff } [\psi]\oplus^{c} [\varphi]=[\varphi] \hspace{137pt}(\text{Lemma} 14)\\ && \text{ iff } [\psi\wedge \varphi]=[\varphi] \hspace{98.5pt}(\text{by the definition of } \oplus^{c})\\ && \text{ iff } K_{\psi\wedge \varphi} \overline{\varphi}, K_{\varphi}\overline{(\psi\wedge \varphi)}\in {\Gamma}_{0} \hspace{52pt}(\text{by the definition of } [\varphi])\\ & &\text{ iff } K_{\psi\wedge \varphi} \overline{\varphi}, K_{\varphi}\overline{(\psi\wedge \varphi)}\in {\Gamma} \hspace{107pt}(\text{Lemma} 11) \end{array} $$

Suppose t^c(ψ) ≤^ct^c(φ). Then, by the observation above, we have $K_{\psi \wedge \varphi } \overline {\varphi }$, $K_{\varphi }\overline {(\psi \wedge \varphi )}$∈Γ. By Ax8, we also have that $K_{\psi \wedge \varphi }\overline {\psi }\in {\Gamma }$. Then, $K_{\psi \wedge \varphi }\overline {\psi } \wedge K_{\varphi }\overline {(\psi \wedge \varphi )}\in {\Gamma }$. Thus, by Ax6, we obtain that $K_{\varphi }\overline {\psi }\in {\Gamma }$. For the other direction, suppose that $K_{\varphi }\overline {\psi }\in {\Gamma }$. By Ax8, we also have $K_{\varphi }\overline {\varphi }\in {\Gamma }$. Therefore, by Ax7, we obtain that $K_{\varphi }\overline {(\psi \wedge \varphi )}\in {\Gamma }$. By Ax8, we already have that $K_{\psi \wedge \varphi } \overline {\varphi }\in {\Gamma }$. Therefore, following the same steps above, we obtain that t^c(ψ) ≤^ct^c(φ). □

Lemma 16 (Existence Lemma for $\Box$)

For every mcs${\Gamma }\in \mathcal {X}$and$\varphi \in \mathcal {EL}_{-}^{*}$, if $\Box \varphi \not \in {\Gamma }$then there is a mcs Δ such that${\Gamma }\sim _{\Box } {\Delta }$andφ∉Δ.

Proof

Standard, see, e.g., [8, Lemma 4.20]. □

Lemma 17 (Truth Lemma)

Let Γ₀be a mcs ofEL^∗and${\mathcal{M}}^{c}= \langle W^{c}, @_{{\Gamma }_{0}}, \mathbb {T}^{c}, \oplus ^{c}, v^{c}, t^{c} \rangle$be the canonical model for Γ₀. Then, for all Γ ∈ W^c and $\varphi \in \mathcal {EL}_{-}^{*}$,

$$\mathcal{M}^{c}, {\Gamma}\vDash \varphi \text{ iff } \varphi\in {\Gamma}.$$

Proof

The proof is by induction on the structure of φ. The cases for the atomic formulae, Booleans, and $\Box \psi$ are standard, where the case for $\Box \psi$ uses Lemma 16. Toward showing the cases for φ := K_ψχ and φ := Aψ, suppose inductively that the statement holds for ψ and χ.

Case φ := K_ψχ:

$$ \begin{array}{llll} \mathcal{M}^{c}, {\Gamma} \vDash K_{\psi}\chi \text{ iff } t^{c}(\chi)\leq^{c} t^{c}(\psi) \text{ and } \forall {\Delta}\in W^{c} (\text{ if } \mathcal{M}^{c}, {\Delta} \vDash \psi \text{ then } \mathcal{M}^{c}, {\Delta} \vDash \chi)\\ \hspace{263pt}(\text{by the semantics})\\ \text{ iff } K_{\psi}\overline{\chi} \in {\Gamma} \text{ and } \forall {\Delta}\in W^{c} (\text{ if } \mathcal{M}^{c}, {\Delta} \vDash \psi \text{ then } \mathcal{M}^{c}, {\Delta} \vDash \chi)\hspace{35pt}(\text{Lemma~15})\\ \text{ iff } K_{\psi}\overline{\chi} \in {\Gamma} \text{ and } \forall {\Delta}\in W^{c} (\text{ if } \psi \in {\Delta} \text{ then } \chi\in {\Delta}) \hspace{11pt}(\text{by the induction hypothesis})\\ \text{ iff } K_{\psi}\overline{\chi} \in {\Gamma} \text{ and } \forall {\Delta}\in W^{c} (\psi \supset \chi \in {\Delta}) \hspace{123pt}(\text{Lemma~8})\\ \text{ iff } K_{\psi}\overline{\chi} \in {\Gamma} \text{ and } \Box(\psi \supset \chi) \in {\Gamma} \hspace{65pt}(\text{Lemma~16, the definition of } W^{c})\\ \text{ iff } (K_{\psi}\overline{\chi}\wedge \Box(\psi \supset \chi)) \in {\Gamma} \hspace{175pt}(\text{Lemma}~8)\\ \text{ iff } K_{\psi}\chi \in {\Gamma} \hspace{202.5pt}(\mathsf{Ax}4 \text{ and Lemma } 8) \end{array} $$

Case φ := Aψ:

$$ \begin{array}{ll} \mathcal{M}^{c}, {\Gamma} \vDash A\varphi & \text{ iff } \mathcal{M}^{c}, @_{{\Gamma}_{0}}\vDash \varphi \hspace{134pt}(\text{by the semantics})\\ & \text{ iff } \varphi \in @_{{\Gamma}_{0}} \hspace{149pt}(\text{IH, Lemma~12.(1)})\\ & \text{ iff } A\varphi \in {\Gamma} \hspace{169pt}(\text{Lemma 12.(4)}) \end{array}$$

□

Corollary 18

EL^∗is a sound and complete axiomatization of$\mathcal {EL}_{-}^{*}$with respect to the class of epistemic models.

Proof

We prove only completeness since soundness is a matter of routine validity check. Let $\not \vdash _{\mathsf {EL}^{*}}\varphi$. This means that {¬φ} is consistent, and by Lemma 9, can be extended to a mcs Γ₀. Then, by Lemma 17, we obtain that ${\mathcal{M}}^{c}, {\Gamma }_{0}\not \vDash \varphi$, where ${\mathcal{M}}^{c}$ is the canonical model for Γ₀. □

B.1.1 The (static) Logic of Knowledge Over $\mathcal {EL}^{+}_{-}$

We now provide a sound and complete axiomatization for the fragment $\mathcal {EL}^{+}_{-}$ of $\mathcal {EL}^{*}_{-}$ without the actuality operator.

Theorem 19

EL⁺given in Table 2is a sound and complete axiomatization of$\mathcal {EL}^{+}_{-}$with respect to the class of epistemic models.

Table 2 Axiomatization EL⁺ (over $\mathcal {EL}^{+}_{-}$)

The proof of Theorem 19 follows similarly to the proof of Corollary 18 except that we need to replace Lemma 10 by the following lemma.

Lemma 20

For all${\Gamma }\in \mathcal {X}$, there exists a$@_{\Gamma }\in \mathcal {X}$such that${\Gamma } \sim _{\Box } @_{\Gamma }$and$1\wedge {\dots } \wedge n\in @_{\Gamma }$.

Proof

Let ${\Gamma }\in \mathcal {X}$. Consider the set $\{\varphi \in \mathcal {EL}^{+}_{-} : \Box \varphi \in {\Gamma }\}\cup \{1\wedge {\dots } \wedge n\}$ and suppose it is inconsistent. This implies that there are finitely many $\varphi _{1}, \dots , \varphi _{k}\in \{\varphi \in \mathcal {EL}^{+}_{-} : \Box \varphi \in {\Gamma }\}$ such that $\vdash _{\mathsf {EL}^{+}} (\varphi _{1}\wedge {\dots } \wedge \varphi _{k})\supset \neg (1\wedge {\dots } \wedge n)$. Then, by $\mathsf {S5}_{\Box }$, we obtain $\vdash _{\mathsf {EL}^{+}} (\Box \varphi _{1}\wedge {\dots } \wedge \Box \varphi _{k})\supset \Box \neg (1\wedge {\dots } \wedge n)$. As each $\Box \varphi _{i}\in {\Gamma }$ for 1 ≤ i ≤ k, by Lemma 8, we have $\Box \neg (1\wedge {\dots } \wedge n)\in {\Gamma }$. This means that $\neg \Diamond (1\wedge {\dots } \wedge n)\in {\Gamma }$, contradicting consistency of Γ due to Ax1 and Lemma 8.(3). Therefore, by Lemma 9, $\{\varphi \in \mathcal {EL}^{+}_{-} : \Box \varphi \in {\Gamma }\}\cup \{1\wedge {\dots } \wedge n\}$ extends to a $\mathcal {EL}^{+}_{-}$-maximally consistent set @_Γ such that $1\wedge {\dots } \wedge n\in @_{\Gamma }$. As $\{\varphi \in \mathcal {EL}^{+}_{-} : \Box \varphi \in {\Gamma }\}\subseteq @_{\Gamma }$, we also have ${\Gamma }\sim _{\Box } @_{\Gamma }$. □

Lemma 20 guarantees the existence of an appropriate actual world $@_{{\Gamma }_{0}}$ in the canonical model for a maximally EL⁺-consistent set Γ₀.

B.2 The Logic of Knowledge Update Over $\mathcal {EL}^{*}$

This section presents soundness and completeness results for the dynamic language $\mathcal {EL}^*$ of epistemic update. We axiomatize two classes of epistemic update models that represent maximal and minimal knowledge updates, respectively. These are the two extreme cases that an epistemic update function μ (given in Definition 1) can interpret: the former is modelled similarly to the so-called public announcements [25, 49, 50], and the latter represents an agent who learns the new piece of information without merging its intension with her prior information state.

Definition 4 (Maximal Epistemic Update Model)

A maximal epistemic update model is a tuple ${\mathcal{M}}= \langle W, @, \mathbb {T}, \oplus , \mu , v, t \rangle$ where $\langle W, @, \mathbb {T}, \oplus , v, t \rangle$ is an epistemic model and $\mu : \mathbb {I}_{@} \times \mathbb {I}_{@} \to \mathbb {I}_{@}$ is an update function such that μ(P,Q) = P ∩ Q.

Theorem 21

A sound and complete axiomatization$\mathsf {EUL}^{*}_{\max \limits }$ of $\mathcal {EL}^*$with respect to the class of maximal epistemic update models is obtained by adding toEL^∗the set of axioms and rules in Table 3.

Table 3 Reduction axioms and inference rules for [i : φ] for maximal epistemic update models

Definition 5 (Minimal Epistemic Update Model)

A minimal epistemic update model is a tuple ${\mathcal{M}}= \langle W, @, \mathbb {T}, \oplus , \mu , v, t \rangle$ where $\langle W, @, \mathbb {T}, \oplus , v, t \rangle$ is an epistemic model and $\mu : \mathbb {I}_{@} \times \mathbb {I}_{@} \to \mathbb {I}_{@}$ is an update function such that μ(P,Q) = Q.

Theorem 22

A sound and complete axiomatization$\mathsf {EUL}^{*}_{min}$of$\mathcal {EL}^*$with respect to the class of minimal epistemic update models is obtained by replacing axiom (R_i) in Table 3by

$$ [i: \varphi] i \equiv A\varphi \supset (\varphi\wedge (i\vee \neg i)). $$

B.2.1 Proofs of Theorems 21 and 22

The proofs of Theorems 21 and 22 are by the so-called standard DEL-technique completeness via reduction (or translation), as briefly explained in Section 4.4. For a detailed presentation of completeness by reduction, we refer the reader to [62, Chapter 7.4].

Definition 6 (Complexity measure for $\mathcal {EL}^*$)

The complexity c(φ) of a formula $\varphi \in \mathcal {EL}^*$ is a natural number recursively defined as

$$ \begin{array}{@{}rcl@{}} c(i)=c(p) &=&1\\ c(\neg\varphi) =c(\Box \varphi)= c(A \varphi)&=&c(\varphi)+1\\ c(\varphi\wedge\psi) & = & 1+ max\{c(\varphi), c(\psi)\}\\ c(K_{\varphi} \psi) & = & 4 + c(\varphi) + c(\psi)\\ c([i: \varphi]\psi) & = & (3 + |\mathcal{A}\mathcal{T}(\varphi)|+c(\varphi))\cdot (|\mathcal{A}\mathcal{T}(\psi)| + c(\psi)), \end{array} $$

where $|\mathcal {A}\mathcal {T}(\varphi )|$ is the number of elements in $\mathcal {A}\mathcal {T}(\varphi )$.

Lemma 23

For all$\varphi , \psi , \chi \in \mathcal {EL}^*$and$i, x\in \mathcal {A}\mathcal {T}$,

• (1) c(φ) > c(ψ) ifψis a proper subformula ofφ,

• (2) $c([i: \varphi ] i) > c(A\varphi \supset (i\wedge \varphi ))$,

• (3) $c([i: \varphi ] x) > c(A\varphi \supset (x\wedge (i\vee \neg i)))$, for i≠x

• (4) $c([i: \varphi ] \neg \psi ) > c(A\varphi \supset \neg [i: \varphi ] \psi )$,

• (5) c([i : φ](ψ ∧ χ)) > c([i : φ]ψ ∧ [i : φ]χ),

• (6) $c([i: \varphi ] \Box \psi ) > c(A\varphi \supset \Box [i: \varphi ] \psi )$,

• (7) $c([i: \varphi ] A\psi ) > c(A\varphi \supset A[i: \varphi ] \psi )$,

• (8) $c([i: \varphi ] K_{\psi }\chi ) > c(A\varphi \supset (K_{\psi }\overline {\chi } \wedge \Box [i: \varphi ](\psi \supset \chi )))$,

• (9) $c([i: \varphi ] K_{\psi }\chi ) > c(A\varphi \supset (K_{\psi \wedge \varphi }\overline {\chi } \wedge \Box [i: \varphi ](\psi \supset \chi )))$,

• (10) $c([i: \varphi ] K_{\psi }\chi ) > c(A\varphi \supset (K_{\psi }\overline {(\chi \wedge \varphi )} \wedge \Box [i: \varphi ](\psi \supset \chi )))$,

• (11) $c([i: \varphi ] K_{\psi }\chi ) > c(A\varphi \supset (K_{\psi \wedge \varphi }\overline {(\chi \wedge \varphi )} \wedge \Box [i: \varphi ](\psi \supset \chi )))$.

Proof

Follows by easy calculations using Definition 6. And, obviously, (8)–(10) follow from (11). □

Definition 7 (Translation $f: \mathcal {EL}^* \rightarrow \mathcal {EL}_{-}^{*}$)

The translation $f: \mathcal {EL}^* \rightarrow \mathcal {EL}_{-}^{*}$ is defined as follows:

$$ \begin{array}{@{}rcl@{}} f(x) & = & x\\ f(\neg \varphi) &= & \neg f(\varphi)\\ f(\Box \varphi) &= & \Box f(\varphi)\\ f(\varphi\wedge \psi) &= & f(\varphi) \wedge f(\psi)\\ f(K_{\varphi}\psi) &= & K_{f(\varphi)}f(\psi)\\ f([i: \varphi] i) & = & f(A\varphi \supset (i\wedge \varphi))\\ f([i: \varphi] x) & = & f(A\varphi \supset (x\wedge (i\vee\neg i))), \text{ for } x\not = i\\ f([i: \varphi] \neg\psi) &= & f(A\varphi \supset \neg [i: \varphi] \psi)\\ f([i: \varphi](\psi\wedge \chi)) & = &f([i: \varphi]\psi \wedge[i: \varphi]\chi),\\ f([i: \varphi] \Box\psi) &= & f(A\varphi \supset \Box[i: \varphi] \psi)\\ f([i: \varphi] A\psi) &= & f(A\varphi \supset A[i: \varphi] \psi)\\ f([i: \varphi] K_{\psi}\chi) & = & f(A\varphi \supset (K_{\psi}\overline{\chi}\wedge \Box [i: \varphi](\psi\supset \chi))), \text{ if } i\not \in \mathcal{A}\mathcal{T}(\psi)\cup \mathcal{A}\mathcal{T}(\chi) \\ \ & = & f(A\varphi \supset (K_{\psi\wedge \varphi}\overline{\chi} \wedge \Box [i: \varphi](\psi\supset \chi))), \text{ if } i\in \mathcal{A}\mathcal{T}(\psi)\backslash \mathcal{A}\mathcal{T}(\chi)\\ \ & = & f(A\varphi \supset (K_{\psi}\overline{(\chi\wedge \varphi)} \wedge \Box [i: \varphi](\psi\supset \chi))), \text{ if } i\in \mathcal{A}\mathcal{T}(\chi)\backslash \mathcal{A}\mathcal{T}(\psi)\\ \ & = & f(A\varphi \supset (K_{\psi\wedge \varphi }\overline{(\chi\wedge \varphi)} \wedge \Box [i: \varphi](\psi\supset \chi))), \text{ if } i\in \mathcal{A}\mathcal{T}(\chi)\cap \mathcal{A}\mathcal{T}(\psi)\\ f([i: \varphi] [j: \psi]\chi) & = &f([i: \varphi] f([j: \psi]\chi)). \end{array} $$

We need the following lemma in order to be able to use the derived replacement of equivalents rule given in Theorem 7.(1) in the completeness proofs of $\mathsf {EUL}^{*}_{\max \limits }$ and $\mathsf {EUL}^{*}_{min}$. For this lemma to go through, it is crucial that the reduction axioms R_i and R_x≠i have occurrences of each element in $\mathcal {A}\mathcal {T}(\varphi )$ and i on the right-hand-side of the equivalences, where φ is the sentence inside the dynamic operator and i is the updated frame of mind.

Lemma 24

For all$\varphi \in \mathcal {EL}^*$, $\mathcal {A}\mathcal {T}(\varphi )=\mathcal {A}\mathcal {T}({f(\varphi )})$.

Proof

The proof follows by an easy c-induction on the structure of φ and uses Lemma 23. Note that the case for φ := [i : ψ]χ requires subinduction on χ. □

Lemma 25

For all$\varphi \in \mathcal {EL}^*$, $\vdash _{\mathsf {EUL}^{*}_{\max \limits }} \varphi \equiv f(\varphi )$.

Proof

The proof follows by c-induction on the structure of φ and uses Lemma 23. Cases for the atomic formulae, the Boolean connectives, and $\Box$ are elementary. Here we only show the cases for φ := K_ψχ and φ := [i : ψ]χ, where the latter requires subinduction on χ. Suppose inductively that $\vdash _{\mathsf {EUL}^{*}_{\max \limits }} \psi \equiv f(\psi )$, for all ψ with c(ψ) < c(φ).

Case φ := K_ψχ

By Lemma 23.(1) and the induction hypothesis (IH), we have $\vdash _{\mathsf {EUL}^{*}_{\max \limits }} \psi \equiv f(\psi )$. Moreover, by Lemma 24, we have $\mathcal {A}\mathcal {T}(\psi )=\mathcal {A}\mathcal {T}({f(\psi )})$. Therefore, by Ax8 in Table 1, we obtain $\vdash _{\mathsf {EUL}^{*}_{\max \limits }} K_{\psi }\overline {f(\psi )}$ and $\vdash _{\mathsf {EUL}^{*}_{\max \limits }} K_{f(\psi )}\overline {\psi }$. Then, by Theorem 7.(1), we obtain $\vdash _{\mathsf {EUL}^{*}_{\max \limits }} K_{\psi }\chi \equiv K_{f(\psi )}\chi$. Similarly, we also have $\vdash _{\mathsf {EUL}^{*}_{\max \limits }} \chi \equiv f(\chi )$ and $\mathcal {A}\mathcal {T}(chi)=\mathcal {A}\mathcal {T}({f(\chi )})$, thus, $\vdash _{\mathsf {EUL}^{*}_{\max \limits }} K_{\chi }\overline {f(\chi )}$ and $\vdash _{\mathsf {EUL}^{*}_{\max \limits }} K_{f(\chi )}\overline {\chi }$. Then, by Theorem 7.(1) again, we obtain $\vdash _{\mathsf {EUL}^{*}_{\max \limits }} K_{f(\psi )}\chi \equiv K_{f(\psi )} f(\chi )$. Therefore, by CPL, we conclude that $\vdash _{\mathsf {EUL}^{*}_{\max \limits }} K_{\psi }\chi \equiv K_{f(\psi )} f(\chi )$, with K_f(ψ)f(χ) = f(K_ψχ) by Definition 7.

Case φ := [i : ψ]χ: we prove only the cases χ := x with x≠i, and χ := [j : 𝜃]α. All the other cases follow similarly by using the corresponding reduction axioms, Lemma 23, and Definition 7.

Subcase χ := x, for x≠i

• 1. $\vdash _{\mathsf {EUL}^{*}_{\max \limits }} [i: \psi ]x \equiv (A\varphi \supset (x\wedge (i\vee \neg i))) \qquad \qquad \qquad ~~~~ R_{x\not =i}$

• 2. $\vdash _{\mathsf {EUL}^{*}_{\max \limits }} (A\varphi \supset (x\wedge (i\vee \neg i)))\equiv f(A\varphi \supset (x\wedge (i\vee \neg i))) ~~~~\text {Lemma} 23.(3), IH$

And, $f(A\varphi \supset (x\wedge (i\vee \neg i))) =f([i: \psi ]x )$ by Definition 7.

Subcase χ := [j : 𝜃]α By Lemma 23.(1) and IH, we know that ⊩ [j : 𝜃]α ≡ f([j : 𝜃]α)

• 1 $\vdash _{\mathsf {EUL}^{*}_{\max \limits }} [j: \theta ]\alpha \equiv f([j: \theta ]\alpha )$ Lemma 23.(1), IH

• 2 $\vdash _{\mathsf {EUL}^{*}_{\max \limits }} [i: \psi ]([j: \theta ]\alpha \equiv f([j: \theta ]\alpha ))$ Nec_[]

• 3. $\vdash _{\mathsf {EUL}^{*}_{\max \limits }} [i: \psi ][j: \theta ]\alpha \equiv [i: \psi ]f([j: \theta ]\alpha )$ R_¬, R_∧

• 4. $\vdash _{\mathsf {EUL}^{*}_{\max \limits }} [i: \psi ]f([j: \theta ]\alpha ) \equiv f([i: \psi ]f([j: \theta ]\alpha ))$ IH

• 5 $\vdash _{\mathsf {EUL}^{*}_{\max \limits }} [i: \psi ][j: \theta ]\alpha \equiv f([i: \psi ]f([j: \theta ]\alpha ))$ 3, 4, CPL

And, f([i : ψ]f([j : 𝜃]α)) = f([i : ψ][j : 𝜃]α) by Definition 7. □

Proof of Theorem 21

Let $\varphi \in \mathcal {EL}^*$ such that $\not \vdash _{\mathsf {EUL}^{*}_{\max \limits }} \varphi$. By Lemma 25, there is $\psi \in \mathcal {EL}_{-}^{*}$ such that $\vdash _{\mathsf {EUL}^{*}_{\max \limits }} \varphi \equiv \psi$. Therefore, $\not \vdash _{\mathsf {EUL}^{*}_{\max \limits }} \psi$. As $\psi \in \mathcal {EL}_{-}^{*}$ and $\mathsf {EL}^{*}\subseteq \mathsf {EUL}^{*}_{\max \limits }$, we also have $\not \vdash _{\mathsf {EL}^{*}}\psi$. Then, by Corollary 18, there is an epistemic model ${\mathcal{M}}=\langle W, @, \mathbb {T}, \oplus , v, t \rangle$ and w ∈ W such that ${\mathcal{M}}, w\not \vDash \psi$. Observe that the tuple $\langle W, @, \mathbb {T}, \oplus , \mu , v, t\rangle$, where $\mu : \mathbb {I}_{@} \times \mathbb {I}_{@} \to \mathbb {I}_{@}$ such that μ(P,Q) = P ∩ Q, is a maximal epistemic update model. Then, by Lemma 5, we have $\langle W, @, \mathbb {T}, \oplus , \mu , v, t\rangle , w\not \vDash \psi$. Therefore, by the soundness of $\mathsf {EUL}^{*}_{\max \limits }$ with respect to maximal epistemic update models, we obtain that $\langle W, @, \mathbb {T}, \oplus , \mu , v, t\rangle , w\not \vDash \varphi$. □

Proof of Theorem 22

Follows the same steps as the proof of Theorem 21. We need to replace every occurrence of $[i: \varphi ] i \equiv A\varphi \supset (i\wedge \varphi )$ by $[i: \varphi ] i \equiv A\varphi \supset (\varphi \wedge (i\vee \neg i))$. □