Abstract
We introduce Arbitrary Public Announcement Logic with Memory (APALM), obtained by adding to the models a ‘memory’ of the initial states, representing the information before any communication took place (“the prior”), and adding to the syntax operators that can access this memory. We show that APALM is recursively axiomatizable (in contrast to the original Arbitrary Public Announcement Logic, for which the corresponding question is still open). We present a complete recursive axiomatization, that includes a natural finitary rule, and study this logic’s expressivity and the appropriate notion of bisimulation. We then examine Group Announcement Logic with Memory (GALM), the extension of APALM obtained by adding to its syntax group announcement operators, and provide a complete finitary axiomatization (again in contrast to the original Group Announcement Logic, for which the only known axiomatization is infinitary). We also show that, in the memory-enhanced context, there is a natural reduction of the so-called coalition announcement modality to group announcements (in contrast to the memory-free case, where this natural translation was shown to be invalid).
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Acknowledgements
We thank the two anonymous reviewers whose insightful comments and suggestions helped improve this manuscript. We also thank our editor for his effort, time, and patience on the revision process. Versions of the present work were presented at the following conferences and workshops: The 25rd Workshop on Logic, Language and Computation (WoLLIC) 2018, Department of Mathematics, Universidad de Los Andes; Workshop on New Directions in Reasoning about Belief and Knowledge 2018, ILLC, University of Amsterdam; Logic and Interactive Rationality Seminar of the Amsterdam Dynamics Group 2018, ILLC, University of Amsterdam. We thank the audiences of these events for their helpful questions and remarks. Aybüke Özgün acknowledges support form the project ‘The Logic of Conceivability’, funded by the European Research Council (ERC CoG), Grant Number 681404.
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Appendix
Appendix
1.1 A.1 Definition of the complexity measure for the Proofs of Lemmas 1 and 20
In some of our inductive proofs, we need a complexity measure on formulas that is different from the standard one based on subformula complexity. The standard notion requires only that formulas are more complex than their subformulas, while we also need that ♢φ and 〈G〉φ are more complex than 〈𝜃〉φ for all \(\theta \in {{\mathscr{L}}_{-\Diamond }}\). To the best of our knowledge, such a complexity measure was first introduced in [5] for the original APAL language from [6]. A similar measure is also used in [7] and has later been introduced for topological versions of APAL in [11, 39, 40]. Definitions below are given for our largest language \({\mathscr{L}}_{G}\). Their counterparts for \({\mathscr{L}}\) are obtained simply by eliminating the clauses for 〈G〉φ.
Definition 1 (Subformula)
Given a formula \(\varphi \in {{\mathscr{L}}_{G}}\), the set Sub(φ) of subformulas of φ is recursively defined as
Any formula in Sub(φ) −{φ} is called a proper subformula of φ.
Definition 2 (Size of formulas in \({\mathscr{L}}_{G}\))
The size s(φ) of formula \(\varphi \in {\mathscr{L}}_{G}\) is a natural number recursively defined as:
Definition 3 (♢,G-Depth of formulas in \({\mathscr{L}}_{G}\))
The ♢,G-depthd(φ) of formula \(\varphi \in {\mathscr{L}}_{G}\) is a natural number recursively defined as:
Finally, we define our intended complexity relation < as lexicographic merge of ♢,G-depth and size, exactly as in [5]:
Definition 4
For any \(\varphi , \psi \in {\mathscr{L}}_{G}\), we put
1.2 A.2 Proofs of results in Section ??
Proof Proof of Proposition 9
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from \(\vdash \varphi \leftrightarrow \psi \), infer \( \vdash [\theta ]\varphi \leftrightarrow [\theta ]\psi \): Follows directly by (K!) and (Nec!).
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\(\langle \theta \rangle 0 \leftrightarrow (0 \wedge U\theta )\): Follows from the definition of 〈𝜃〉0 := ¬[𝜃]¬0 and the axiom (R¬.)
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\(\langle \theta \rangle \psi \leftrightarrow (\theta \wedge [\theta ] \psi )\): Follows from the definition 〈𝜃〉ψ := ¬[𝜃]¬ψ and the axiom (R¬.)
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\(\Box \varphi \to [\rho ] \varphi \) (\(\rho \in {{\mathscr{L}}_{-\Diamond }}\) arbitrary):
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\(\vdash \Box \varphi \leftrightarrow [\top ]\Box \varphi \) (R[⊤])
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\(\vdash [\top ]\Box \varphi \rightarrow [\top \wedge \rho ]\varphi \), (for arbitrary \(\rho \in {{\mathscr{L}}_{-\Diamond }}\)) ([!]\(\Box \)-elim)
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\(\vdash \! [\top \!\wedge \! \rho ] \varphi \!\rightarrow \! [\rho ]\varphi \), (for arbitrary \(\rho \in {{\mathscr{L}}_{-\Diamond }}\)) \((\vdash \!(\top \!\wedge \! \rho )\!\leftrightarrow \! \rho \) and (RE))
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\(\vdash \Box \varphi \rightarrow [\rho ]\varphi \), (for arbitrary \(\rho \in {{\mathscr{L}}_{-\Diamond }})\) (1-3, CPL)
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from \(\vdash \chi \to [p]\varphi , \text {infer} \vdash \chi \to \Box \varphi (p \not \in P_{\chi }\cup P_{\varphi })\):
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⊩ χ → [p]φ (assumption)
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⊩ χ → [⊤][p]φ (R[⊤])
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⊩ χ → [〈⊤〉p]φ (R[!])
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⊩ χ → [⊤∧ p]φ (Prop.9.3, Rp, RE)
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\(\vdash \chi \to [\top ]\Box \varphi \) (p∉Pχ ∪ Pφ and [!]\(\Box \)-intro)
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\(\vdash \chi \to \Box \varphi \) (R[⊤])
-
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all S4 axioms and rules for \(\Box \): The derivation of the necessitation rule for \(\Box \), (Nec\(_{\Box }\)), easily follows from (Nec!) and Prop. 9.5. The T-axiom for \(\Box \) follows from Prop. 9.4 and R[⊤]. For the K-axiom:
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\(\vdash (\Box (\varphi {\rightarrow } \psi ) \wedge \Box \varphi ) {\rightarrow } ([p](\varphi {\rightarrow }\psi ) {\wedge } [p]\varphi )\) (p∉Pφ ∪ Pψ, Prop. 9.4)
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\(\vdash ([p](\varphi \rightarrow \psi ) \wedge [p]\varphi ) \rightarrow [p] \psi \) (K!)
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\(\vdash (\Box (\varphi \rightarrow \psi ) \wedge \Box \varphi ) \rightarrow [p] \psi \) (1, 2, CPL)
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\(\vdash (\Box (\varphi \rightarrow \psi ) \wedge \Box \varphi ) \rightarrow \Box \psi \) (p∉Pφ ∪ Pψ, Prop. 9.5)
For the 4-axiom:
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\(\vdash \Box \varphi \rightarrow [p\wedge q] \varphi \) (for some p,q∉Pφ, Prop. 9.4)
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\(\vdash \Box \varphi \rightarrow [p]\Box \varphi \) (\([!]\Box \)-intro)
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\(\vdash \Box \varphi \rightarrow \Box \Box \varphi \) (p∉Pφ, Prop. 9.5)
-
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\((\varphi \to \psi )^{0} \leftrightarrow (\varphi ^{0} \to \psi ^{0})\): This is straightforward by the set of axioms called Equivalences with0.
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\(\vdash \varphi ^{00} \leftrightarrow \varphi ^{0}\):
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\(\vdash 0 \to (\varphi \leftrightarrow \varphi ^{0})\) (0-eq)
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\(\vdash (0 \to (\varphi \leftrightarrow \varphi ^{0}))^{0}\) (Nec0)
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\(\vdash 0^{0} \to (\varphi \leftrightarrow \varphi ^{0})^{0}\) (Prop.9.7)
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\(\vdash 0^{0} \to (\varphi ^{0} \leftrightarrow \varphi ^{00})\) (Prop.9.7)
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⊩ 00 (Ax0)
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\(\vdash \varphi ^{0} \leftrightarrow \varphi ^{00}\) (4, 5, MP)
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\(\Box \varphi ^{0} \leftrightarrow \varphi ^{0}\) and \(\varphi ^{0} \leftrightarrow \Diamond \varphi ^{0}\): From left-to-right direction of both cases follow from the T-axiom for \(\Box \). From right-to-left direction we will first prove that \(\vdash \varphi ^{0} \to \Box \varphi ^{0}\):
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⊩ φ0 → (p → φ0) (for \(p\notin P_{\varphi ^{0}}\), CPL)
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⊩ φ0 → [p]φ0 (R0)
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\(\vdash \varphi ^{0} \to \Box \varphi ^{0}\) (Prop. 9.5)
For ⊩♢φ0 → φ0, we have:
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\(\vdash (\neg \varphi )^{0} \to \Box (\neg \varphi )^{0}\) (by the above result)
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\(\vdash \neg \varphi ^{0} \to \Box \neg \varphi ^{0}\) (Eq\(^{0}_{\neg }\))
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⊩♢φ0 → φ0 (contraposition of 2)
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\(\vdash (\Box \varphi )^{0} \to \Box \varphi ^{0}\)
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\(\vdash \Box \varphi \to \varphi \) (S4 for \(\Box \))
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\(\vdash (\Box \varphi \to \varphi )^{0}\) (Nec0)
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\(\vdash (\Box \varphi )^{0} \to \varphi ^{0}\) (Prop.9.7, 2, MP)
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\(\vdash (\Box \varphi )^{0} \to \Box \varphi ^{0}\) (Prop.9.9)
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⊩ (0 ∧♢φ0) → φ
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\(\vdash 0 \to (\varphi ^{0} \leftrightarrow \varphi )\) (0-eq)
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⊩ 0 → (φ0 → φ) (CPL)
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⊩ 0 → (♢φ0 → φ) (Prop.9.9)
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⊩ (0 ∧♢φ0) → φ (CPL)
-
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⊩ φ → (0 ∧♢φ)0
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\(\vdash (\Box \neg \varphi )^{0} \to \neg \varphi \) (Imp\(^{0}_{\Box }\))
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\(\vdash \neg \neg \varphi \to \neg (\Box \neg \varphi )^{0}\) (contraposition of 1)
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\(\vdash \neg \neg \varphi \to (\neg \Box \neg \varphi )^{0}\) (Eq\(^{0}_{\neg }\))
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⊩ φ → (♢φ)0 (the defn. of \(\Box \), CPL)
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⊩ φ → (00 ∧ (♢φ)0) (Ax0)
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⊩ φ → (0 ∧♢φ)0 (Eq\(^{0}_{\wedge }\))
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⊩ φ → ψ0 if and only if ⊩ (0 ∧♢φ) → ψ From left-to-right: Suppose ⊩ φ → ψ0 and show: ⊩ (0 ∧♢φ) → ψ.
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⊩ (0 ∧♢ψ0) → ψ (Prop.9.11)
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⊩♢φ →♢ψ0 (by assumption and Nec\(_{\Box }\))
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⊩ (0 ∧♢φ) → (0 ∧♢ψ0) (2 and CPL)
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⊩ (0 ∧♢φ) → ψ (1-3, CPL)
From right-to-left: Suppose ⊩ (0 ∧♢φ) → ψ and show ⊩ φ → ψ0.
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⊩ φ → (0 ∧♢φ)0 (Prop.9.12)
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⊩ (0 ∧♢φ) → ψ (assumption)
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⊩ ((0 ∧♢φ) → ψ)0 (Nec0)
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⊩ (0 ∧♢φ)0 → ψ0 (Prop.9.7)
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⊩ φ → ψ0 (1-4, CPL)
-
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\([\theta ] (\varphi \wedge \psi ) \leftrightarrow ([\theta ]\varphi \wedge [\theta ]\psi )\): Follows from (K!) and (Nec!).
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\([\theta ][p] \varphi \leftrightarrow [\theta \wedge p] \varphi \)
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\(\vdash [\theta ][p]\varphi \leftrightarrow [\langle \theta \rangle p]\varphi \) (R[!])
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\(\vdash [\langle \theta \rangle p]\varphi \leftrightarrow [\theta \wedge [\theta ]p]\varphi \) (Prop.9.3, RE)
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\(\vdash [\theta \wedge [\theta ]p]\varphi \leftrightarrow [\theta \wedge (\theta \to p)]\varphi \) (Rp, RE)
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\(\vdash [\theta \wedge (\theta \to p)]\varphi \leftrightarrow [\theta \wedge p]\varphi \) (CPL, RE)
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\(\vdash [\theta ][p]\varphi \leftrightarrow [\theta \wedge p]\varphi \) (1-4, CPL)
-
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\([\theta ]\bot \leftrightarrow \neg \theta \)
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\(\vdash [\theta ]\bot \leftrightarrow [\theta ](p\wedge \neg p)\) (the defn. of ⊥)
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\(\vdash [\theta ](p\wedge \neg p) \leftrightarrow [\theta ]p \wedge [\theta ]\neg p\) (Prop.9.14)
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\(\vdash [\theta ]p \wedge [\theta ]\neg p \leftrightarrow ((\theta \to p) \wedge (\theta \to \neg (\theta \to p)))\) (Rp, R¬)
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\(\vdash ((\theta \to p) \wedge (\theta \to \neg (\theta \to p))) \leftrightarrow \neg \theta \) (CPL)
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\(\vdash [\theta ]\bot \leftrightarrow \neg \theta \) (1-4, CPL)
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□
1.3 A.3 Proofs of results in Section ??
Proof Proof of Proposition 22
The proof is by <-induction on φ, using Lemma 20 and the following induction hypothesis (IH): for all ψ < φ and all models \({\mathcal M}= (W^0, W, \sim _1, \ldots , \sim _n, \|\cdot \|)\), we have \([\![{\psi }]\!]\subseteq W\). The base cases φ := p, φ := ⊤, and φ := 0 are straightforward by the semantics given in Defn.3. The inductive cases for Booleans are immediate. Similarly, the following cases make use of the corresponding semantic clause in Defn. 3.
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Case φ := ψ0: \( [\![{\psi ^{0}}]\!]=[\![\psi ]\!]_{{\mathscr{M}}^{0}}\cap W \subseteq W\).
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Case φ := Kiψ: \([\![{K_{i}\psi }]\!] = \{w\in W: w_{i} \subseteq [\![{\varphi }]\!]\}\subseteq W\).
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Case φ := Uψ: [ [Uψ] ] ∈{∅,W}, thus \([\![{U\psi }]\!] \subseteq W\).
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Case φ := 〈𝜃〉ψ: Since 𝜃 < 〈𝜃〉ψ (Lemma 20.1), by the IH on 𝜃, we have that \([\![\theta ]\!]\subseteq W\). Moreover, since ψ < 〈𝜃〉ψ (Lemma 20.1), by the IH on ψ, we also have that \([\![{\psi }]\!]_{{\mathscr{M}}|[\![{\theta }]\!]} \subseteq [\![{\theta }]\!]\) (recall that \({\mathcal M}|[\![{\theta }]\!]= (W^0, [\![{\theta }]\!], \sim _1, \ldots , \sim _n, \|\cdot \|) {[\![{\theta }]\!] }\)). Therefore, by Defn. 3, we obtain that \([\![{\langle \theta \rangle \psi }]\!]=[\![{\psi }]\!]_{{\mathscr{M}}|[\![{\theta }]\!]} \subseteq [\![{\theta }]\!]\subseteq W\).
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Case φ := ♢ψ: By Lemma 20.2, it follows that for each \(\theta \in {\mathscr{L}}_{-\Diamond }\), 〈𝜃〉ψ < ♢ψ. Then, by the IH, we have that for all \(\theta \in {{\mathscr{L}}_{-\Diamond }}\), \([\![\langle \theta \rangle \psi ]\!]\subseteq W\). Thus \(\bigcup \{[\![\langle \theta \rangle \psi ]\!] : \theta \in {{\mathscr{L}}_{-\Diamond }}\} \subseteq W\), i.e., \([\![{\Diamond \psi }]\!] \subseteq W\).
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Case φ := 〈G〉ψ: By Lemma 20.3, it follows that for each \(\theta \in {\mathscr{L}}_{-\Diamond }\), 〈𝜃〉ψ < 〈G〉ψ. Then, by the IH, we have that for all \(\theta _{i}\in {{\mathscr{L}}_{-\Diamond }}\), \([\![{\langle {\bigwedge _{i\in G} K_{i} \theta _{i}}\rangle \psi }]\!]\subseteq W\). Thus \(\bigcup \{[\![{\langle \bigwedge _{i\in G} K_{i} \theta _{i} \rangle \psi ]\!]} : \theta _{i}\in {{\mathscr{L}}_{-\Diamond }} \}\subseteq W\), i.e., \([\![{\langle G \rangle \psi }]\!] \subseteq W\).
□
1.4 A.4 Proofs of results in Section ??
Proof Proof of Proposition 38
-
1.
Let \({\mathscr{M}}=(W^0, \mathcal {A}, \sim _1, \ldots , \sim _n, \|\cdot \|)\) be a standard pseudo-model. Then, \(A\in \mathcal {A}\) implies \(A{=}[\![{\theta }]\!]^{PS}_{W^{0}}\subseteq W^{0}\) for some 𝜃, hence \({\mathscr{M}}_{A}{=}(W^0, \mathcal {A}, \sim _1, \ldots , \sim _n,\) ∥⋅∥) is a model whenever A in non-empty.
-
(a)
The proof is by <-induction from Lemma 20. The base cases and the inductive cases for Booleans are straightforward.
Case φ := ψ0. We have \([\![ \psi ^{0}]\!]^{PS}_{A}=[\![{\psi }]\!]^{PS}_{W^{0}} \cap A= [\![{\psi }]\!]_{{{\mathscr{M}}^{0}_{A}}}\cap A=[\![ \psi ^{0}]\!]_{{\mathscr{M}}_{A}} \) (by Defn.27, IH, and Defn. 3).
Case φ := Kiψ. We have \([\![ K_{i}\psi ]\!]^{PS}_{A}= \{w\in A: {w^{A}_{i}} \subseteq [\![{\psi }]\!]^{PS}_{A}\} =\{w\in A: w_{i} \subseteq [\![{\psi }]\!]_{{\mathscr{M}}_{A}}\}=[\![ K_{i}\psi ]\!]_{{\mathscr{M}}_{A}}\) (by Defn.27, IH, and Defn. 3).
Case φ := Uψ. By Definitions 3 and 27, we have:
$$ [\![ U\psi]\!]_{\mathcal{M}_{A}} = \begin{cases} A \quad \text{if} [\![{\psi}]\!]_{\mathcal{M}_{A}}=A \\ \emptyset \quad \text{otherwise}\\ \end{cases} [\![ U\psi]\!]^{PS}_{A} = \begin{cases} A \quad \text{if} [\![{\psi}]\!]^{PS}_{A} =A\\ \emptyset \quad \text{otherwise} \end{cases} $$By IH, \([\![{\psi }]\!]^{PS}_{A}=[\![{\psi }]\!]_{{\mathscr{M}}_{A}}\), therefore, \([\![ U\psi ]\!]^{PS}_{A}=[\![ U\psi ]\!]_{{\mathscr{M}}_{A}}\).
Case φ:=〈ψ〉χ. By Defn. 3, we know that \([\![ \langle \psi \rangle \chi ]\!]_{{\mathscr{M}}_{A}} {=} [\![ \chi ]\!]_{{\mathscr{M}}_{A}|[\![{\psi }]\!]_{{\mathscr{M}}_{A}}}\). Now consider the relativized model \({\mathscr{M}}_{A}|[\![{\psi }]\!]_{{\mathscr{M}}_{A}}{=}(W^0, [\![{\psi }]\!]_{{\mathscr{M}}_A}, \sim _1, \ldots , \sim _n,\) ∥⋅∥). By Lemma 20.1 and IH, we have \([\![{\psi }]\!]_{{\mathscr{M}}_{A}}=[\![{\psi }]\!]^{PS}_{A}\). Moreover, by the definition of standard pseudo-models, we know that \(A=[\![{\theta }]\!]^{PS}_{W^{0}}\) for some \(\theta \in {{\mathscr{L}}_{-\Diamond }}\). Therefore, \([\![{\psi }]\!]_{{\mathscr{M}}_{A}}=[\![{\psi }]\!]^{PS}_{A}=[\![{\psi }]\!]^{PS}_{[\![{\theta }]\!]^{PS}_{W^{0}}}=[\![{\langle \theta \rangle ]\!]\psi }^{PS}_{W^{0}}\). Therefore, \([\![{\psi }]\!]_{{\mathscr{M}}_{A}}\in \mathcal {A}\). We then have
$$ [\![ \langle \psi \rangle \chi]\!]_{\mathcal{M}_{A}} = [\![ \chi]\!]_{\mathcal{M}_{[\![{\psi}]\!]_{\mathcal{M}_A}}} = [\![ \chi]\!]_{\mathcal{M}_{[\![{\psi}]\!]^{PS}_{A}}} = [\![ \chi]\!]^{PS}_{[\![{\psi}]\!]^{PS}_{A}} =[\![ \langle \psi\rangle \chi]\!]^{PS}_{A}, $$by the semantics and IH on ψ and on χ (since \([\![{\psi }]\!]^{PS}_{A}\in \mathcal {A}\)). Case φ := ♢ψ. By Defn.3, Lemma 20.2, IH, the fact that \({\mathscr{M}}\) is a standard pseudo model, and Lemma 37.1 - applied in this order - we obtain the following equivalences:
$$ [\![\Diamond\psi]\!]_{\mathcal{M}_A} = \bigcup \{ [\![ \langle\chi\rangle \psi]\!]_{\mathcal{M}_A} : {\chi\in \mathcal{L}_{-\Diamond}}\} = \bigcup \{ [\![ \langle \chi \rangle \psi]\!]^{PS}_A : {\chi\in \mathcal{L}_{-\Diamond}}\} = [\![ \Diamond \psi]\!]^{PS}_A. $$Case φ := 〈G〉ψ. By Defn.21, Lemma 20.3, IH, the fact that \({\mathscr{M}}\) is a standard pseudo model, and Lemma 37.2 - applied in this order - we obtain the following equivalences:
$$ \begin{array}{@{}rcl@{}} [\![ \langle G \rangle\psi]\!]_{\mathcal{M}_A} &=& \bigcup \{[\![ \langle {\bigwedge_{i\in G} K_i \theta_i}\rangle\varphi]\!]_{\mathcal{M}_A} : \{\theta_i : i\in G\}\subseteq \mathcal{L}_{-\Diamond} \} \\ &=& \bigcup\{[\![ \langle \bigwedge_{i\in G}K_i\theta_i\rangle \varphi]\!]^{PS}_A : \{\theta_i: i\in G\} \subseteq \mathcal{L}_{-\Diamond}\}. \end{array} $$Therefore \([\![ \langle G \rangle \psi ]\!]_{{\mathscr{M}}_{A}}=[\![ \langle G \rangle \psi ]\!]^{PS}_{A}\).
-
(b)
By part (a), \([\![ \varphi ]\!]_{{{\mathscr{M}}^{0}_{A}}}=[\![ \varphi ]\!]_{{\mathscr{M}}_{W^{0}}}=[\![ \varphi ]\!]^{PS}_{W^{0}}\) for all φ. Since \({\mathscr{M}}\) is standard, we have \(A=[\![{\theta }]\!]^{PS}_{W^{0}}=[\![{\theta }]\!]_{{{\mathscr{M}}^{0}_{A}}}\) for some \(\theta \in {{\mathscr{L}}_{-\Diamond }}\), so \({\mathscr{M}}_{A}\) is an a-model.
-
2.
Let \({\mathscr{M}}=(W^0, W, \sim _1, \ldots , \sim _n, \|\cdot \|)\) be an a-model. Since \(\mathcal {A}=\{[\![{\theta }]\!]_{{\mathscr{M}}^{0}} : \theta \in {\mathscr{L}}_{-\Diamond }\}\subseteq \mathcal {P}(W^{0})\), the model \({\mathscr{M}}'=(W^0, \mathcal {A}, \sim _1, \ldots , \sim _n, \|\cdot \|)\) is a pre-model. Therefore, the semantics given in Defn. 27 is defined on \({\mathscr{M}}'=(W^0, \mathcal {A}, \sim _1, \ldots , \sim _n, \|\cdot \|)\).
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(a)
By Proposition 35, it suffices to prove that the pre-model \({\mathscr{M}}'=(W^0, \mathcal {A}, \sim _1, \ldots , \sim _n, \|\cdot \|)\) is standard, i.e. that \(\{[\![{\theta }]\!]_{{\mathscr{M}}^{0}} : \theta \in {{\mathscr{L}}_{-\Diamond }}\}=\{[\![{\theta }]\!]^{PS}_{W^{0}} : \theta \in {{\mathscr{L}}_{-\Diamond }}\}\). For this, we need to show that for every a-model \({\mathscr{M}}=(W^0, W, \sim _1, \ldots , \sim _n, \|\cdot \|)\), we have \([\![{\theta }]\!]_{{\mathscr{M}}}=[\![{\theta }]\!]^{PS}_{W}\) for all \(\theta \in {{\mathscr{L}}_{-\Diamond }}\).
We prove this by subformula induction on 𝜃. The base cases and the inductive cases for Booleans are straightforward.
Case 𝜃 := ψ0. Then \([\![ \psi ^{0}]\!]_{{\mathscr{M}}} = [\![{\psi }]\!]_{{\mathscr{M}}^{0}}\cap W = [\![{\psi }]\!]^{PS}_{W^{0}} \cap W= [\![ \psi ^{0}]\!]^{PS}_{W}\) (by Defn. 3, IH, and Defn. 27).
Case 𝜃 := Kiψ. We have \([\![ K_{i}\psi ]\!]_{{\mathscr{M}}} =\{w\in W: w_{i} \subseteq [\![{\psi }]\!]_{{\mathscr{M}}}\}= \{w\in W: {w^{W}_{i}} \subseteq [\![{\psi }]\!]^{PS}_{W}\} =[\![ K_{i}\psi ]\!]^{PS}_{W}\) (by Defn. 3, IH, and Defn. 27).
Case 𝜃 := Uψ. By Definitions 3 and 27, we have:
By IH, \([\![{\psi }]\!]^{PS}_{W}=[\![{\psi }]\!]_{{\mathscr{M}}}\), therefore, \([\![ U\psi ]\!]^{PS}_{W}=[\![ U\psi ]\!]_{{\mathscr{M}}}\).
Case 𝜃 := 〈ψ〉χ. By Definition 3, we know that \([\![ \langle \psi \rangle \chi ]\!]_{{\mathscr{M}}} = [\![ \chi ]\!]_{{\mathscr{M}}|{[\![{\psi }]\!]_{{\mathscr{M}}}}}\). Now consider the relativized model \({\mathscr{M}}|{[\![{\psi }]\!]_{{\mathscr{M}}}}=(W^0, {[\![{\psi }]\!]_{{\mathscr{M}}}}, \sim _1, \ldots , \sim _n, \|\cdot \|)\). By Lemma 20.1. and IH on ψ, we have \([\![{\psi }]\!]_{{\mathscr{M}}}=[\![{\psi }]\!]^{PS}_{W}\). Moreover, by the definition of a-models, we know that \(W=[\![{\theta }]\!]_{{\mathscr{M}}^{0}}\) for some \(\theta \in {{\mathscr{L}}_{-\Diamond }}\). Therefore, \([\![{\psi }]\!]_{{\mathscr{M}}}=[\![{\psi }]\!]_{{\mathscr{M}}^{0}|[\![{\theta }]\!]_{{\mathscr{M}}^{0}}}=[\![\langle \theta \rangle \psi ]\!]_{{\mathscr{M}}^{0}}\). Hence, since \(\langle \theta \rangle \psi \in {{\mathscr{L}}_{-\Diamond }}\), the model \({\mathscr{M}}|{[\![{\psi }]\!]_{{\mathscr{M}}}}\) is also an a-model obtained by updating the initial model \({\mathscr{M}}^{0}\) by 〈𝜃〉ψ. We then have
\([\![ \langle \psi \rangle \chi ]\!]_{{\mathscr{M}}} = [\![ \chi ]\!]_{{\mathscr{M}}|{[\![{\psi }]\!]_{{\mathscr{M}}}}}\) (by Defn. 3) \(= [\![ \chi ]\!]_{{\mathscr{M}}|{[\![{\psi }]\!]^{PS}_{W}}}\) (by IH on ψ) \(= [\![ \chi ]\!]^{PS}_{[\![{\psi }]\!]^{PS}_{W}}\) (by IH on χ, \({\mathscr{M}}|{[\![{\psi }]\!]_{{\mathscr{M}}}}\) is an a-model) \(=[\![ \langle \psi \rangle \chi ]\!]^{PS}_{W}\) (by Defn. 27).
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(b)
The proof of this part follows by <-induction on φ (where < is as in Lemma 20). All the inductive cases are similar to ones in the above proof, except for the cases φ := ♢ψ and φ := 〈G〉ψ, shown below.
Case φ := ♢ψ. By Defn. 3, Lemma 20.2, IH, the fact that \({\mathscr{M}}'\) is a standard pseudo model, and Lemma 37.2. - applied in that order - we obtain the following equivalences:
$$ \begin{array}{@{}rcl@{}} [\![ \Diamond\psi]\!]_{\mathcal{M}} &=& \bigcup \{ [\![ \langle \chi\rangle \psi]\!]_{\mathcal{M}} : {\chi\in \mathcal{L}_{-\Diamond}}\}\\ &=& \bigcup \{ [\![ \langle \chi\rangle \psi]\!]^{PS}_W : {\chi\in \mathcal{L}_{-\Diamond}}\} = [\![ \Diamond \psi]\!]^{PS}_W. \end{array} $$Case φ := 〈G〉ψ. By Defn. 21, Lemma 20.3, IH, the fact that \({\mathscr{M}}'\) is a standard pseudo model and Lemma 37.2 - applied in that order - we obtain the following: equivalences,
$$ \begin{array}{@{}rcl@{}} [\![ \langle G \rangle\psi]\!]_{\mathcal{M}} &=& \bigcup \{[\![ \langle {\bigwedge_{i\in G} K_i \theta_i}\rangle\varphi]\!]_{\mathcal{M}} : \{\theta_i : i\in G\}\subseteq \mathcal{L}_{-\Diamond} \}\\ &=& \bigcup\{[\![ \langle \bigwedge_{i\in G}K_i\theta_i\rangle \varphi]\!]^{PS}_W : \{\theta_i: i\in G\} \subseteq \mathcal{L}_{-\Diamond}\}. \end{array} $$Therefore, \([\![ \langle G \rangle \psi ]\!]_{{\mathscr{M}}} = [\![ \langle G \rangle \psi ]\!]^{PS}_W\).
□
1.5 A.5 Proofs of results in Section ??
Proof Proof of Lemma 43
We proceed by induction on the structure of s ∈ NF. For s := 𝜖, take ψ := ⊤ and 𝜃 := ⊤, then it follows from the axiom R[⊤]. For the inductive cases we will verify only \(s:=s^{\prime }, \bullet ^0\); \(s:=s^{\prime }, \eta \to \); \(s:=s^{\prime }, U\); and \(s:=s^{\prime }, \rho \). The case \(s:=s^{\prime }, K_i\) is analogous to the case \(s:=s^{\prime }, U\).
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Case \(s:=s^{\prime },\bullet ^0\)
\(\vdash [s^{\prime },\bullet ^0]\varphi \text { iff}~ \vdash [s^{\prime }]\varphi ^0\) (by Defn. 41) \( \text {iff}~ \vdash \psi ^{\prime } \to [\theta ^{\prime }]\varphi ^0\) (for some \(\psi ^{\prime }\in {\mathscr{L}}_G\) and \(\theta ^{\prime }\in {\mathscr{L}}_{-\Diamond }\), by IH) \( \text {iff}~ \vdash \psi ^{\prime }\to (\theta ^{\prime } \to \varphi ^0)\) (by R0) \( \text {iff}~ \vdash (\psi ^{\prime } \wedge \theta ^{\prime }) \to \varphi ^0 \)\( \text {iff}~ \vdash (0 \wedge \Diamond (\psi ^{\prime } \wedge \theta ^{\prime })) \to \varphi \) (by Prop. 9.13) iff ⊩ ψ → [𝜃]φ (where \(\psi := 0 \wedge \Diamond (\psi ^{\prime } \wedge \theta ^{\prime })\in {\mathscr{L}}_G\) and \(\theta :=\top \in {\mathscr{L}}_{-\Diamond }\)).
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Case \(s:=s^{\prime }, \eta \to \)
\(\vdash [s^{\prime },\eta \to ]\varphi \text { iff}~ \vdash [s^{\prime }](\eta \to \varphi )\) (by Defn. 41) \( \text {iff}~ \vdash \psi ^{\prime } \to [\theta ^{\prime }] (\eta \to \varphi )\) (for some \(\psi ^{\prime }\in {\mathscr{L}}_G\) and \(\theta ^{\prime }\in {\mathscr{L}}_{-\Diamond }\), by IH) \( \text {iff}~ \vdash \psi ^{\prime } \to ([\theta ^{\prime }]\eta \to [\theta ^{\prime }]\varphi )\) (by K!) \( \text {iff}~ \vdash (\psi ^{\prime } \wedge [\theta ^{\prime }]\eta ) \to [\theta ^{\prime }]\varphi \)iff ⊩ ψ → [𝜃]φ (where \(\psi :=\psi ^{\prime } \wedge [\theta ^{\prime }]\eta \in {\mathscr{L}}_G\) and \(\theta :=\theta ^{\prime }\in {\mathscr{L}}_{-\Diamond }\)).
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Case \(s:=s^{\prime },U\)
\(\vdash [s^{\prime },U]\varphi \text { iff}~ \vdash [s^{\prime }]U\varphi \) (by Defn. 41) \( \text {iff}~ \vdash \psi ^{\prime } \to [\theta ^{\prime }]U\varphi \) (for some \(\psi ^{\prime }\in {\mathscr{L}}_G\) and \(\theta ^{\prime }\in {\mathscr{L}}_{-\Diamond }\), by IH) \( \text {iff}~ \vdash \psi ^{\prime } \to (\theta ^{\prime } \to U[\theta ^{\prime }]\varphi )\) (by RU) \( \text {iff}~ \vdash (\psi ^{\prime } \wedge \theta ^{\prime }) \to U[\theta ^{\prime }]\varphi \)\( \text {iff}~ \vdash E(\psi ^{\prime } \wedge \theta ^{\prime }) \to [\theta ^{\prime }]\varphi \) (pushing U back with its dual E, since U is an S5 modality) iff ⊩ ψ → [𝜃]φ (\(\psi :=E(\psi ^{\prime } \wedge \theta ^{\prime })\in {\mathscr{L}}_G\) and \(\theta :=\theta ^{\prime }\in {\mathscr{L}}_{-\Diamond }\)).
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Case \(s:=s^{\prime },\rho \)
\(\vdash [s^{\prime },\rho ]\varphi \text { iff}~ \vdash [s^{\prime }][\rho ]\varphi \) (by Defn. 41) \( \text {iff}~ \vdash \psi ^{\prime } \to [\theta ^{\prime }][\rho ]\varphi \) (by IH) \( \text {iff}~ \vdash \psi ^{\prime } \to [\langle \theta ^{\prime }\rangle \rho ] \varphi \) (by R[!]) iff ⊩ ψ → [𝜃]φ (where \(\psi :=\psi ^{\prime }\in {\mathscr{L}}_G\) and \(\theta :=\langle \theta ^{\prime }\rangle \rho \in {\mathscr{L}}_{-\Diamond }\))
In each case, it is easy to see that \(P_{\psi }\cup P_{\theta }\subseteq P_{s}\). □
Proof Proof of Lemma 49 (Lindenbaum’s Lemma)
The proof proceeds by constructing an increasing chain \({\varGamma }_0\subseteq {\varGamma }_1\subseteq {\dots } \subseteq {\varGamma }_n\subseteq \dots \) of witnessed theories, where Γ0 := Γ, and each Γi is recursively defined. Since we have to guarantee that each Γi is witnessed, we follow a two-fold construction, where Γ0 = Γ0+ := Γ. Let γ0,γ1,…,γn,… be an enumeration of all pairs of the form γi = (si,φi) consisting of any si ∈ NF and any formula \(\varphi _i\in {\mathscr{L}}_G\). Let (sn,φn) be the nth pair in the enumeration. We then set
Note that the empty string 𝜖 is in NF and for every \(\psi \in {\mathscr{L}}_G\) we have 〈𝜖〉ψ := ψ by the definition of possibility forms. Therefore, the above enumeration of pairs includes every formula ψ of \({\mathscr{L}}_G\) in the form of its corresponding pair (𝜖,ψ). By Lemma 46, each Γn+ is witnessed. Then, if φn is of the form φn := ♢𝜃 for some \(\theta \in {\mathscr{L}}_G\), there exists a p ∈ Prop such that Γn+ is consistent with 〈sn〉〈p〉𝜃 (since Γn+ is witnessed). Similarly, if φn is of the form φn := 〈G〉𝜃 for some \(\theta \in {\mathscr{L}}_G\), there exists \(\{p_i: i\in G\}\subseteq Prop\) such that Γn+ is consistent with \(\langle s_n\rangle \langle \bigwedge _{i\in G} K_i p_i\rangle \theta \). We then define
where p ∈ Prop and \(\{p_i:i\in G\}\subseteq Prop\) such that Γn+ is consistent with 〈sn〉〈p〉𝜃 or consistent with \(\langle s_n\rangle \langle \bigwedge _{i\in G} K_i p_i \rangle \theta \), respectively. Again by Lemma 46, it is guaranteed that each Γn is witnessed. Now consider the union \(T_{{\varGamma }}=\bigcup _{n\in \mathbb {N}}{\varGamma }_n\). By Lemma 47, we know that TΓ is a theory. To show that TΓ is witnessed, first let s ∈ NF and \(\psi \in {\mathscr{L}}_G\) and suppose 〈s〉♢ψ is consistent with TΓ. The pair (s,♢ψ) appears in the above enumeration of all pairs, thus (s,♢ψ) := (sm,φm) for some \(m\in \mathbb {N}\). Hence, 〈s〉♢ψ := 〈sm〉φm. Then, since 〈s〉♢ψ is consistent with TΓ and \({\varGamma }_m\subseteq T_{{\varGamma }}\), we know that 〈s〉♢ψ is in particular consistent with Γm. Therefore, by the above construction, 〈s〉〈p〉ψ ∈Γm+ 1 for some p ∈ Prop such that Γm+ is consistent with 〈s〉〈p〉ψ. Thus, as TΓ is consistent and \({\varGamma }_{m+1}\subseteq T_{{\varGamma }}\), we have that 〈s〉〈p〉ψ is also consistent with TΓ. Thus 〈s〉〈p〉ψ is also consistent with TΓ for some p ∈ Prop. Now, let us check the witnessing condition for 〈G〉. Let \(G\subseteq \mathcal {A}\mathcal {G} \), s ∈ NF, and \(\psi \in {\mathscr{L}}_G\) and suppose that 〈s〉〈G〉ψ is consistent with TΓ. The pair (s,〈G〉ψ) appears in the above enumeration of all pairs, thus (s,〈G〉ψ) := (sm,φm) for some \(m\in \mathbb {N}\). Hence, 〈s〉〈G〉ψ := 〈sm〉φm. Then, since 〈s〉〈G〉ψ is consistent with TΓ and \({\varGamma }_m\subseteq T_{{\varGamma }}\), we know that 〈s〉〈G〉ψ is in particular consistent with Γm. Therefore, by the above construction, \(\langle s\rangle \langle \bigwedge _{i\in G}K_i p_i\rangle \psi \in {\varGamma }_{m+1}\) for some \(\{p_i: i\in G\}\subseteq Prop\) such that Γm+ is consistent with \(\langle s\rangle \langle \bigwedge _{i\in G}K_i p_i \rangle \psi \). Thus, as TΓ is consistent and \({\varGamma }_{m+1}\subseteq T_{{\varGamma }}\), we have that \(\langle s\rangle \langle \bigwedge _{i\in G}K_i p_i\rangle \psi \) is also consistent with TΓ. Hence, we conclude that TΓ is witnessed. Finally, TΓ is also maximal by construction: otherwise there would be a witness theory T such that \(T_{{\varGamma }}\subsetneq T\). This implies that there exists \(\varphi \in {\mathscr{L}}_G\) with φ ∈ T but φ∉TΓ. Then, by the construction of TΓ, we obtain Γi ⊩¬φ for all \(i\in \mathbb {N}\). Therefore, since \(T_{{\varGamma }} \subseteq T\), we have T ⊩¬φ. Hence, since φ ∈ T, we conclude T ⊩⊥ (contradicting T being consistent). □
Proof Proof of Lemma 50 (Extension Lemma)
Let \(\theta \in {\mathscr{L}}_G\) and assume that {0,𝜃} is a theory. Moreover, let γ0,γ1,…,γn,… an enumeration of all pairs of the form (sn,φn) consisting of any sn ∈ NF, and every formula \(\varphi _n\in {\mathscr{L}}_G\) of the form φn := ♢ψ or φn := 〈G〉ψ with \(\psi \in {\mathscr{L}}_G\). We will recursively construct a chain of initial theories \({\varGamma }_0\subseteq {\dots } \subseteq {\varGamma }_n \subseteq \dots \) such that
-
1.
Γ0 = {0,𝜃},
-
2.
Pn := {p ∈ P : p occurs in Γn} is finite for every \(n\in \mathbb {N}\), and
-
3.
for every γn := (sn,φn) with sn ∈ NF and \(\varphi _n \in {\mathscr{L}}_G\), if \( {\varGamma }_n \nvdash \neg \langle s_n \rangle \varphi _n\) where φn := ♢ψ then there is pm “fresh” such that 〈sn〉〈pm〉ψ ∈Γn+ 1, and, if \( {\varGamma }_n \nvdash \neg \langle s_n \rangle \varphi _n \) where φn := 〈G〉ψ for some \(G\subseteq \mathcal {AG}\) then there is \(\{p_{m_i} : i\in G\}\) where \(p_{m_i}\) is “fresh” for every i ∈ G such that \( \langle s_n \rangle \langle \bigwedge _{i\in G}K_i p_{m_i} \rangle \psi \in {\varGamma }_{n+1} \). Otherwise we will define Γn+ 1 = Γn.
For every γn, let \(P^{\prime }(n):=\{p\in P^{\prime } \ | \ p \text { occurs either in } s_n\text { or } \varphi _n\}\). Clearly every \(P^{\prime }(n)\) is always finite. We now construct an increasing chain of initial theories recursively. We set Γ0 := {0,𝜃}, and let
where m, mi are, in each case, the least natural number greater than the indices in Pn ∪ P(n), i.e., pm, \(p_{m_i}\) for all i ∈ G are fresh in each case (since Pn ∪ P(n) is finite and Prop is countably infinite, we always have enough fresh propositional variables). We now show that \({\varGamma }:=\bigcup _{n\in \mathbb {N}}{\varGamma }_n\) is an initial witnessed theory. First show that Γ is a theory. By Lemma 47, it suffices to show by induction that every Γn is a theory. We are given that Γ0 is a theory. For the inductive step suppose Γn is consistent but Γn+ 1 is not. Hence, Γn≠Γn+ 1 and moreover Γn+ 1 ⊩⊥.Then, Γn+ 1 = Γn ∪{〈sn〉〈pm〉ψ} (when φn := ♢ψ) or \({\varGamma }_{n+1}= {\varGamma }_n \cup \{ \langle s_n \rangle \langle \bigwedge _{i\in G}K_i p_{m_i} \rangle \psi \}\) (when φn := 〈G〉ψ). Here we will only check the latter case since the former case is analogous. Since \({\varGamma }_{n+1}= {\varGamma }_n \cup \{ \langle s_n \rangle \langle \bigwedge _{i\in G}K_i p_{m_i} \rangle \psi \}\) we have \({\varGamma }_n \vdash [s_n] [\bigwedge _{i\in G}K_i p_{m_i}] \neg \psi \). Therefore there exists \(\{\theta _1,\ldots , \theta _k\}\subseteq {\varGamma }_n\) such that \(\{\theta _1,\ldots , \theta _k\}\vdash [s_n] [\bigwedge _{i\in G}K_i p_{m_i}] \neg \psi \). Let \(\theta =\bigwedge _{1\leq i \leq k} \theta _i\). Then \(\vdash \theta \rightarrow [s_n][\bigwedge _{i\in G}K_i p_{m_i}] \neg \psi \), so \(\vdash [\theta {\rightarrow }, s_n][\bigwedge _{i\in G}K_i p_{m_i}] \neg \psi \) with \(p_{m_i}\notin \mathrm {P}_{\theta } \cup \mathrm {P}_{s_n} \cup \mathrm {P}_{\varphi _n}\) for every i ∈ G. Thus, by the admissible rule in Lemma 44.2, we obtain ⊩ [𝜃→,sn][G]¬ψ, i.e., ⊩ 𝜃 → [sn][G]¬ψ. Therefore, 𝜃 ⊩¬〈sn〉〈G〉ψ. Since \(\{\theta _1,\ldots , \theta _k\}\subseteq {\varGamma }_n\), we therefore have Γn ⊩¬〈sn〉〈G〉ψ. But, this would mean Γn = Γn+ 1, contradicting our assumption (that Γn+ 1≠Γn). Therefore Γn+ 1 is consistent and thus a theory. Hence, by Lemma 47, Γ is a theory. Condition (3) above implies that Γ is also witnessed. Then, by Lindenbaum’s Lemma (Lemma 49), there is a maximal witnessed theory TΓ such that \({\varGamma }\subseteq T_{{\varGamma }}\). Moreover, since \(0\in {\varGamma } \subseteq T_{{\varGamma }}\), the set TΓ is in fact a maximal witnessed initial theory. □
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Baltag, A., Özgün, A. & Sandoval, A.L.V. Arbitrary Public Announcement Logic with Memory. J Philos Logic 52, 53–110 (2023). https://doi.org/10.1007/s10992-022-09664-6
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DOI: https://doi.org/10.1007/s10992-022-09664-6