A Uniform Logic of Information Dynamics

A Uniform Logic of Information Dynamics⋆ Wesley H. Holliday, Tomohiro Hoshi, and Thomas F. Icard, III Logical Dynamics Lab, Center for the Study of Language and Information Cordura Hall, 210 Panama Street, Stanford, CA 94305 Department of Philosophy, Building 90, Stanford University, Stanford, CA 94305 Abstract Unlike standard modal logics, many dynamic epistemic logics are not closed under uniform substitution. A distinction therefore arises between the logic and its substitution core, the set of formulas all of whose substitution instances are valid. The classic example of a non-uniform dynamic epistemic logic is Public Announcement Logic (PAL), and a well-known open problem is to axiomatize the substitution core of PAL. In this paper we solve this problem for PAL over the class of all relational models with infinitely many agents, PAL-Kω , as well as standard extensions thereof, e.g., PAL-Tω , PAL-S4ω , and PAL-S5ω . We introduce a new Uniform Public Announcement Logic (UPAL), prove completeness of a deductive system with respect to UPAL semantics, and show that this system axiomatizes the substitution core of PAL. Keywords: dynamic epistemic logic, Public Announcement Logic, schematic validity, substitution core, uniform substitution 1 Introduction One of the striking features of many of the dynamic epistemic logics [28,19,13,9,4] studied in the last twenty years is the failure of closure under uniform substitution in these systems. Given a valid principle of information dynamics in such a system, uniformly substituting complex epistemic formulas for atomic sentences in the principle may result in an invalid instance. Such failures of closure under uniform substitution turn out to reveal insights into the nature of information change [1,7,11,24,8]. They also raise the question: what are the more robust principles of information dynamics that are valid in all instances, that are schematically valid? Even for the simplest system of dynamic epistemic logic, Public Announcement Logic (PAL) [28], the answer has been unknown. In van Benthem’s “Open Problems in Logical Dynamics” [3], Question 1 is whether the set of schematic validities of PAL is axiomatizable. 1 ⋆ In T. Bolander, T. Braüner, S. Ghilardi, and L. Moss, eds., Advances in Modal Logic, Volume 9, 348-367, College Publications, 2012. 1 Dynamic epistemic logics are not the only non-uniform modal logics to have been studied. Other examples include Buss’s [16] modal logic of “pure provability,” Åqvist’s [10] two- 2 A Uniform Logic of Information Dynamics In this paper, we give an axiomatization of the set of schematic validities—or substitution core—of PAL over the class of all relational models with infinitely many agents, PAL-Kω , as well as standard extensions thereof, e.g., PAL-Tω , PAL-S4ω , and PAL-S5ω . After reviewing the basics of PAL in §1.1, we introduce the idea of Uniform Public Announcement Logic (UPAL) in §1.2, prove completeness of a UPAL deductive system in §3 with respect to alternative semantics introduced in §2, and show that it axiomatizes the substitution core of PAL in §4. In §5, we demonstrate our techniques with examples, and in §6 we conclude by discussing extensions of these techniques to other logics. Although much could be said about the conceptual significance of UPAL as a uniform logic of information dynamics, here we only present the formal results. For conceptual discussion of PAL, we refer the reader to the textbooks [9,4]. Our work here supports a theme of other recent work in dynamic epistemic logic: despite its apparent simplicity, PAL and its variants prove to be a rich source for mathematical investigation (see, e.g., [3,2,25,24,22,32,26,5,23,33]). 1.1 Review of PAL We begin our review of PAL with the language we will use throughout. Definition 1.1 For a set At of atomic sentences and a set Agt of agent symbols with |Agt| = κ, the language LκPAL is generated by the following grammar: ϕ ::= ⊤ | p | ¬ϕ | (ϕ ∧ ϕ) | ✸a ϕ | ⟨ϕ⟩ϕ, where p ∈ At and a ∈ Agt. We define ✷a ϕ as ¬✸a ¬ϕ and [ϕ]ψ as ¬⟨ϕ⟩¬ψ. • Sub(ϕ) is the set of subformulas of ϕ; • At(ϕ) = At ∩ Sub(ϕ); • Agt(ϕ) = {a ∈ Agt | ✸a ψ ∈ Sub(ϕ) for some ψ ∈ LκPAL }; • An(ϕ) = {χ ∈ LκPAL | ⟨χ⟩ψ ∈ Sub(ϕ) for some ψ ∈ LκPAL }. We will be primarily concerned with the language Lω PAL with infinitely many agents, which leads to a more elegant treatment than LnPAL for some arbitrary finite n. In §6 we will briefly discuss the single-agent and finite-agent cases. We will consider two interpretations of LκPAL , one now and one in §2. The standard interpretation uses the following models and truth definition. Definition 1.2 Models for PAL are tuples of the form M = ⟨W, {Ra }a∈Agt , V ⟩, where W is a non-empty set, Ra is a binary relation on W , and V : At → P(W ). dimensional modal logic (see [31]), Carnap’s [17] modal system for logical necessity (see [12,30]), an epistemic-doxastic logic proposed by Halpern [21], and the full computation tree logic CTL∗ (see [29]). Among propositional logics, inquisitive logic [27,18] is a non-uniform example. In some of these cases, the schematically valid fragment—or substitution core— turns out to be another known system. For example, the substitution core of Carnap’s system C is S5 [30], and the substitution core of inquisitive logic is Medvedev Logic [18, §3.4]. Holliday, Hoshi, and Icard 3 Definition 1.3 Given a PAL model M = ⟨W, {Ra }a∈Agt , V ⟩ with w ∈ W , ϕ, ψ ∈ LκPAL , and p ∈ At, we define M, w ! ϕ as follows: M, w M, w M, w M, w M, w M, w ! ⊤; !p ! ¬ϕ !ϕ∧ψ ! ✸a ϕ ! ⟨ϕ⟩ψ iff iff iff iff iff w ∈ V (p); M, w ! ϕ; M, w ! ϕ and M, w ! ψ; ∃v ∈ W : wRa v and M, v ! ϕ; M, w ! ϕ and M|ϕ , w ! ψ, where M|ϕ = ⟨W|ϕ , {Ra|ϕ }a∈Agt , V|ϕ ⟩ is the model such that W|ϕ = {v ∈ W | M, v ! ϕ}; ∀a ∈ Agt: Ra|ϕ = Ra ∩ (W|ϕ × W|ϕ ); ∀p ∈ At: V|ϕ (p) = V (p) ∩ W|ϕ . We use the notation !ϕ"M = {v ∈ W | M, v ! ϕ}. For a class of models C, ThLκPAL (C) is the set of formulas of LκPAL that are valid over C. For the following statements, we use the standard nomenclature for normal modal logics, e.g., K, T, S4, and S5 for the unimodal logics and Kκ , Tκ , S4κ , and S5κ for their multimodal versions with |Agt| = κ (assume κ countable). Let Mod(Lκ ) be the class of all models of the logic Lκ , so Mod(Kκ ) is the class of all models, Mod(Tκ ) is the class of models with reflexive Ra relations, etc. We write Lκ for the Hilbert-style deductive system whose set of theorems is Lκ , and for any deductive system S, we write ⊢S ϕ when ϕ is a theorem of S. Theorem 1.4 (PAL Axiomatization [28]) Let PAL-Lκ be the system extending Lκ with the following rule and axioms: 2 i. (replacement) ψ↔χ ϕ(ψ/p) ↔ ϕ(χ/p) ii. (atomic reduction) ⟨ϕ⟩p ↔ (ϕ ∧ p) iii. (negation reduction) ⟨ϕ⟩¬ψ ↔ (ϕ ∧ ¬⟨ϕ⟩ψ) iv. (conjunction reduction) ⟨ϕ⟩(ψ ∧ χ) ↔ (⟨ϕ⟩ψ ∧ ⟨ϕ⟩χ) v. (diamond reduction) ⟨ϕ⟩✸a ψ ↔ (ϕ ∧ ✸a ⟨ϕ⟩ψ). For all ϕ ∈ LκPAL , ⊢PAL-Kκ ϕ iff ϕ ∈ ThLκPAL (Mod(Kκ )). The same result holds for Tκ /Tκ , S4κ /S4κ , and S5κ /S5κ in place of Kκ /Kκ . 2 If L contains the rule of uniform substitution, then we must either restrict this rule so that κ in PAL-Lκ we can only substitute into formulas ϕ with An(ϕ) = ∅, or remove the rule and add for each axiom of Lκ all substitution instances of that axiom with formulas in Lκ PAL . Either way, we take the rules of modus ponens and ✷a -necessitation from Lκ to apply in PAL-Lκ to all formulas. Finally, for ϕ, ψ ∈ Lκ PAL and p ∈ At(ϕ), ϕ(ψ/p) is the formula obtained by replacing all occurrences of p in ϕ by ψ. For alternative axiomatizations of PAL, see [32,33]. 4 A Uniform Logic of Information Dynamics Although we have taken diamond operators as primitive for convenience in later sections, typically the PAL axiomatization is stated in terms of box operators by replacing axiom schemas ii - v by the following: [ϕ]p ↔ (ϕ → p); [ϕ]¬ψ ↔ (ϕ → ¬[ϕ]ψ); [ϕ](ψ ∧ χ) ↔ ([ϕ]ψ ∧ [ϕ]χ); [ϕ]✷a ψ ↔ (ϕ → ✷a [ϕ]ψ). 1.2 Introduction to UPAL As noted above, one of the striking features of PAL is that it is not closed under uniform substitution. In the terminology of Goldblatt [20], PAL is not a uniform modal logic. For example, the valid atomic reduction axiom has invalid substitution instances, e.g., ⟨p⟩✷a p ↔ (p∧✷a p). Given this observation, a distinction arises between PAL and its substitution core, defined as follows. Definition 1.5 A substitution is any σ : At → LκPAL ; and (·)σ : LκPAL → LκPAL is the extension such that (ϕ)σ is obtained from ϕ by replacing each p ∈ At(ϕ) by σ(p) [14, Def. 1.18]. The substitution core of ThLκPAL (C) is the set {ϕ ∈ LκPAL : (ϕ)σ ∈ ThLκPAL (C) for all substitutions σ}. Formulas in the substitution core of ThLκPAL (C) are schematically valid over C. Examples of formulas that are in ThLκPAL (Mod(Kκ )) but are not in the substitution core of ThLκPAL (Mod(Kκ )) include the following (for κ ≥ 1): 3 [p]p [p]✷a p [p](p → ✷a p) [p ∧ ¬✷a p]¬(p ∧ ¬✷a p) ✷a p → [p]✷a p ✷a p → [p](p → ✷a p) ✷a (p → q) → (⟨q⟩✷a r → ⟨p⟩✷a r) (⟨p⟩✷a r ∧ ⟨q⟩✷a r) → ⟨p ∨ q⟩✷a r. We discuss the epistemic significance of such failures of uniformity in [23]. Burgess [15] explains the logical significance of uniformity as follows: The standard aim of logicians at least from Russell onward has been to characterize the class [of] all formulas all of whose instantiations are true. Thus, though Russell was a logical atomist, when he endorsed p ∨ ∼ p as [a] law of logic, he did not mean to be committing himself only to the view that the disjunction of any logically atomic statement with its negation is true, but rather to be committing himself to the view that the disjunction of any statement whatsoever with its negation is true . . . . This has remained the standard employment of statement letters ever since, not only among Russell’s successors in the classical tradition, but also among the great majority of formal logicians who have thought classical logic to be in need of additions and/or amendments, including C. I. Lewis, the founder of modern modal logic. With such an understanding of the role of statement letters, it is clear that if A is a law of logic, and B is any substitution in A, then B also is a law of logic . . . . Thus it is that the rule of substitution applies not 3 The first two principles in the second column are schematically valid over transitive singleagent models, but not over all single-agent models or over transitive multi-agent models. Holliday, Hoshi, and Icard 5 only in classical logic, but in standard, Lewis-style modal logics (as well as in intuitionistic, temporal, relevance, quantum, and other logics). None of this is meant to deny that there may be circumstances where it is legitimate to adopt some other understanding of the role of statement letters. If one does so, however, it is indispensable to note the conceptual distinction, and highly advisable to make a notational and terminological distinction. (147-148) In PAL, an atomic sentence p has the same truth value at any pointed models M, w and M|ϕ , w, whereas a formula containing a modal operator may have different truth values at M, w and M|ϕ , w, which is why uniform substitution does not preserve PAL-validity. Hence in PAL an atomic sentence cannot be thought of as a propositional variable in the ordinary sense of something that stands in for any proposition. By contrast, if we consider the substitution core of PAL as a logic in its own right, for which semantics will be given in §2, then we can think of the atomic sentences as genuine propositional variables. The distinction between PAL and its substitution core leads to Question 1 in van Benthem’s list of “Open Problems in Logical Dynamics” [3]: Question 1 ([2,3,4]) Is the substitution core of PAL axiomatizable? To answer this question, we will introduce a new framework of Uniform Public Announcement Logic (UPAL), which we use to prove the following result. Theorem 1.6 (Axiomatization of the PAL Substitution Core) Let UPAL-Lκ be the system extending Lκ with the following rules and axioms: 4 ϕ for any substitution σ (ϕ)σ ϕ [p]ϕ 1. (uniformity) 2. (necessitation) 3. (extensionality) 4. (distribution) [p](q → r) → ([p]q → [p]r) 5. (p-seriality) p → ⟨p⟩⊤ 6. (truthfulness) ⟨p⟩⊤ → p 7. (⊤-reflexivity) p → ⟨⊤⟩p 8. (functionality) ⟨p⟩q → [p]q 9. (pa-commutativity) ⟨p⟩✸a q → ✸a ⟨p⟩q 10. (ap-commutativity) ✸a ⟨p⟩q → [p]✸a q 11. (composition) ⟨p⟩⟨q⟩r ↔ ⟨⟨p⟩q⟩r. ϕ↔ψ ⟨ϕ⟩p ↔ ⟨ψ⟩p 4 As in PAL-L , in UPAL-L we take the rules of modus ponens and ✷ -necessitation from L κ κ a κ to apply to all formulas in Lκ PAL . 6 A Uniform Logic of Information Dynamics For all ϕ ∈ Lω PAL , ⊢UPAL-Kω ϕ iff ϕ is in the substitution core of ThLωPAL (Mod(Kω )). The same result holds for Tω /Tω , S4ω /S4ω , and S5ω /S5ω in place of Kω /Kω , with only minor adjustments to the proof (see note 5). Theorem 1.7 (Axiomatization of the PAL Substitution Core cont.) 1. ⊢UPAL-Tω ϕ iff ϕ is in the substitution core of ThLωPAL (Mod(Tω )); 2. ⊢UPAL-S4ω ϕ iff ϕ is in the substitution core of ThLωPAL (Mod(S4ω )); 3. ⊢UPAL-S5ω ϕ iff ϕ is in the substitution core of ThLωPAL (Mod(S5ω )). Unless the specific base system Lκ matters, we simply write ‘UPAL’ and ‘PAL’. It is easy to check that all the axioms of PAL except atomic reduction are derivable in UPAL, and the rule of replacement is an admissible rule in UPAL. Another system with the same theorems as UPAL, but presented in a format closer to that of the typical box version of PAL, is the following (with ⊥ := ¬⊤): I. (uniformity) ϕ for any substitution σ (ϕ)σ II. (RE) ϕ↔ψ [p]ϕ ↔ [p]ψ III. ([ ]-extensionality) ϕ↔ψ [ϕ]p ↔ [ψ]p IV. (N) [p]⊤ V. (⊤-reflexivity) [⊤]p → p VI. (⊥-reduction) [p]⊥ ↔ ¬p VII. (¬-reduction) [p]¬q ↔ (p → ¬[p]q) VIII. (∧-reduction) [p](q ∧ r) ↔ ([p]q ∧ [p]r) IX. (✷a -reduction) [p]✷a q ↔ (p → ✷a [p]q) X. ([ ]-composition) [p][q]r ↔ [p ∧ [p]q]r. We have formulated UPAL as in Theorem 1.6 to make clear the correspondence between axioms and the semantic conditions in Definition 2.3 below, as well as to make clear the specific properties used in the steps of our main proof. 2 Semantics for UPAL In this section we introduce semantics for Uniform Public Announcement Logic, for which the system of UPAL is shown to be sound and complete in §3. Definition 2.1 Models for UPAL are tuples M of the form ⟨M, {Ra }a∈Agt , {Rϕ }ϕ∈LκPAL , V⟩, where M is a non-empty set, Ra and Rϕ are binary relations on M , and V : At → P(M ). Holliday, Hoshi, and Icard 7 Unlike in the PAL truth definition, in the UPAL truth definition we treat ⟨ϕ⟩ like any other modal operator. Definition 2.2 Given a UPAL model M = ⟨M, {Ra }a∈Agt , {Rϕ }ϕ∈LκPAL , V⟩ with w ∈ M , ϕ, ψ ∈ LκPAL , and p ∈ At, we define M, w " ϕ as follows: M, w M, w M, w M, w M, w M, w " ⊤; "p " ¬ϕ "ϕ∧ψ " ✸a ϕ " ⟨ϕ⟩ψ iff iff iff iff iff w ∈ V(p); M, w " ϕ; M, w " ϕ and M, w " ψ; ∃v ∈ M : wRa v and M, v " ϕ; ∃v ∈ M : wRϕ v and M, v " ψ. We use the notation ∥ϕ∥M = {v ∈ M | M, v " ϕ}. Instead of giving the ⟨ϕ⟩ operators a special truth clause, we ensure that they behave in a PAL-like way by imposing constraints on the Rϕ relations in Definition 2.3 below. Wang and Cao [33] have independently proposed a semantics for PAL in this style, with respect to which they prove that PAL is complete. The difference comes in the specific constraints for UPAL vs. PAL. Definition 2.3 A UPAL model M = ⟨M, {Ra }a∈Agt , {Rϕ }ϕ∈LκPAL , V⟩ is legal iff the following conditions hold for all ψ, χ ∈ LκPAL , w, v ∈ M , and a ∈ Agt: (extensionality) if ∥ψ∥M = ∥χ∥M , then Rψ = Rχ ; (ψ-seriality) if w ∈ ∥ψ∥M , then ∃v: wRψ v; (truthfulness) if wRψ v, then w ∈ ∥ψ∥M ; (⊤-reflexivity) wR⊤ w; (functionality) if wRψ v, then for all u ∈ M , wRψ u implies u = v; (ψa-commutativity) if wRψ v and vRa u, then ∃z: wRa z and zRψ u; (aψ-commutativity) if wRa v, vRψ u and w ∈ ∥ψ∥M , then ∃z: wRψ z and zRa u; (composition) R⟨ψ⟩χ = Rψ ◦ Rχ . In §4, we will also refer to weaker versions of the first and third conditions: (extensionality for ϕ) if ψ, χ ∈ An(ϕ) ∪ {⊤} and ∥ψ∥M = ∥χ∥M , then Rψ = Rχ ; (truthfulness for ϕ) if ψ ∈ An(ϕ) ∪ {⊤} and wRψ v, then w ∈ ∥ψ∥M . It is easy to see that each of the axioms of UPAL in Theorem 1.6 corresponds to the condition of the same name written in boldface in Definition 2.3. 3 Completeness of UPAL In this section, we take our first step toward proving Theorem 1.6 by proving: 8 A Uniform Logic of Information Dynamics Theorem 3.1 (Soundness and Completeness) The system of UPAL-Kω given in Theorem 1.6 is sound and complete for the class of legal UPAL models. Soundness is straightforward. To prove completeness, we use the standard canonical model argument. Definition 3.2 The canonical model Mc = ⟨M c , {Rca }a∈Agt , {Rcϕ }ϕ∈LκPAL , V c ⟩ is defined as follows: 1. M c = {Γ | Γ is a maximally UPAL-Kω -consistent set}; 2. ΓRca ∆ iff ψ ∈ ∆ implies ✸a ψ ∈ Γ; 3. ΓRcϕ ∆ iff ψ ∈ ∆ implies ⟨ϕ⟩ψ ∈ Γ; 4. V c (p) = {Γ ∈ M c | p ∈ Γ}. The following fact, easily shown, will be used in the proof of Lemma 3.5. Fact 3.3 For all Γ ∈ M c , ϕ ∈ LκPAL , if ⟨ϕ⟩⊤ ∈ Γ, then {ψ | ⟨ϕ⟩ψ ∈ Γ} ∈ M c . The proof of the truth lemma is completely standard [14, §4.2]. Lemma 3.4 (Truth) For all Γ ∈ M c and ϕ ∈ LκPAL , Mc , Γ " ϕ iff ϕ ∈ Γ. To complete the proof of Theorem 3.1, we need only check the following. Lemma 3.5 (Legality) Mc is a legal model. c c Proof. Suppose ∥ϕ∥M = ∥ψ∥M , so by Lemma 3.4 and the properties of maximally consistent sets, ϕ ↔ ψ ∈ Γ for all Γ ∈ M c . Hence ⊢UPAL-Kω ϕ ↔ ψ, for if ¬(ϕ ↔ ψ) is UPAL-Kω -consistent, then ¬(ϕ ↔ ψ) ∈ ∆ for some ∆ ∈ M c , contrary to what was just shown. It follows that for any α ∈ LκPAL , ⊢UPAL-Kω ⟨ϕ⟩α ↔ ⟨ψ⟩α, given the extensionality and uniformity rules of UPALKω . Hence if Γ1 Rcϕ Γ2 , then for all α ∈ Γ2 , ⟨ϕ⟩α ∈ Γ1 and ⟨ψ⟩α ∈ Γ1 by the consistency of Γ1 , which means Γ1 Rcψ Γ2 . The argument in the other direction is the same, whence Rcϕ = Rcψ . Mc satisfies extensionality. Suppose Γ1 Rc⟨ϕ⟩ψ Γ2 , so for all α ∈ Γ2 , ⟨⟨ϕ⟩ψ⟩α ∈ Γ1 . Hence ⟨ϕ⟩⟨ψ⟩α ∈ Γ1 given the composition axiom and uniformity rule of UPAL-Kω , so ⟨ϕ⟩⊤ ∈ Γ1 by normal modal reasoning with the distribution axiom. It follows by Fact 3.3 and Definition 3.2.3 that there is some Σ1 such that Γ1 Rϕ Σ1 and ⟨ψ⟩α ∈ Σ1 , and by similar reasoning that there is some Σ2 such that Σ1 Rψ Σ2 and α ∈ Σ2 . Hence Γ2 ⊆ Σ2 , so Γ2 = Σ2 given that Γ2 is maximal. Therefore, Rc⟨ϕ⟩ψ ⊆ Rcϕ ◦ Rcψ . The argument in the other direction is similar. Mc satisfies composition. We leave the other legality conditions to the reader. ✷ 4 Bridging UPAL and PAL In this section, we show that UPAL axiomatizes the substitution core of PAL. It is easy to check that all of the axioms of UPAL are PAL schematic validities, and all of the rules of UPAL preserve schematic validity, so UPAL derives only PAL schematic validities. To prove that UPAL derives all PAL schematic validities, Holliday, Hoshi, and Icard 9 we show that if ϕ is not derivable from UPAL, so by Theorem 3.1 there is a legal UPAL model falsifying ϕ, then there is a substitution τ and a PAL model falsifying (ϕ)τ , in which case ϕ is not schematically valid over PAL models. Proposition 4.1 For any ϕ ∈ Lω PAL , if there is a legal UPAL model M = ⟨M, {Ra }a∈Agt , {Rψ }ψ∈LωPAL , V⟩ with w0 ∈ M such that M, w0 " ϕ, then there is a PAL model N = ⟨N0 , {Sa }a∈Agt , U ⟩ with w0 ∈ N0 and a substitution τ such that N , w0 ! (ϕ)τ . Our first step in proving Proposition 4.1 is to show that we can reduce ϕ to a certain simple form, which will help us in constructing the substitution τ . Definition 4.2 The set of simple formulas is generated by the grammar ϕ ::= ⊤ | p | ¬ϕ | ϕ ∧ ϕ | ✸a ϕ | ⟨ϕ⟩p, where p ∈ At and a ∈ Agt. Proposition 4.3 For every ϕ ∈ LκPAL , there is a simple formula ϕ′ ∈ LκPAL that is equivalent to ϕ over legal UPAL models (and all PAL models). Proof. The proof is similar to the standard PAL reduction argument [9, §7.4], only we do not perform atomic reduction steps, and we use the composition axiom of UPAL to eliminate consecutive occurrences of dynamic operators. ✷ By Proposition 4.3, given that M is legal, we can assume that ϕ is simple. Before constructing N and τ , we show that our initial model M can be transformed into an intermediate model N that satisfies a property (part 2 of Lemma 4.4) that we will take advantage of in our proofs below. We will return to the role of this property in relating UPAL to PAL in Example 5.2 and §6. For what follows, we need some new notation. First, let ! Ra ; RAgt = a∈Agt R∗ is the reflexive transitive closure of R; and R(w) = {v ∈ M | wRv}. Lemma 4.4 For any legal model M = ⟨M, {Ra }a∈Agt , {Rϕ }ϕ∈LωPAL , V⟩ with w0 ∈ M such that M, w0 " ϕ, there is a model N = ⟨N, {Sa }a∈Agt , {Sϕ }ϕ∈LωPAL , U⟩ with w0 ∈ N such that 1. N, w0 " ϕ; 2. if α, β ∈ An(ϕ) ∪ {⊤} and ∥α∥N ̸= ∥β∥N , then ∗ ∗ ∥α∥N ∩ SAgt (w0 ) ̸= ∥β∥N ∩ SAgt (w0 ). 3. N satisfies ⊤-reflexivity, functionality, extensionality for ϕ and truthfulness for ϕ. Proof. Consider some α, β ∈ An(ϕ) ∪ {⊤} such that ∥α∥M ̸= ∥β∥M . Hence there is some v ∈ M such that M, v " α ↔ β. Let M′ be exactly like M 10 A Uniform Logic of Information Dynamics except that for some x ̸∈ Agt(ϕ), w0 R′x v. 5 Then it is easy to show that for all ψ ∈ Sub(ϕ) and u ∈ M , M′ , u " ψ iff M, u " ψ. Hence M′ , w0 " ϕ and M′ , v " α ↔ β. Then given w0 R′x v, we have ′ ′ M ∥α∥M ∩ R′∗ ∩ R′∗ Agt (w0 ). Agt (w0 ) ̸= ∥β∥ Finally, one can check that M′ satisfies ⊤-reflexivity, functionality, extensionality for ϕ and truthfulness for ϕ by the construction. By repeating this procedure, starting now with M′ , for each of the finitely many α and β as described above, one obtains a model N as described in Lemma 4.4 ✷ Obtaining N from M as in Lemma 4.4, we now define our PAL model ∗ N = ⟨N0 , {Sa }a∈Agt , U ⟩. Let N0 = SAgt (w0 ); for some z ̸∈ Agt(ϕ), let Sz be the universal relation on N0 ; and for each a ∈ Agt with a ̸= z, let Sa be the restriction of Sa to N0 . We will define the valuation U after constructing the substitution τ . The following facts will be used in the proof of Lemma 4.8. Fact 4.5 1. For all a ∈ Agt and w ∈ N0 , Sa (w) = Sa (w). 2. if ∥α∥N ∩ N0 = ∥β∥N ∩ N0 , then for all u ∈ N0 , N, u " ⟨α⟩χ iff N, u " ⟨β⟩χ. Proof. Part 1 is obvious. For part 2, if ∥α∥N ∩ N0 = ∥β∥N ∩ N0 , then ∥α∥N = ∥β∥N by Lemma 4.4.2, so Sα = Sβ by Lemma 4.4.3 (extensionality for ϕ).✷ Remark 4.6 There is another way of transforming the UPAL model M = ⟨M, {Ra }a∈Agt , {Rϕ }ϕ∈LωPAL , V⟩ into a PAL model N sufficient for our purposes. First, let N = ⟨N, {Sa }a∈Agt , {Sϕ }ϕ∈LωPAL , U⟩ be exactly like M except that for some z ̸∈ Agt(ϕ), Sz is the universal relation on N , and observe that N satisfies the conditions of Lemma 4.4. Second, take N = ⟨N0 , {Sa }a∈Agt , U ⟩ such that N0 = N , Sa = Sa , and U is defined as below, and observe that Fact 4.5 holds. Then the proof can proceed as below. The difference is that this approach takes the domain of the PAL model to be the entire domain of the UPAL model N, with Sz as the universal relation on this entire domain, whereas our approach takes the domain of the PAL model to be just that of the “epistemic submodel” ∗ generated by w0 in N, SAgt (w0 ), with Sz as the universal relation on this set. We prefer the latter approach because it allows us to work with smaller PAL models when we carry out the construction with concrete examples as in §5. 5 As noted after Theorem 1.6, we can modify our proof for other models classes. For example, for the class of models with equivalence relations, in this step we can define R′x to be the smallest equivalence relation extending Rx such that w0 R′x v. Note that since α, β ∈ An(ϕ) ∪ {⊤} and x ̸∈ Agt(ϕ), no matter how we define R′x , the following claim in the text still holds. Holliday, Hoshi, and Icard 11 To construct τ (p) for p ∈ At(ϕ), let B1 , . . . , Bm be the sequence of all Bi such that ⟨Bi ⟩p ∈ Sub(ϕ), and let B0 := ⊤. For 0 ≤ i, j ≤ m, if ∥Bi ∥N ∩ N0 = ∥Bj ∥N ∩ N0 , delete one of Bi or Bj from the list (but never B0 ), until there is no such pair. Call the resulting sequence A0 , . . . , An , and define s(i) = {j | 0 ≤ j ≤ n and ∥Aj ∥N ∩ N0 # ∥Ai ∥N ∩ N0 }. Extend the language with new variables p0 , . . . , pn and a0 , . . . , an , and define τ (p) = γ0 ∧ · · · ∧ γn such that " ¬✷z aj ) → pi . γi := (✷z ai ∧ j∈s(i) Having extended the language for each p ∈ At(ϕ), define the valuation U for N0 such that for each p ∈ At(ϕ), U (p) = U (p) ∩ N0 , and for the new variables: (a) U (pi ) = {w ∈ N0 | ∃u: wSAi u and u ∈ U(p)}; (b) U (ai ) = ∥Ai ∥N ∩ N0 . Hence: (a) !pi "N = {w ∈ N0 | ∃u: wSAi u and u ∈ U(p)}; (b) !ai "N = ∥Ai ∥N ∩ N0 . Note that it follows from (a) and the UPAL truth definition that (c) !pi "N = ∥⟨Ai ⟩p∥N ∩ N0 . Using these facts, we will show that N, w0 " ϕ implies N , w0 ! τ (ϕ). Lemma 4.7 For all 0 ≤ i ≤ n, !τ (p)"N|ai = !pi "N . Proof. We first show that for 0 ≤ i, j ≤ n, i ̸= j: (i) !γi "N|ai = !pi "N|ai ; (ii) !γj "N|ai = !ai "N|ai (= N0|ai ). For (i), we claim that !✷z ai ∧ " ¬✷z ak "N|ai = N0|ai . k∈s(i) Since ai is atomic, !✷z ai "N|ai = N0|ai . By definition of the s function and (b), for all k ∈ s(i), !ak "N # !ai "N , so !¬✷z ak "N|ai = N0|ai . Hence the claimed equation holds, so !γi "N|ai = !pi "N|ai given the structure of γi . For (ii), we claim that for j ̸= i, " ¬✷z ak "N|ai = ∅. !✷z aj ∧ k∈s(j) 12 A Uniform Logic of Information Dynamics By construction of the sequence A0 , . . . , An for p and (b), !aj "N ̸= !ai "N . Hence if not !ai "N # !aj "N , then !ai "N ̸⊆ !aj "N , so !✷z aj "N|ai = ∅ because Sz is the universal relation on N0 . If !ai "N # !aj "N , then by (b) and the definition of s, i ∈ s(j); since ai is atomic, !¬✷z ai "N|ai = ∅. In either case the claimed equation holds, so !γj "N|ai = N0|ai given the structure of γj . Given the construction of τ , (i) and (ii) imply: # !τ (p)"N|ai = !γi "N|ai ∩ !γj "N|ai = !pi "N|ai ∩ !ai "N|ai = !pi "N , j̸=i where the last equality holds because !pi "N ⊆ !ai "N , which follows from (a), (b), and the fact that N satisfies truthfulness for ϕ. ✷ We now establish the connection between the UPAL model N on the one hand and the PAL model N and substitution τ on the other. Lemma 4.8 For all simple subformulas χ of ϕ, !(χ)τ "N = ∥χ∥N ∩ N0 . Proof. By induction on χ. For the base case, we must show !(p)τ "N = ∥p∥N ∩ N0 for p ∈ At(ϕ). By construction of the sequence A0 , . . . , An for p, A0 = ⊤, so ∥A0 ∥N ∩N0 = N0 . Then by (b), !a0 "N = N0 , and hence !(p)τ "N|a0 !p0 "N by Lemma 4.7 {w ∈ N0 | ∃u : wSA0 u and u ∈ U(p)} by (a) {w ∈ N0 | w ∈ U(p)} by ⊤-reflexivity and functionality = ∥p∥N ∩ N0 . !(p)τ "N = = = = The boolean cases are straightforward. Next, we must show !(✷a ϕ)τ "N = ∥✷a ϕ∥N ∩ N0 . For the inductive hypothesis, !(ϕ)τ "N = ∥ϕ∥N ∩ N0 , so !(✷a ϕ)τ "N = = = = = = !✷a (ϕ)τ "N {w ∈ N0 | Sa (w) ⊆ !(ϕ)τ "N } {w ∈ N0 | Sa (w) ⊆ ∥ϕ∥N ∩ N0 } {w ∈ N0 | Sa (w) ⊆ ∥ϕ∥N } given Sa ⊆ N0 × N0 {w ∈ N0 | Sa (w) ⊆ ∥ϕ∥N } by Fact 4.5.1 N ∥✷a ϕ∥ ∩ N0 . Finally, we must show !(⟨Bi ⟩p)τ "N = ∥⟨Bi ⟩p∥N ∩ N0 . For the inductive hypothesis, !(Bi )τ "N = ∥Bi ∥N ∩ N0 . By construction of the sequence A0 , . . . , An for p ∈ At(ϕ), there is some Aj such that (⋆) Therefore, ∥Bi ∥N ∩ N0 = ∥Aj ∥N ∩ N0 . !(Bi )τ "N = ∥Aj ∥N ∩ N0 = !aj "N by (b), Holliday, Hoshi, and Icard 13 and hence !(⟨Bi ⟩p)τ "N = = = = = = !⟨(Bi )τ ⟩(p)τ "N !⟨aj ⟩(p)τ "N !(p)τ "N|aj !pj "N ∥⟨Aj ⟩p∥N ∩ N0 ∥⟨Bi ⟩p∥N ∩ N0 by Lemma 4.7 by (c) given (⋆) and Fact 4.5.2. The proof by induction is complete. ✷ With the following fact, we complete the proof of Proposition 4.1. Fact 4.9 N , w0 ! (ϕ)τ . Proof. Immediate from Lemma 4.8 given N, w0 " ϕ. 5 ✷ Examples In this section, we work out two examples illustrating how the techniques of §4 allow us to find, for any formula ϕ that is valid but not schematically valid in PAL, a PAL model that falsifies a substitution instance of ϕ. The proof in §4 shows that all we need to do is find a legal UPAL model falsifying ϕ. However, since legal UPAL models are generally large, we would like to instead find a small UPAL model falsifying ϕ, from which we can read off a PAL model that falsifies a substitution instance of ϕ. In fact, we can always do so provided that the model satisfies a weaker condition than legality. For a given ϕ ∈ LκPAL , we say that a UPAL model M is ϕ-legal iff it satisfies all of the legality conditions of Definition 2.3 when we replace ψ-seriality with: (ψ-seriality for ϕ) if ψ ∈ An(ϕ) ∪ {⊤} and w ∈ ∥ψ∥M , then ∃v: wRψ v. Hence in a ϕ-legal model, we can let all of the infinitely many Rψ relations irrelevant to ϕ be empty, which makes constructing ϕ-legal models easier. With this new notion, we can state a simple method for finding a PAL model that falsifies a substitution instance of the non-schematically valid ϕ: Step 1. Transform ϕ into an equivalent simple formula ϕ′ . Step 2. ϕ′ . Find a ϕ′ -legal pointed UPAL model M, w0 such that M, w0 " Step 3. Obtain N and τ from M, w0 as in §4 so that N , w0 ! (ϕ′ )τ . Since ϕ ↔ ϕ′ is schematically valid in PAL, we have N , w ! (ϕ)τ , as desired. The key to this method is that the construction in §4 also establishes the following variant of Proposition 4.1: Proposition 5.1 For any simple ϕ ∈ Lω PAL , if there is a ϕ-legal UPAL model M = ⟨M, {Ra }a∈Agt , {Rψ }ψ∈LωPAL , V⟩ with w0 ∈ M such that M, w0 " ϕ, then there is a PAL model N = ⟨N0 , {Sa }a∈Agt , U ⟩ with w0 ∈ N0 and a substitution τ such that N , w0 ! (ϕ)τ . 14 A Uniform Logic of Information Dynamics This proposition holds because if ϕ is already simple, then the only properties of M used in the proof of Fact 4.9 are ⊤-reflexivity, functionality, extensionality for ϕ and truthfulness for ϕ, which are part of ϕ-legality. Finally, if ϕ does not contain any occurrence of a dynamic operator in the scope of any other, then we can simply skip Step 1 and do Steps 2 and 3 for ϕ itself. One can check that the construction in §4 works not only with a simple formula, but more generally with any formula with the scope restriction. Example 5.2 Consider the PAL-valid formula ϕ := [p]p, which is already simple. Let us try to falsify ϕ in a ϕ-legal UPAL model. The obvious first try is M in Fig. 1, which is indeed a ϕ-legal UPAL model, in which all Ra relations are empty. (We simplify the diagrams by omitting all reflexive R⊤ loops.) However, M has an un-PAL-like property: although ∥⊤∥M ∩ R∗Agt (w0 ) = ∥p∥M ∩ R∗Agt (w0 ), we have w0 R⊤ w0 but not w0 Rp w0 . (See §6 for why this is un-PAL-like.) To eliminate this property, we modify M to N = ⟨N, {Sa }a∈Agt , {Sψ }ψ∈LωPAL , U⟩ in Fig. 1 as in Lemma 4.4. 6 Next, following the procedure in §4, we obtain the PAL model N = ⟨N0 , {Sa }a∈Agt , U ⟩ in Fig. 1 and the substitution τ given below. x0 w0 x0 p N|τ (p) p0 , a 0 , a 1 Rp w0 p M Sp w0 p Sz p0 , a 0 , a1 w1 w0 p N Sz a0 w1 N Fig. 1. UPAL and PAL Models for Example 5.2 Where A0 := ⊤, A1 := p, and a0 , a1 , p0 , and p1 are the new atoms, we define the valuation U in N such that: U (a0 ) = ∥A0 ∥N ∩ N0 = {w0 , w1 }; U (a1 ) = ∥A1 ∥N ∩ N0 = {w0 }; U (p0 ) = {w ∈ N0 | ∃u: wSA0 u and u ∈ U(p)} = {w0 }; U (p1 ) = {w ∈ N0 | ∃u: wSA1 u and u ∈ U(p)} = ∅. Defining the function s such that s(i) = {j | 0 ≤ j ≤ n and ∥Aj ∥N ∩ N0 # ∥Ai ∥N ∩ N0 }, 6 In fact, the construction of Lemma 4.4 would connect w to x by R , but note that we z 0 0 can always connect w0 to a new point falsifying α ↔ β (in this case, ⊤ ↔ p) instead. Holliday, Hoshi, and Icard 15 we have s(0) = {1} and s(1) = ∅. Defining τ (p) = γ0 ∧ · · · ∧ γn such that " ¬✷z aj ) → pi , γi := (✷z ai ∧ j∈s(i) we have τ (p) = ((✷z a0 ∧ ¬✷z a1 ) → p0 ) ∧ (✷z a1 → p1 ). Observe: !(✷z a0 ∧ ¬✷z a1 ) → p0 "N = {w0 }; !✷z a1 → p1 "N = {w0 , w1 }; !τ (p)"N = {w0 }. Hence N|τ (p) is the model displayed in the upper-right in Fig. 1. Observe: !(✷z a0 ∧ ¬✷z a1 ) → p0 "N|τ (p) = {w0 }; !✷z a1 → p1 "N|τ (p) = ∅; !τ (p)"N|τ (p) = ∅. Hence N , w0 ! ([p]p)τ , so our starting formula ϕ is not schematically valid over PAL models. Example 5.3 Consider the PAL-valid formula ϕ := [p ∧ ¬✷b p]¬(p ∧ ¬✷b p). 7 Let us try to falsify ϕ in a ϕ-legal UPAL model. The obvious first try is the model A in Fig. 2. However, A is not ϕ-legal, since it violates ψb-commutativity for ψ := p ∧ ¬✷b p. By modifying A to N = ⟨N, {Sa }a∈Agt , {Sψ }ψ∈LωPAL , U⟩ in Fig. 2, we obtain a ϕ-legal UPAL model with N, w0 " ϕ. (In this case, the transformation of Lemma 4.4 is uncecessary, since the condition of Lemma 4.4.2 is already satisfied by N.) Following the procedure of §4, we obtain the PAL model N = ⟨N0 , {Sa }a∈Agt , U ⟩ in Fig. 3 and the substitution τ given below. Where A0 := ⊤, A1 := p ∧ ¬✷b p, and a0 , a1 , p0 , and p1 are the new atoms, we define the valuation U in N such that: U (a0 ) = ∥A0 ∥N ∩ N0 = {w0 , w1 , w2 }; U (a1 ) = ∥A1 ∥N ∩ N0 = {w0 , w1 }; U (p0 ) = {w ∈ N0 | ∃u: wSA0 u and u ∈ U(p)} = {w0 , w1 }; U (p1 ) = {w ∈ N0 | ∃u: wSA1 u and u ∈ U(p)} = {w0 }. Defining the function s as before, we have s(0) = {1} and s(1) = ∅. Since this is the same s as in Example 5.2, the substitution is also the same: τ (p) = ((✷z a0 ∧ ¬✷z a1 ) → p0 ) ∧ (✷z a1 → p1 ). 7 Here we could transform ϕ := [p ∧ ¬✷b p]¬(p ∧ ¬✷b p) into the simple ϕ′ := (p ∧ ¬✷b p) → ¬([p ∧ ¬✷b p]p ∧ ((p ∧ ¬✷b p) → ¬((p ∧ ¬✷b p) → ✷b [p ∧ ¬✷b p]p))), but as noted before Example 5.2, if ϕ does not contain any occurrence of a dynamic operator in the scope of any other, then we can skip Step 1 and do Steps 2 and 3 for ϕ itself. 16 A Uniform Logic of Information Dynamics x0 p x1 x0 p x1 Sp∧¬✷b p p w1 Rp∧¬✷b p w0 p Rb w1 w0 Sb p A w2 N Fig. 2. UPAL Models for Example 5.3 Note that since the construction of N from N is such that Sz = Sb , we can simply take ✷z to be ✷b in τ (p), so that Agt((ϕ)τ ) = Agt(ϕ) = {b}. p0 , a 0 , a 1 p0 , a 0 , a 1 p w1 p w1 Sz = Sb w0 p Sz = Sb p0 , p1 , a0 , a1 w2 a0 w0 p p0 , p1 , a0 , a1 N N|(p∧¬✷b p)τ Fig. 3. PAL Models for Example 5.3 Observe: !(✷z a0 ∧ ¬✷z a1 ) → p0 "N = {w0 , w1 }; !✷z a1 → p1 "N = {w0 , w1 , w2 }; !τ (p)"N = {w0 , w1 }; !τ (p) ∧ ¬✷b τ (p)"N = {w0 , w1 }. Hence N|(p∧¬✷b p)τ is the model displayed on the right in Fig. 3. Observe: !(✷z a0 ∧ ¬✷z a1 ) → p0 "N|(p∧¬✷b p)τ = {w0 , w1 }; !✷z a1 → p1 "N|(p∧¬✷b p)τ = {w0 }; Holliday, Hoshi, and Icard 17 !τ (p)"N|(p∧¬✷b p)τ = {w0 }; !τ (p) ∧ ¬✷b τ (p)"N|(p∧¬✷b p)τ = {w0 }. Hence N , w0 ! ([p ∧ ¬✷b p]¬(p ∧ ¬✷b p))τ , so our starting formula ϕ is not schematically valid over PAL models. We invite the reader to work out other examples using UPAL, starting from the other valid but not schematically valid PAL principles mentioned in §1.2. 6 Discussion In this paper, we have shown that UPAL axiomatizes the substitution core of PAL with infinitely many agents. In this final section, we briefly discuss the axiomatization question for the single-agent and finite-agent cases. For a given language and class of models, the key question is how close we can come to expressing that two formulas are co-extensional in the epistemic submodel generated by the current point. For example, this condition is + expressed by the formula ✷+ a (ϕ ↔ ψ) (where ✷a α := α ∧ ✷a α) in single-agent PAL over transitive models. In this case, we get a new schematic validity in PAL: (inner extensionality) ✷+ a (ϕ ↔ ψ) → (⟨ϕ⟩α ↔ ⟨ψ⟩α). The corresponding legality condition for UPAL models is: (inner extensionality) if ∥ϕ∥M ∩ Ra (w) = ∥ψ∥M ∩ Ra (w), then wRϕ v iff wRψ v, which does not follow from any of the other legality conditions. For multiple agents, we cannot in general express the co-extensionality of two formulas in the epistemic submodel generated by the current point; however, if we allow our models to be non-serial, then we do get related schematic validities for the single and finite-agent cases that are not derivable in UPAL-Kn (where the antecedent can be written using ✷a operators and ⊥): 8 (FPE) “all RAgt -paths from the current point are of length ≤ n” → (E n (ϕ ↔ ψ) → (⟨ϕ⟩α ↔ ⟨ψ⟩α)), where E 0 α := α ∧ " ✷a α and E n α := α ∧ E 0 E n−1 α. a∈Agt The corresponding legality condition for UPAL is: (FPE) 8 if R∗Agt (w) is path-finite and ∥ϕ∥M ∩ R∗Agt (w) = ∥ψ∥M ∩ R∗Agt (w), then wRϕ v iff wRψ v, The (FPE) axioms are also schematically valid over serial models, because the antecedent is always false, but then they are also derivable using the seriality axiom ✸a ⊤. 18 A Uniform Logic of Information Dynamics where R∗Agt (w) is path-finite just in case every RAgt -path from w ends in a dead-end point in finitely many steps. This shows why the axiomatization of the substitution core of PAL-Kω is more elegant than that of PAL-Kn : with infinitely many agents we cannot express the “everybody knows” modality E, so we do not need to add to UPAL the infinitely many FPE axioms. Finally, if we consider PAL with the standard common knowledge operator C, then we can express co-extensionality in the generated epistemic submodel using the formula C(ϕ ↔ ψ), in which case we get the new schematic validity (common extensionality) C(ϕ ↔ ψ) → (⟨ϕ⟩α ↔ ⟨ψ⟩α). The corresponding legality condition in UPAL is: (common extensionality) if ∥ϕ∥M ∩ R∗Agt (w) = ∥ψ∥M ∩ R∗Agt (w), then wRϕ v iff wRψ v. We leave it to future work to give analyses for the above languages analogous to the analysis we have given here for Lω PAL . A natural next step is to axiomatize the substitution core of the system of PAL-RC [6] with relativized common knowledge. Relativized common knowledge C(ϕ, ψ) is interpreted in UPAL models exactly as in PAL models. We conjecture that UPAL together with the relativized common knowledge reduction axiom ⟨p⟩C(q, r) ↔ C(⟨p⟩q, ⟨p⟩r), the common extensionality axiom above, and the appropriate base logic (see [6]) axiomatizes the substitution core of PAL-RC with finitely or infinitely many agents over any of the model classes we have discussed. Indeed, it can be shown using arguments similar to those of §4 that the set of formulas in the language LκPAL-RC that are valid over legal UPAL models with common extensionality is exactly the substitution core of PAL-RC. Hence it only remains to prove that the extended system just described—call it UPAL-RC—is sound and complete for this model class. Such a proof requires a finite canonical model construction to deal with common knowledge, and we cannot go into the details here. 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