{"id":"6abc27af-b558-4866-8103-e7bed79e59fa","arxiv_id":"2411.15775","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A base-extension semantics for public announcement logic is defined via updates to modal relations on bases, and it is proved sound and complete.","lead":"The paper defines a base-extension semantics for public announcement logic, in which a public announcement updates the relations between proof bases instead of deleting possible worlds. This is the first such semantics for a dynamic epistemic logic and it is tested on the classic card game and muddy children puzzles.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4's proof is circular: it derives ⊢ϕ↔t(ϕ) from ⊩ϕ↔t(ϕ) via Theorem 3, which only goes from provability to validity, so the completeness claim is unproven as written.","rationale":"The reader's verdict is CONDITIONAL, and I agree with that assessment, but I identify a different load-bearing concern than the reader's stated weakest assumption. The reader focused on Lemma 6, the construction of effective updates, which is indeed essential and only sketched. However, the proof of Theorem 4 contains a clear invalid inference: it uses Theorem 3 (soundness) in reverse to claim ⊢ϕ↔t(ϕ) from ⊩ϕ↔t(ϕ). This is a circular step in the proof of the paper's central completeness result. The gap is likely fixable because the PAL axioms are equivalences, so a syntactic proof of ⊢ϕ↔t(ϕ) should be routine, and the semantic Lemma 21 is then only needed to transfer validity of ϕ to validity of t(ϕ). But as written, the main theorem is unproven. This does not change the verdict: the paper is promising and the issues are probably repairable, so conditional acceptance remains appropriate. The reader's rationale did mention the Theorem 4 step, so our agreement is partial rather than full.","tokens_in":32300,"tokens_out":9672,"duration_ms":87010,"concrete_test":"Attempt an independent inductive proof of ⊢PAL ϕ↔t(ϕ) using only the axioms and rules of Definition 8 and the complexity measure of Definition 26. If the derivation succeeds, Theorem 4 can be repaired by replacing the invalid step with this syntactic equivalence, and the completeness claim stands. If the derivation fails (for example, because some PAL axiom is not derivable as an equivalence in the presence of necessitation), then the completeness theorem is false or requires a different argument. As a supplementary check, formalize the proof of Theorem 4 in a proof assistant; the step 'By Theorem 3 and Lemma 21, we get ⊢ϕ↔t(ϕ)' should be rejected by type-checking, confirming the circularity.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 4, the completeness of the base-extension semantics for PAL, contains an invalid step. It assumes ⊩ϕ and then states: 'By Theorem 3 and Lemma 21, we get ⊢ϕ↔t(ϕ).' This inference is not licensed. Theorem 3 is soundness: it permits passing from ⊢ψ to ⊩ψ, never from ⊩ψ to ⊢ψ. Lemma 21 gives the semantic equivalence ⊩ϕ↔t(ϕ), so the only valid conclusion is ⊩t(ϕ) (and, jointly with the assumption, ⊩ϕ). To obtain ⊢ϕ from ⊢t(ϕ), one needs the syntactic equivalence ⊢PAL ϕ↔t(ϕ), which the paper does not derive. It cannot be obtained from Theorem 3 and Lemma 21 without assuming the completeness that Theorem 4 is supposed to establish. The later appeal to S5 completeness for t(ϕ) is also insufficient unless ⊢PAL ϕ↔t(ϕ) is available. Thus the central completeness claim is not established by the argument as written. This is likely repairable: the PAL axioms in Definition 8 are equivalences, and an induction on the complexity measure of Definition 26 should yield ⊢PAL ϕ↔t(ϕ). But the text as it stands is circular, and the main theorem depends on this step. I focus on this rather than on Lemma 6 because the Theorem 4 gap is an unambiguous logical error in the central claim, whereas Lemma 6, though sketched, is at least an attempted construction.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a base-extension semantics (B-eS) for public announcement logic (PAL). Extending the authors' earlier B-eS for multi-agent S5, it evaluates formulas at bases (sets of atomic rules) connected by 'modal relations', and it interprets announcements [φ]ψ as updates of those modal relations rather than as deletions of worlds. Because restricting a modal relation to bases that agree on φ need not yield a modal relation, the authors introduce 'effective updates' (Definition 17) and allow the same announcement to produce several possible updated relation sets, so that [φ] is a genuinely box-like operator. The main theorems are Theorem 3 (soundness: ⊢φ implies ⊩φ) and Theorem 4 (completeness: ⊩φ implies ⊢φ), the latter via a translation t: PAL → S5. The paper closes with an analysis of the three-player card game and the muddy children puzzle; a distinctive claim is that the muddy-children reasoning requires bases to support negative information ('Anne is not muddy') rather than merely failing to support 'Anne is muddy'.","tokens_in":32468,"tokens_out":46209,"duration_ms":366507,"significance":"If the technical results can be repaired, this is the first base-extension semantics for a dynamic epistemic logic and a nontrivial extension of proof-theoretic semantics from static modal logics to epistemic actions. The innovation of evaluating announcements against a set of 'effective updates' is well motivated by the concrete failure example in Section 5 and gives announcements a genuinely intensional reading. The two case studies are a strength: Propositions 2 and 3 turn the muddy-children example into testable claims about which base rules must be explicit, and the authors are careful to disclaim strong minimality. There are no machine-checked proofs or executable artifacts; the paper's value is conceptual, and its central claims are currently gated because the completeness proof contains an unlicensed step and the update construction (Lemma 6) is verified only in sketch form.","major_comments":[{"comment":"The proof of Theorem 4, the completeness of the base-extension semantics for PAL, is circular as written. After assuming ⊩φ, the proof states 'By Theorem 3 and Lemma 21, we get ⊢φ↔t(φ)'. Theorem 3 is soundness (⊢ψ ⇒ ⊩ψ), and Lemma 21 gives only the semantic equivalence ⊩φ↔t(φ); their conjunction licenses at most ⊩t(φ), never ⊢φ↔t(φ). To conclude ⊢φ from ⊢S5 t(φ), the syntactic translation lemma ⊢PAL φ↔t(φ) is required, and it is neither stated nor proved; it cannot be derived from the semantic lemma without assuming the very completeness that Theorem 4 is meant to establish. The repair is straightforward and the infrastructure is present: the axioms of Definition 8 are biconditionals, and the complexity inequalities of Lemma 20 are exactly those needed for an induction on c(φ) (Definition 26) proving ⊢PAL φ↔t(φ). The proof of Theorem 4 should be rewritten around this syntactic lemma. Note also that citing Theorem 1 for the S5 completeness of t(φ) is imprecise: since PAL-validity of announcement-free formulas is stated relative to update sets (Definition 15), Theorem 2 is the relevant completeness result, and the coincidence of the two validity notions on announcement-free formulas should be stated explicitly.","section":"Section 5, proof of Theorem 4"},{"comment":"Lemma 6 is the linchpin of the update semantics: it asserts that the relation R+_{A|φ} built in Definition 24 is an S5-modal relation and an update set, and it underpins Corollary 1, Lemma 7, Lemma 17, and hence the soundness of the announcement axioms. As written, its proof is a sketch with several unverifiable steps. (a) The Euclidean case is dismissed with 'follows analogously'; because clause (3) of Definition 24 requires compatibility of two bijections f and g, the Euclidean composition argument needs to be written out as it is (partially) for transitivity. (b) In the proof of modal-relation condition (c) of Definition 11, the text writes δπ+(B+) = {δ | δ ⊂ S*_{A|φ} and B+ ∈ δ}, but Definition 23 defines the elements of δπ+(·) as collections of blocks π+_i of π+; under the definition as written, the displayed identity is not meaningful, and neither is the instruction to obtain C′ 'by simply adding rules'. (c) The verification of the update-set condition claims that (C′′−r), ((C′′−r)−r′), and (C′′−r′) belong to π+_i, but membership in π+_i requires membership in the domain/range of S*_{a|φ}; the proof does not show that removing rules from bases in S*_{a|φ} preserves this membership, which is precisely where condition (d) of Definition 11 must be invoked. (d) The proof uses the symbol δw(C′), which is never defined in the main text. The construction of Definitions 21–24 and the verification of Lemma 6 should be presented in full.","section":"Section 5, Lemma 6 and Definitions 20–24"},{"comment":"The definition of support for Kaφ after an announcement is type-incoherent as written. The clause reads: 'if Δ is non-empty, then for all C⊇B, R′_A∈RA|ΔC, updates C+ of C and C′ s.t. R′_aC+C′, ⊩^Δ_{C′,RA}φ'. By Definition 17, R′_a is a relation on the extended base set Ω_{P′}, so the target C′ is an updated base in Ω_{P′}; but the support relation in the conclusion is evaluated with the original update set RA, which is defined on Ω_P. The support relation of Definition 18 is only defined for bases in the language of the ambient update set, so the clause as written has no defined meaning. The intended reading is presumably ⊩^Δ_{C′,R′_A}φ, which would be consistent with Lemma 8 (where support after an effective update is evaluated with the updated relations in the subscript); alternatively the paper must explain how RA is extended to Ω_{P′} and why the choice of the particular effective update does not matter. Because the proof of the announcement-and-knowledge axiom in Lemma 18 unpacks this clause directly, the definition and that proof need to be aligned and stated precisely.","section":"Definition 18, clause for Kaφ with non-empty Δ"},{"comment":"Lemma 17, the equivalence used to prove the announcement-and-knowledge axiom [φ]K_aψ ↔ φ→K_a[φ]ψ in Lemma 18, is not proved as written. The two conditions are stated with different and partially implicit quantifier structures, and neither direction is checked against Definition 17 clause 3. In the direction from 1 to 2, the proof derives from (1) a particular pair C⊇B and D with R_aCD, applies condition (d) to obtain X⊆D with R_aBX, and then concludes the universal statement 'for all X s.t. R_aBX and Y⊇X, if ⊩_{Y,RA}φ then ⊩^φ_{Y,RA}ψ'; the step from the particular D to the universal quantification over all Y⊇X is not justified, and the proof never passes from the given X of (2) to the C,D of (1). In the direction from 2 to 1, the proof starts from X (with R_aBX) and Y⊇X, obtains C⊇B with R_aCY by symmetry and condition (c), and asserts that an effective update R′_A connects C+ and Y+; by Definition 17 clause 3 this requires ⊩_{C,RA}φ in addition to R_aCY and ⊩_{Y,RA}φ, which is not established. A correct proof of 2→1 would instead apply condition (d) to R_aCD with the subset B⊆C, yielding X⊆D with R_aBX, and then invoke (2) with Y=D using ⊩_{D,RA}φ, which is available from Definition 17 clause 3. Please restate Lemma 17 with explicit quantifier prefixes and supply both directions.","section":"Section 5, Lemma 17"},{"comment":"The soundness and completeness theorems of this paper rest on the S5 base-extension results, but the proof layer below Lemma 6 is not fully available to the reader. Theorem 1 is imported from the authors' companion paper [7]; Theorem 2, the update-set variant that the PAL semantics actually uses, delegates both directions to [7], with the update-set adaptation of one direction relegated to Appendix A. The appendix is itself a sketch: it repeats the hand-wave 'The Euclidean case follows analogously', it invokes an undefined atom q_w ('By Lemma 5, we know there is a maximally-consistent C ⊇ A_w s.t. ⊩C q_w'), and the verification of the final update-set clause for sub- and supersets is compressed into a single paragraph whose key claim (reachable bases must have δ_W of equal length) is asserted rather than proved. Since Theorem 4 applies S5 completeness to the translation t(φ) and since Lemma 6's proof explicitly mimics the appendix construction, the completeness chain is only as solid as this appendix. The authors should either re-prove the S5 update-set results in full or restructure the presentation so that the foundational layer is self-contained and verifiable.","section":"Section 4 (Theorems 1–2) and Appendix A"}],"minor_comments":[{"comment":"Two consecutive sentences refer to 'the first [6]' and 'the second [6]', although they describe two different papers; the second citation should presumably be [7].","section":"Section 1, third paragraph"},{"comment":"The Baltag–Moss–Solecki entry is corrupted: 'S/suppress lawomir Solecki' should read 'Sławomir Solecki'.","section":"Reference list, item [1]"},{"comment":"The clause for implication is written as '⊩_{B,RA} ϕ→ψ iff ϕ ⊩_{B,RA} ψ', which silently invokes the nonempty-Γ clause (with Γ={ϕ}) defined only afterwards; reordering the clauses, or writing '{ϕ} ⊩_{B,RA} ψ', would remove a real source of confusion.","section":"Definitions 12, 15, and 18"},{"comment":"The last sentence of Definition 18 quantifies over 'sets of S5-modal relations RA', although both the opening of that definition and Definition 15 restrict attention to update sets; this appears to be a slip.","section":"Definition 18, final validity clause"},{"comment":"The conclusion of Proposition 1 is written with the turnstile '⊢B012,RA [¬1a]Kc(0a∧1b∧2c)'; since the claim is about support at a base, the symbol should be '⊩B012,RA'.","section":"Section 6.1, Proposition 1"},{"comment":"The proof of Lemma 20 contains typographical errors: 'c([ϕ]ψ→χ)' should read 'c([ϕ](ψ→χ))', and 'c([ϕ][ψ]χ) = (2+c(ϕ))×((2+c(ψ))×c(χ)' together with the following displayed line are missing closing parentheses.","section":"Section 5, Lemma 20"},{"comment":"The text contains a number of small language slips, e.g., 'any formula that is false on a Kripke modal' (should be 'Kripke model'), 'so used to proof the same' (should be 'prove'), and Lemma 14 begins 'We proof this by induction' (should be 'prove').","section":"Section 2.2 and Section 5"},{"comment":"The definition of C+ (the update of a base) is barely comprehensible as printed; the ranges of the indices in 'p′_0,...,p′_i ⇒ p_∅' and in '{p_1,...,p_n}\\{p_k}' and the role of the added rules should be explained in prose, since this construction is the heart of Lemma 6.","section":"Definition 22"}],"recommendation":"major_revision","confidential_remarks":"The manuscript continues the authors' own research program on base-extension semantics for modal logic, and the self-citations to [6] and [7] are programmatic rather than gratuitous. My main concern for the editorial process is verifiability: the central results depend on the companion paper [7] (in press) and on the sketch-level Appendix A and Lemma 6; I would ask the handling editor to confirm the status of [7] and to consider having the update construction checked by a second referee. There are also visible editing artifacts (corrupted reference [1], undefined q_w, inconsistent turnstile usage) that suggest a careful proofreading pass. The paper is a good fit for the journal's scope in proof-theoretic semantics."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a serious paper and worth refereeing, but the completeness proof as written doesn't go through. The main contribution—base-extension semantics for PAL via 'effective updates'—is real and new relative to the cited literature. The observation that announcements stop being partial functions in this semantics is a genuinely interesting conceptual difference from Kripke semantics, and the paper makes good use of it. The muddy children discussion earns its keep: the point that a base must support 'Anne is not muddy' rather than merely fail to support 'Anne is muddy' is a nice illustration of inferentialist explicitness.\n\nThe central construction is the effective update of Definition 24. Lemma 6, which says the construction yields S5-modal relations, is essential and is only sketched. The prose says 'the Euclidean case follows analogously,' but for a construction this intricate that is not enough. A referee should ask for a full proof or a formalization. Lemma 17, needed for the announcement-knowledge axiom, is also hard to parse and its two directions don't obviously line up; the proof as written seems to lean on condition (d) in a way that isn't justified.\n\nThe bigger problem is Theorem 4. The proof says 'By Theorem 3 and Lemma 21, we get ⊢ϕ↔t(ϕ).' Theorem 3 is soundness (provability implies validity); it licenses passing from ⊢ to ⊩, not the reverse. Lemma 21 gives the semantic equivalence. So the step is simply not licensed. The completeness claim rests on this. This looks repairable—the PAL axioms are equivalences, and an induction on the paper's own complexity measure should yield the syntactic equivalence—but as it stands the central theorem is unproven.\n\nAlso worth noting: the S5 base-extension soundness and completeness are imported from the authors' own [7], without re-proof. That is not fatal, but it makes verification of the foundational layer hinge on an unformalized prior paper. The PAL-specific content is new, so this is not a circularity in the premises, only a heavy delegation.\n\nBottom line: the approach is promising, the examples are instructive, and the effective-update semantics is a solid contribution to proof-theoretic semantics for DEL. But a reader cannot verify the central claim from this manuscript alone. Send it to a good referee, and tell the authors to fix Theorem 4 and spell out Lemma 6. If they do, this could be a useful paper.","headline":"Genuinely new base-extension semantics for PAL with a real conceptual payoff, but the completeness proof has an unlicensed step and the key update lemma is sketched—worth refereeing, needs repair.","tokens_in":33172,"tokens_out":1932,"would_cite":true,"duration_ms":17858,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["03B45","03B42"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper presents a base-extension semantics for public announcement logic and proves it sound and complete with respect to the PAL axiomatic system.","keywords":["public announcement logic","base-extension semantics","proof-theoretic semantics","modal logic","inferentialism","S5","dynamic epistemic logic","muddy children puzzle"],"falsifier":"A concrete way to test the claim would be to take a small update set (for example, bases $B',C',B,C$ with $\\phi$ holding at $B',B,C$ but not $C'$, as in the paper's Figure 9), compute the relation $R^+_{a|\\phi}$ of Definition 24, and check directly whether it satisfies the Euclidean condition and conditions (c) and (d) of Definition 11 for all subsets and supersets; a single failure would refute Lemma 6 and collapse the soundness theorem. Alternatively, one could search for a PAL formula that is valid in the Kripke semantics but not supported by the base-extension semantics, which would refute Theorem 4.","tokens_in":31918,"feed_emoji":"📣","tokens_out":5478,"duration_ms":48919,"temperature":0.7,"pith_summary":"The paper extends base-extension semantics, a proof-theoretic semantics in which validity is defined by induction over bases of atomic rules, to public announcement logic (PAL). Its central goal is to show that announcement operators can be given a fully inferentialist account: an announcement updates the modal relations between bases rather than deleting worlds, and the resulting support relation is sound and complete with respect to the standard PAL axiomatic system (Theorems 3 and 4). A sympathetic reader should care because this establishes that dynamic epistemic reasoning about knowledge and public communication does not require Kripke truth-conditions; it can be grounded in atomic inference. The paper also reads the three-player card game and the muddy children puzzle through this semantics, showing that the inferentialist treatment forces all modelling assumptions to be made explicit as base rules.","feed_headline":"Sound and complete proof-theoretic semantics for announcements","feed_subtitle":"Meaning is use: announcements update relations between bases, putting dynamic epistemic logic on inferentialist ground.","key_machinery":"The machinery is the notion of a modal relation on bases (Definition 11), a binary relation $R_a$ between bases satisfying conditions (a)--(d) that control interaction with the subset relation and separate consistent from inconsistent bases; together with update sets (Definition 14), which add reachability and finiteness structure so that updates can always be performed. Announcements act on these relations: Definition 17 defines an effective update $R'_A$ of $R_A$ by $\\phi$ at $B$ as an update set on an extended atom language whose reachable base pairs mirror the old relation restricted to bases supporting $\\phi$, and Definition 24 constructs such an update explicitly using new atoms $p_\\emptyset,p_i$ and the function $\\delta_{\\pi^+}$. The support relation (Definition 18) then evaluates $[\\phi]\\psi$ by quantifying over supersets and effective updates. This construction carries the soundness proof because Lemma 6 and Lemma 7 guarantee the updated relations are still modal relations and genuinely effective.","core_discovery":"The paper claims that for every formula $\\phi$ of PAL, $\\vdash\\phi$ iff $\\Vdash\\phi$: the arguments in the paper establish soundness (Theorem 3, every provable formula is supported at every base under every update set) and completeness (Theorem 4, every supported formula is provable in the PAL axiomatization, via the standard translation $t$ from PAL to S5 combined with S5 completeness). The key move is to treat an announcement $[\\phi]$ not as deleting bases but as transforming the modal relations $R_a$ into a family of possible 'effective updates' $R_a|_{\\phi}^{B}$, depending on the base $B$ at which the announcement is evaluated. Announcements therefore cease to be partial functions in this semantics, yet Lemma 8 shows all effective updates agree on formulae in the original language, so announcement validity is still well defined.","pith_inferences":["If soundness and completeness hold for PAL, the same update-relations technology is a plausible starting point for action model logic, where agents' uncertainty about which action occurred could be encoded as uncertainty between candidate modal relations.","The dependence of effective updates on the base suggests an intensional notion of informational equivalence: two updates can be semantically equivalent for the original language while differing on atoms added during the update; making this precise could connect to unsuccessful announcements and common knowledge.","The authors' explicit positive encoding of negative facts in the muddy children example invites a testable modelling principle: in base-extension models of epistemic puzzles, every frame assumption that a modeller would 'build into' the Kripke valuation must appear as base rules or positive support conditions, otherwise announcements misfire.","It would be a natural extension to check whether the muddy-children observation scales: for $n$ children and $m$ muddy children, requiring explicit support of all negated $m_j$ at the appropriate bases should let the $m$-step solution be reproduced in this semantics."],"forward_implications":["The base-extension semantics for PAL is sound and complete, so proof-theoretic validity captures the full dynamic logic of truthful public announcements.","Announcements in this semantics are not unique updates: the same announcement at different bases produces different modal relations, but Lemma 8 ensures that any choice of effective update gives the same verdict on formulae in the original language.","The card-game and muddy-children analyses provide minimal base set-ups: any extension of those bases preserves the expected conclusions, and the muddy-children case shows that 'not muddy' must be positively supported rather than merely absent.","The completeness proof by translation gives a compositional reduction of PAL back to S5, so the dynamic announcement axioms are exactly what is needed to eliminate announcement operators in the base-extension setting."],"supporting_citations":[{"why":"Supplies the multi-agent S5 base-extension semantics, modal relations, and soundness/completeness result that this paper extends to PAL.","marker":"[7]"},{"why":"Provides the earlier base-extension semantics for K, KT, K4, and S4 on which the S5 treatment builds.","marker":"[6]"},{"why":"Supplies the standard Kripke semantics and axiomatization of PAL, the two worked examples, and the translation-based completeness strategy.","marker":"[4]"},{"why":"Introduces the logic of public announcements and the update operation that the base-extension semantics reconstructs at the level of modal relations.","marker":"[1]"},{"why":"Gives the base-extension semantics for classical logic whose validity conditions and monotonicity lemmas are reused for the propositional part.","marker":"[16]"},{"why":"Provides the inferential semantics and maximal-consistency lemmas (Lindenbaum-style extension) used in the soundness and completeness proofs.","marker":"[8]"}],"fun_headline_variants":["Base-extension semantics for public announcement logic","Announcements as base updates: sound and complete","Inferentialist semantics for dynamic epistemic logic","Proof-theoretic semantics for public announcements","Announcements transform bases: sound and complete logic"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper assumes that the explicit update construction of Definition 24, applied to any existing update set at a base where the announced formula holds, always produces relations satisfying the four modal-relation conditions of Definition 11 and the update-set conditions of Definition 14; the proof of this (Lemma 6) is a sketch, with the Euclidean case left as 'analogous'.","fun_headline_variants_meta":{"raw":{"variants":["Base-extension semantics for public announcement logic","Announcements as base updates: sound and complete","Inferentialist semantics for dynamic epistemic logic","Proof-theoretic semantics for public announcements","Announcements transform bases: sound and complete logic"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001153,"raw_usage":{"total_tokens":4748,"prompt_tokens":884,"completion_tokens":3864,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":500,"completion_tokens_details":{"reasoning_tokens":3795}},"tokens_in":500,"tokens_out":3864,"duration_ms":26839,"temperature":1.0,"reasoning_tokens":3795,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:56:29.615133+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete way to test the claim would be to take a small update set (for example, bases $B',C',B,C$ with $\\phi$ holding at $B',B,C$ but not $C'$, as in the paper's Figure 9), compute the relation $R^+_{a|\\phi}$ of Definition 24, and check directly whether it satisfies the Euclidean condition and conditions (c) and (d) of Definition 11 for all subsets and supersets; a single failure would refute Lemma 6 and collapse the soundness theorem. Alternatively, one could search for a PAL formula that is valid in the Kripke semantics but not supported by the base-extension semantics, which would refute Theorem 4.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the multi-agent S5 base-extension semantics, modal relations, and soundness/completeness result that this paper extends to PAL."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the earlier base-extension semantics for K, KT, K4, and S4 on which the S5 treatment builds."},{"cited_title":"Dynamic Epistemic Logic","cited_arxiv_id":null,"evidence_quote":"Supplies the standard Kripke semantics and axiomatization of PAL, the two worked examples, and the translation-based completeness strategy."},{"cited_title":"Moss, and S/suppress lawomir Solecki","cited_arxiv_id":null,"evidence_quote":"Introduces the logic of public announcements and the update operation that the base-extension semantics reconstructs at the level of modal relations."},{"cited_title":"Classical logic without bivalence","cited_arxiv_id":null,"evidence_quote":"Gives the base-extension semantics for classical logic whose validity conditions and monotonicity lemmas are reused for the propositional part."},{"cited_title":"On an inferential semantics for classical logic","cited_arxiv_id":null,"evidence_quote":"Provides the inferential semantics and maximal-consistency lemmas (Lindenbaum-style extension) used in the soundness and completeness proofs."}],"review_version":1}