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arxiv: 1907.09105 · v1 · pith:X4XLLO3Pnew · submitted 2019-07-22 · 💻 cs.LO

How to Agree without Understanding Each Other: Public Announcement Logic with Boolean Definitions

Malvin Gattinger (Bernoulli Institute , University of Groningen) , Yanjing Wang (Department of Philosophy , Peking University) This is my paper

Pith reviewed 2026-05-24 17:59 UTC · model grok-4.3

classification 💻 cs.LO
keywords public announcement logicepistemic logicboolean definitionsknowledge and meaningconservative extensionmulti-agent agreement
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The pith

Public announcement logic can be extended so agents track both truth and meaning of propositions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper introduces a conservative extension of public announcement logic that lets agents have knowledge about both the truth values and the meanings of propositions. Meanings are captured by Boolean definitions attached to atomic propositions. The result is that one can model situations where an agent understands what a proposition means but does not know if it is true, or where agents agree on a fact without sharing the same understanding of the terms involved. A complete axiomatization is provided for this extended logic.

Core claim

We present a conservative extension of Public Announcement Logic (PAL) in which agents have knowledge or belief about both the truth values and the meanings of propositions. We give a complete axiomatization of PAL with Boolean Definitions and discuss various examples. An agent may understand a proposition without knowing its truth value or the other way round. Moreover, multiple agents can agree on something without agreeing on its meaning and vice versa.

What carries the argument

Boolean definitions that represent the meanings of propositions, added to public announcement logic while preserving its original semantics.

If this is right

  • Agents can separate understanding a proposition from knowing whether it holds.
  • Multiple agents may reach agreement on truth without sharing the same definitions of the terms.
  • The new logic has a complete axiomatization.
  • The extension does not change the validity of formulas from the original public announcement logic.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This separation could help analyze communication failures where parties use the same words differently.
  • Applications might include modeling legal or scientific disputes over terminology.
  • Further work could explore how public announcements interact with updates to these definitions.

Load-bearing premise

That meanings of propositions can be represented by Boolean definitions in a way that keeps the extension conservative and admits a complete axiomatization without altering the underlying PAL semantics.

What would settle it

Finding a formula that is valid in standard PAL but not in the extension, or an invalid formula that becomes valid, or an axiom that is not sound.

Figures

Figures reproduced from arXiv: 1907.09105 by Malvin Gattinger (Bernoulli Institute, Peking University), University of Groningen), Yanjing Wang (Department of Philosophy.

Figure 1
Figure 1. Figure 1: Knowing without Understanding p q p := q ¬p ¬q p := q i [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Understanding different parts p ¬q r p := r p ¬q ¬r p := p i p q ¬r p := q j [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Consensus with misunderstanding Example 4 (Consensus with misunderstanding). As in the Konnyaku Mondo example, two agents can agree on something but actually have different beliefs about what it means. In the model from [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: A simple substitution tree and the resulting circular formula. [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: A definition chain and the resulting occurrence substitution tree. The right branch yields a [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
read the original abstract

In standard epistemic logic, knowing that p is the same as knowing that p is true, but it does not say anything about understanding p or knowing its meaning. In this paper, we present a conservative extension of Public Announcement Logic (PAL) in which agents have knowledge or belief about both the truth values and the meanings of propositions. We give a complete axiomatization of PAL with Boolean Definitions and discuss various examples. An agent may understand a proposition without knowing its truth value or the other way round. Moreover, multiple agents can agree on something without agreeing on its meaning and vice versa.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper presents a conservative extension of Public Announcement Logic (PAL) incorporating Boolean definitions to represent the meanings of propositions. This allows agents to have knowledge or belief about both the truth values and the meanings of propositions separately. The manuscript claims to give a complete axiomatization of the resulting logic (PAL with Boolean Definitions) and discusses examples in which agents may understand a proposition without knowing its truth value (or vice versa), and in which multiple agents can agree on a proposition without agreeing on its meaning.

Significance. If the claimed conservative extension and complete axiomatization hold, the work would be a useful addition to dynamic epistemic logic by explicitly separating knowledge of truth from understanding of meaning. This distinction is relevant for modeling communication and agreement scenarios. The paper ships a complete axiomatization (a strength when proofs are supplied) and preserves the original PAL semantics via reduction axioms, which is the standard route to conservativeness.

major comments (1)
  1. [Axiomatization section (and abstract)] The central claim is a complete axiomatization of the extension, yet the available manuscript text supplies no derivation details, soundness or completeness proofs, or concrete examples of the new reduction axioms. This is load-bearing for the main result and prevents verification of the conservativeness claim.
minor comments (2)
  1. [Semantics] The weakest assumption—that Boolean definitions can be added as static model components while keeping the extension conservative—would benefit from an explicit statement of how definitions interact with public announcements (e.g., whether they are announcement-invariant).
  2. [Syntax] Notation for the new operators (knowledge/belief about meanings) should be introduced with a clear comparison table to standard PAL operators.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for highlighting the importance of verifying the axiomatization and conservativeness claims. We address the single major comment below.

read point-by-point responses

  1. Referee: [Axiomatization section (and abstract)] The central claim is a complete axiomatization of the extension, yet the available manuscript text supplies no derivation details, soundness or completeness proofs, or concrete examples of the new reduction axioms. This is load-bearing for the main result and prevents verification of the conservativeness claim.

    Authors: The referee is correct that the submitted manuscript does not contain the soundness and completeness proofs, derivation details, or worked examples of the new reduction axioms. These elements are necessary to substantiate the central claim. In the revised version we will add a self-contained section that states the full axiom system, proves soundness with respect to the given semantics, proves completeness via the standard reduction-axiom strategy for PAL-style logics, and supplies concrete examples of the new reduction axioms in action. This addition will make the conservativeness argument verifiable. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper constructs a conservative extension of standard Public Announcement Logic by adjoining Boolean definitions as static model components separate from the valuation, then supplies reduction axioms that eliminate the new operators while leaving the underlying PAL semantics unchanged. This is the standard route to conservativeness in dynamic epistemic logics and does not reduce any central claim to a fitted parameter, self-definition, or load-bearing self-citation. The provided abstract and description contain no equations or steps that equate a derived quantity to its own input by construction, and no uniqueness theorems or ansatzes are imported from prior work by the same authors. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The work relies on standard background from epistemic logic; the new element is the Boolean definitions mechanism introduced to capture meanings.

axioms (1)
  • standard math Standard axioms and semantics of Public Announcement Logic
    The paper states the extension is conservative, so it inherits the prior PAL axioms and rules.
invented entities (1)
  • Boolean definitions for proposition meanings no independent evidence
    purpose: To allow agents to have separate knowledge or belief about the meaning of a proposition versus its truth value
    Introduced in the paper as the key modeling device for the extension; no independent evidence outside the logic itself is mentioned.

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Reference graph

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