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arxiv: 2506.16901 · v2 · submitted 2025-06-20 · 💰 econ.TH

Do You Know What I Mean? A Syntactic Representation for Differential Bounded Awareness

Ani Guerdjikova , Evan Piermont , John Quiggin This is my paper

Pith reviewed 2026-05-19 08:01 UTC · model grok-4.3

classification 💰 econ.TH
keywords differential bounded awarenesssyntactic representationtranslation operatorsexpressiveness comparisonjoint state spacemulti-agent communicationsubjective languages
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The pith

Translation operators between languages allow embedding of each agent's subjective state space into a joint language and state space when specific conditions hold.

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

The paper develops a syntactic method for agents who represent the world differently because of unequal awareness. It introduces translation operators that map propositions from one language into another by finding the closest possible meaning given the limits of what the target language can express. These operators lead to necessary and sufficient conditions under which all individual representations fit inside one common state space and one common language. A reader would care because real communication often occurs without everyone sharing the same facts or vocabulary, and this framework supplies a precise way to compare how much each agent can say and what follows for their decisions.

Core claim

We define translation operators between two languages which provide a 'best approximation' for the meaning of propositions in the target language subject to its expressive power. We show that, in general, the translation operators preserve some, but not all, logical operations. We derive necessary and sufficient conditions for the existence of a joint state space and a joint language, in which the subjective state spaces of each agent, and their individual languages, may be embedded. This approach allows us to compare languages with respect to their expressiveness and thus, with respect to the properties of the associated state space.

What carries the argument

Translation operators that map propositions from one agent's language to another's by selecting the best approximation allowed by the target's expressive power; these operators enable the embedding of separate subjective languages into a single joint language and state space.

If this is right

  • Translation operators preserve some logical operations but fail to preserve others.
  • Languages become comparable by how much they can express about the underlying state space.
  • Properties of each agent's state space, such as which events are distinguishable, can be ranked through the joint embedding.
  • The existence of a joint structure is fully characterized by the behavior of the translation operators.

Where Pith is reading between the lines

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

  • The same translation method could be applied to study how contracts or regulations must be written when parties hold unequal information about possible states.
  • Laboratory experiments could restrict subjects' vocabularies in a controlled way and check whether observed behavior matches the predicted embedding conditions.
  • The framework suggests a route for extending static awareness models to repeated interactions where agents gradually learn or expand each other's languages.

Load-bearing premise

That best-approximation translation operators can be rigorously and consistently defined between any pair of languages according to the expressive power of the target.

What would settle it

A concrete pair of languages for which no consistent best-approximation translation operators exist, or a case in which the stated necessary and sufficient conditions are satisfied yet no joint state space and language can be constructed that contains both subjective representations.

Figures

Figures reproduced from arXiv: 2506.16901 by Ani Guerdjikova, Evan Piermont, John Quiggin.

Figure 1
Figure 1. Figure 1: Joint state-space and the events corresponding to elementary [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Embeddings of the individual state-spaces into the joint state [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
read the original abstract

Without the assumption of complete, shared awareness, it is necessary to consider communication between agents who may entertain different representations of the world. A syntactic (language-based) approach provides powerful tools to address this problem. In this paper, we define translation operators between two languages which provide a `best approximation' for the meaning of propositions in the target language subject to its expressive power. We show that, in general, the translation operators preserve some, but not all, logical operations. We derive necessary and sufficient conditions for the existence of a joint state space and a joint language, in which the subjective state spaces of each agent, and their individual languages, may be embedded. This approach allows us to compare languages with respect to their expressiveness and thus, with respect to the properties of the associated state space.

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

2 major / 2 minor

Summary. The paper develops a syntactic framework for agents with differential bounded awareness. It defines translation operators between languages that map propositions from one to a 'best approximation' in the target language given its expressive power. These operators are shown to preserve some but not all logical operations. The central result derives necessary and sufficient conditions for a joint state space and joint language into which each agent's subjective state space and language embed, permitting expressiveness comparisons across languages.

Significance. If the embedding conditions are rigorously established, the framework offers a useful syntactic tool for modeling communication under incomplete shared awareness in epistemic game theory. The derivation of necessary and sufficient conditions for joint structures is potentially valuable for comparing languages and associated state spaces. No machine-checked proofs or reproducible code are present, but the syntactic approach itself is a constructive contribution if the translation operators are made fully operational.

major comments (2)
  1. [§3] §3 (Translation Operators): The 'best approximation' provided by the translation operator is introduced without an explicit metric, ordering, or unique selection criterion that is preserved under embedding. As a result, multiple candidate translations may satisfy the definition, potentially yielding non-isomorphic joint languages and rendering the necessary-and-sufficient conditions dependent on an arbitrary choice rather than uniquely determined by the setup.
  2. [§5] §5 (Joint State Space and Language): The necessity and sufficiency claims for the existence of the joint structure rest directly on the properties of the translation operators. Without a canonical definition of 'best approximation' that guarantees uniqueness or invariance of the embedding, the derived conditions risk being non-unique or sensitive to the particular choice of operator, which is load-bearing for the central theorem.
minor comments (2)
  1. [Abstract] The abstract states that operators 'preserve some but not all logical operations' without indicating which operations are preserved; a short illustrative example or table in §4 would clarify this for readers.
  2. [§2] Notation for languages, propositions, and state spaces should be introduced with a dedicated preliminary subsection before the main definitions to avoid forward references.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed comments, which help clarify important aspects of the framework. We respond to each major comment below and outline the revisions we will implement.

read point-by-point responses

  1. Referee: [§3] §3 (Translation Operators): The 'best approximation' provided by the translation operator is introduced without an explicit metric, ordering, or unique selection criterion that is preserved under embedding. As a result, multiple candidate translations may satisfy the definition, potentially yielding non-isomorphic joint languages and rendering the necessary-and-sufficient conditions dependent on an arbitrary choice rather than uniquely determined by the setup.

    Authors: We agree that the definition of the translation operator in Section 3 relies on the notion of 'best approximation' subject to expressive power without specifying an explicit metric or ordering. This leaves open the possibility of multiple valid translations. In the revised manuscript we will augment Section 3 with a canonical partial order on propositions, defined via the inclusion of expressible atomic formulas and syntactic complexity within the target language. We will prove that this ordering selects a unique best approximation under standard assumptions on the languages and that any two operators satisfying the approximation property induce isomorphic embeddings. These additions will be accompanied by an example illustrating uniqueness. revision: yes

  2. Referee: [§5] §5 (Joint State Space and Language): The necessity and sufficiency claims for the existence of the joint structure rest directly on the properties of the translation operators. Without a canonical definition of 'best approximation' that guarantees uniqueness or invariance of the embedding, the derived conditions risk being non-unique or sensitive to the particular choice of operator, which is load-bearing for the central theorem.

    Authors: The referee is correct that the necessity and sufficiency result in Section 5 is load-bearing on the translation operators. Once the canonical ordering is introduced in Section 3, we will add a supporting lemma in Section 5 establishing that the existence of the joint state space and joint language is invariant across any translation operators that respect the best-approximation property under the new ordering. The proof will show that different choices yield equivalent expressiveness comparisons and isomorphic joint structures, thereby preserving the original necessary-and-sufficient conditions. revision: yes

Circularity Check

0 steps flagged

Derivation of embedding conditions builds from explicitly defined translation operators without reduction to inputs by construction

full rationale

The paper introduces translation operators as a primitive that maps propositions while providing a best approximation subject to expressive power, then derives necessary and sufficient conditions for a joint state space and joint language into which individual state spaces and languages embed. No quoted step equates the derived conditions to the definition of the operators themselves, nor does any load-bearing premise reduce via self-citation or fitted parameter. The construction is self-contained: the operators are stipulated, their preservation properties are shown, and the embedding conditions follow as a separate existence result. This matches the default case of an independent mathematical derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The framework rests on introducing translation operators and joint embeddings with no fitted numerical parameters; it relies on background logic but adds new constructs without independent empirical handles.

axioms (1)
  • standard math Standard properties of logical operations (conjunction, disjunction, negation) in propositional languages
    Invoked when stating that translation operators preserve some but not all logical operations.
invented entities (2)
  • Translation operator no independent evidence
    purpose: Provide best approximation of proposition meanings between languages of differing expressive power
    Newly defined construct to handle differential awareness syntactically.
  • Joint state space and joint language no independent evidence
    purpose: Common structure into which subjective state spaces and languages can be embedded
    Existence conditions derived to enable expressiveness comparisons.

pith-pipeline@v0.9.0 · 5668 in / 1302 out tokens · 60365 ms · 2026-05-19T08:01:33.085174+00:00 · methodology

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Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages · 1 internal anchor

  1. [1]

    = l∗ 3 ∧ (¬l∗ 4), π (le

    work page

  2. [2]

    = l∗ 1 ∧ (¬l∗ 4), π (le

    work page

  3. [3]

    vers” is mapped to “earth worms and no parasitic worms

    = l∗ 2 ∧ (¬l∗ 4): “vers” is mapped to “earth worms and no parasitic worms”; similarly, “moutons 16 blancs (noirs)” is mapped to “white (black) sheep and no parasitic worms”, capturing unawareness of parasitic worms. “Sheep” in ¯L is mapped to “white or black sheep”, ¯π ¯le 3 = l∗ 1 ∨ l∗ 2, representing coarsening of this category. Associate with each of t...

    work page

  4. [4]

    ∧ π ¯le 1 , l∗ 3 = π (le

    work page

  5. [5]

    L is a combi- nation of a coarsening and restriction of ¯L

    ∧ π (¯t). 3.3.2 A Joint State Space The conditions for L∗ to be a joint language for L and ¯L can be equivalently captured via conditions on their state-spaces / algebra of events. Write W = W (L) and ¯W = W ¯L for the state spaces of the two languages. Lemma 4. A language L∗ is a joint language for L and ¯L if and only if there exist two embeddings σ : B...

    work page 1967

  6. [6]

    T + ¯L→L ¯le 1 = T − ¯L→L ¯le 1 = le

    = ¯π ¯le 1 = l∗e 3 , both translation operators map le 1 to ¯le 1 and vice-versa. T + ¯L→L ¯le 1 = T − ¯L→L ¯le 1 = le

    work page

  7. [7]

    worms” in ¯L are translated to “vers

    In general, the correspondence between the two languages is not one-to-one, and transla- tion has to approximate the meaning of propositions. E.g., T + ¯L→L ¯le 1 ∨ ¯le 2 = T − ¯L→L ¯le 1 ∨ ¯le 2 = le 1 (“worms” in ¯L are translated to “vers” in L, generat- ing a biased meaning, because the subsistence farmer is unaware of parasitic worms). Translating “m...

    work page

  8. [8]

    = ¯le 3, whereas the inner translation is the contradiction, T − ¯L→L (le

    work page

  9. [9]

    white sheep

    = f, because ¯L lacks a hyponym (a more specific term) for a “white sheep”. □ 23 4.1 Duality of the Translation Operators Consider two languages L and ¯L and let TL→¯L be a translation operator. The dual of TL→¯L, T d L→¯L, is defined by: T d L→¯L (l) = ¬ (TL→¯L (¬l)) The inner and outer translation are dual if L and ¯L are pure coarsenings of a joint lan...

    work page

  10. [10]

    T + L→¯L and T + ¯L→L satisfy preservation of contradictions and conjunctions, as well as outer consistency of translation

    work page

  11. [11]

    T − L→¯L and T − ¯L→L satisfy preservation of contradictions and disjunctions, as well as inner consistency of translation; and

    work page

  12. [12]

    minimal” state space, which is consistent with the outer translations T + L→¯L and T + ¯L→L. It cannot exhibit

    T + L→¯L and T − L→¯L, as well as T + ¯L→L and T − ¯L→L satisfy restricted duality. We construct the joint state space in two steps. First, we use the outer translation operators to construct the minimal joint state space ˜Q for which T + L→¯L and T + ¯L→L satisfy the definition in (6). Next, we do the same for the inner translation operators, T − L→¯L an...

    work page

  13. [13]

    intersect

    ⊆ ˆBL∗ (l∗ 2) Finally, each proposition in the languages L and ¯L can be identified with an event in the joint state space Q, and thus, with a proposition in L∗. The maps π and ¯π can be defined as π (l) = ˆB−1 L∗ (σ (bl)) ¯π ¯l = ˆB−1 ¯L∗ (¯σ (b¯l)) which trivially satisfies the properties in Definition 1. 6 A Second Look at Comparative Awareness In Coro...

    work page

  14. [14]

    Q, and by T −,Q L→¯L, T −,Q ¯L→L, T +,Q L→¯L, T +,Q ¯L→L the inner and outer translations w.r.t

    Denote by E −,Q L→¯L, E −,Q ¯L→L, E+,Q L→¯L, E+,Q ¯L→L the inner and outer event mappings w.r.t. Q, and by T −,Q L→¯L, T −,Q ¯L→L, T +,Q L→¯L, T +,Q ¯L→L the inner and outer translations w.r.t. Q. Lemma 23. The inner events defined with respect to the joint state space Q exactly coincides with those on ˆQ defined in the proof of Proposition 3: E −,Q L→¯L ...

    work page

  15. [15]

    Let Q be the joint state space constructed in the proof of Proposition 4. By Lemma 1, for the algebra 2 Q, we can find an axiomatized language, (Λ ∗, L∗) such that its equivalence classes are given by: L∗ = n l∗ | ˆBL∗ (l∗) ∈ 2Q o and such that the implication relation is given by l∗ 1 ⇒L∗ l∗ 2 iff ˆBL∗ (l∗

    work page

  16. [16]

    Why Bother With Syntax?

    ⊆ ˆBL∗ (l∗ 2) Each proposition in the languages L and ¯L can be identified with an event in the joint state space Q, and thus, with a proposition in L∗. The maps π and ¯π can be defined as π (l) = ˆB−1 L∗ (σ (bl)) ¯π ¯l = ˆB−1 ¯L∗ (¯σ (b¯l)) which trivially satisfies the properties in Definition 1. Since we have shown that the translation operators T − L→...

    work page internal anchor Pith review Pith/arXiv arXiv 2021