Standard RSA: No Scope Distinction

Standard RSA computes P(w | u) without any notion of scope. It treats "every student read some book" as an atomic utterance and computes a single distribution over worlds.

RSA gives one answer, but there are two legitimate readings (global vs local EXH).

def Comparisons.RSAExhExpressivity.standardRSAGame : RSA.IBR.InterpGame

Standard RSA interpretation game for embedded SI. Note: This game has NO scope parameter - it's scope-blind.

Equations

• One or more equations did not get rendered due to their size.

The Expressivity Gap: Formal Statement

The key observation is that standard RSA, by treating utterances atomically, must give the same probability to worlds that EXH would distinguish by scope.

The distinguishing worlds are w_SA and w_AS:

• Global EXH: allows these (prob > 0)
• Local EXH: excludes these (prob = 0)
• Standard RSA L0: allows these (literal meaning satisfied)

This means standard RSA cannot implement local EXH - it always "leaks" probability to worlds that local EXH would exclude.

def Comparisons.RSAExhExpressivity.distinguishingWorlds : List RSA.Core.EmbeddedSI.EmbeddedSIWorld

The worlds that distinguish global from local EXH. These are true under global EXH but false under local EXH.

Equations

• Comparisons.RSAExhExpressivity.distinguishingWorlds = [RSA.Core.EmbeddedSI.EmbeddedSIWorld.SA, RSA.Core.EmbeddedSI.EmbeddedSIWorld.AS]

theorem Comparisons.RSAExhExpressivity.globalExh_allows_SA : RSA.Core.EmbeddedSI.globalExhMeaning RSA.Core.EmbeddedSI.EmbeddedSIWorld.SA = true

Global EXH allows the distinguishing worlds

theorem Comparisons.RSAExhExpressivity.globalExh_allows_AS : RSA.Core.EmbeddedSI.globalExhMeaning RSA.Core.EmbeddedSI.EmbeddedSIWorld.AS = true

theorem Comparisons.RSAExhExpressivity.localExh_excludes_SA : RSA.Core.EmbeddedSI.localExhMeaning RSA.Core.EmbeddedSI.EmbeddedSIWorld.SA = false

Local EXH excludes the distinguishing worlds

theorem Comparisons.RSAExhExpressivity.localExh_excludes_AS : RSA.Core.EmbeddedSI.localExhMeaning RSA.Core.EmbeddedSI.EmbeddedSIWorld.AS = false

theorem Comparisons.RSAExhExpressivity.standardRSA_includes_SA : RSA.Core.EmbeddedSI.embeddedMeaning RSA.Core.EmbeddedSI.EmbeddedSIMessage.everySome RSA.Core.EmbeddedSI.EmbeddedSIWorld.SA = true

Standard RSA L0 includes SA (literal meaning satisfied)

theorem Comparisons.RSAExhExpressivity.standardRSA_includes_AS : RSA.Core.EmbeddedSI.embeddedMeaning RSA.Core.EmbeddedSI.EmbeddedSIMessage.everySome RSA.Core.EmbeddedSI.EmbeddedSIWorld.AS = true

theorem Comparisons.RSAExhExpressivity.expressivity_gap : ∃ (w : RSA.Core.EmbeddedSI.EmbeddedSIWorld), RSA.Core.EmbeddedSI.localExhMeaning w = false ∧ RSA.Core.EmbeddedSI.globalExhMeaning w = true ∧ RSA.Core.EmbeddedSI.embeddedMeaning RSA.Core.EmbeddedSI.EmbeddedSIMessage.everySome w = true

The expressivity gap exists.

There exists a world that is:

1. Excluded by local EXH (prob 0)
2. Included by global EXH (prob > 0)
3. Included by standard RSA literal meaning

This shows standard RSA can only express global, not local EXH.

How Compositional RSA Resolves This

The solution is to make scope a latent variable that the listener infers:

L1(w, scope | u) ∝ P(w) × P(scope) × S1(u | w, scope)

Now the listener can infer either:

• Global scope interpretation (SA, AS possible)
• Local scope interpretation (only SS possible)

This is exactly what ScontrasPearl2021 does for "every horse didn't jump". The scope ambiguity model lifts interpretation to a latent variable.

Compositional RSA = Standard RSA + Scope as Latent Variable.

structure Comparisons.RSAExhExpressivity.CompositionalRSAScenario : Type

Compositional RSA scenario: scope is a latent variable

• world : RSA.Core.EmbeddedSI.EmbeddedSIWorld
• scope : RSA.Core.EmbeddedSI.ExhScope

def Comparisons.RSAExhExpressivity.compositionalMeaning (config : CompositionalRSAScenario) : Bool

Compositional RSA can express both readings

Equations

• Comparisons.RSAExhExpressivity.compositionalMeaning config = RSA.Core.EmbeddedSI.exhScopedMeaning config.scope config.world

theorem Comparisons.RSAExhExpressivity.compositionalRSA_local_excludes_SA : compositionalMeaning { world := RSA.Core.EmbeddedSI.EmbeddedSIWorld.SA, scope := RSA.Core.EmbeddedSI.ExhScope.local_ } = false

Compositional RSA with local scope excludes SA

theorem Comparisons.RSAExhExpressivity.compositionalRSA_global_allows_SA : compositionalMeaning { world := RSA.Core.EmbeddedSI.EmbeddedSIWorld.SA, scope := RSA.Core.EmbeddedSI.ExhScope.global } = true

Compositional RSA with global scope allows SA

The IBR Perspective

@cite{franke-2011} shows that IBR (the α→∞ limit of RSA) equals exhMW. But this is still SCOPE-BLIND - it's exhMW applied to the WHOLE sentence.

The IBR/exhMW analysis of "every student read some book":

• Alternatives: {"every some", "every all"}
• exhMW excludes worlds where stronger alternatives are true
• Result: excludes only w_AA (where "every all" is true)
• This matches GLOBAL EXH, not local EXH

Even IBR (the limit of RSA) is scope-blind. To get local readings, scope must be a latent variable.

theorem Comparisons.RSAExhExpressivity.ibr_excludes_AA : RSA.Core.EmbeddedSI.embeddedMeaning RSA.Core.EmbeddedSI.EmbeddedSIMessage.everyAll RSA.Core.EmbeddedSI.EmbeddedSIWorld.AA = true

IBR excludes AA (where "every all" is true)

theorem Comparisons.RSAExhExpressivity.ibr_keeps_SA : RSA.Core.EmbeddedSI.embeddedMeaning RSA.Core.EmbeddedSI.EmbeddedSIMessage.everyAll RSA.Core.EmbeddedSI.EmbeddedSIWorld.SA = false

IBR keeps SA (where "every all" is false)

theorem Comparisons.RSAExhExpressivity.ibr_is_global_not_local : RSA.Core.EmbeddedSI.embeddedMeaning RSA.Core.EmbeddedSI.EmbeddedSIMessage.everyAll RSA.Core.EmbeddedSI.EmbeddedSIWorld.AA = true ∧ RSA.Core.EmbeddedSI.embeddedMeaning RSA.Core.EmbeddedSI.EmbeddedSIMessage.everyAll RSA.Core.EmbeddedSI.EmbeddedSIWorld.SA = false ∧ RSA.Core.EmbeddedSI.embeddedMeaning RSA.Core.EmbeddedSI.EmbeddedSIMessage.everyAll RSA.Core.EmbeddedSI.EmbeddedSIWorld.AS = false ∧ RSA.Core.EmbeddedSI.embeddedMeaning RSA.Core.EmbeddedSI.EmbeddedSIMessage.everyAll RSA.Core.EmbeddedSI.EmbeddedSIWorld.SS = false ∧ RSA.Core.EmbeddedSI.globalExhMeaning RSA.Core.EmbeddedSI.EmbeddedSIWorld.AA = false ∧ RSA.Core.EmbeddedSI.globalExhMeaning RSA.Core.EmbeddedSI.EmbeddedSIWorld.SA = true ∧ RSA.Core.EmbeddedSI.localExhMeaning RSA.Core.EmbeddedSI.EmbeddedSIWorld.SA = false

IBR prediction matches global EXH, not local EXH.

IBR (exhMW) excludes worlds where a stronger alternative is true. "every all" is only true at AA, so IBR excludes only AA. This gives exactly the global EXH reading, not the local one.

The Expressivity Hierarchy

1. Standard RSA (scope-blind):

   • Treats utterances atomically
   • Cannot distinguish scope positions
   • In the α→∞ limit, equals global EXH (exhMW)

2. IBR / exhMW (scope-blind):

   • Deterministic limit of RSA
   • Still scope-blind
   • Implements global EXH only

3. Compositional RSA (scope-aware):

   • Lifts scope to a latent variable
   • Can express both global and local readings
   • Listener infers scope jointly with world

4. EXH operator (fully compositional):

   • Can be inserted at any scope position
   • Gives different meanings at different positions
   • Compositional RSA approximates this

Standard RSA ⊂ Compositional RSA ≈ EXH. Standard RSA cannot express local exhaustification. The RSA → IBR → exhMW chain only captures global readings. For local readings, the scope-aware approach of ScontrasPearl2021, LexicalUncertainty, or compositional RSA is needed.

theorem Comparisons.RSAExhExpressivity.hierarchy_is_strict : ∃ (w : RSA.Core.EmbeddedSI.EmbeddedSIWorld), RSA.Core.EmbeddedSI.embeddedMeaning RSA.Core.EmbeddedSI.EmbeddedSIMessage.everySome w = true ∧ compositionalMeaning { world := w, scope := RSA.Core.EmbeddedSI.ExhScope.local_ } = false

The expressivity hierarchy is strict: Standard RSA < Compositional RSA

Witnessed by the existence of a world that compositional RSA can exclude (with local scope) but standard RSA cannot.