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Linglib.Phenomena.ScalarImplicatures.Embedded.Basic

RSA Embedded Scalar Implicatures: Simplified Model (For Analysis) #

@cite{bergen-levy-goodman-2016} @cite{geurts-2010} @cite{potts-etal-2016}

This file implements a simplified 2-lexicon model to analyze why minimal Lexical Uncertainty models fail to derive embedded implicature patterns.

Status #

The ℚ-based RSA evaluation infrastructure (RSA.Eval, boolToRat, LURSA) has been removed. Type definitions and the model limitation analysis are preserved. RSA computations need to be re-implemented using the new RSAConfig framework.

This File's Purpose #

Demonstrates that a minimal 2-lexicon, 3-world model gives inverted predictions, motivating the richer structure in the full model.

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@[reducible, inline]
abbrev Phenomena.ScalarImplicatures.Embedded.Simplified.EmbeddedWorld :
Type

World states for embedded scalar scenarios use the canonical SomeAllWorld from Phenomena.ScalarImplicatures.Basic: .none (nobody solved any problems), .someNotAll (someone solved some-but-not-all), .all (someone solved all).

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Instances For
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    inductive Phenomena.ScalarImplicatures.Embedded.Simplified.DEUtterance :
    Type

    Utterances for DE context: "No one solved {some/all} problems"

    We need scalar alternatives for RSA to reason about informativity.

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      @[implicit_reducible]
      instance Phenomena.ScalarImplicatures.Embedded.Simplified.instDecidableEqDEUtterance :
      DecidableEq DEUtterance
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      @[implicit_reducible]
      instance Phenomena.ScalarImplicatures.Embedded.Simplified.instReprDEUtterance :
      Repr DEUtterance
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      def Phenomena.ScalarImplicatures.Embedded.Simplified.instReprDEUtterance.repr :
      DEUtterance → ℕ → Std.Format
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      • One or more equations did not get rendered due to their size.
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        def Phenomena.ScalarImplicatures.Embedded.Simplified.instInhabitedDEUtterance.default :
        DEUtterance
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          @[implicit_reducible]
          instance Phenomena.ScalarImplicatures.Embedded.Simplified.instInhabitedDEUtterance :
          Inhabited DEUtterance
          Equations

          The Lexical Uncertainty Model #

          Each lexicon L assigns meanings to "some":

          The listener reasons over which lexicon the speaker is using.

          source
          def Phenomena.ScalarImplicatures.Embedded.Simplified.lexBaseMeaning :
          DEUtterance → EmbeddedWorld → Bool

          Base lexicon meaning: "some" = at-least-one

          "No one solved some problems" under L_base:

          • True only when nobody solved any problems
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            def Phenomena.ScalarImplicatures.Embedded.Simplified.lexRefinedMeaning :
            DEUtterance → EmbeddedWorld → Bool

            Refined lexicon meaning: "some" = some-but-not-all

            "No one solved some problems" under L_refined:

            • True when nobody solved "some-but-not-all"
            • This is TRUE when someone solved ALL (they didn't solve "some-but-not-all")!
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              inductive Phenomena.ScalarImplicatures.Embedded.Simplified.UEUtterance :
              Type

              Utterances for UE context

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                @[implicit_reducible]
                instance Phenomena.ScalarImplicatures.Embedded.Simplified.instDecidableEqUEUtterance :
                DecidableEq UEUtterance
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                def Phenomena.ScalarImplicatures.Embedded.Simplified.instReprUEUtterance.repr :
                UEUtterance → ℕ → Std.Format
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                • One or more equations did not get rendered due to their size.
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                  @[implicit_reducible]
                  instance Phenomena.ScalarImplicatures.Embedded.Simplified.instReprUEUtterance :
                  Repr UEUtterance
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                  @[implicit_reducible]
                  instance Phenomena.ScalarImplicatures.Embedded.Simplified.instInhabitedUEUtterance :
                  Inhabited UEUtterance
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                  def Phenomena.ScalarImplicatures.Embedded.Simplified.instInhabitedUEUtterance.default :
                  UEUtterance
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                    def Phenomena.ScalarImplicatures.Embedded.Simplified.lexBaseUEMeaning :
                    UEUtterance → EmbeddedWorld → Bool

                    Base lexicon meaning for UE: "some" = at-least-one

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                      def Phenomena.ScalarImplicatures.Embedded.Simplified.lexRefinedUEMeaning :
                      UEUtterance → EmbeddedWorld → Bool

                      Refined lexicon meaning for UE: "some" = some-but-not-all

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                      Instances For

                        Analysis of Results #

                        With α = 1 and uniform priors, the simplified 2-lexicon model gives INVERTED predictions compared to the empirical pattern. This motivates the need for richer model structure.

                        DE Context ("No one solved some"):

                        UE Context ("Someone solved some"):

                        Why This Happens #

                        The key asymmetry is world coverage:

                        In DE:

                        L_refined makes the utterance true in MORE worlds, so even though L_base is more informative, L_refined gets extra probability mass.

                        What Potts et al. Actually Does #

                        The paper succeeds because of richer model structure:

                        1. Multiple refinable items: Not just "some", but also proper names, predicates like "scored" vs "aced" (equation 14)

                        2. Richer world space: 3 players × 3 outcomes = 10 equivalence classes

                        3. Message alternatives: Full cross-product of quantifiers and predicates

                        4. Low λ = 0.1: Speaker nearly uniform, so implicatures emerge from lexicon structure, not informativity