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Linglib.Theories.Semantics.Lexical.Noun.Binominal

Binominal Noun Phrase Semantics #

Cross-linguistic semantic composition rules for binominal (N₁-of-N₂) constructions, connecting the taxonomy in Core.Lexical.Binominal to the semantic theories for gradable nouns and quantizing nouns.

Evaluative BNP Semantics (Stage 4) #

Evaluative BNPs (that idiot of a doctor) compose N₁ as a gradable predicate with N₂ as a restricting predicate (@cite{ten-wolde-2023}, @cite{morzycki-2009}). The denotation is conjunctive: x must satisfy both N₂ (be a doctor) and POS(N₁) (be d-idiotic for d ≥ θ).

Evaluative Modifier Semantics (Stage 5) #

EMs (a hell of a game) bleach N₁ to an evaluative modifier: N₁ no longer contributes its own lexical predicate but applies an evaluative measure function (@cite{nouwen-2024}) to a contextually- determined property of N₂.

Binominal Intensifier Semantics (Stage 6) #

BIs (a hell of a good time) bleach N₁ further to a degree word: [N₁ of a] intensifies a following adjective, composing via intensifiedMeaning from @cite{nouwen-2024}.

Quantizing ↔ Pseudo-partitive Bridge #

All quantizing noun classes (@cite{scontras-2014}) — container nouns, atomizers, and measure terms — instantiate pseudo-partitive binominals in @cite{ten-wolde-2023}'s taxonomy.

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def Semantics.Lexical.Noun.Binominal.ebnpSemantics {Entity : Type} (n₁ : GradableNouns.GradableNoun Entity) (n₂ : Entity → Bool) :
Entity → Bool

Evaluative BNP semantics: N₁ ascribes a gradable property to N₂.

"that idiot of a doctor" = the doctor x such that idiot(x) ≥ θ_idiot. This is precisely GradableNoun.pos applied to N₂, where N₁ provides the measure function and contextual standard.

This composition rule applies cross-linguistically: English of, French de, Italian di, Dutch van evaluative BNPs all have the same truth conditions.

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    theorem Semantics.Lexical.Noun.Binominal.ebnp_requires_n₂ {Entity : Type} (n₁ : GradableNouns.GradableNoun Entity) (n₂ : Entity → Bool) (x : Entity) :
    ebnpSemantics n₁ n₂ x = true → n₂ x = true

    Evaluative BNPs are conjunctive: the referent must satisfy both N₂ (doctor) and N₁'s positive form (idiot above threshold).

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    theorem Semantics.Lexical.Noun.Binominal.ebnp_requires_n₁_pos {Entity : Type} (n₁ : GradableNouns.GradableNoun Entity) (n₂ : Entity → Bool) (x : Entity) :
    ebnpSemantics n₁ n₂ x = true → n₁.pos x = true

    Evaluative BNPs entail N₁'s positive form.

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    def Semantics.Lexical.Noun.Binominal.quantizingToOfBinominal :
    Probabilistic.Measurement.QuantizingNounClass → Core.Lexical.Binominal.OfBinominalType

    Map a QuantizingNounClass to the OfBinominalType it instantiates.

    All quantizing nouns (@cite{scontras-2014}) are pseudo-partitive in @cite{ten-wolde-2023}'s taxonomy: N₁ quantizes, N₂ is the semantic head.

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      theorem Semantics.Lexical.Noun.Binominal.all_quantizing_are_pseudopartitive (c : Probabilistic.Measurement.QuantizingNounClass) :
      quantizingToOfBinominal c = Core.Lexical.Binominal.OfBinominalType.pseudoPartitive

      Every quantizing noun class maps to pseudo-partitive.

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      theorem Semantics.Lexical.Noun.Binominal.quantizing_n₂_headed (c : Probabilistic.Measurement.QuantizingNounClass) :
      (quantizingToOfBinominal c).head = Core.Lexical.Binominal.BNPHead.n₂

      Pseudo-partitive is N₂-headed, consistent with the quantizing noun semantics where N₂ denotes the measured stuff.

      "That idiot of a doctor" #

      End-to-end chain: lexical entry → gradable noun theory → EBNP composition → concrete truth conditions.

      Using the Person type and exampleIdiot from GradableNouns:

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      def Semantics.Lexical.Noun.Binominal.isDoctor :
      GradableNouns.Person → Bool

      A doctor predicate for the worked example.

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        theorem Semantics.Lexical.Noun.Binominal.george_is_idiot_doctor :
        ebnpSemantics GradableNouns.exampleIdiot isDoctor GradableNouns.Person.george = true

        George is "that idiot of a doctor": he is a doctor (true) and his idiocy degree (8) exceeds the standard (3).

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        theorem Semantics.Lexical.Noun.Binominal.sarah_is_idiot_doctor :
        ebnpSemantics GradableNouns.exampleIdiot isDoctor GradableNouns.Person.sarah = true

        Sarah is also "an idiot of a doctor": doctor (true), idiocy 4 ≥ 3.

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        theorem Semantics.Lexical.Noun.Binominal.floyd_not_idiot_doctor :
        ebnpSemantics GradableNouns.exampleIdiot isDoctor GradableNouns.Person.floyd = false

        Floyd is not "an idiot of a doctor": he is not a doctor.

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        theorem Semantics.Lexical.Noun.Binominal.george_idiot_independent :
        GradableNouns.exampleIdiot.pos GradableNouns.Person.george = true

        George would be an idiot even if he weren't a doctor — the gradable noun threshold is independent of the N₂ restriction. This confirms that ebnpSemantics is genuinely conjunctive.

        Stage 5: Evaluative Modifier #

        At stage 5 on @cite{ten-wolde-2023}'s grammaticalization cline, N₁ has bleached from a full gradable predicate (EBNP) to an evaluative modifier: a hell of a game ≈ an extremely good game. N₁ no longer contributes its own lexical predicate — instead, [N₁ of a] applies an evaluative measure function (@cite{nouwen-2024}) to a contextually-determined property of N₂.

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        def Semantics.Lexical.Noun.Binominal.emSemantics {Entity : Type} {max : ℕ} (eval : Adjective.Intensification.EvaluativeMeasure max) (contextualDegree : Entity → Core.Scale.Degree max) (θ_eval : Core.Scale.Threshold max) (n₂ : Entity → Bool) :
        Entity → Bool

        EM semantics: N₁ as evaluative modifier of a contextual property of N₂.

        "a hell of a game" = the game x such that the contextually-salient property of x (e.g., quality) exceeds the evaluative threshold set by N₁.

        Unlike EBNP, N₁ does not contribute its own lexical predicate — it has bleached to a pure evaluative function. The contextualDegree parameter represents the pragmatically-determined property being evaluated.

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          theorem Semantics.Lexical.Noun.Binominal.em_requires_n₂ {Entity : Type} {max : ℕ} (eval : Adjective.Intensification.EvaluativeMeasure max) (cdeg : Entity → Core.Scale.Degree max) (θ : Core.Scale.Threshold max) (n₂ : Entity → Bool) (x : Entity) :
          emSemantics eval cdeg θ n₂ x = true → n₂ x = true

          EM preserves the N₂ restriction.

          Stage 6: Binominal Intensifier #

          At stage 6, N₁ has fully grammaticalized to a degree word: a hell of a good time ≈ a very good time. The [N₁ of a] unit modifies a following adjective rather than N₂ directly. Semantics composes via intensifiedMeaning from @cite{nouwen-2024}: the adjective's positive form AND N₁'s evaluative threshold must both be exceeded.

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          def Semantics.Lexical.Noun.Binominal.biSemantics {Entity : Type} {max : ℕ} (eval : Adjective.Intensification.EvaluativeMeasure max) (adjDegree : Entity → Core.Scale.Degree max) (θ_adj θ_eval : Core.Scale.Threshold max) (n₂ : Entity → Bool) :
          Entity → Bool

          BI semantics: N₁ as degree intensifier of a following adjective.

          "a hell of a good time" = the time x such that good(x) is intensified: both the adjective threshold (good enough) and the evaluative threshold (hell-level) are exceeded.

          This directly instantiates @cite{nouwen-2024}'s intensifiedMeaning: the adjective's positive form AND the evaluative measure must both hold.

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            theorem Semantics.Lexical.Noun.Binominal.bi_requires_n₂ {Entity : Type} {max : ℕ} (eval : Adjective.Intensification.EvaluativeMeasure max) (adjDeg : Entity → Core.Scale.Degree max) (θ_adj θ_eval : Core.Scale.Threshold max) (n₂ : Entity → Bool) (x : Entity) :
            biSemantics eval adjDeg θ_adj θ_eval n₂ x = true → n₂ x = true

            BI preserves the N₂ restriction.

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            theorem Semantics.Lexical.Noun.Binominal.bi_entails_em {Entity : Type} {max : ℕ} (eval : Adjective.Intensification.EvaluativeMeasure max) (adjDeg : Entity → Core.Scale.Degree max) (θ_adj θ_eval : Core.Scale.Threshold max) (n₂ : Entity → Bool) (x : Entity) :
            biSemantics eval adjDeg θ_adj θ_eval n₂ x = true → emSemantics eval adjDeg θ_eval n₂ x = true

            BI entails EM when the adjective degree is the contextual property.

            This formalizes the grammaticalization claim: BI is a special case of EM where the contextual property is fixed to a specific adjective, and an additional threshold (the adjective's standard) is imposed. BI strengthens EM by adding the adjective's positive form as a conjunct.

            "A hell of a doctor" / "A hell of a good doctor" #

            Extends the §3 worked example (Person, isDoctor) with EM and BI:

            Uses muHorrible 10 as the evaluative measure for hell: μ_hell(d) = |d − 5|, peaking at extreme degrees.

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            def Semantics.Lexical.Noun.Binominal.doctorQuality :
            GradableNouns.Person → Core.Scale.Degree 10

            Quality measure for EM: contextually-determined goodness as a doctor. George (9) is an outstanding doctor; Sarah (6) is decent; Floyd (3) is poor.

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              def Semantics.Lexical.Noun.Binominal.hellEval :
              Adjective.Intensification.EvaluativeMeasure 10

              Evaluative measure for hell: peaks at extreme degrees. μ_hell(d) = |d − 5| — uses @cite{nouwen-2024}'s extreme-peaking profile. hell is valence-neutral (positive in a hell of a good time, negative in the hell of war); the extreme-peaking shape captures its intensity regardless of polarity.

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                theorem Semantics.Lexical.Noun.Binominal.george_hell_of_doctor :
                emSemantics hellEval doctorQuality (Core.Scale.thr 3 ⋯) isDoctor GradableNouns.Person.george = true

                George is "a hell of a doctor": doctor ✓, quality (9) yields μ_hell(9) = |9−5| = 4 > 3 = θ_eval ✓.

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                theorem Semantics.Lexical.Noun.Binominal.sarah_not_hell_of_doctor :
                emSemantics hellEval doctorQuality (Core.Scale.thr 3 ⋯) isDoctor GradableNouns.Person.sarah = false

                Sarah is not "a hell of a doctor": quality (6) yields μ_hell(6) = |6−5| = 1, not > 3.

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                theorem Semantics.Lexical.Noun.Binominal.george_hell_of_good_doctor :
                biSemantics hellEval doctorQuality (Core.Scale.thr 5 ⋯) (Core.Scale.thr 3 ⋯) isDoctor GradableNouns.Person.george = true

                George is "a hell of a good doctor": good(9 > 5) ✓ AND μ_hell(9) = 4 > 3 ✓. BI compounds both thresholds.

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                theorem Semantics.Lexical.Noun.Binominal.sarah_not_hell_of_good_doctor :
                biSemantics hellEval doctorQuality (Core.Scale.thr 5 ⋯) (Core.Scale.thr 3 ⋯) isDoctor GradableNouns.Person.sarah = false

                Sarah is not "a hell of a good doctor": good(6 > 5) ✓ but μ_hell(6) = 1, not > 3. She's good, but not hell-level good.

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                theorem Semantics.Lexical.Noun.Binominal.george_bi_entails_em :
                emSemantics hellEval doctorQuality (Core.Scale.thr 3 ⋯) isDoctor GradableNouns.Person.george = true

                BI → EM entailment holds in the worked example: George's BI truth entails his EM truth (by bi_entails_em).

                EBNP and EM are categorically independent #

                EBNP uses N₁'s own gradable predicate (GradableNoun.pos); EM uses a bleached evaluative measure function (EvaluativeMeasure). These are different semantic types, so neither entails the other in general.

                The worked example witnesses both directions of independence:

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                theorem Semantics.Lexical.Noun.Binominal.ebnp_not_entails_em :
                ebnpSemantics GradableNouns.exampleIdiot isDoctor GradableNouns.Person.sarah = true ∧ emSemantics hellEval doctorQuality (Core.Scale.thr 3 ⋯) isDoctor GradableNouns.Person.sarah = false

                EBNP does not entail EM: Sarah satisfies ebnpSemantics (she is an idiot-doctor) but fails emSemantics (she is not a hell-of-a-doctor).