[TBD18]: How Projective Is Projective Content?

The paper's central contribution is the Gradient Projection Principle (GPP):

"If content C is expressed by a constituent embedded under an entailment-canceling operator, then C projects to the extent that it is not at-issue."

It makes gradient the binary Projection Principle of the pragmatic account ([STBR10], [Rob12]) — "projects iff not at-issue". This file formalizes the principle and its structural consequences, not the experimental tables, taking the tight reading of "to the extent that": projection degree equals not-at-issueness. The empirical claim is a gradient correlation with item-level variance, not this identity (cf. the dependency's MonotoneAntiCorrelation docstring).

Main definitions

• gppProjection — the GPP map, the complement of at-issueness (Set.Icc.symm).
• pottsProjection — [Pot05]'s rival: CI projects maximally, at-issueness-blind.
• both are run against the pooled per-expression data (Generalizations.Projectivity.allData).

Main results

• gppProjection_antitone — the GPP as order-reversal.
• gpp_excludes_atIssue — recovers the binary Projection Principle as a threshold collapse.
• gpp_below_potts_of_atIssue, gpp_eq_potts_iff_backgrounded — contra [Pot05]: the accounts agree only on fully-backgrounded content.
• gpp_beats_potts_below_diagonal — on low-projectivity items the GPP beats Potts.

Implementation notes

Degrees and thresholds are the Discourse.AtIssueness scales on Set.Icc (0 : ℚ) 1; the GPP map is Set.Icc.symm. Potts's maximal projection is grounded in Pragmatics.Expressives.TwoDimProp.ci_projects_through_neg.

The Gradient Projection Principle

def TonhauserBeaverDegen2018.gppProjection (ai : Discourse.AtIssueness.AtIssuenessDegree) : Discourse.AtIssueness.ProjectivityDegree

The GPP map: projection degree is the complement of at-issueness — content projects to the extent it is not at-issue ([TBD18]).

Equations

• TonhauserBeaverDegen2018.gppProjection ai = Set.Icc.symm ai

theorem TonhauserBeaverDegen2018.gppProjection_antitone : Antitone gppProjection

The GPP as order-reversal: more at-issue content is no more projective.

theorem TonhauserBeaverDegen2018.gppProjection_zero : gppProjection 0 = 1

Fully not-at-issue content (at-issueness 0) projects maximally.

theorem TonhauserBeaverDegen2018.gppProjection_one : gppProjection 1 = 0

Fully at-issue content (at-issueness 1) does not project.

Recovering the binary Projection Principle

The binary principle ([STBR10]) — projects iff not at-issue — is the threshold collapse of the gradient GPP.

theorem TonhauserBeaverDegen2018.gpp_projects_iff (ai : Discourse.AtIssueness.AtIssuenessDegree) (θ : Discourse.AtIssueness.ProjectivityThreshold) : Discourse.AtIssueness.isProjective (gppProjection ai) θ ↔ ↑ai < ↑(Set.Icc.symm θ)

The GPP projects past θ iff at-issueness is below the complementary threshold.

theorem TonhauserBeaverDegen2018.gpp_excludes_atIssue (ai : Discourse.AtIssueness.AtIssuenessDegree) (θ : Discourse.AtIssueness.ProjectivityThreshold) : ¬(Discourse.AtIssueness.isAtIssue ai (Set.Icc.symm θ) ∧ Discourse.AtIssueness.isProjective (gppProjection ai) θ)

The binary Projection Principle: never both at-issue and projecting at complementary thresholds.

Contra Potts

[Pot05] predicts CI content (appositives, NRRCs, expressives) projects maximally and obligatorily — its CI dimension is unchanged by every entailment-canceling operator. The GPP ties projection to at-issueness, so any at-issue content projects below the ceiling; the two agree only for fully-backgrounded content.

def TonhauserBeaverDegen2018.pottsProjection : Discourse.AtIssueness.AtIssuenessDegree → Discourse.AtIssueness.ProjectivityDegree

[Pot05]'s prediction: CI content projects maximally (degree 1), regardless of at-issueness.

Equations

• TonhauserBeaverDegen2018.pottsProjection x✝ = 1

@[simp]

theorem TonhauserBeaverDegen2018.pottsProjection_val (ai : Discourse.AtIssueness.AtIssuenessDegree) : ↑(pottsProjection ai) = 1

theorem TonhauserBeaverDegen2018.potts_atIssue_blind (ai₁ ai₂ : Discourse.AtIssueness.AtIssuenessDegree) : pottsProjection ai₁ = pottsProjection ai₂

Potts's prediction is at-issueness-blind — the same for all content, which the GPP denies.

theorem TonhauserBeaverDegen2018.potts_ci_invariant_under_neg {W : Type u_1} (p : Pragmatics.Expressives.TwoDimProp W) : p.neg.ci = p.ci

Potts's maximal projection abstracts the operator-invariance of the CI dimension: negation leaves CI content unchanged ([Pot05]).

theorem TonhauserBeaverDegen2018.gpp_below_potts_of_atIssue {ai : Discourse.AtIssueness.AtIssuenessDegree} (h : 0 < ↑ai) : ↑(gppProjection ai) < ↑(pottsProjection ai)

Contra [Pot05]: any at-issue content (at-issueness > 0) projects strictly below Potts's ceiling — the structural form of "appositives are not maximally projective".

theorem TonhauserBeaverDegen2018.gpp_eq_potts_iff_backgrounded (ai : Discourse.AtIssueness.AtIssuenessDegree) : gppProjection ai = pottsProjection ai ↔ ai = 0

The GPP and Potts agree iff the content is fully backgrounded (at-issueness 0).

theorem TonhauserBeaverDegen2018.appositive_potts_independent : Pragmatics.Expressives.appositiveProperties.independent = true

Potts files appositives in the independent CI dimension — the source of the maximal-projection prediction the GPP refines.

The GPP as a MonotoneAntiCorrelation

Discourse.AtIssueness.MonotoneAntiCorrelation (built for this paper, consumed by Studies/SolstadBott2024) bundles anti-correlated pairs; the GPP produces one from any list of at-issueness values.

def TonhauserBeaverDegen2018.gppAntiCorrelation (ais : List ℚ) : Discourse.AtIssueness.MonotoneAntiCorrelation

Any list of at-issueness values, paired with their GPP projection, forms a MonotoneAntiCorrelation.

Equations

• TonhauserBeaverDegen2018.gppAntiCorrelation ais = { pairs := List.map (fun (a : ℚ) => { atIssueness := a, projectivity := 1 - a }) ais, anticorrelated := ⋯ }

Illustrations from the paper

The paper's qualitative findings instantiate the GPP: stated as hypotheses on at-issueness, the projectivity ordering follows from gppProjection_antitone.

theorem TonhauserBeaverDegen2018.only_no_more_projective_than_nrrc {onlyAI nrrcAI : Discourse.AtIssueness.AtIssuenessDegree} (h : nrrcAI ≤ onlyAI) : gppProjection onlyAI ≤ gppProjection nrrcAI

Since only is more at-issue than an NRRC, the GPP predicts it projects no more ([TBD18]).

theorem TonhauserBeaverDegen2018.appositive_not_maximally_projective {apposAI : Discourse.AtIssueness.AtIssuenessDegree} (h : 0 < ↑apposAI) : ↑(gppProjection apposAI) < 1

At-issue appositive content projects sub-maximally — the GPP reading of the central result against [Pot05].

Predicting against the data

gppProjection and pottsProjection map at-issueness to predicted projectivity; the paper's per-expression means are pooled in Generalizations.Projectivity.allData (artifact-sourced rows in Data.Examples.TonhauserBeaverDegen2018). The means are continuous, so per-row predictions are computed over allData (string paperFeatures and ℚ do not reduce in the kernel); the provable content is each account's systematic error.

theorem TonhauserBeaverDegen2018.gpp_errs_off_diagonal (d : Generalizations.Projectivity.ProjectionDatum) (h : ↑d.projectivity ≠ ↑d.notAtIssueness) : 0 < Generalizations.Projectivity.predictionError gppProjection d

The GPP errs on any content whose projectivity differs from its not-at-issueness — the off-diagonal rows (establish below it, occasion verbs above it).

theorem TonhauserBeaverDegen2018.potts_errs_subceiling (d : Generalizations.Projectivity.ProjectionDatum) (h : ↑d.projectivity < 1) : 0 < Generalizations.Projectivity.predictionError pottsProjection d

Potts over-predicts every content below the ceiling (projectivity < 1).

theorem TonhauserBeaverDegen2018.gpp_beats_potts_below_diagonal (d : Generalizations.Projectivity.ProjectionDatum) (h1 : ↑d.projectivity < ↑d.notAtIssueness) (h2 : ↑d.notAtIssueness < 1) : Generalizations.Projectivity.predictionError gppProjection d < Generalizations.Projectivity.predictionError pottsProjection d

Below both its not-at-issueness and the ceiling, the GPP is strictly closer to the observation than Potts — the low-projectivity items the paper highlights.