Gärtner and Gyuris (2017): On Delimiting the Space of Bias Profiles for Polar Interrogatives

This file formalizes [GG17]'s space of bias profiles. Following [Sud13], a polar interrogative clause type is biased evidentially (by compelling contextual evidence for p, against p, or neither) and epistemically (by the speaker's expectation that p, that not p, or neither), and separately for each of the three polar questions it can express, the positive question and the negative questions with inside and outside negation ([Lad81]). A bias profile chooses a nonempty set of the three values for each of the six cells, so there are 7⁶ = 117649 profiles (card_profile). The paper's delimiting principles are predicates on profiles, and the sizes of the spaces they leave are theorems: the principles that constrain cells separately count as products over cells (card_cellwise), those relating a positive question to its negative counterparts count as sums over the positive question's value (card_perDim, card_byReading), giving 46656 for Convexity, 2744 for Narrow Epistemic Choice, 512 for Static Complementarity with Convexity, 729 and 4096 for the two versions of Polarity Match, 63504 for PPQ ≠ NPQ, and 33856 and 56536 for distributive and collective Quantitative Markedness. Avoid Disagreement entails Convexity and excludes Narrow Epistemic Choice (not_avoidDisagreement_of_narrowEpistemic), the paper's reason to confine Polarity Match to the evidential dimension. Appendix A's five total profiles from [Sud13] and [Gyu17] are checked against every principle (sample_staticComplementarity), and the infelicity judgments of examples (1), (2), (8), and (9) follow from the profiles (rows_felicity).

Implementation notes

The paper's three questions are readings, Option Polarity: PPQ is none, IN-NPQ the inner reading of a negative interrogative, double-checking ¬p (some .negative), and ON-NPQ the outer reading, double-checking p (some .positive); the six cells and the seven choices are enumerated types, so that every decision the kernel makes ranges over a literal list. The Hungarian e-interrogative cannot express an inside-negation question, so it is a partial profile kept apart from the five total ones; the paper's Czech and Romero comparisons belong to the later papers' studies.

References

• [GG17]
• [Sud13]
• [Gyu17]
• [Lad81]
• [BG00]

inductive GartnerGyuris2017.Choice : Type

A bias choice: one of the seven nonempty sets of states of evidence or expectation, for p (+), against p (−), or neither (%).

• pos : Choice
• neg : Choice
• neut : Choice
• posNeut : Choice
• posNeg : Choice
• negNeut : Choice
• all : Choice

@[instance_reducible]

instance GartnerGyuris2017.instDecidableEqChoice : DecidableEq Choice

Equations

• GartnerGyuris2017.instDecidableEqChoice x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[instance_reducible]

instance GartnerGyuris2017.instFintypeChoice : Fintype Choice

Equations

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

def GartnerGyuris2017.Choice.toFinset : Choice → Question.BiasValue

The states a choice admits.

Equations

• GartnerGyuris2017.Choice.pos.toFinset = {1}
• GartnerGyuris2017.Choice.neg.toFinset = {-1}
• GartnerGyuris2017.Choice.neut.toFinset = {0}
• GartnerGyuris2017.Choice.posNeut.toFinset = {1, 0}
• GartnerGyuris2017.Choice.posNeg.toFinset = {1, -1}
• GartnerGyuris2017.Choice.negNeut.toFinset = {-1, 0}
• GartnerGyuris2017.Choice.all.toFinset = Finset.univ

theorem GartnerGyuris2017.Choice.toFinset_sudo : pos.toFinset = Question.BiasValue.plus Polarity.positive ∧ neg.toFinset = Question.BiasValue.plus Polarity.negative ∧ neut.toFinset = Question.BiasValue.neutral ∧ posNeut.toFinset = Question.BiasValue.minus Polarity.negative ∧ negNeut.toFinset = Question.BiasValue.minus Polarity.positive

Five of the seven choices are the bias values of [Sud13]: [+positive], [+negative], [neutral], [−negative] and [−positive].

theorem GartnerGyuris2017.card_choice : Fintype.card Choice = 7

inductive GartnerGyuris2017.Dimension : Type

The dimension a bias is drawn from.

• evidential : Dimension
• epistemic : Dimension

@[instance_reducible]

instance GartnerGyuris2017.instDecidableEqDimension : DecidableEq Dimension

Equations

• GartnerGyuris2017.instDecidableEqDimension x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[instance_reducible]

instance GartnerGyuris2017.instFintypeDimension : Fintype Dimension

Equations

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

inductive GartnerGyuris2017.Cell : Type

The six cells of a profile: a reading of a polar question in a bias dimension.

• ppqEv : Cell
• ppqEp : Cell
• inEv : Cell
• inEp : Cell
• onEv : Cell
• onEp : Cell

@[instance_reducible]

instance GartnerGyuris2017.instDecidableEqCell : DecidableEq Cell

Equations

• GartnerGyuris2017.instDecidableEqCell x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[instance_reducible]

instance GartnerGyuris2017.instFintypeCell : Fintype Cell

Equations

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

def GartnerGyuris2017.Cell.reading : Cell → Option Polarity

The reading of a cell.

Equations

• GartnerGyuris2017.Cell.ppqEv.reading = none
• GartnerGyuris2017.Cell.ppqEp.reading = none
• GartnerGyuris2017.Cell.inEv.reading = some Polarity.negative
• GartnerGyuris2017.Cell.inEp.reading = some Polarity.negative
• GartnerGyuris2017.Cell.onEv.reading = some Polarity.positive
• GartnerGyuris2017.Cell.onEp.reading = some Polarity.positive

def GartnerGyuris2017.Cell.dim : Cell → Dimension

The dimension of a cell.

Equations

• GartnerGyuris2017.Cell.ppqEv.dim = GartnerGyuris2017.Dimension.evidential
• GartnerGyuris2017.Cell.inEv.dim = GartnerGyuris2017.Dimension.evidential
• GartnerGyuris2017.Cell.onEv.dim = GartnerGyuris2017.Dimension.evidential
• GartnerGyuris2017.Cell.ppqEp.dim = GartnerGyuris2017.Dimension.epistemic
• GartnerGyuris2017.Cell.inEp.dim = GartnerGyuris2017.Dimension.epistemic
• GartnerGyuris2017.Cell.onEp.dim = GartnerGyuris2017.Dimension.epistemic

def GartnerGyuris2017.Cell.of : Option Polarity → Dimension → Cell

The cell of a reading in a dimension.

Equations

• GartnerGyuris2017.Cell.of none GartnerGyuris2017.Dimension.evidential = GartnerGyuris2017.Cell.ppqEv
• GartnerGyuris2017.Cell.of none GartnerGyuris2017.Dimension.epistemic = GartnerGyuris2017.Cell.ppqEp
• GartnerGyuris2017.Cell.of (some Polarity.negative) GartnerGyuris2017.Dimension.evidential = GartnerGyuris2017.Cell.inEv
• GartnerGyuris2017.Cell.of (some Polarity.negative) GartnerGyuris2017.Dimension.epistemic = GartnerGyuris2017.Cell.inEp
• GartnerGyuris2017.Cell.of (some Polarity.positive) GartnerGyuris2017.Dimension.evidential = GartnerGyuris2017.Cell.onEv
• GartnerGyuris2017.Cell.of (some Polarity.positive) GartnerGyuris2017.Dimension.epistemic = GartnerGyuris2017.Cell.onEp

@[reducible, inline]

abbrev GartnerGyuris2017.Profile : Type

A bias profile: a choice for each cell.

Equations

• GartnerGyuris2017.Profile = (GartnerGyuris2017.Cell → GartnerGyuris2017.Choice)

theorem GartnerGyuris2017.card_profile : Fintype.card Profile = 117649

Counting principles

@[reducible, inline]

abbrev GartnerGyuris2017.Cellwise (A : Cell → Choice → Prop) (p : Profile) : Prop

A principle that constrains each cell on its own.

Equations

• GartnerGyuris2017.Cellwise A p = ∀ (c : GartnerGyuris2017.Cell), A c (p c)

theorem GartnerGyuris2017.card_cellwise (A : Cell → Choice → Prop) [(c : Cell) → DecidablePred (A c)] : Fintype.card { p : Profile // Cellwise A p } = ∏ c : Cell, Fintype.card { x : Choice // A c x }

A cellwise principle leaves the product over cells of the choices it allows there.

def GartnerGyuris2017.byDim (p : Profile) (d : Dimension) : Option Polarity → Choice

A profile by dimension.

Equations

• GartnerGyuris2017.byDim p d q = p (GartnerGyuris2017.Cell.of q d)

def GartnerGyuris2017.pair (p : Profile) (q : Option Polarity) : Choice × Choice

A reading's evidential and epistemic choices.

Equations

• GartnerGyuris2017.pair p q = (p (GartnerGyuris2017.Cell.of q GartnerGyuris2017.Dimension.evidential), p (GartnerGyuris2017.Cell.of q GartnerGyuris2017.Dimension.epistemic))

def GartnerGyuris2017.readingEquiv {X : Type u_1} (R : X → X → Prop) : { g : Option Polarity → X // R (g none) (g (some Polarity.negative)) ∧ R (g none) (g (some Polarity.positive)) } ≃ (x : X) × { y : X // R x y } × { y : X // R x y }

A principle relating the positive question's value to each negative question's value factors through the positive question's value.

Equations

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

theorem GartnerGyuris2017.card_reading {X : Type u_1} [Fintype X] [DecidableEq X] (R : X → X → Prop) [DecidableRel R] : Fintype.card { g : Option Polarity → X // R (g none) (g (some Polarity.negative)) ∧ R (g none) (g (some Polarity.positive)) } = ∑ x : X, Fintype.card { y : X // R x y } * Fintype.card { y : X // R x y }

def GartnerGyuris2017.dimEquiv : Profile ≃ (Dimension → Option Polarity → Choice)

Profiles as functions from dimensions to reading triples.

Equations

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

theorem GartnerGyuris2017.card_perDim (Q : (Option Polarity → Choice) → Prop) [DecidablePred Q] : Fintype.card { p : Profile // ∀ (d : Dimension), Q (byDim p d) } = Fintype.card { g : Option Polarity → Choice // Q g } ^ 2

A principle imposed within each dimension counts as the square of the per-dimension count.

def GartnerGyuris2017.readingsEquiv : Profile ≃ (Option Polarity → Choice × Choice)

Profiles as functions from readings to pairs of choices.

Equations

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

theorem GartnerGyuris2017.card_byReading (R : Choice × Choice → Choice × Choice → Prop) [DecidableRel R] : Fintype.card { p : Profile // R (pair p none) (pair p (some Polarity.negative)) ∧ R (pair p none) (pair p (some Polarity.positive)) } = ∑ x : Choice × Choice, Fintype.card { y : Choice × Choice // R x y } * Fintype.card { y : Choice × Choice // R x y }

A principle relating each negative question's pair of choices to the positive question's.

The delimiting principles (section 2)

@[reducible, inline]

abbrev GartnerGyuris2017.NoUniformity (p : Profile) : Prop

Section 2.1, No Uniformity: not every cell makes the same choice.

Equations

• GartnerGyuris2017.NoUniformity p = ¬∃ (x : GartnerGyuris2017.Choice), ∀ (c : GartnerGyuris2017.Cell), p c = x

def GartnerGyuris2017.unireadingEquiv : Choice ≃ { p : Profile // ∃ (x : Choice), ∀ (c : Cell), p c = x }

The uniform profiles are the choices.

Equations

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

theorem GartnerGyuris2017.card_noUniformity : Fintype.card { p : Profile // NoUniformity p } = 117642

@[reducible, inline]

abbrev GartnerGyuris2017.PPQNeqNPQ (p : Profile) : Prop

Section 2.2, PPQ ≠ NPQ: in each dimension, negation changes the bias.

Equations

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

theorem GartnerGyuris2017.card_ppqNeqNpq : Fintype.card { p : Profile // PPQNeqNPQ p } = 63504

@[reducible, inline]

abbrev GartnerGyuris2017.MarkednessDistributive (p : Profile) : Prop

Section 2.3.1, distributive Quantitative Markedness (11a): in each dimension the positive question has at least as many options as each negative question.

Equations

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

theorem GartnerGyuris2017.card_markednessDistributive : Fintype.card { p : Profile // MarkednessDistributive p } = 33856

def GartnerGyuris2017.size (h : Choice × Choice) : ℕ

The number of bias options a reading has across both dimensions.

Equations

• GartnerGyuris2017.size h = Finset.card h.1.toFinset + Finset.card h.2.toFinset

@[reducible, inline]

abbrev GartnerGyuris2017.MarkednessCollective (p : Profile) : Prop

Section 2.3.1, collective Quantitative Markedness (11b): across both dimensions the positive question has at least as many options as each negative question.

Equations

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

theorem GartnerGyuris2017.card_markednessCollective : Fintype.card { p : Profile // MarkednessCollective p } = 56536

@[reducible, inline]

abbrev GartnerGyuris2017.QualitativeMarkedness : Profile → Prop

Section 2.3.2, generalized Qualitative Markedness: the neutral value belongs to the positive question's cells and to no negative question's cell.

Equations

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

theorem GartnerGyuris2017.card_qualitativeMarkedness : Fintype.card { p : Profile // QualitativeMarkedness p } = 1296

@[reducible, inline]

abbrev GartnerGyuris2017.AvoidDisagreement : Profile → Prop

Section 2.4, Avoid Disagreement: no negative value for a positive question, no positive value for a negative one.

Equations

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

theorem GartnerGyuris2017.card_avoidDisagreement : Fintype.card { p : Profile // AvoidDisagreement p } = 729

@[reducible, inline]

abbrev GartnerGyuris2017.DontRuleOutAgreement : Profile → Prop

Section 2.4, Don't Rule Out Agreement: every positive-question cell admits the positive value and every negative-question cell the negative one.

Equations

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

theorem GartnerGyuris2017.card_dontRuleOutAgreement : Fintype.card { p : Profile // DontRuleOutAgreement p } = 4096

@[reducible, inline]

abbrev GartnerGyuris2017.Convex (x : Choice) : Prop

Section 2.5: a choice is convex when it does not skip the neutral value between the positive and the negative one, that is, when it is not {+, −}.

Equations

• GartnerGyuris2017.Convex x = (x ≠ GartnerGyuris2017.Choice.posNeg)

@[reducible, inline]

abbrev GartnerGyuris2017.Convexity : Profile → Prop

Section 2.5, Convexity: every cell is convex.

Equations

• GartnerGyuris2017.Convexity = GartnerGyuris2017.Cellwise fun (x : GartnerGyuris2017.Cell) => GartnerGyuris2017.Convex

theorem GartnerGyuris2017.card_convexity : Fintype.card { p : Profile // Convexity p } = 46656

@[reducible, inline]

abbrev GartnerGyuris2017.NarrowChoice (x : Choice) : Prop

Section 2.6: the epistemic options the sample exhibits, {+} and {+, −, %}.

Equations

• GartnerGyuris2017.NarrowChoice x = (x = GartnerGyuris2017.Choice.pos ∨ x = GartnerGyuris2017.Choice.all)

@[reducible, inline]

abbrev GartnerGyuris2017.NarrowEpistemic : Profile → Prop

Section 2.6, Narrow Epistemic Choice: every epistemic cell is {+} or {+, −, %}.

Equations

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

theorem GartnerGyuris2017.card_narrowEpistemic : Fintype.card { p : Profile // NarrowEpistemic p } = 2744

@[reducible, inline]

abbrev GartnerGyuris2017.StaticComplementarity : Profile → Prop

Section 2.7, Static Complementarity: epistemic cells take the narrow options and evidential cells the remaining ones.

Equations

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

theorem GartnerGyuris2017.card_staticComplementarity_convexity : Fintype.card { p : Profile // StaticComplementarity p ∧ Convexity p } = 512

Relations among the principles (section 3.1.2)

theorem GartnerGyuris2017.Convexity_of_avoidDisagreement {p : Profile} (h : AvoidDisagreement p) : Convexity p

Avoid Disagreement entails Convexity: a cell without one of the poles cannot be {+, −}.

theorem GartnerGyuris2017.not_avoidDisagreement_of_narrowEpistemic {p : Profile} (h : NarrowEpistemic p) : ¬AvoidDisagreement p

Narrow Epistemic Choice gives every negative question the positive epistemic value, which Avoid Disagreement forbids; Polarity Match can only be meant evidentially.

Appendix A: the sample

def GartnerGyuris2017.mk (ppqEv ppqEp inEv inEp onEv onEp : Choice) : Profile

A profile from its six cells, in the order positive, inside-negation, outside-negation question, each evidential then epistemic.

Equations

• GartnerGyuris2017.mk ppqEv ppqEp inEv inEp onEv onEp GartnerGyuris2017.Cell.ppqEv = ppqEv
• GartnerGyuris2017.mk ppqEv ppqEp inEv inEp onEv onEp GartnerGyuris2017.Cell.ppqEp = ppqEp
• GartnerGyuris2017.mk ppqEv ppqEp inEv inEp onEv onEp GartnerGyuris2017.Cell.inEv = inEv
• GartnerGyuris2017.mk ppqEv ppqEp inEv inEp onEv onEp GartnerGyuris2017.Cell.inEp = inEp
• GartnerGyuris2017.mk ppqEv ppqEp inEv inEp onEv onEp GartnerGyuris2017.Cell.onEv = onEv
• GartnerGyuris2017.mk ppqEv ppqEp inEv inEp onEv onEp GartnerGyuris2017.Cell.onEp = onEp

def GartnerGyuris2017.englishV1 : Profile

[1] English V1-interrogative ([Sud13]).

Equations

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

def GartnerGyuris2017.japaneseNull : Profile

[2] Japanese ∅-interrogative ([Sud13]).

Equations

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

def GartnerGyuris2017.japaneseNo : Profile

[3] Japanese no-interrogative ([Sud13]).

Equations

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

def GartnerGyuris2017.japaneseDesho : Profile

[4] Japanese desho-interrogative ([Sud13]).

Equations

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

def GartnerGyuris2017.hungarianFallRise : Profile

[5] Hungarian fall-rise interrogative ([Gyu17]).

Equations

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

def GartnerGyuris2017.hungarianE : Cell → Option Choice

[6] Hungarian e-interrogative ([Gyu17]): it cannot express an inside-negation question, so its profile is partial.

Equations

• GartnerGyuris2017.hungarianE GartnerGyuris2017.Cell.ppqEv = some GartnerGyuris2017.Choice.neut
• GartnerGyuris2017.hungarianE GartnerGyuris2017.Cell.ppqEp = some GartnerGyuris2017.Choice.all
• GartnerGyuris2017.hungarianE GartnerGyuris2017.Cell.inEv = none
• GartnerGyuris2017.hungarianE GartnerGyuris2017.Cell.inEp = none
• GartnerGyuris2017.hungarianE GartnerGyuris2017.Cell.onEv = some GartnerGyuris2017.Choice.neut
• GartnerGyuris2017.hungarianE GartnerGyuris2017.Cell.onEp = some GartnerGyuris2017.Choice.pos

def GartnerGyuris2017.sample : List Profile

The five total profiles of the sample.

Equations

• GartnerGyuris2017.sample = [GartnerGyuris2017.englishV1, GartnerGyuris2017.japaneseNull, GartnerGyuris2017.japaneseNo, GartnerGyuris2017.japaneseDesho, GartnerGyuris2017.hungarianFallRise]

theorem GartnerGyuris2017.hungarianFallRise_eq_englishV1 : hungarianFallRise = englishV1

The Hungarian fall-rise interrogative shares the English profile.

theorem GartnerGyuris2017.sample_noUniformity_markednessCollective (p : Profile) : p ∈ sample → NoUniformity p ∧ MarkednessCollective p

No Uniformity and collective Quantitative Markedness hold throughout the sample.

theorem GartnerGyuris2017.sample_markednessDistributive (p : Profile) : p ∈ sample → (MarkednessDistributive p ↔ p ≠ japaneseNull ∧ p ≠ japaneseNo)

Distributive Quantitative Markedness fails for the Japanese ∅- and no-interrogatives, whose outside-negation questions have more evidential options than their positive questions.

theorem GartnerGyuris2017.sample_ppqNeqNpq : (∀ p ∈ sample, PPQNeqNPQ p ↔ p ≠ japaneseNull ∧ p ≠ japaneseDesho) ∧ hungarianE Cell.ppqEv = hungarianE Cell.onEv

PPQ ≠ NPQ fails within the sample for the Japanese ∅- and desho-interrogatives, whose inside-negation question shares a choice with the positive question, and the paper's own counterexample is the Hungarian e-interrogative, whose positive and outside-negation questions share the evidential anti-bias {%}.

theorem GartnerGyuris2017.sample_polarityMatch (p : Profile) : p ∈ sample → ¬AvoidDisagreement p ∧ (DontRuleOutAgreement p ↔ p = japaneseDesho)

Avoid Disagreement fails throughout the sample, while Don't Rule Out Agreement holds only for the desho-interrogative.

theorem GartnerGyuris2017.sample_convexity_narrowEpistemic (p : Profile) : p ∈ sample → Convexity p ∧ (NarrowEpistemic p ↔ p ≠ japaneseDesho)

Convexity holds throughout the sample; Narrow Epistemic Choice fails only for the desho-interrogative, whose negative questions carry {−}.

theorem GartnerGyuris2017.sample_staticComplementarity (p : Profile) : p ∈ sample → (StaticComplementarity p ↔ p ≠ japaneseNo ∧ p ≠ japaneseDesho)

Static Complementarity fails exactly for the Japanese no- and desho-interrogatives, the six violations of (26).

Examples (1), (2), (8), (9): infelicity from the profiles

inductive GartnerGyuris2017.Construction : Type

The clause types whose examples the paper judges.

• englishV1 : Construction
• hungarianE : Construction

@[instance_reducible]

instance GartnerGyuris2017.instDecidableEqConstruction : DecidableEq Construction

Equations

• GartnerGyuris2017.instDecidableEqConstruction x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def GartnerGyuris2017.Construction.cell : Construction → Cell → Option Choice

A clause type's choice for a cell.

Equations

• GartnerGyuris2017.Construction.englishV1.cell x✝ = some (GartnerGyuris2017.englishV1 x✝)
• GartnerGyuris2017.Construction.hungarianE.cell x✝ = GartnerGyuris2017.hungarianE x✝

structure GartnerGyuris2017.Row : Type

An example: the clause type, the question it expresses, and the bias value the scenario fixes in one dimension, with the judgment.

• construction : Construction
• reading : Option Polarity
• dimension : Dimension
• value : SignType
• judgment : Judgment

def GartnerGyuris2017.instDecidableEqRow.decEq (x✝ x✝¹ : Row) : Decidable (x✝ = x✝¹)

Equations

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

@[instance_reducible]

instance GartnerGyuris2017.instDecidableEqRow : DecidableEq Row

Equations

• GartnerGyuris2017.instDecidableEqRow = GartnerGyuris2017.instDecidableEqRow.decEq

def GartnerGyuris2017.Row.ofDatum (ex : Datum) : Option Row

Equations

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

def GartnerGyuris2017.rows : List Row

The judged examples (1), (2a), (2b), (8), (9a), (9b).

Equations

• GartnerGyuris2017.rows = List.filterMap GartnerGyuris2017.Row.ofDatum GartnerGyuris2017.Examples.all

theorem GartnerGyuris2017.rows_felicity (r : Row) : r ∈ rows → (r.judgment = Judgment.acceptable ↔ ∃ x ∈ r.construction.cell (Cell.of r.reading r.dimension), r.value ∈ x.toFinset)

An example is felicitous exactly when the scenario's bias value is among the clause type's options for that cell: the judgments follow from the profiles.