Tieu, Križ and Chemla (2019): children's homogeneity

The trucks are blue and The trucks are not blue are both neither true nor false when some but not all of the trucks are blue, a GAP context. Tieu, Križ and Chemla test four- and five-year-old French-speaking children on such sentences and on the not all implicature of some, against Magri's account, on which the definite means some and reaches all by exhaustifying twice, the not all implicature being a step of that computation.

A child may instead read the definite as a quantifier, existential or universal, scoping under or over negation. Such a construal is classical, and a reading is the supervaluation over its construals: the homogeneous reading over the existential and the universal construal, as in the paper's statement of the supervaluation account, and the scope-ambiguous universal of Experiment 2 over the two scopes of the universal.

Main results

• value_of_isGap: in a GAP context the readings take the values of Figure 2 and Table 8; value_of_subset, value_of_disjoint: elsewhere they agree, which the control trials check.
• value_homogeneous: the homogeneous reading is Križ's trivalent plural.
• binaryPattern_quantified_bijective: the four quantified readings give the four possible pairs of binary judgments, one of them the wide-scope existential that no child showed.
• binaryPattern_eq_wideScopeUniversal, gapPattern_injective: binary judgments conflate the homogeneous reading with the wide-scope universal, and ternary ones separate them.
• homogeneous_weakNegation: the homogeneous reading with weak negation is the universal one.
• definite_in_gap_iff_some_in_all: on Magri's account a participant accepts the positive definite in a GAP context exactly when they accept some where every object has the property.
• implicaturePattern_of_isGap: the account predicts the existential pattern without the implicature and the homogeneous one with it.

Implementation notes

A world is the set of objects with the property. Magri's account is the double strengthening of Studies/Magri2014, and a participant computes the not all implicature exactly when all is among their alternatives to some (.strong ∈ A), since the paper's argument rests on the two inferences sharing their alternatives. A binary judgment accepts exactly the true sentences, and Table 8's rewards 1, 0 and −1 are the three truth values. Not formalized are Table 8's partial-truth group, which on the definite responds as the homogeneous one; the ternary coding of the implicature trials, after Katsos and Bishop; and the statistics. In Experiment 1, 16 of 24 children were homogeneous and 8 existential, and 6 of the homogeneous children and 5 of 22 adults lacked the implicature; in Experiment 2 the homogeneous group without the implicature had 5 of 24 children and 2 of 25 adults.

TODO

The printed Table 2 has the universal groups accept the positive sentence in a GAP context, against Figure 2. The printed Table 8 gives the scope-ambiguous and wide-scope universal groups GAP rewards that contradict their definitions beside it, and fixes rows for all under negation and some in a GAP context that no definition determines. The definitions are followed.

References

• [TKC19]
• [Mag14]
• [Kri16]
• [Spe13b]
• [KB11]

def TieuKrizChemla2019.IsGap {Atom : Type u_1} (x w : Finset Atom) : Prop

A GAP context is one where some but not all objects of the plurality have the property, Figure 1.

Equations

• TieuKrizChemla2019.IsGap x w = ((∃ a ∈ x, a ∈ w) ∧ ∃ a ∈ x, a ∉ w)

Construals

inductive TieuKrizChemla2019.Force : Type

The force of a construal of the plural definite is existential, (6), or universal, (7).

• existential : Force
• universal : Force

@[instance_reducible]

instance TieuKrizChemla2019.instDecidableEqForce : DecidableEq Force

Equations

• TieuKrizChemla2019.instDecidableEqForce x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[instance_reducible]

instance TieuKrizChemla2019.instFintypeForce : Fintype Force

Equations

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

def TieuKrizChemla2019.Force.gq {Atom : Type u_1} : Force → Quantifier.GQ Atom

The determiner of a force is some or every.

Equations

• TieuKrizChemla2019.Force.existential.gq = Quantifier.GQ.some
• TieuKrizChemla2019.Force.universal.gq = Quantifier.GQ.every

inductive TieuKrizChemla2019.Scope : Type

A construal of the definite scopes under or over sentential negation.

• low : Scope
• wide : Scope

@[instance_reducible]

instance TieuKrizChemla2019.instDecidableEqScope : DecidableEq Scope

Equations

• TieuKrizChemla2019.instDecidableEqScope x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[instance_reducible]

instance TieuKrizChemla2019.instFintypeScope : Fintype Scope

Equations

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

def TieuKrizChemla2019.Scope.neg {Atom : Type u_1} : Scope → Quantifier.GQ Atom → Quantifier.GQ Atom

Negation at a scope takes the outer negation of a quantifier scoping under it and the inner negation of one scoping over it.

Equations

• TieuKrizChemla2019.Scope.low.neg x✝ = x✝ᶜ
• TieuKrizChemla2019.Scope.wide.neg x✝ = x✝.innerNeg

structure TieuKrizChemla2019.Construal : Type

A bivalent construal of the plural definite is a force with a scope relative to negation.

• force : Force

  The quantificational force.

• scope : Scope

  The scope relative to negation.

@[instance_reducible]

instance TieuKrizChemla2019.instDecidableEqConstrual : DecidableEq Construal

Equations

• TieuKrizChemla2019.instDecidableEqConstrual = TieuKrizChemla2019.instDecidableEqConstrual.decEq

def TieuKrizChemla2019.instDecidableEqConstrual.decEq (x✝ x✝¹ : Construal) : Decidable (x✝ = x✝¹)

Equations

• TieuKrizChemla2019.instDecidableEqConstrual.decEq { force := a, scope := a_1 } { force := b, scope := b_1 } = if h : a = b then h ▸ if h : a_1 = b_1 then h ▸ isTrue ⋯ else isFalse ⋯ else isFalse ⋯

@[instance_reducible]

instance TieuKrizChemla2019.instFintypeConstrual : Fintype Construal

Equations

• TieuKrizChemla2019.instFintypeConstrual = Fintype.ofEquiv ((_ : TieuKrizChemla2019.Force) × TieuKrizChemla2019.Scope) TieuKrizChemla2019.Construal.proxyTypeEquiv

def TieuKrizChemla2019.Construal.gq {Atom : Type u_1} (c : Construal) : Polarity → Quantifier.GQ Atom

The quantifier a construal gives the definite at a polarity.

Equations

• c.gq Polarity.positive = c.force.gq
• c.gq Polarity.negative = c.scope.neg c.force.gq

def TieuKrizChemla2019.Construal.Holds {Atom : Type u_1} (x : Finset Atom) (c : Construal) (p : Polarity) (w : Finset Atom) : Prop

Under a construal, the Xs are P at a polarity holds at the world w, the objects with the property.

Equations

• TieuKrizChemla2019.Construal.Holds x c p w = c.gq p (fun (x_1 : Atom) => x_1 ∈ x) fun (x : Atom) => x ∈ w

@[instance_reducible]

instance TieuKrizChemla2019.instDecidableHoldsOfDecidableEq {Atom : Type u_1} (x : Finset Atom) [DecidableEq Atom] (c : Construal) (p : Polarity) (w : Finset Atom) : Decidable (Construal.Holds x c p w)

Equations

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

Readings

inductive TieuKrizChemla2019.Reading : Type

The readings a participant may assign the plural definite are the quantified readings, among them the existential and the universal of Figure 2, the wide-scope universal of Table 8 and the wide-scope existential of the empty fourth group; the homogeneous reading; and the universal ambiguous in scope relative to negation, Table 8.

• quantified (c : Construal) : Reading

  The definite as a quantifier at a scope.

• homogeneous : Reading

  The definite with a truth-value gap, (1)–(2).

• scopeAmbiguous : Reading

  The definite as all, ambiguous in scope relative to negation.

def TieuKrizChemla2019.instDecidableEqReading.decEq (x✝ x✝¹ : Reading) : Decidable (x✝ = x✝¹)

Equations

• One or more equations did not get rendered due to their size.
• TieuKrizChemla2019.instDecidableEqReading.decEq (TieuKrizChemla2019.Reading.quantified a) (TieuKrizChemla2019.Reading.quantified b) = if h : a = b then h ▸ isTrue ⋯ else isFalse ⋯
• TieuKrizChemla2019.instDecidableEqReading.decEq (TieuKrizChemla2019.Reading.quantified c) TieuKrizChemla2019.Reading.homogeneous = isFalse ⋯
• TieuKrizChemla2019.instDecidableEqReading.decEq (TieuKrizChemla2019.Reading.quantified c) TieuKrizChemla2019.Reading.scopeAmbiguous = isFalse ⋯
• TieuKrizChemla2019.instDecidableEqReading.decEq TieuKrizChemla2019.Reading.homogeneous (TieuKrizChemla2019.Reading.quantified c) = isFalse ⋯
• TieuKrizChemla2019.instDecidableEqReading.decEq TieuKrizChemla2019.Reading.homogeneous TieuKrizChemla2019.Reading.homogeneous = isTrue ⋯
• TieuKrizChemla2019.instDecidableEqReading.decEq TieuKrizChemla2019.Reading.scopeAmbiguous (TieuKrizChemla2019.Reading.quantified c) = isFalse ⋯
• TieuKrizChemla2019.instDecidableEqReading.decEq TieuKrizChemla2019.Reading.scopeAmbiguous TieuKrizChemla2019.Reading.scopeAmbiguous = isTrue ⋯

@[instance_reducible]

instance TieuKrizChemla2019.instDecidableEqReading : DecidableEq Reading

Equations

• TieuKrizChemla2019.instDecidableEqReading = TieuKrizChemla2019.instDecidableEqReading.decEq

@[instance_reducible]

instance TieuKrizChemla2019.instFintypeReading : Fintype Reading

Equations

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

def TieuKrizChemla2019.Reading.construals : Reading → Finset Construal

The construals a reading supervaluates over. The homogeneous reading is true when (8a) and (8b) both are and false when neither is, and the scope-ambiguous one is true or false according to where the universal takes scope.

Equations

• One or more equations did not get rendered due to their size.
• (TieuKrizChemla2019.Reading.quantified c).construals = {c}

theorem TieuKrizChemla2019.Reading.construals_nonempty (r : Reading) : r.construals.Nonempty

def TieuKrizChemla2019.Reading.value {Atom : Type u_1} (x : Finset Atom) [DecidableEq Atom] (r : Reading) (p : Polarity) : Finset Atom → SignType

The value of the definite sentence at a polarity under a reading is the supervaluation over the reading's construals.

Equations

• TieuKrizChemla2019.Reading.value x r p w = SignType.supervaluation r.construals fun (x_1 : TieuKrizChemla2019.Construal) => TieuKrizChemla2019.Construal.Holds x x_1 p w

def TieuKrizChemla2019.Reading.gapPattern : Reading → SignType × SignType

The values of the positive and the negative sentence in a GAP context under each reading, Figure 2 and the group definitions of Table 8.

Equations

• (TieuKrizChemla2019.Reading.quantified { force := TieuKrizChemla2019.Force.existential, scope := TieuKrizChemla2019.Scope.low }).gapPattern = (1, -1)
• (TieuKrizChemla2019.Reading.quantified { force := TieuKrizChemla2019.Force.existential, scope := TieuKrizChemla2019.Scope.wide }).gapPattern = (1, 1)
• (TieuKrizChemla2019.Reading.quantified { force := TieuKrizChemla2019.Force.universal, scope := TieuKrizChemla2019.Scope.low }).gapPattern = (-1, 1)
• (TieuKrizChemla2019.Reading.quantified { force := TieuKrizChemla2019.Force.universal, scope := TieuKrizChemla2019.Scope.wide }).gapPattern = (-1, -1)
• TieuKrizChemla2019.Reading.homogeneous.gapPattern = (0, 0)
• TieuKrizChemla2019.Reading.scopeAmbiguous.gapPattern = (-1, 0)

def TieuKrizChemla2019.Reading.binaryPattern (r : Reading) : Bool × Bool

A binary judgment accepts exactly the true sentences, collapsing the intermediate reward into the minimal one.

Equations

• r.binaryPattern = Prod.map (fun (v : SignType) => decide (v = 1)) (fun (v : SignType) => decide (v = 1)) r.gapPattern

Predictions

theorem TieuKrizChemla2019.Construal.holds_of_isGap {Atom : Type u_1} {x w : Finset Atom} (hw : IsGap x w) (c : Construal) (p : Polarity) : Holds x c p w ↔ (c.force = Force.existential ↔ p = Polarity.positive ∨ c.scope = Scope.wide)

In a GAP context the existential holds of the property and of its negation and the universal of neither, so a construal holds exactly when it is existential at the positive polarity or over negation.

theorem TieuKrizChemla2019.Construal.holds_of_subset {Atom : Type u_1} {x w : Finset Atom} (hx : x.Nonempty) (hw : x ⊆ w) (c : Construal) (p : Polarity) : Holds x c p w ↔ p = Polarity.positive

Where every object has the property, every construal makes the positive sentence true and the negative one false.

theorem TieuKrizChemla2019.Construal.holds_of_disjoint {Atom : Type u_1} {x w : Finset Atom} (hx : x.Nonempty) (hw : Disjoint x w) (c : Construal) (p : Polarity) : Holds x c p w ↔ p = Polarity.negative

Where no object has the property, every construal makes the positive sentence false and the negative one true.

theorem TieuKrizChemla2019.value_of_isGap {Atom : Type u_1} {x w : Finset Atom} [DecidableEq Atom] (hw : IsGap x w) (r : Reading) : (Reading.value x r Polarity.positive w, Reading.value x r Polarity.negative w) = r.gapPattern

In a GAP context every reading takes the values of Figure 2 and Table 8.

theorem TieuKrizChemla2019.value_of_subset {Atom : Type u_1} {x w : Finset Atom} [DecidableEq Atom] (hx : x.Nonempty) (hw : x ⊆ w) (r : Reading) : (Reading.value x r Polarity.positive w, Reading.value x r Polarity.negative w) = (1, -1)

Where every object has the property all readings make the positive sentence true and the negative one false, as the clearly true and clearly false controls of both experiments require.

theorem TieuKrizChemla2019.value_of_disjoint {Atom : Type u_1} {x w : Finset Atom} [DecidableEq Atom] (hx : x.Nonempty) (hw : Disjoint x w) (r : Reading) : (Reading.value x r Polarity.positive w, Reading.value x r Polarity.negative w) = (-1, 1)

Where no object has the property all readings make the positive sentence false and the negative one true.

@[simp]

theorem TieuKrizChemla2019.value_quantified {Atom : Type u_1} {x : Finset Atom} [DecidableEq Atom] (c : Construal) (p : Polarity) (w : Finset Atom) : Reading.value x (Reading.quantified c) p w = SignType.ofProp (Construal.Holds x c p w)

A quantified reading is classical.

theorem TieuKrizChemla2019.value_negative_of_low {Atom : Type u_1} {x : Finset Atom} [DecidableEq Atom] {r : Reading} (h : ∀ c ∈ r.construals, c.scope = Scope.low) (w : Finset Atom) : Reading.value x r Polarity.negative w = -Reading.value x r Polarity.positive w

A reading whose construals all scope under negation negates by strong Kleene negation.

theorem TieuKrizChemla2019.value_homogeneous {Atom : Type u_1} {x : Finset Atom} [DecidableEq Atom] (hx : x.Nonempty) : Reading.value x Reading.homogeneous Polarity.positive = Homogeneity.barePlural (fun (x1 : Atom) (x2 : Finset Atom) => x1 ∈ x2) x

The homogeneous reading is the trivalent plural barePlural, the supervaluation over the objects of the plurality.

theorem TieuKrizChemla2019.isGap_iff_not_homogeneous {Atom : Type u_1} {x w : Finset Atom} (hx : x.Nonempty) : IsGap x w ↔ ¬Homogeneity.Homogeneous {{w : Finset Atom | ∃ a ∈ x, a ∈ w}, {w : Finset Atom | x ⊆ w}} w

A GAP context is where (8a) and (8b) disagree, so where the presupposition that all or none of the objects have the property fails.

theorem TieuKrizChemla2019.homogeneous_weakNegation {Atom : Type u_1} {x : Finset Atom} [DecidableEq Atom] (hx : x.Nonempty) (w : Finset Atom) : -SignType.ofProp (Reading.value x Reading.homogeneous Polarity.positive w = 1) = Reading.value x (Reading.quantified { force := Force.universal, scope := Scope.low }) Polarity.negative w

A participant who reads the definite homogeneously but reverses the positive judgment to obtain the negative one, weak negation (fn 23), responds as a universal participant.

theorem TieuKrizChemla2019.domainRestriction_of_isGap {Atom : Type u_1} {x w : Finset Atom} [DecidableEq Atom] (hw : IsGap x w) : Reading.value ({x ∈ x | x ∈ w}) Reading.homogeneous Polarity.positive w = 1 ∧ Reading.value ({x ∈ x | x ∉ w}) Reading.homogeneous Polarity.negative w = 1

A child who restricts the plurality to the objects that verify the sentence, (19) and (20), accepts both sentences in a GAP context, the pattern of the wide-scope existential, which no child showed.

Binary and ternary judgments

theorem TieuKrizChemla2019.gapPattern_injective : Function.Injective Reading.gapPattern

Ternary judgments separate all readings in a GAP context, Experiment 2's design.

theorem TieuKrizChemla2019.binaryPattern_quantified_bijective : Function.Bijective fun (c : Construal) => (Reading.quantified c).binaryPattern

The four quantified readings give the four possible pairs of binary judgments in a GAP context, the three of Figure 2 and the wide-scope existential of the empty fourth group.

theorem TieuKrizChemla2019.binaryPattern_eq_wideScopeUniversal (r : Reading) : r ∈ {Reading.homogeneous, Reading.scopeAmbiguous} → r.binaryPattern = (Reading.quantified { force := Force.universal, scope := Scope.wide }).binaryPattern

Binary judgments conflate the homogeneous reading and the scope-ambiguous universal with the wide-scope universal, all three rejecting both sentences in a GAP context.

The implicature account

def TieuKrizChemla2019.mates (A : Finset Magri2014.Item) : Magri2014.Item → Finset Magri2014.Item

A participant's Horn-mates are those of Magri's primal configuration with A as the alternatives to some. Magri's are {.mystery, .strong} (mates_primal), with all among them.

Equations

• TieuKrizChemla2019.mates A = Function.update Magri2014.primalMates Magri2014.Item.weak A

theorem TieuKrizChemla2019.mates_primal : mates {Magri2014.Item.mystery, Magri2014.Item.strong} = Magri2014.primalMates

theorem TieuKrizChemla2019.exh_mates_weak_of_notMem {W : Type u_2} [Fintype W] [DecidableEq W] {wk st : Finset W} {A : Finset Magri2014.Item} (hA : Magri2014.Item.strong ∉ A) : Magri2014.exh (mates A) (Magri2014.primal wk st) Magri2014.Item.weak = wk

Without all among its alternatives, some excludes nothing.

theorem TieuKrizChemla2019.exh_mates_weak_of_mem {W : Type u_2} [Fintype W] [DecidableEq W] {wk st : Finset W} {A : Finset Magri2014.Item} (hA : Magri2014.Item.strong ∈ A) (h : (wk \ st).Nonempty) : Magri2014.exh (mates A) (Magri2014.primal wk st) Magri2014.Item.weak = wk \ st

With all among its alternatives, some is strengthened to some but not all, (10).

theorem TieuKrizChemla2019.strengthened_mates_mystery {W : Type u_2} [Fintype W] [DecidableEq W] {wk st : Finset W} {A : Finset Magri2014.Item} : Magri2014.strengthened (mates A) (Magri2014.primal wk st) Magri2014.Item.mystery = Exhaustification.innocent.exh {Magri2014.exh (mates A) (Magri2014.primal wk st) Magri2014.Item.weak} wk

The strengthened definite (11) is computed from the exhaustified some (10). The inner exhaustification of the definite is vacuous and the outer one denies the exhaustified some, so the not all implicature is a sub-computation of the homogeneity implicature.

theorem TieuKrizChemla2019.strengthened_mates_mystery_of_notMem {W : Type u_2} [Fintype W] [DecidableEq W] {wk st : Finset W} {A : Finset Magri2014.Item} (hA : Magri2014.Item.strong ∉ A) : Magri2014.strengthened (mates A) (Magri2014.primal wk st) Magri2014.Item.mystery = wk

Without all among the alternatives the definite keeps its existential meaning.

theorem TieuKrizChemla2019.strengthened_mates_mystery_of_mem {W : Type u_2} [Fintype W] [DecidableEq W] {wk st : Finset W} {A : Finset Magri2014.Item} (hA : Magri2014.Item.strong ∈ A) (h : st ⊆ wk) (hne : st.Nonempty) : Magri2014.strengthened (mates A) (Magri2014.primal wk st) Magri2014.Item.mystery = st

With all among the alternatives the definite is universal, (11).

theorem TieuKrizChemla2019.strengthened_mates_not_mystery {W : Type u_2} [Fintype W] [DecidableEq W] {wk st : Finset W} {A : Finset Magri2014.Item} (h : st ⊆ wk) : Magri2014.strengthened (mates A) (compl ∘ Magri2014.primal wk st) Magri2014.Item.mystery = wkᶜ

Under negation nothing is strengthened and the definite means none, whatever the alternatives.

def TieuKrizChemla2019.someWorlds {Atom : Type u_1} (x : Finset Atom) [DecidableEq Atom] [Fintype Atom] : Finset (Finset Atom)

Some and the definite hold where some object of the plurality has the property, the weak pole of the scale.

Equations

• TieuKrizChemla2019.someWorlds x = {w : Finset Atom | ∃ a ∈ x, a ∈ w}

def TieuKrizChemla2019.allWorlds {Atom : Type u_1} (x : Finset Atom) [DecidableEq Atom] [Fintype Atom] : Finset (Finset Atom)

All holds where every object of the plurality has the property, the strong pole.

Equations

• TieuKrizChemla2019.allWorlds x = {x_1 : Finset Atom | x ⊆ x_1}

@[simp]

theorem TieuKrizChemla2019.mem_someWorlds {Atom : Type u_1} {x w : Finset Atom} [DecidableEq Atom] [Fintype Atom] : w ∈ someWorlds x ↔ ∃ a ∈ x, a ∈ w

@[simp]

theorem TieuKrizChemla2019.mem_allWorlds {Atom : Type u_1} {x w : Finset Atom} [DecidableEq Atom] [Fintype Atom] : w ∈ allWorlds x ↔ x ⊆ w

theorem TieuKrizChemla2019.allWorlds_subset_someWorlds {Atom : Type u_1} {x : Finset Atom} [DecidableEq Atom] [Fintype Atom] (hx : x.Nonempty) : allWorlds x ⊆ someWorlds x

theorem TieuKrizChemla2019.isGap_iff_mem_sdiff {Atom : Type u_1} {x w : Finset Atom} [DecidableEq Atom] [Fintype Atom] : IsGap x w ↔ w ∈ someWorlds x \ allWorlds x

A GAP context is where the weak pole holds without the strong, the account's gap.

theorem TieuKrizChemla2019.definite_in_gap_iff_some_in_all {Atom : Type u_1} {x w : Finset Atom} [DecidableEq Atom] [Fintype Atom] {A : Finset Magri2014.Item} {w' : Finset Atom} (hw : IsGap x w) (hw' : x ⊆ w') : w ∈ Magri2014.strengthened (mates A) (Magri2014.primal (someWorlds x) (allWorlds x)) Magri2014.Item.mystery ↔ w' ∈ Magri2014.exh (mates A) (Magri2014.primal (someWorlds x) (allWorlds x)) Magri2014.Item.weak

Since the not all implicature is a sub-computation of homogeneity, a participant accepts the positive definite in a GAP context exactly when they accept some where every object has the property, whatever their alternatives. The homogeneous participants of both experiments who accept some there contradict the account.

def TieuKrizChemla2019.implicaturePattern {Atom : Type u_1} (x : Finset Atom) [DecidableEq Atom] [Fintype Atom] (A : Finset Magri2014.Item) (w : Finset Atom) : Bool × Bool

The account's binary judgments of the positive and the negative definite at a world, for a participant whose alternatives to some are A.

Equations

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

theorem TieuKrizChemla2019.implicaturePattern_of_isGap {Atom : Type u_1} {x w : Finset Atom} [DecidableEq Atom] [Fintype Atom] {A : Finset Magri2014.Item} (hw : IsGap x w) : implicaturePattern x A w = (if Magri2014.Item.strong ∈ A then Reading.homogeneous else Reading.quantified { force := Force.existential, scope := Scope.low }).binaryPattern

The account predicts the existential pattern without the implicature and the homogeneous one with it, the two groups of Table 2 it allows.