Chemla & Schlenker (2012): Incremental vs. symmetric accounts of presupposition projection

This file formalizes the paper's predictions and its experimental results. The paper takes a trivalent theory in which a sentence is presuppositionally acceptable when its supervaluation is defined at every world of the context (its (9), the symmetric condition, SupervaluationAcceptable of Schlenker's appendix on the symmetric version), and an incremental condition that also completes the sentence after each trigger by every good final without triggers (its (10), the incremental version). The Incremental Theory lets (10) alone govern projection; the Strict Symmetric Theory lets (9) alone govern it; the Mixed Theory prefers (10) but lets a presupposition satisfied only by (9) through at a cost.

The experiments test sentences with the anaphoric trigger too in if-, or- and unless-sentences, with the clause that can satisfy the presupposition before the trigger (canonical order) or after it (inverse order) ((48)–(49), target). In the canonical order both conditions predict the conditional presupposition if p, q ((13a)–(15a), supervaluationAcceptable_incremental_canonical, supervaluationAcceptable_symmetric_canonical). In the inverse order the incremental condition predicts the unconditional presupposition q (supervaluationAcceptable_incremental_inverse), while the symmetric condition still predicts if p, q ((13b)–(15b), supervaluationAcceptable_symmetric_inverse).

The results are Data/Experiments/ChemlaSchlenker2012. The conditional inference is endorsed more than the unconditional one for every tested sentence (Tables 4 and 7, experiment1_conditional_gt_unconditional, experiment4_conditional_gt_unconditional), including the inverse order, where the Incremental Theory predicts the unconditional presupposition and so an unconditional inference (supervaluationAcceptable_incremental_inverse_subset) and the symmetric condition does not (exists_supervaluationAcceptable_symmetric_inverse_not_subset). And every target sentence of Experiment 2 is rated above the control that forces local accommodation with too (Table 5, experiment2_targets_gt_control), so the inverse order is not unacceptable.

Implementation notes

• Conditionals are material implications and unless F, G is if not F, G, as the paper assumes (§1.4).
• The properties of too that the paper's argument uses to separate the Incremental from the Mixed Theory's predictions for inferences (§2.2, §2.3) are not formalized, and neither is the Mixed Theory's cost, so the fit of the theories to the ratings is not stated here.

References

• [CS12]
• [Sch09a]

The predictions of §1.4

def ChemlaSchlenker2012.unlessThen {Atom : Type u_1} (F G : Presupposition.Formula Atom) : Presupposition.Formula Atom

Unless F, G means if not F, G (§1.4).

Equations

• ChemlaSchlenker2012.unlessThen F G = Presupposition.Formula.bin Presupposition.Connective.cond F.not G

def ChemlaSchlenker2012.target {Atom : Type u_1} (p q q' : Atom) : Construction → Order → Presupposition.Formula Atom

The target sentence of a construction in an order, with the clause p that can satisfy the presupposition q of the trigger qq' ((48)–(49), p. 200).

Equations

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

theorem ChemlaSchlenker2012.supervaluationAcceptable_incremental_canonical {Atom : Type u_1} {W : Type u_2} (I : Atom → Set W) {p q q' : Atom} {C : Set W} (k : Construction) : Schlenker2009.SupervaluationAcceptable I Schlenker2009.Version.incremental C (target p q q' k Order.canonical) ↔ C ⊆ {w : W | w ∈ I p → w ∈ I q}

In the canonical order the incremental condition predicts the conditional presupposition if p, q ((13a)–(15a), p. 185).

theorem ChemlaSchlenker2012.supervaluationAcceptable_symmetric_canonical {Atom : Type u_1} {W : Type u_2} (I : Atom → Set W) {p q q' : Atom} {C : Set W} (k : Construction) : Schlenker2009.SupervaluationAcceptable I Schlenker2009.Version.symmetric C (target p q q' k Order.canonical) ↔ C ⊆ {w : W | w ∈ I p → w ∈ I q}

In the canonical order the symmetric condition predicts the conditional presupposition if p, q ((13a)–(15a), p. 185).

theorem ChemlaSchlenker2012.supervaluationAcceptable_incremental_inverse {Atom : Type u_1} {W : Type u_2} (I : Atom → Set W) {p q q' : Atom} {C : Set W} (k : Construction) : Schlenker2009.SupervaluationAcceptable I Schlenker2009.Version.incremental C (target p q q' k Order.inverse) ↔ C ⊆ I q

In the inverse order the incremental condition predicts the unconditional presupposition q ((13b)–(15b), p. 185).

theorem ChemlaSchlenker2012.supervaluationAcceptable_symmetric_inverse {Atom : Type u_1} {W : Type u_2} (I : Atom → Set W) {p q q' : Atom} {C : Set W} (k : Construction) : Schlenker2009.SupervaluationAcceptable I Schlenker2009.Version.symmetric C (target p q q' k Order.inverse) ↔ C ⊆ {w : W | w ∈ I p → w ∈ I q}

In the inverse order the symmetric condition still predicts the conditional presupposition if p, q ((13b)–(15b), p. 185).

theorem ChemlaSchlenker2012.supervaluationAcceptable_incremental_inverse_subset {Atom : Type u_1} {W : Type u_2} (I : Atom → Set W) {p q q' : Atom} {C : Set W} (k : Construction) (h : Schlenker2009.SupervaluationAcceptable I Schlenker2009.Version.incremental C (target p q q' k Order.inverse)) : C ⊆ I q

In the inverse order, any context that the Incremental Theory accepts entails the unconditional presupposition (§2.3.2).

theorem ChemlaSchlenker2012.exists_supervaluationAcceptable_symmetric_inverse_not_subset {Atom : Type u_1} {W : Type u_2} (I : Atom → Set W) {p q q' : Atom} (k : Construction) {w : W} (hp : w ∉ I p) (hq : w ∉ I q) : Schlenker2009.SupervaluationAcceptable I Schlenker2009.Version.symmetric {w} (target p q q' k Order.inverse) ∧ ¬{w} ⊆ I q

The symmetric condition accepts the inverse order in a context that does not entail the unconditional presupposition, one with a world where p and q both fail.

The results

theorem ChemlaSchlenker2012.experiment1_conditional_gt_unconditional (r : Experiment1Inference) : r ∈ experiment1Inferences → r.inference = Inference.conditional → ∃ r' ∈ experiment1Inferences, r'.construction = r.construction ∧ r'.order = r.order ∧ r'.inference = Inference.unconditional ∧ r'.mean.toRat < r.mean.toRat

In Experiment 1, for every tested sentence, the conditional inference is endorsed more than the unconditional one (Table 4, p. 206).

theorem ChemlaSchlenker2012.experiment4_conditional_gt_unconditional (r : Experiment4Inference) : r ∈ experiment4Inferences → r.inference = Inference.conditional → ∃ r' ∈ experiment4Inferences, r'.construction = r.construction ∧ r'.order = r.order ∧ r'.inference = Inference.unconditional ∧ r'.mean.toRat < r.mean.toRat

In Experiment 4, for every tested sentence, the conditional inference is endorsed more than the unconditional one (Table 7, p. 212).

theorem ChemlaSchlenker2012.experiment2_targets_gt_control (r : Experiment2Target) : r ∈ experiment2Targets → (experiment2Controls Control.localAccommodationToo).mean.toRat < r.mean.toRat

Every target sentence of Experiment 2 is rated above the control sentence that forces local accommodation with too (Table 5, p. 209).