Chemla (2009): presuppositions of quantified sentences

Chemla's first experiment finds the universal inference from a trigger in the nuclear scope of each and no endorsed far more often than from the scope of the numerical quantifiers (Data/Experiments/Chemla2009a). One amendment the paper considers keeps the universal presupposition that the transparency theory predicts for every quantifier, and explains why it is not accommodated after the numerical quantifiers by how hard it is to derive (§4.1.2). Appendix F makes the measure precise. Under no, a single instance of transparency's condition, at the negation of the presupposition, entails the universal presupposition. Under less than 3, the appendix states, no Boolean combination of the restrictor and the presupposition does, and it leaves open a proof that no single instance does.

A quantifier derives its universal presupposition in one step when such an instance entails it in every model (OneStep). This holds exactly when the quantifier tells one end of each row of its tree of numbers from every other point of the row (oneStep_iff): the end where no individual of the restrictor is in the scope, as no and some do, or the end where all of them are, as each and not every do. So among the quantifiers of Experiment 1 the universal presupposition is one step away under exactly each and no, the two whose universal inference is robust (oneStep_tree_iff).

Implementation notes

• A Boolean combination of the restrictor and the presupposition is a set of pairs whose membership is a function of membership in the two; the appendix speaks of conjunctions and disjunctions of the two and their negations.

References

• [Che09a]
• [Sch07]

def Chemla2009a.Determiner.tree : Determiner → Quantifier.NumberTree

The quantifier of a determiner of Experiment 1, on the tree of numbers.

Equations

• Chemla2009a.Determiner.each.tree = Quantifier.NumberTree.all
• Chemla2009a.Determiner.no.tree = Quantifier.NumberTree.no
• Chemla2009a.Determiner.lessThan3.tree = Quantifier.NumberTree.cardinal {b : ℕ | b < 3}
• Chemla2009a.Determiner.moreThan3.tree = Quantifier.NumberTree.cardinal {b : ℕ | 3 < b}
• Chemla2009a.Determiner.exactly3.tree = Quantifier.NumberTree.cardinal {3}

def Chemla2009a.OneStep (q : Quantifier.NumberTree) : Prop

Appendix F: transparency derives the universal presupposition of a trigger in the nuclear scope of q in one step when, for some Boolean combination of the restrictor and the presupposition, the instance of its condition at that combination entails in every model that every individual of the restrictor satisfies the presupposition.

Equations

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

theorem Chemla2009a.oneStep_iff {q : Quantifier.NumberTree} : OneStep q ↔ (∀ (a b : ℕ), 0 < b → (q a b ↔ ¬q (a + b) 0)) ∨ ∀ (a b : ℕ), 0 < a → (q a b ↔ ¬q 0 (a + b))

Appendix F: transparency derives the universal presupposition in one step exactly when q tells the end of each row where no individual is in the scope from every other point of the row, or the end where every individual is.

theorem Chemla2009a.oneStep_tree_iff (d : Determiner) : OneStep d.tree ↔ (experiment1Groups d).group = QuantifierGroup.eachOrNo

Appendix F: among the quantifiers of Experiment 1, transparency derives the universal presupposition in one step from exactly those pooled with each and no.