Tieu, Bill, Romoli and Crain (2020): Testing Theories of Plural Meanings

This file formalizes the three theories of the multiplicity inference of the English plural that Tieu, Bill, Romoli and Crain test experimentally, and the predictions on which they diverge. Emily fed giraffes conveys that she fed more than one, (1), while Emily didn't feed giraffes conveys that she fed none, (2), and the inference likewise disappears in conditional antecedents and questions, (3)–(4). The ambiguity approach of Farkas and de Swart gives the plural a weak reading, one or more, and a strong reading, more than one, and selects the stronger by the Strongest Meaning Hypothesis, (5)–(7). The implicature approach of Spector and of Zweig gives the plural the weak meaning and derives the multiplicity inference by exhaustification against the singular alternative, (13)–(15), which is entailed rather than excludable under negation, (16)–(17). The homogeneity approach of Križ, extended to bare plurals, makes the plural predicate undefined of a single giraffe, so the sentence is true of more than one, false of none and undefined otherwise, (20)–(24). All three predict the monotonicity pattern, readings_positive and readings_negative. They diverge in a context where Emily fed exactly one giraffe, (27): the implicature approach makes the positive sentence literally true with a false implicature and the negative one false, while the ambiguity approach makes both false and the homogeneity approach both undefined, singular_context. The implicature approach alone derives the multiplicity inference by the mechanism of the not all inference of some, (29), the same exhaustifier against a stronger alternative, exhIE_some, whence its uniformity prediction, (28), that children compute fewer of both.

Implementation notes

The worlds are the number of giraffes fed, so the readings are sets of natural numbers and negation is complementation. The implicature approach is the innocent-exclusion exhaustifier of Semantics.Exhaustification.InnocentExclusion with the singular alternative; the paper's versions differ in how they derive that alternative, footnote 4, and agree on the result. The Strongest Meaning Hypothesis picks the least reading under entailment, IsLeast. Homogeneity is a trivalent predicate whose negation is Neg.neg. The experiments are not formalized: in Experiment 1, a truth-value judgment task, adults rejected positive plural sentences after a story with one animal fed far more often than four- and five-year-olds did, and accepted the negative ones only moderately; in Experiment 2 children computed fewer multiplicity inferences and fewer not all implicatures than adults, and the two rates were correlated within children; in Experiment 3, a ternary judgment task, adults gave the positive plural in a singular context an intermediate reward and the negative plural the minimal one. The examples are the rows of Data.Examples.TieuEtAl2020.

References

• [TBRC20]
• [FdS10]
• [Spe07a]
• [Zwe09]
• [Kri15b]
• [Fox07]

Plural meanings over the number of giraffes fed

def TieuEtAl2020.weak : Set ℕ

The weak reading is one or more (5a).

Equations

• TieuEtAl2020.weak = {n : ℕ | 1 ≤ n}

def TieuEtAl2020.strong : Set ℕ

The strong reading is more than one, the multiplicity inference (5b).

Equations

• TieuEtAl2020.strong = {n : ℕ | 2 ≤ n}

def TieuEtAl2020.singular : Set ℕ

The singular alternative is exactly one (14).

Equations

• TieuEtAl2020.singular = {1}

The implicature approach (section 1.2.2)

theorem TieuEtAl2020.exhIE_weak : Exhaustification.exhIE {weak, singular} weak = strong

Exhaustifying the weak plural against its singular alternative yields the multiplicity inference, (13)–(15).

theorem TieuEtAl2020.exhIE_compl_weak : Exhaustification.exhIE {weakᶜ, singularᶜ} weakᶜ = weakᶜ

Under negation the singular alternative is entailed by the negated plural, so nothing is excluded and the sentence conveys that no giraffe was fed, (16)–(17).

theorem TieuEtAl2020.exhIE_some {k : ℕ} (hk : 2 ≤ k) : Exhaustification.exhIE {{n : ℕ | 1 ≤ n}, {k}} {n : ℕ | 1 ≤ n} = {n : ℕ | 1 ≤ n ∧ n ≠ k}

The not all implicature of some of the k giraffes is the same exhaustifier against the stronger alternative all (29), the mechanism the uniformity prediction (28) rests on.

The ambiguity approach (section 1.2.1)

The Strongest Meaning Hypothesis (7) prefers, among the readings of a plural sentence, the one entailing all the others: the least reading under inclusion, IsLeast.

theorem TieuEtAl2020.isLeast_strong : IsLeast {weak, strong} strong

In a positive sentence the strong reading is preferred, (5).

theorem TieuEtAl2020.isLeast_compl_weak : IsLeast {weakᶜ, strongᶜ} weakᶜ

Under negation the negated weak reading is preferred, (6).

The homogeneity approach (section 1.2.3)

def TieuEtAl2020.homogeneous (n : ℕ) : SignType

The plural sentence is true of a plurality of giraffes, false of none and undefined otherwise, (20)–(22).

Equations

• TieuEtAl2020.homogeneous n = if 2 ≤ n then 1 else if n = 0 then -1 else 0

theorem TieuEtAl2020.homogeneous_eq_one_iff {n : ℕ} : homogeneous n = 1 ↔ 2 ≤ n

theorem TieuEtAl2020.neg_homogeneous_eq_one_iff {n : ℕ} : -homogeneous n = 1 ↔ n = 0

Negation leaves undefinedness untouched, so the negated sentence is true of no giraffe fed, (23)–(24).

Predictions (section 1.3)

theorem TieuEtAl2020.readings_positive : Exhaustification.exhIE {weak, singular} weak = strong ∧ IsLeast {weak, strong} strong ∧ {n : ℕ | homogeneous n = 1} = strong

All three approaches derive the multiplicity inference of a positive plural sentence.

theorem TieuEtAl2020.readings_negative : Exhaustification.exhIE {weakᶜ, singularᶜ} weakᶜ = weakᶜ ∧ IsLeast {weakᶜ, strongᶜ} weakᶜ ∧ {n : ℕ | -homogeneous n = 1} = weakᶜ

All three approaches make a negated plural sentence convey that none was fed.

theorem TieuEtAl2020.singular_context : (1 ∈ weak ∧ 1 ∉ Exhaustification.exhIE {weak, singular} weak ∧ 1 ∉ Exhaustification.exhIE {weakᶜ, singularᶜ} weakᶜ) ∧ (1 ∉ strong ∧ 1 ∉ weakᶜ) ∧ homogeneous 1 = 0 ∧ -homogeneous 1 = 0

In the singular context (27) the implicature approach makes the positive sentence literally true but its enriched meaning false and the negative sentence false, an asymmetry. The ambiguity approach makes both false and the homogeneity approach both undefined.