Bar-Lev (2025): Homogeneity, underspecification, and ambiguity in plural predication #
Plural predication is homogeneous: if only half of the four kids laughed, neither the four kids laughed nor its negation can be truthfully uttered. Bar-Lev traces homogeneity to pluralization by a trivalent version of Link's star operator, which makes a plural predicate true of a sum of pluralities that satisfy it and false of a plurality no part of which does. Relativized to a cover of the parts that matter in the context, the operator yields three meaning types, his Table 1. A cover by all the parts gives an underspecified reading with total homogeneity, as with lift the table. A cover by the atoms gives a distributive reading with homogeneity over the atoms, and a cover by the plurality itself a collective reading with no homogeneity, as with weigh exactly 250 lbs.
The truth and falsity conditions of the trivalent star are the two ends of the readings of the
plural predication (sUnion_readings_coverStar), which is how the paper's appendix places the
account in Križ and Spector's framework. The models check the paper's contrasts. When half of the
kids lift the table together, the four kids (didn't) lift the table has no truth value
(lift_half_gap), where homogeneity over the atoms would make the negation true
(distributive_half_false). Lifting individually, together, or in two halves makes the sentence
true (lift_true), so that the four kids didn't lift the table but they did together is
contradictory. These two people don't weigh 250 lbs is true both when they weigh 125 lbs each and
when they weigh 250 lbs each, which needs the two minimal covers (weigh_ambiguous), since under
a cover by all the parts the positive sentence is true in both situations
(weigh_true_of_power_cover).
Implementation notes #
Pluralities are nonempty finite sets of individuals, sum is union, and the atoms are the
singletons. A cover is a set of pluralities with no empty one; the minimal covers of (24b) are the
atoms of x and {x} itself (eq_atoms_of_minimal, eq_singleton_of_minimal). A sentence is
true where all of its readings are and false where none is; Bar-Lev states this trivalence as a
presupposition, which the readings leave open.
TODO #
- The co-distributive two-place operator of §6.1 and upward homogeneity by overlap, §6.2.
- The context dependence of cover choice argued for in §7.
References #
Covers and the trivalent star (§2, §4, §5) #
Plural predication relative to a cover C, (25) and (30), holds of a plurality that is a sum
of pluralities in C that satisfy P.
Equations
- BarLev2025.coverStar C P z = {w : W | Mereology.AlgClosure (fun (y : Finset α) => y ∈ C ∧ P y w) z}
Instances For
By the trivalent star, (12) and (31b), the plural predication is false exactly where no part
of x in the cover satisfies P.
A minimal cover containing the atoms of x, (24b-i), consists of the atoms of x.
A minimal cover containing x, (24b-ii), is {x}.
Under a cover by all the parts of x, (32), the reading is underspecified, ⋆P itself, and
homogeneity is total: the sentence is false only where no part of x satisfies P.
Under the cover by the atoms of x, (33), the reading is distributive and homogeneity
concerns the atoms: the sentence is false only where no atom of x satisfies P.
Under the cover by x itself, (34), the reading is collective and there is no homogeneity:
the sentence is true where x satisfies P and false elsewhere.
lift the table: underspecification and total homogeneity (§2, §3) #
Equations
- BarLev2025.instDecidableEqLifting x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
Which pluralities of kids lifted the table in each situation.
Equations
- BarLev2025.lifted x✝ BarLev2025.Lifting.individually = (x✝.card = 1)
- BarLev2025.lifted x✝ BarLev2025.Lifting.together = (x✝ = Finset.univ)
- BarLev2025.lifted x✝ BarLev2025.Lifting.halves = (x✝ = {0, 1} ∨ x✝ = {2, 3})
- BarLev2025.lifted x✝ BarLev2025.Lifting.half = (x✝ = {0, 1})
Instances For
Equations
- One or more equations did not get rendered due to their size.
A homogeneity presupposition on the atoms, Gajewski's distributivity operator (7), makes the sentence false in the same situation, so that its negation would be true.
Under a cover by all the parts, the four kids lifted the table is true whether they lifted it individually, together, or in two halves, (13) and the intermediate situation of §3.
When half of the kids lift the table together and the other half does nothing, (3), the sentence is neither true nor false, so neither it nor its negation can be truthfully uttered.
weigh exactly 250 lbs: ambiguity (§3) #
The two people of (17) weigh 125 lbs each or 250 lbs each.
Instances For
Equations
- BarLev2025.instDecidableEqWeighing x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
Which pluralities of the two people weigh exactly 250 lbs.
Equations
- BarLev2025.weighs250 x✝ BarLev2025.Weighing.each125 = (x✝ = Finset.univ)
- BarLev2025.weighs250 x✝ BarLev2025.Weighing.each250 = (x✝.card = 1)
Instances For
Equations
- One or more equations did not get rendered due to their size.
Under a cover by all the parts, these two people weigh 250 lbs would be true in both situations, so its negation (17) could be true in neither.
With the minimal covers, (17) is true in both situations: the cover by the atoms makes the positive sentence false when each weighs 125 lbs, and the cover by the pair makes it false when each weighs 250 lbs.
The paper's judgments #
The judgments of (3) are the model's: neither (3a) nor its negation (3b) is true when half of the kids lift the table and the other half does nothing, and (3a) is true when each half lifts it.
The four kids didn't lift the table but they did together, (13), is contradictory: lifting the table together makes the negated sentence false.
These two people don't weigh 250 lbs but they do together (individually), (17), is consistent: the negated sentence is true when together they weigh 250 lbs, by the cover by the atoms, and when each weighs 250 lbs, by the cover by the pair.