Plural Semantics

Origin. Godehard Link, "The logical analysis of plurals and mass terms: a lattice-theoretical approach" (in Bäuerle, Schwarze, and von Stechow, eds., Meaning, Use, and Interpretation of Language, de Gruyter, 1983), which is the paper the whole field descends from. Sharvy (1980) supplied the maximality analysis of the definite article that Link's σ-operator generalizes; Scha (1981) identified cumulative readings; Landman (1989, "Groups I and II") added group formation; Schwarzschild's Pluralities (Kluwer, 1996) replaced fixed distributivity with contextually supplied covers. Boolos (1984) argued separately that plural quantification needs no plural objects at all.

Models. The domain of individuals is a complete join semilattice: the atoms are the ordinary singular individuals, and every set of them has a sum, which is what a plural noun phrase denotes. Part-of is the lattice order, defined from sum. A predicate then applies either to a sum directly, giving a collective reading, or to each part of it, giving a distributive one, and the difference is a fact about the predicate and the operator applied to it rather than an ambiguity in the noun phrase. The definite plural is the maximal sum satisfying the noun, so "the students" refers to a single object with the students as parts.

Formalism.

Plural domain: D* = closure of D under sum. a ⊕ b: sum of a and b. Atoms and sums. Mereological structure.

Sum operation: ⊕: associative, commutative, idempotent. Join semilattice. a ≤ b iff a ⊕ b = b. Part-of relation.

Plural predicates: *P: pluralized P. *P(x) iff x can be divided into parts each satisfying P. Distributive: applies to atoms. Collective: applies to sum.

Distributive reading: "The students smiled." Each student smiled. ∀x[x ≤ σy.student(y) → smiled(x)]. Predicate distributes.

Collective reading: "The students gathered." The group gathered. gathered(σy.student(y)). Predicate applies to sum.

Covers: Cov(X) = partition-like structure. Intermediate between collective and distributive. "The students lifted three tables." Various cover readings.

σ-operator: σx.P(x) = maximal sum of P's. Definite plural. "The students" = sum of all students. Unique maximal P.

Cumulative readings (Scha 1981): "Three boys ate five apples." Neither reading is scopal: the boys total three, the apples five. Written with the binary star: **R(x,y), the closure of R under pointwise sum. No quantifier outscopes the other.

Groups (Landman 1989): ↑x turns a sum into an atom, its group. Needed where a plurality is an argument in its own right — "The boys and the girls competed against each other." The group is an atom, so it cannot be distributed into.

Reciprocals: "The girls know each other" is strong; "The prisoners released each other" is not. Dalrymple, Kanazawa, Kim, Mchombo, and Peters (1998): the Strongest Meaning Hypothesis. The reciprocal takes the logically strongest reading the predicate and context allow.

Symbols.

Metatheory. The plural domain is a complete join semilattice with atoms — the free structure generated by the singular individuals — so ⊕ is associative, commutative, and idempotent, and the part-of order is definable from it rather than primitive. Sharvy (1980) showed that the definite article is uniformly maximality, taking the supremum in this lattice, which gives the singular and plural definites one entry rather than two. Link's star operator closes a predicate under sum, and its binary version delivers Scha's cumulative readings without giving either quantifier scope over the other. Schwarzschild (1996) generalized distributivity to quantification over a contextually supplied cover, of which the atomic partition and the one-cell cover are the extremes. Boolos (1984) argued that plural quantification is expressively monadic second-order yet ontologically innocent — a rival to the lattice approach, not a variant of it.

Applies to. Plural NPs. Distributivity. Reciprocals. Collectivity.

Limitations. The cover variable is unconstrained, so the theory generates readings no speaker reports: any division of a plurality is admissible, and the semantics supplies no principle saying which are available. Groups are stipulated rather than derived — the lattice provides sums but no criterion of group identity, so Landman's ↑ is an extra primitive whose behaviour under coordination and part-of must be fixed by hand. Object mass nouns are a standing problem: "furniture" denotes discrete objects yet patterns as a mass term, which undercuts the atomic/non-atomic split the ontology rests on. The morphological correlate the theory presumes is not general, since classifier and general-number languages mark no singular–plural contrast on the noun. And the framework is static and extensional, so plural anaphora, dependent plurals, and the interaction with events all require a dynamic or event-based supplement.

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