The basic system

Ontology The first thing in our model is ∆, the domain for entities. I will define the sum operator t as a function from sets of entities to entities such that for every non-empty setφsuch thatφ⊂∆,tX ∈∆. Similarly to Landman, I will writeXtY

for t{X, Y}. The part-of relation ≤ is defined as follows: X ≤Y iff X tY =Y. ≤

is a partial ordering over ∆.

An atom is an element x ∈ ∆ such that there is no element Y ∈ ∆ such that

Y ≤xand Y 6=x. We can define ∆a to be the subset of ∆ that includes all and only

the atoms. I will use lower case letters x, y, z for variables that range over ∆a, and

upper case letters X, Y, Z for variables that range over all of ∆ (including ∆a).

We can define the cardinality operator || over members of ∆ as follows: for any

X, |X| is the number of members in the set {x ∈∆a :x≤ X}. In other words, |X|

is the number of atoms that make up X. ∆ has the following properties:

1. h∆,ti is an i-join semilattice. This means that for any X, Y ∈ ∆ there is also Z ∈ ∆ such that XtY = Z, and there is no W ∈ ∆ such that X ≤ W

and Y ≤ W, unless Z ≤W. In other words, for every two elements in ∆, the smallest element of which X and Y are both parts must also be in ∆3_.

2. In addition to being an i-join semilattice, ∆ is also distributive, which means that if X≤Y tZ, then eitherX ≤Y, or X ≤Z, or ∃Y0 ≤Y &∃Z0 ≤Z such that X =Y0tZ0. In other words, if X is part of the sum of Y and Z, every part of X is part ofY or part of Z.

3. ∆ also has the property of being witnessed. IfX ≤Y andX 6=Y, then∃Z ≤Y

such that there is no W such that W ≤ X and W0 ≤Z. Informally, if X is a proper part of Y, then the rest of Y is also an element of ∆.

4. The final property that ∆ has is that it is atomic. This means that if there is more than one element in ∆, then there is no element 0 such that for every element X ∈∆, 0≤ X. This means that if there is more than one element in ∆, then there is more than one atom.

Taken together, these properties mean that h∆,ti is a mereology (or, using Landman’s terminology, a part-whole structure). It is equivalent to a complete boolean algebra with the 0 cut off.

In addition to entities, the ontology includes events. The domain of events is Σ. A sum operator t is also defined over events, and hΣ,ti is a mereology that shares

all the properties of ∆ listed above. The domain of atomic events is Σa. I will write

variables that range over Σa ase, e0, e00 and so forth, and variables over Σ asE, E0, E00.

Events and individuals are related via role predicates, such as agent (ag) or

theme (th). Role predicates are partial functions from events to atomic entities. Ifx

is the agent of e, then Landman writesag(e) = x. Unlike Landman, I will follow the

more traditional notation introduced by Parsons (1990), writing ag(e)(x) instead of

ag(e) = x. I will use Θ as a variable over role predicates. Like Landman, I assume

role uniqueness: if Θ(e)(x) is true, then there is no y such thaty 6=x and Θ(e)(y) is true.

Verbal predicates are taken to be sets of events. Nominal predicates denote sets of individuals.

Type shifters Landman makes use of several type shifters in his theory, most of which are not directly relevant for current purposes. However, a couple of the more basic type shifters are necessary in order to derive basic sentence meanings. The first is the argument lifting type shifterlift, which comes in several versions:

(208) a. Intransitive LIFT: λXλE[...] ⇒λψhhe,ti,tiλE[ψ(λX[...])]

b. Transitive LIFT:λXλY λE[...]⇒ λψhhe,ti,tiλY λE[ψ(λX[...])]

In addition, we need a rule of existential closure over events:

(209) Existential Closure (EC): λE[....]⇒ ∃E[....]

Group formation One of the crucial elements in Landman’s theory is the group- making operators↑. For every elementX in ∆, there is an element xin ∆a such that

↑X =x, with the special provision that iffX ∈∆a, ↑X =X.

Landman (2000), following Schwarzschild (1992, 1996), actually has a more com- plex version of↑than I have given above, which is designed to disallow iterative group formation (i.e., the group forming operator is not defined over sums that include a

group atom among their parts, such as ↑(xt ↑(ytz)). For simplicity’s sake, I will not concern myself with this issue here. Nothing that follows hinges on whether or not iterative group formation is possible.

Pluralization The second important tool is the ∗, or pluralizing, operator. The star operator takes a set of entities or events and returns its closure under sum. For example, ∗{x, y, z}= {x, y, z, xty, xtz, y tz, xtytz}.

The ∗ operator is extended to apply to role predicates, creating what Landman calls plural roles. To extend ∗ to relations, we can define a sum operation over roles

ttas follows: hE, XitthE0, Yi=hEtE0, XtYi. ∗Θ is the closure of Θ under tt. It is worthwhile noting that while Landman uses the singular roles in the definition of the plural roles, he never uses them in the actual semantics of sentences. Rather, as he points out, if x and e are atoms, then ∗Θ(e)(x) = Θ(e)(x).

By the assumption that thematic roles are unique, and since any two elements only have one sum, it follows that uniqueness also holds for pluralized roles: if ∗Θ(E)(X) is true, then there is no Y such that Y 6=X and ∗Θ(E)(Y) is true.