The Distributivity Constraint does not specify anything about compositional im- plementations. This section shows one possible way to incorporate it into a compositional framework, namely by using the LFs presented in Chapter 3. I implement the constraint as a selectional restriction on the words that occur neces- sarily as part of a distributive construction. Following McCawley (1968) and Singh (2007), I assume that selectional restrictions are a kind of lexical presupposition. For example, infor-adverbials, the presupposition is attached to the wordfor; in each-constructions, to the word each; in pseudopartitives, to the word of. In a framework that uses partial truth values in the style of Karttunen and Peters (1979), Muskens (1996) or Heim and Kratzer (1998), a lexical presupposition is expressed as a restriction on the possible input values of a function. The expressionλx:ϕ . ψ represents the partial function that is defined for allxsuch that its lexical presup- positionϕholds, and that returnsψwherever the function is defined. Sentences are interpreted as pairs of propositions: an assertion and a presupposition. These presuppositions must be fulfilled in order for the sentence to have a truth value. I assume that the global presupposition of a sentence is the conjunction of all the
lexical presuppositions associated with its lexical items. That is, all presuppositions project straight to the top. Sentences whose global presupposition is true have the same truth value as their assertion; sentences where it is false lack a truth value. Denotations of lexical items that carry a presupposition are represented as partial functions that are undefined whenever this lexical presupposition is false.
This account is obviously too simple to model presupposition projection and accommodation in a serious way. I ignore the finer details of presupposition projection for two reasons. First, none of the examples I consider in this work involve more than one clause, so there would be little room for presuppositions to project other than going to the top. Second, the kinds of presuppositions I model are not based on contingent factual information. This would make it difficult to interpret them in any other way than by projecting them to the top even in multiclause contexts. For example,*John thinks that Bill ran a mile for an hourhas the presupposition thatrun a mileis atelic. Ifrun a milecould be locally satisfied, we would expect this sentence to have the interpretationJohn thinks that Bill ran a mile in an hour and that running a mile is atelic. But it is not a contingent fact that the predicaterun a mileis telic. Believing that it is atelic would require the hearer to believe that as soon as one starts running a mile, one has already run a mile. If John knows the meaning ofrunand ofa mile, he should not be able to hold this kind of belief.
One might similarly ask whether it is possible to accommodate the kinds of presuppositions I assume. I do not have much to say about this question. However, in Chapter 8, we will consider various flavors of distributive operators. As we will see, insertion of such an operator in the scope of a presupposition- carrying distributive item can be seen as a repair strategy, somewhat similar to presupposition accommodation.
The following lexical entries all embody the Distributivity Constraint. They are identical to the skeletal entries in Chapter 3, except that I have added a lexical presupposition. The letters K, S, and M stand for Key, Share, and Map respectively. For clarity, the entry foreachanticipates the effect of Definition (58) and replaces ε(K)withPureAtom.
(59) JofK=λShαtiλMhαβiλKhβtiλbhαi :SRM,ε(K)(S). S(b) ∧ K(M(b))
(60) JforK=λKhitiλMhviiλShvtiλe:SRM,ε(K)(S). S(e) ∧ K(M(e))
(61) JeachK=λShvtiλMhveiλKhetiλe :SRM, PureAtom(S). S(e) ∧ K(M(e)) The skeletal LFs from Chapter 3 are repeated in Figures 4.1 through 4.4 with these entries added. These full LFs are my official proposal. The types of these entries reflect the specific order in which they combine with their constituents according to these LFs. This order is not essential and could be easily changed
without consequences for the theory.
For ease of reference, here is the translation of a temporalfor-adverbial: (62) Jfor an hourK
=λPhvtiλe :SRτ,ε(λt[hours(t)=1])(P).
P(e) ∧ hours(τ(e)) = 1
This translation is obtained as a result of combining the entry forforin (60) withan hourand with a [runtime] head as shown in Figure 4.3.
Also for ease of reference, I spell out and expand the results of the computations in Figures 4.1 through 4.4.
(63) Jthree liters of waterK=
λx:SRvolume,ε(λd[liters(d) = 3])(λx[water(x)]).
[water(x) ∧ liters(volume(x)) = 3]
(This function is defined if and only ifwaterhas stratified reference with
respect to dimension parametervolumeand granularity parameter set to
a very small value compared with three liters. If defined, it is true of any water amount whose volume measures three liters. )
(64) Jthree hours of walkingK= Jwalk for three hoursK=
λe :SRτ,ε(λt[hours(t) = 3])(λe[∗walk(e)]).
[∗walk(e) ∧ hours(τ(e)) = 3]
(This function is defined if and only ifwalkhas stratified reference with respect to dimension parameterτ and granularity parameter set to a very small value compared with three hours. If defined, it is true of any walking event whose runtime measures three hours. )
(65) Jthree boys each walkedK= λe :SRag, PureAtom(λe[∗walk(e)]).
[∗walk(e) ∧ ∗boy(∗ag(e)) ∧ |∗ag(e)|= 3]
(This function is defined if and only ifwalkhas stratified reference with respect to dimension parameteragentand granularity parameterPureAtom. If defined, it is true of any walking event whose plural agent is three boys.) The definedness conditions of these functions expand as in (66) through (68). I assume that these definedness conditions are all fulfilled as a matter of world knowledge or lexical semantics. The expansions in (66) through (68) can be understood as meaning postulates. Their paraphrases should make it clear that this is a plausible assumption.
(66) SRvolume,ε(λd[liters(d) = 3])(λx[water(x)]) ⇔ ∀x[water(x)→x∈∗λy water(y)∧ ε(λd[liters(d) = 3])(volume(y)) ]
(There is a way of dividing every water amount exhaustively into parts (“strata”) which are water amounts and whose volumes are very small compared to three liters.)
(67) SRτ,ε(λt[hours(t) = 3])(λe[∗walk(e)])
⇔ ∀e[∗walk(e)→e∈∗λe0
∗
walk(e0)∧
ε(λt[hours(t) = 3])(τ(e0))
]
(There is a way of dividing every walking event exhaustively into parts (“strata”) which are walking events and whose runtimes are very small compared to three hours.)
(68) SRag, PureAtom(λe[∗walk(e)])
⇔ ∀e[∗walk(e)→e∈∗λe0
∗
walk(e0)∧ PureAtom(ag(e0))
]
(There is a way of dividing every walking event exhaustively into parts (“strata”) which are walking events and whose agents are pure atoms.)