Kristina Liefke (LMU Munich) and Markus Werning (RUB Bochum)

1. Introduction. Partee (2006) conjectures a formal semantics (hereafter, single-type se- mantics; STS) which interprets CPs and referential DPs in the same type: parametrized properties of situations (type o). Partee’s semantics contrasts with Montague semantics and its recent contenders (dubbed dual- or multi-type semantics; DTS) which assume dis- tinct types for the semantic values of referential DPs (i.e. individuals; type e) and CPs (i.e. propositions, truth-values, or sets of assignment functions; type t). Partee’s conjecture is motivated by results in event semantics and discourse representation theory, which support the indirect uni-directional shiftability between the types e and t. However, none of these results gives STS an explicit modeling advantage over DTS. Our paper compensates for this shortcoming. In particular, it identifies a number of new, stronger, arguments for the adoption of STS which display this semantics’ greater unificatory and explanatory power. 2. Arguments. Our arguments for STS improve upon the strength and scope of Partee’s original argument. They are based on the ability of STS to simply and uniformly accom- modate many e-to-t and t-to-e type-shifting phenomena ( the uniformity argument) and to explain the semantic relations between DPs and CPs ( the entailment argu- ment) and the truth-evaluability of DP-fragments ( the assertoricity argument): 2.1. The uniformity argument is based on the observation that STS uniformly and stan- dardly accommodates many phenomena which require di↵erent non-standard (i.e. type-sh- ifting) accommodation strategies in DTS. These phenomena include the neutrality of some verbs between a DP- and a CP-complement (cf. (1)) (Kim & Sag 2005) and the possibility of coordinating/equating DPs with CPs in some contexts ((2),(3)) (Bayer 1996; Potts 2002).

(1) a. Pat remembered rdpBills.

b. Pat remembered rcpthat Bill was waiting for hers.

(2) Pat rememberedrdpBills and rcpthat he was waiting for hers.

(3) rdpThe problems was rcpthat Pat was avoiding Bills.

Since equation is typically restricted to expressions of type e, the modeling of (3) in DTS requires the non-standard interpretation of CPs in the standard type of referential DPs, e. Since coordination is restricted to expressions of the same conjoinable type, the modeling of (2) in DTS requires the non-standard interpretation of referential DPs in some CP-type. The complement-neutrality of verbs (cf. (1)) can be accommodated through either non- standard interpretation. Since STS interprets the arguments of the main verbs from (1) to (3) in the same type, o, it enables a uniform, standard interpretation of these phenomena. 2.2. The entailment argument for STS is based on the observation that STS, but not DTS, straightforwardly explains the obtaining of semantic relations between CPs and ref- erential DPs. Such relations are exemplified by the relation between the complements of the verb remember from (1a) and (1b): Intuitively, the information of the CP that Bill was waiting for her [i.e. Pat] from (1b) includes the information of the DP Bill in any commu- nicative context in which (1b) is uttered. In a particular remembering context in whose as- sociated remembered situation Bill is waiting for Pat, the information of the DP also in- cludes the information of the CP, such that the DP is equivalent to the CP in this context. The context-general inclusion of the DP in the CP is witnessed by the intuitive redun- dancy of the DP conjunct in (2), by the possibility of replacing the coordinator and in (2) by the specifier viz. (in (4)), and by the difficulty of negating only the DP conjunct in (2)

2

(in (5a)). The context-specific inclusion of the CP in the DP is witnessed by the fact that the negation of the CP conjunct in (2) (in (5b)) is only difficult for those remembered sit- uations in which Bill’s waiting for Pat is a property of Bill.

(4) Pat rememberedrdpBills, viz. rcpthat he was waiting for hers.

(5) a. #_{Pat did not remember} _{r}

dpBills, but remembered rcpthat he was waiting for iiihers.

b. ?_{Pat remembered}_{r}

dpBills, but did not remember rcpthat he was waiting for
iiiher_{s.}

DTS explain the semantic relations between CPs and DPs through non-standard mech-
anisms like ellipsis or flexible DP-typing. STS explains these relations without resort to
such mechanisms. To make this possible, it analyzes the single basic type o as the type for
functions from contextually specified situations (type s) to sets of situations (type _{xs, ty).}
In particular, STS interprets the CP that Bill was waiting for Pat as a function from situ-
ations to the set of extensions of these situations which are inhabited by Bill and in which
Bill is waiting for Pat (in [1]). Below, extensions of a situation are the situations that are
inhabited by all individuals which inhabit the original situation and in which all inhabi-
tants of the original situation have the properties which they have in the original situation.
In [1], ‘iÑ j’ and ‘E pbill, jq’ assert that the situation j is an extension of the situation i
resp. that j is inhabited by Bill. The behavior of Ñ and E is specified in (Liefke 2014).

[1] i j. i_{Ñ j ^ pE pbill, jq ^ wait pbill, pat, jqq}

[2] i j. iÑ j ^ E pbill, jq

In STS, the DP Bill is interpreted as a function from contextually specified situations to the set of extensions of these situations which are inhabited by Bill (in [2]).

In virtue of the above, the interpretation of a referential DP will, at each situation, be a superset of the interpretation of each upward-entailing CP containing the DP. If the CP is further true in the situation, the CP has the same interpretation as the DP in this sit- uation. The former explains the intuitive redundancy of (2) and the semantic deviance of (5a). The latter explains the deviance of (5b) for some particular remembered situation. 2.3. The assertoricity argument. The identical interpretation of DPs and CPs in STS also explains the fact (observed in (Stainton 2006)) that utterances of DP-fragments (e.g. (6a)) have intuitive truth- and falsity-conditions. Thus, in the communicative context from (6), the utterance of (6a) is intuitively true if (6b) is true, and false if (6b) is false.

(6) Mary knows that Bill is waiting for Pat at the front exit and that Pat is trying to avoid him. When Pat moves towards this exit, Mary warns her by saying (a):

a. _{r}dpBills

b. rdpBills is waiting at the front exit.

To explain the truth-evaluability of DP-fragments, STS does not use the special mecha-
nisms which are used in DTS. Instead, since (6a) has the same interpretation as (6b) in the
context described in (6), it inherits the truth- and falsity-conditions of (6b) in this context.
In STS, the truth of a CP or DP p which is interpreted at a situation is defined via
the membership of the world(-part) of evaluation w in the interpretation-at- of p. In par-
ticular, if w _{P JpKp q, p-interpreted-at- is true at w. If w R JpKp q, p-interpreted-at- is}
false at w. Since – as we assume – the situation specified by (6) is a part of the actual
world @ and since Bill is waiting for Pat at the front exit in @, (6b) and (6a) are both true
at @. This corresponds to the reported intuitions about the truth of (6a) and (6b).

3. Objections and Replies. Admittedly, there are several phenomena which can be used to argue against STS. These include the fact that many linguistic contexts do not allow the grammatical substitution of a referential DP (or of a CP) by a CP (resp. by a DP), and that most contexts do not allow the meaning-preserving replacement of a CP by its DP-nominalization. The final part of our paper presents and refutes these arguments. 4. Conclusion. Arguments for STS draw on di↵erent kinds of phenomena (syntactic vs. semantic) and are directed at di↵erent goals (unification vs. explanation). Seeming sup- port against STS can be refuted with reference to some standard verbal properties. References.

Bayer, S. 1996. The coordination of unlike categories, Language 72/3, 579–616.

Kim, J. and I. Sag. 2005. It-extraposition in English: A constraint-based approach, HPSG’2005, Universi- dade de Lisboa, 2005.

Liefke, K. 2014. A Single-Type Semantics for Natural Language, Doctoral dissertation, Tilburg Center for Logic and Philosophy of Science.

Partee, B. 2006. Do we need two basic types?, Snippets, Vol. 20, 2009.

Potts, C. 2002. The lexical semantics of parenthetical -as and appositive -which, Syntax 5/1, 55–88. Stainton, R. 2006. Words and Thoughts: Subsentences, ellipsis and the philosophy of language, Oxford

Mandarin dou: the common core of distributivity, maximality, and EVEN

Mingming Liu, Rutgers University

We present a unified analysis of Mandarin dou as an alternative sensitive (sentential) operator whose semantics equals to Karttunen & Peters’ (1979) EVEN (1). Different ‘uses’ of dou are analyzed by associating dou with different types of alternative sets: even-dou involves non- entailment-based alternative sets, while distributive-dou entailment-based ones.

(1) _{Jdou(π )K is defined iff ∀q ∈ A lt (π )[¬(π = q) → π ≺}likelyq], if defined,Jdou(π )K = Jπ K
An illustration of our proposal is provided by analyzing (2), ambiguous between (2a) and (2b).
(2) San-ge
three-CL
xuesheng
student
dou
DOU
mai.le
buy.ASP
shi.ben
ten.CL
shu.
book

a. EVEN-dou: ‘A group of three students together bought 10 books, which is unlikely.’ b. DISTRIBUTIVE-dou: ‘The three students each bought 10 books.’

Under (2a), dou adds an even-flavor and the sentence is interpreted collectively (the collective- cumulative distinction is irrelevant to our discussion), while in (2b) dou is even-less but triggers a distributive effect (Lin 1998) and a maximality effect (Giannakidou & Cheng 2006, Xiang 2008), indicated by the each and the in the gloss respectively.

Towards the analysis: The prejacent of dou under (2a) is analyzed as the πeven in (3) and

its alternatives in (4), with collective predication analyzed as predication over groups, using Landman’s (1989) ↑ which turns sums into atoms.

(3) _{Jπ}evenK =

∧_{∃X[students(X) ∧ |X| = 3 ∧ bought.ten.books(↑ X)]}

(4) A lt(JπevenK) = {

∧_{∃X[students(X) ∧ |X| = n ∧ bought.ten.books(↑ X)] : n ∈ N}+_{}}

The prejacent of dou under (2b) is analyzed as the πeven.less in (5) and its alternatives in (6),

with distributive predication captured using Link’s 1983 (covert) Dist (9) on VP.
(5) _{Jπ}even.lessK =

∧_{∃X[students(X) ∧ |X| = 3 ∧ Dist(bought.ten.books)(X)]}

(6) A lt(Jπeven.lessK) = {

∧_{∃X[students(X)∧|X| = n∧Dist(bought.ten.books)(X)] : n ∈ N}+_{}}

(7) _{JDist K = λ Pλ X ∀y[y ≤}atomX → P(y)]

Notice that a covert Dist in Chinese is independently justified by (8) where dou is absent but a distributive reading is possible and strongly preferred for every speaker consulted.

(8) [Context: Among these kids, I asked who drew two pictures, and you say:] Jieke Jack he and Lisi Lisi hua draw le ASP liang two fu. CL

‘Jack and Lisi each drew two pictures.’

The distributive effect of even-less-dou in (2b) involves trivializing dou’s even flavor in a distributive context (cf. Liao 2011), where the prejacent could logically entail all the other alternatives: that three students each bought ten books ⊂ that two students each bought ten books.

Since entailment is stronger than likelihood (Crniˇc 2011), dou’s even-presupposition is triv- ialized (= without adding an even-flavor) because it is weaker than the assertion and automati- cally satisfied. Thus, we get an even-less-dou (= distributive-dou) in (2b).

Alternatively, under a collective construal in (2a), dou’s prejacent does not entail its alter- natives: that a group of three students together bought ten books has nothing to do with that a group of two students together bought ten books; thus the even-presupposition remains intact and we get the even-dou.

Overall, distributive-dou is just a even-less-dou; since even-less-dou happens in a distribu- tive context where the prejacent could entail its alternatives, we have the correlation between douand distributivity.

Dou’s maximality effect: in (2b), three students is interpreted as definite/maximal (Liu 1997, Cheng 2009); in other words, (2b) is felicitous only if there are exactly three students in the context.

This follows from our analysis of dou as EVEN. For dou’s prejacent (5) to entail all

the other alternatives in (6) (to satisfy dou’s presuppostion in (1)), there can only be three students in the context: if there were four students, there would be a relevant alternative q=∧∃X[students(X) ∧ |X| = 4 ∧ Dist(bought.ten.books)(X)]; q entails π, making dou’s presuppostion unsatisfiable. In general, (2b) is felicitous only if there are exactly three students in the context. The maximality/definiteness effect is explained.

Improvement over previous accounts: Previsously, Lin 1998 analyzes dou directly as Link’s distributive operator on VP. While this captures even-less-dou’s distributive effect, it fails to capture its maximality effect.

(9) _{Jdou}LinK = λ Pλ X ∀y[y ≤atomX → P(y)]

On the other hand, Giannakidou & Cheng 2006 and Xiang 2008 takes dou to be a maximal- ity operator (Sharvy/Link’s the essentially). While this captures its maximality effect but the distributive effect is lost.

(10) _{Jd ou}G&CK = λ P.σ xP(x)

Our analysis of dou encapsulates both Lin (1998) and Giannakidou & Cheng (2006), by treating dou asEVEN.

Extension: Our proposal for Chinese dou can be extended to English all. All has both a maximizing effect (Dowty 1987, Brisson 2003, cf. all the boys vs. the boys, and all three boys vs. three boys) and a distributive effect (Champollion 2010, witness that all blocks cu- mulative/collective readings and is incompatible with certain collective predicates such as be numerous). Both effects can be explained by treating all as an alternative sensitive (sentential) operator. With a system such as Malamud’s 2012 where sentences with plural definites trigger alternatives, the all in (11) forces both maximality and distributivity.

(11) _{Jall(π )K is defined iff ∀q ∈ A lt (π )[¬(π = q) → π ⊂ q], if defined, Jall(π )K = Jπ K}

Selected References: Champollion 10 Parts of a whole.UPenn Thesis. Giannakidou & Cheng 06 (in)definiteness, polarity and free choice JoS. Karttunen & Peters 79 Conventional implica- tures. Landman 89 Groups L&P. Liao 11 Alternatives and Exhaustification.Harvard thesis.Lin 98 Distributivity in Chinese NALS. Xiang 08 Maximality and scalar inferences JEAL.