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\title[Singularity and plurality of discourse reference to worlds]{%
  Singularity and plurality of discourse reference to worlds\thanks{%
    This work was supported by National Science Foundation, Award No. 2116972, ``Research on conditional and modal language'' (Magdalena Kaufmann, PI; Stefan Kaufmann, co-PI). For discussions and comments, I am indebted to Magdalena Kaufmann, Teruyuki Mizuno, Yoshiki Fujiwara, Giulio Ciferri Muramatsu, Yuya Noguchi, Floris Roelofsen,  Yasutada Sudo, Yimei Xiang, Yusuke Yagi, Alessandro Zucchi, four anonymous reviewers of SALT 32, as well as my audience at Many-time-zone Reading Group (December 2021), Theoretical Linguistics at Keio (TaLK) Semantics Conference (March 2022), UConn Meaning Group (April 2022), and SALT 32 (June 2022).   
  }
}

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\author[Yang]{%
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  \saltauthor{Muyi Yang \\ \institute{University of Connecticut}}%
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\begin{abstract}  
The referential analysis of conditionals has recently been put forth as an alternative of the Kratzer-style restrictor analysis (\citealt{Schein01}, \citealt{Schlenker04}, among others). Under this analysis, conditional antecedents are definite descriptions of worlds/situations. This paper explores a widely accepted assumption of the referential analysis, namely that conditional antecedents refer to \textit{plural} objects. I show that the singularity/plurality of conditional antecedents can  correlate with whether the conditional expresses modal or adverbial quantification. I use this correlation to motivate an analysis where conditional antecedents are \textit{number-neutral} by default, but can be forced to denote \textit{singular} referents. This idea is formally implemented within the dynamic framework by \cite{Brasoveanu10}. 
\end{abstract}

\begin{keywords}
  Japanese, discourse reference, plurality, modality, adverbs of quantification 
  \end{keywords}

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\section{Introduction}
\label{section-introduction}
In formal semantics, conditionals are standardly treated  as  quantificational constructions, with a quantificational operator being restricted by the antecedent (\citealt{Lewis75}, \citealt{Kratzer86}). 
However, the recent literature has put forth an alternative analysis, namely \textbf{the referential analysis of conditionals} (\citealt{Schein01}, \citealt{Schlenker04}, among others; see also \citealt{Stalnaker68} for an early predecessor). The hallmark of this analysis is that  conditional antecedents are definite descriptions of possible worlds or situations. To illustrate, consider the conditional in \Next.  Under the referential analysis, the antecedent  of \Next refers to the world(s) where Mary comes, in the same way that \textit{the}-phrases refer to individuals;  the consequent ascribes to this referent that John is coming at some future point. 

\ex. If Mary comes, John will come, too.




This paper explores a widely accepted assumption of the referential analysis, namely that conditional antecedents refer to \textit{plural} objects (argued explicitly by \citealt{Schein01} and \citealt{Schlenker04}; adopted later by \citealt{Brasoveanu10},  \citealt{EbertEbertHinterwimmer14}, among others). As noted by \cite{Schlenker04}, plurality is needed to capture  adverbs of quantification (henceforth Q-adverbs) in conditionals.\footnote{
For an additional argument of plurality, Schlenker cites \citeauthor{Schein01}'s (\citeyear{Schein01}) observations about conditionals with iterated \textit{if}-clauses, such as \Next[a]. Schein notes that \Next[a] is interpreted similarly to \Next[b], whose antecedent expresses the conjunction of S$_1$ and S$_2$. He argues that we can easily derive the interpretation of \Next[a] by analyzing   conditional antecedents as \textit{plural} definite descriptions: S$_1$ refers to a plural situation  such that all of its atomic parts verify  the property of having a vaccinated donkey, and that situation  gets restricted further by S$_2$, i.e.\ by the property that the donkey has vitamin deficiency. The resulting interpretation of iterated \textit{if}-clauses is equivalent with that of \Next[a], which refers to a plural situation that verifies the property of having a  donkey that is vaccinated and has  a vitamin deficiency. 

\ex. \a. [$_{S_1}$ If a donkey is vaccinated], then [$_{S_2}$ if it has a vitamin deficiency], it usually faints.\\ \hspace*{\fill} (\citealt{Barker97}: 202 (12), cited in  \citealt{Schein01}: 408 (18))
	\b. If a donkey is vaccinated \textit{and} has a vitamin deficiency, it usually faints. 

However, in languages with multiple forms of conditional connectives, the judgments of conditionals with iterated antecedents are extremely subtle; I will thus leave the examination of Schein's argument  to future research. See also fn.\ \ref{footnote-Bizet} for an  argument of plurality by \cite{Lewis73} from counterfactuals.

}  The reasoning is as follows.\footnote{
Note that Schlenker himself does not provide a detailed argumentation  of the plural view. Hence, what  I outline below is a stepwise rendition of  Schlenker's point,  which strikes me as necessary in order to conduct an in-depth evaluation of the plural view. 
} As shown by \cite{deSwart95},  Q-adverbs cannot be restricted by \textit{when}-clauses that describe unique events. This  can be illustrated by   the contrast between \Next[a] and \Next[b]: 


\ex. \label{deSwart} 
	\a. *When Anil died, his wife usually killed herself.
	\b. When an Indian died, his wife usually killed herself.\label{deSwart-good}  \vspace{-1mm} \flushright  (\citealt{deSwart95}: 178 (22)) 

\noindent Crucially, the verb \textit{die} describes a non-repeatable event. Hence, when \textit{die} applies to an individual, as in \Last[a], the \textit{when}-clause  describes a unique event, i.e.\ that of Anil dying. In contrast, when \textit{die} applies to an indefinite subject, as in \Last[b], the \textit{when}-clause describes multiple events, each involving an Indian dying. The contrast in the acceptability of \Last[a] and \Last[b] thus indicates that the Q-adverb \textit{usually} must be restricted by \textit{when}-clauses that describe multiple events. This can be  taken to indicate the constraint that Q-adverbs require non-singleton restrictors. 

In addition, it is well-known that the restrictors of Q-adverbs can also be provided by English \textit{if}-clauses   (\citealt{Lewis75}). For instance, in \Next, \textit{usually} quantifies over the situations described by the \textit{if}-clause, as shown by the paraphrase provided below: 

\ex. If Mary comes, John usually comes, too. \\
	 Roughly: Most situations where M comes are situations where J comes.\label{English-Q-adverb}

\noindent Since we have shown in \ref{deSwart} that \textit{usually} requires non-singleton restrictors, it is plausible to conclude that the \textit{if}-clause of \Last describes multiple events, too.\footnote{\label{footnote-Bizet}
An even earlier argument of the plural view comes from \citeauthor{Lewis73}'s  (\citeyear{Lewis73}) discussion of the Bizet/Verdi counterfactuals (in response to \citealt{Stalnaker68}):
\ex. \a. If Bizet and Verdi had been compatriots, Bizet would have been Italian. 
	\b. If Bizet and Verdi had been compatriots, Verdi would have been French.
	
Based on his intuition that \Last[a] and \Last[b] are false, Lewis argues that the  antecedents of the two conditionals pick out multiple closest  worlds: one where Bizet and Verdi were Italian, and one where they were  French. I leave it to future work to  examine how the semantics of conditionals developed in this paper fares with counterfactuals, and in particular, with the Bizet/Verdi  cases.
	
}


However, the standard assumption of the referential analysis of conditionals goes one step further---it is assumed that \textit{if}-clauses \textit{always} refer to pluralities. Under this assumption, \textit{if}-clauses refer to plural objects not only in conditionals that express adverbial quantification, such as \Last (henceforth \textbf{Q-adverbial conditionals}); they are also  assumed to  refer to pluralities in conditionals that express modal quantification (henceforth \textbf{modal conditionals}). \Next provides an example of modal conditionals, which contains the modal auxiliary \textit{might}. Under the standard assumption of the referential analysis, the \textit{if}-clause of \Next refers to a plural object, just like that of \Last. 

\ex.  If Mary comes, John might come, too. 


In this paper, I present a paradigm of Japanese conditionals where  the conditional marker \textit{moshi} is compatible with modal conditionals, but not  Q-adverbial conditionals. I show that the split between Q-adverbial and modal conditionals made by \textit{moshi} challenges the assumption that  both types of conditionals have antecedents that refer to pluralities. To solve this problem, I argue that conditional antecedents can refer to singular objects, and that this  singularity  can be overtly marked. 

The rest of this paper is structured as follows. Sec \ref{section-data} presents the key data. Sec \ref{section-existing-view-by-Frank} introduces an independent assumption about  the difference between Q-adverbial and modal conditionals.  Sec \ref{section-analysis} spells out my formal proposal. Sec \ref{section-conclusion} concludes the paper. 


%%%%%%%%%%
\section{Data}\label{section-data}
\subsection{Basic profile of Japanese conditionals}\label{section-basic-profile}

Japanese conditionals are obligatorily marked by connectives that appear as verbal suffixes (e.g.\ \textit{-(re)ba}, \textit{-tara}) or enclitics  (e.g.\ \textit{to},  \textit{nara}) in antecedent-final position:

 \exg. Mary-ga \{ku-\textbf{reba} / ki-\textbf{tara} / kuru \textbf{to} / kuru \textbf{nara}\}, John-mo kuru.\\
	  	M-\textsc{nom} come-\textsc{reba} {} come-\textsc{tara} {} come \textsc{to} {} come \textsc{nara} J-\textsc{add} come\\
	  `If Mary comes, John also comes.' 



\noindent Conditional antecedents can sometimes also be accompanied by the adverb \textit{moshi}, which normally appears in antecedent-initial position, as in \Next: 
  \exg. \{\textbf{moshi} / $\emptyset$\} Mary-ga ku-reba, John-mo kuru.\\
	  \textsc{moshi} {} {} M-\textsc{nom} come-\textsc{reba} J-\textsc{add} come \\
	  `If Mary comes, John will also come.' 



The rest of this section  examines the distribution of \textit{moshi} in Q-adverbial and modal conditionals. Note that apart from \textit{moshi}, some of the verbal suffixes and enclitics  in conditional antecedents (cf.\ \LLast) also show sensitivity to whether they appear in a Q-adverbial or a modal conditional. To avoid such confounds,  I use   \textit{-(re)ba} in the following, as this connective  has been observed to be compatible with  both types of conditionals (e.g.\ \citealt{Masuoka93, Masuoka06}). 


% establishes the generalization in \Next, which was first observed by \cite{Kaufmann17-handout}:
%\ex. \textbf{Generalization:}\\ 
%	\textit{Moshi} prevents the antecedent clause of a conditional from restricting an adverb of quantification. \label{generalization-moshi}


%%%%%%%%%%
\subsection{Distribution of \textit{moshi}}
The key observation  is that \textit{moshi} exhibits sensitivity to the type of conditionals where it appears: it is compatible with modal conditionals, but resists  Q-adverbial conditionals.  I provide four diagnostics to establish this observation. 

\paragraph{Interaction with Q-adverbs and modals}
Similarly to English \ref{English-Q-adverb},  Japanese conditionals with the Q-adverb \textit{taitei} `usually' in the consequent are interpreted as Q-adverbial conditionals, as shown in \Next:\footnote{
Intuitively, \textit{taitei} `usually' in \ref{no-moshi-itsumo-taitei} quantifies over plane rides, rather than the intervals where I'm on a plane. Hence, the sentence is true just in case, say, at least sixty percent of my plane rides are such that I feel sick, regardless of the duration of each plane ride under consideration. 
}

\exg. hikouki-ni nor-eba, \textbf{taitei}  kibun-ga waruku naru.\\
	plane-\textsc{dat} get.on-\textsc{reba}  usually feeling-\textsc{nom} bad become\\
	`If I get on a plane, I usually feel sick.' \label{no-moshi-itsumo-taitei}\\
 	Roughly:  Most situations where I get on a plane are sit.\ where I feel sick.

\noindent However, as noted by \cite{Kaufmann17-handout} (credited to Ikumi Imani, p.c.), Q-adverbial conditionals become unacceptable when  \textit{moshi} is present. Compare \Next with \Last:
 \exg. ?? \textbf{moshi} hikouki-ni nor-eba,   \textbf{taitei} kibun-ga waruku naru.\\
	\textsc{moshi} plane-\textsc{dat} get.on-\textsc{reba}  usually  feeling-\textsc{nom} bad become\\
	Intended: `If I get on a plane, I usually feel sick.' \label{moshi-itsumo-taitei}
	
	
\noindent In contrast, modal operators are compatible with \textit{moshi}, be they modal auxiliaries (e.g.\ \textit{kamoshirenai} `might' in \Next) or modal adverbials (e.g.\ \textit{tabun} `maybe' in \NNext): % As shown in \Next and \NNext, \textit{moshi} in modal conditionals is optional---it is allowed, though  not required, and it does not add any extra meaning to the sentences when present. 

  \exg. \{\textbf{moshi} / $\emptyset$\} Mary-ga ku-reba, John-mo kuru \textbf{kamoshirenai}.\\
   		\textsc{moshi} {} {} M-\textsc{nom} come-\textsc{reba} J-\textsc{add} come might\\
  		`If Mary shows up, John might show up, too.'\label{moshi-kamoshirenai}

  \exg. \{\textbf{moshi} / $\emptyset$\} Mary-ga ku-reba, \textbf{tabun} John-mo  kuru.\\
   		\textsc{moshi} {} {} M-\textsc{nom} come-\textsc{reba} maybe J-\textsc{add} come \\
  		`If Mary shows up, maybe John will show up, too.'\label{moshi-tabun}



\paragraph{Generic conditionals}
We can further confirm the contrast between \ref{moshi-itsumo-taitei} vs.\ \ref{moshi-kamoshirenai}/\ref{moshi-tabun}  by examining the distribution of \textit{moshi} in  conditionals that contain covert  quantificational operators. 
One such  case are generic conditionals.
Generic conditionals are standardly assumed to contain a covert generic operator, which gets directly restricted by the antecedent clause and interpreted   similarly to the Q-adverb \textit{usually} (\citealt{FarkasSugioka83}, among many others). \cite{Kaufmann17-handout} observes that \textit{moshi} is unacceptable in generic conditionals, as shown in \Next. This provides another instantiation of the incompatibility between \textit{moshi} and Q-adverbial conditionals  observed in \ref{moshi-itsumo-taitei}.  

\exg. \{\#\textbf{moshi} / $\emptyset$\} taiyou-ga shizum-eba, yoru-ni naru.\\
	\textsc{moshi} {} {} sun-\textsc{nom} sink-\textsc{reba} night-\textsc{dat} become\\
 As a description of natural laws: `It becomes night if the sun goes down.' \vspace{-3mm} \flushright (Adapted from \citealt{Kaufmann17-handout}:  12 (53), credited to  Ikumi Imani, p.c.)\label{generic}


\paragraph{Ambiguity between covert \textit{always} and covert \textit{must}}
\Next provides another conditional that does not contain explicit quantificational operators. It is ambiguous between the readings paraphrased in \Next[a] and \Next[b]. 
\exg. Mary-ga ku-reba, John-mo kuru.\\
	M-\textsc{nom} come-\textsc{reba} J-\textsc{add} come\\
	`If Mary comes, John comes, too.' \label{no-modal}
	\a. All situations where Mary comes are situations where John comes. 
	\b. If Mary comes (e.g.\ tomorrow), it must be the case that John will come. 

\noindent Under the reading in \Last[a], the sentence expresses universal quantification over the situations described by the antecedent. I assume with \cite{Lewis75} that this reading can be derived with a covert Q-adverb \textit{always}. In contrast, under the reading in \Last[b], the speaker draws on her knowledge and  describes what is likely to  happen in case Mary comes; in other words, under this reading, the sentence gets interpreted as a modal conditional.  I assume with \cite{Kratzer86} that this interpretation can be derived with a covert epistemic  \textit{must}. Now, observe that when \textit{moshi} is added, the Q-adverbial reading becomes unavailable, as in \Next:

\exg. \textbf{moshi} Mary-ga ku-reba, John-mo kuru.\\
	\textsc{moshi} M-\textsc{nom} come-\textsc{reba} J-\textsc{add} come\\
	`If Mary comes, John comes, too.' 
	\a. \xmark\ All situations where Mary comes are situations where John  comes. 
	\b. If Mary comes (e.g.\ tomorrow), it must be the case that John will come. 

\noindent I take the contrast between \LLast and \Last to be another instantiation of the observation that \textit{moshi} is compatible with  modals conditionals, but not  Q-adverbial conditionals.  


\paragraph{Q-adverbs that allow narrow-scope readings}
At this point, the reader may wonder whether conditionals containing Q-adverbs also allow  modal interpretations in addition to their  Q-adverbial readings. For instance, one might expect \ref{no-moshi-itsumo-taitei} to be acceptable under  the following interpretation: `if I get on a plane (e.g.\ tomorrow), it must be the case that I'll feel sick many times during that flight'. This would be a reading where the Q-adverb \textit{taitei} `usually' takes narrow scope within the consequent, quantifying over the \textit{intervals} of my stay on a plane. However, this reading is unavailable for \ref{no-moshi-itsumo-taitei} (and hence for its counterpart with \textit{moshi} in \ref{moshi-itsumo-taitei}, too), which  I take to indicate  that in conditionals, \textit{taitei} `usually' must be restricted by conditional antecedents. For a systematic investigation of the Q-adverbs that resist the narrow-scope reading, see \citeauthor{deSwart93} (\citeyear{deSwart93}: \S5). 

However, not all Q-adverbs behave like \textit{taitei}.\footnote{
See \cite{Nitta02} for a systematic classification of Japanese frequency adverbs. 
} For instance, Q-adverbs like  \textit{yoku} `often' and \textit{tokidoki} `sometimes' can, but do not have to, be restricted by conditional antecedents, as shown by the two readings of the sentence in \Next:\footnote{
\cite{Geurts04} discusses   English \textit{usually} vs.\ \textit{often}/\textit{sometimes}   as displaying a similar contrast in terms of whether narrow scope in the consequent clause is available; \citeauthor{deSwart93} (\citeyear{deSwart93}: \S5)  discusses these two types of Q-adverbs as ``strong'' and ``weak'' frequency adverbs more generally. 
}

\exg. hikouki-ni nor-eba, \{\textbf{yoku} /  \textbf{tokidoki}\} kibun-ga waruku naru.\\
	plane-\textsc{dat} get.on-\textsc{reba} often {} sometimes feeling-\textsc{nom} bad become\\
	`If I get on a plane, I often/sometimes feel sick.' \label{no-moshi-yoku-tokidoki}
	\a. Many/Some situations where I get on a plane are sit.\ where I feel sick.
	\b. If I get on a plane (e.g.\ tomorrow), I'll feel sick many times/on and off during that flight. 

\noindent Note that similarly to the modal reading of \ref{no-modal}, I assume that the reading in \Last[b] is derived from a covert epistemic \textit{must} that scopes over \textit{yoku} `often' and \textit{tokidoki} `sometimes'. Now, observe that the  effect of \textit{moshi} in \Last is similar to that in \ref{no-modal}, i.e.\ its presence blocks the Q-adverbial reading, as shown in \Next:

\exg. \textbf{moshi}  hikouki-ni nor-eba, \{{yoku} /  {tokidoki}\} kibun-ga waruku naru.\\
	\textsc{moshi} plane-\textsc{dat} get.on-\textsc{reba} often {} sometimes feeling-\textsc{nom} bad become\\
	`If I get on a plane, I often/sometimes feel sick.'  \label{moshi-yoku-tokidoki}
	\a. \xmark\ Many/Some sit.\ where I get on a plane are sit.\ where I feel sick.
	\b. If I get on a plane (e.g.\ tomorrow), I'll feel sick many times/on and off during that flight. 

\noindent The unavailability of the reading in \Last[a] is consistent with our observation so far, i.e.\ \textit{moshi} is compatible with modal conditionals, but not Q-adverbial conditionals. 

%%%%%%%%%%
\subsection{Interim summary} 
What we have seen is the following. 
As noted in the introduction, Q-adverbial conditionals in English were used to motivate the assumption that conditional antecedents denote plural objects. 
In this section, we have observed that conditionals containing \textit{moshi} lack Q-adverbial interpretations. 
It thus seems plausible to conclude that conditional antecedents containing \textit{moshi} do \textit{not}  denote plural referents. 


However, it still remains unexplained why \textit{moshi} is allowed in modal conditionals. Put differently, what is the difference between Q-adverbial and modal conditionals such that \textit{moshi} is allowed in the latter, but not in the former? 
The next section tackles this question by presenting an independent  argument regarding the difference between the two types of conditionals from the literature. 
 
%%%%%%%%%%
\section{Q-adverbial vs.\  modal quantification  in conditionals}\label{section-existing-view-by-Frank}
For concreteness, let us first take a step back and consider how  Q-adverbial and modal conditionals are captured by Kratzer's restrictor analysis. Under the restrictor analysis, both Q-adverbs and modals are restricted by conditional antecedents,  as exemplified by the construals of the Q-adverbial conditional in \Next and the modal conditional in \NNext: 

\ex. If Mary comes, John sometimes comes, too.\\
	$\leadsto$ $\exists s[\textsc{mary-come}(s)] [\textsc{john-come}(s)]$

\ex. If Mary comes, John might come, too. \\
	$\leadsto$ $\exists w[w_@R^{epi}w\ \&\ \textsc{mary-come}(w)][\textsc{john-come}(w)]$

\noindent In words, \LLast expresses existential  quantification over the situations where Mary comes. Likewise, \Last expresses existential quantification over the worlds that (i) are epistemically accessible from the actual world $w_@$ and (ii) verify Mary's coming.  

However, converging evidence from various independent sources has shown that the Kratzer-style  construal for modal conditionals exemplified  in \Last is incorrect (\citealt{Frank96}, \citealt{Zvolenszky02}, \citealt{vonFintelIatridou05}, \citealt{Schwager06}, among many others). Here, I present one piece of  evidence from deontic conditionals found by \cite{Frank96}.  \Next provides a slightly modified version of her example:\footnote{
	\ref{Frank} is a minimal modification of Frank's original \textit{If the new laws for opening hours pass, salespeople will have to work longer} (\citealt{Frank96}: 199), suggested by Magdalena Kaufmann (p.c.)\ in response to complications with \textit{will} pointed out to her by Dan Lassiter [at \textit{Conditionals 2}, Paris, June 2022].
}
\ex. If the new laws for opening hours pass, salespeople \textbf{have to} work longer. \label{Frank}

\noindent Under the restrictor analysis, \Last would be construed as in \Next; that is, it would be analyzed as universal quantification over the worlds that can be deontically accessed from $w_@$ and verify the passing of the new laws:
\ex.  $\forall w[w_@R^{deo}w\ \&\ \textsc{new-laws-pass}(w)] [\textsc{salesppl-work-longer}(w)]$

\noindent Frank points out that this construal makes wrong predictions for \ref{Frank}. 
Suppose that at $w_@$, the actual laws do not require salespeople to work longer, and that the new laws would require them to do so, but those laws have not gone through. Intuitively, the sentence is true in this scenario, but the construal in \Last predicts the sentence to be false. The reasoning is as follows.  According to \Last, the necessity of salespeople's working longer gets evaluated with respect to the \textit{actual} laws; this is so in virtue of  the conjunct $w_@R^{deo}w$ in the restrictive clause, that is, only the worlds that are compatible with the laws at $w_@$ will be in  the quantificational  domain. Since salespeople do not need to work longer at $w_@$ (and presumably, it would be deontically optimal that salespeople do not  work longer than they are supposed to), this means that all worlds in the quantificational domain are such that salespeople do not work longer. As a result, the construal in \Last comes out as false in this scenario. 

More generally, what goes wrong with the construal in \Last is that the criteria that are used to evaluate the deontic modal are independent of the content of the antecedent, namely the passing of the new laws. Intuitively, what we need is a construal where the criteria can reflect the change of laws described by the antecedent, instead of being ``fixed'' to the laws of the actual world. 
A number of solutions have been proposed to implement this intuition (\citealt{Frank96}, \citealt{KaufmannSchwager09}, \citealt{CondoravdiLauer16}, among others; see an overview in \citealt{KaufmannKaufmann15}: 258--261). 
Here, I adopt a common idea that underlies all those proposals, namely that modals can be evaluated pointwise at antecedent worlds. For  \ref{Frank}, this means that  \textit{have to} need not be  restricted by the antecedent clause,  but rather can be evaluated at each world where the new laws have passed. This idea, once properly implemented, will ensure that the necessity of salespeople's  working longer gets evaluated with respect to  the new laws, rather than the actual laws. 

To sum up, we have seen that Q-adverbs and modals in conditionals differ in that Q-adverbs are directly restricted by conditional antecedents (\textit{\`a la} Kratzer), whereas  modals can be interpreted pointwise at antecedent-worlds. This idea will be formalized in Sec \ref{section-analysis}  within a dynamic framework. 


%%%%%%%%%%
\section{Analysis}\label{section-analysis}
%%%%%%%%%%
\subsection{Basic set-up of the framework}

My analysis is couched in \textbf{Intensional Plural Compositional Discourse Representation Theory} (\textbf{Intensional PCDRT}),  a dynamic-semantic framework developed by \cite{Brasoveanu10}. This framework is equipped with the apparatus for dealing with plurality and definite descriptions of worlds, both of which are necessary for analyzing our phenomena of interest. This subsection introduces the basic ingredients of this framework; Sec \ref{section-illustration-individual-drefs} provides a simple illustration of them put to work. 

First, there are four types in the version of Intensional PCDRT that I adopt:   $t$ (truth values),   $e$ (individuals),   $s$ (variable assignments) and  $\omega$ (situations). For simplicity, I assume that maximal situations are possible worlds  (\citealt{Kratzer89}). 


Second, information available at each  point in a discourse is represented as a set of \textbf{information states} (henceforth \textbf{info states}). Differently from traditional dynamic-semantic frameworks (e.g.\ Dynamic Predicate Logic, \citealt{GSV96}),  info state are modeled as \textit{sets of} variable assignments  (hence, type $\langle s,t \rangle$), rather than as variable assignments (cf.\    \citealt{vandenBerg96}). 

Third, this framework facilitates two types of anaphora, anaphora to individuals  and to situations. Specifically, there are two types of \textbf{discourse referents} (\textbf{drefs}) that can be referred back to, namely individual drefs and situation drefs. The former are  functions from variable assignments to individuals (type $\langle s,e\rangle$), and the latter from variable assignments to situations (type $\langle s, \omega\rangle$). 

Fourth, sentences denote  relations between info states. They are thus of type  $\langle st, \langle st, t\rangle \rangle$, which I  abbreviate as type $T$. Formally, sentences are translated as  discourse representation structures (DRSs), written as  $[\textit{newdrefs} | \textit{conditions} ]$:\footnote{
Some sentences do not introduce new drefs. They are thus  \textit{tests}, and are translated into DRSs in the form $[\textit{conditions}]$, defined as $\lambda I_{\langle s,t\rangle}.\lambda J_{\langle s,t\rangle}. I=J\wedge \textit{conditions}J$. 
}  
\ex. $[\textit{newdrefs} | \textit{conditions} ]:=\lambda I_{\langle s,t\rangle}.\lambda J_{\langle s,t\rangle}.I[\textit{newdrefs} ]J \wedge \textit{conditions}J $   %\vspace{-3mm} \flushright  (\citealt{Brasoveanu10}: 453 (17))

\noindent  \Last says that a DRS relates   an input info state $I$ with an output info state $J$ such that (i) with respect to drefs, $I$ and $J$ differ only in terms of the newly introduced ones specified by  \textit{newdrefs}, and (ii) all \textit{conditions} are satisfied in $J$. In the main text, I leave the idea of dref introduction and condition satisfaction at an intuitive level (see  Sec \ref{section-illustration-individual-drefs} for an illustration); their formal definitions  are provided in Appendix. 

Finally, a sentence is true with respect to an info state just in case there is at least one way of successfully updating the info state with the DRS it denotes. 



%Table \ref{table-objects} summarizes the basic types of this framework and some of the frequently used complex objects.
%\begin{table}[htb]
%\begin{tabular}{|l|c|c|c|}\hline
 % Object 		& Type 	& Variables & Constants \\ \hline
% Truth values 	& $t$ 	& 			& \textsc{1}, \textsc{0}\\
% Individuals		& $e$	& $x$, $y$, $\dots$ & \textsc{tom}, \textsc{cat}$_1$, $\dots$ \\
% Variable assignments & $s$ & $i$, $j$, $\dots$  &\\
% Situations 	& $\omega$ 	& $w_1$, $w_2$, $\dots$ & \textbf{w}$_1$, \textbf{w}$_2$, $\dots$\\ \hline
% Info states 	& $\langle s,t\rangle$ & $I$, $J$ ,$\dots$& \\
% Individual drefs	& $\langle s,e\rangle$ & $u$, $v$, $\dots$ & \\
% Situation drefs & $\langle s,\omega \rangle$ & $p$, $q$, $\dots$ & \\
% DRSs		& $\langle st, \langle st, t\rangle \rangle$ (for short, $T$) & $D_1$, $D_2$, $\dots$ & \\ \hline
%\end{tabular}
%\caption{Basic types and frequently used objects}\label{table-objects}
%\end{table}


%%%%%%%%%%
\subsection{Simple illustration: Introduction of individual drefs}\label{section-illustration-individual-drefs}
Let us use the mini-discourse in \Next to see how the system works. By common convention, subscripts indicate that a dref is newly introduced, subscripts indicate anphoric reference; semi-colons stand for dynamic conjunction (see Appendix for the translations of the lexical items).\footnote{
For the sake of illustration, I translate the sentences in \ref{mouse} into extensional, rather than intensional terms. Intensional translations of, say, \ref{translation-mouse-anaphora} would appear as $\lambda q_{\langle s,\omega\rangle}. [\textsc{laughed}_q\{v\}]$; see Appendix for  relevant details of the intensional system.  Throughout, curly braces indicate application to the value of the dref at a particular variable assignment; \textsc{smallcaps} stand for expressions in the translation language, and \textbf{boldface} stands for entities and situations in the model. 
} 

\ex. \label{mouse} \a.  A$^v$ mouse came.  $\leadsto$ $[v|\textsc{sg}\{v\}]; [\textsc{mouse}\{v\}]; [\textsc{came}\{v\}]$  \label{translation-mouse}
	\b.  It$_v$ laughed. $\leadsto$ $[\textsc{laughed}\{v\}]$   \label{translation-mouse-anaphora}
	 


Suppose a model with three individuals:  \textbf{jerry}, who is a mouse that came and laughed, \textbf{mickey}, who is  a mouse but didn't come and didn't laugh, and \textbf{tom}, who is a cat that came and laughed. And suppose that our input info state does not yet contain any information.  Fig.\  1 visualizes the interpretation of \Last[a] in this model.\footnote{\label{footnote-dummy}
Technically, the discourse-initial  info state $I_0$ should be the singleton set of a dummy assignment $i_{\star}$, which assigns to all drefs a dummy value $\star$ that falsifies all lexical relations  (cf.\ \citealt{Brasoveanu10}: 455--456). For instance, when   $\textsc{mouse}\{v\}$  is applied to the discourse-initial info state $\{i_\star\}$, the only assignment in this info state, $i_\star$, does not store a $v$-individual that satisfies the  property of being a mouse; hence, the discourse-initial info state cannot be updated by the DRS $[\textsc{mouse}\{v\}]$. This captures the fact that anaphors cannot be used in discourse-initial utterances, e.g.\ \textit{It$_v$ is a mouse}. 
}

\begin{figure}[t]
	\centering
	\scalebox{.8}{
\begin{tabular}{|c|c|} 
\multicolumn{1}{c}{\small $I_0$}\\ \hline
\\ \hline
\end{tabular} 
\begin{tabular}{ccccccccc} 
&   \multicolumn{1}{c}{\small $I'$} & \multicolumn{1}{c}{\small {$v$}}  &   & \multicolumn{1}{c}{\small $I'$} &  \multicolumn{1}{c}{\small $v$} & & \multicolumn{1}{c}{\small $I'$}  & \multicolumn{1}{c}{\small $v$}\\ \cline{2-3}\cline{5-6}\cline{8-9}
  &  \multicolumn{1}{|c|}{ $i_1$} & \multicolumn{1}{|c|}{{\textbf{jerry}}} & $ \xrightarrow{[\textsc{mouse}\{v\}]}$  &  \multicolumn{1}{|c|}{$i_1$} & \multicolumn{1}{|c|}{ \textbf{jerry}} & $\xrightarrow{[\textsc{came}\{v\}]}$ & \multicolumn{1}{|c|}{$i_1$} & \multicolumn{1}{|c|}{ \textbf{jerry}}\\\cline{2-3}\cline{5-6}\cline{8-9}
   & \multicolumn{1}{c}{\small $I''$} & \multicolumn{1}{c}{\small {$v$}} & & \multicolumn{1}{c}{\small $I''$} &  \multicolumn{1}{c}{\small $v$} \\ \cline{2-3}\cline{5-6}
$\xrightarrow{[v|\textsc{sg}\{v\}]}$  & \multicolumn{1}{|c|}{ $i_2$} & \multicolumn{1}{|c|}{ {\textbf{mickey}} } & $ \xrightarrow{[\textsc{mouse}\{v\}]}$ & \multicolumn{1}{|c|}{$i_2$} & \multicolumn{1}{|c|}{ \textbf{mickey} } & $\xrightarrow{[\textsc{came}\{v\}]}$ & $\emptyset$\\ \cline{2-3}\cline{5-6}
& \multicolumn{1}{c}{\small $I'''$} & \multicolumn{1}{c}{\small {$v$}} \\ \cline{2-3}
& \multicolumn{1}{|c|}{ $i_3$} &  \multicolumn{1}{|c|}{ {\textbf{tom}} } &$ \xrightarrow{[\textsc{mouse}\{v\}]}$  & $\emptyset$\\ \cline{2-3}
\end{tabular} 
}
\caption{Introduction of individual drefs} \label{figure-individual-drefs}
\end{figure}


In the sequence of updates depicted in Fig.\ \ref{figure-individual-drefs}, the first DRS $[v|\textsc{sg}\{v\}]$  is contributed by the indefinite determiner; it specifies the introduction of a new dref $v$ and a \textbf{singularity condition} imposed on $v$. Specifically, it asks us to randomly store at $v$ all individuals in our model, but  to also ensure that $v$ stores exactly one value  in each info state. As a result, $[v|\textsc{sg}\{v\}]$ gives rise to a distributive effect:  the discourse, which originally consists of only one info state (i.e.\ $I_0$), gets split into a \textit{set of} info states (i.e.\ $I'$, $I''$, $I'''$), each storing exactly one individual at $v$. %This  feature of the singularity condition will later play an important role in our analysis of \textit{moshi}.   
Next, the output of $[v|\textsc{sg}\{v\}]$ gets updated  first with  $[\textsc{mouse}\{v\}]$, and then with $[\textsc{came}\{v\}]$. This requires us to check whether the individuals stored at $v$ satisfy the property of being a mouse and having come.  Since \textbf{tom} is not a mouse, the info state $I'''$ is eliminated by  $[\textsc{mouse}\{v\}]$. Since \textbf{mickey} did not come, $I''$ is eliminated by $[\textsc{came}\{v\}]$. In contrast, the $v$-individual stored in $I'$, \textbf{jerry}, is  a mouse that came; hence, $I'$ can be successfully updated by the DRS sequence in \ref{translation-mouse}. Despite not being depicted in the figure, it is easy to see that $I'$ will also pass the test expressed by \ref{translation-mouse-anaphora}.  
Overall, since truth  is defined  by the existence of output info states, we correctly predict the mini-discourse in  \ref{mouse} to be true in the given model. 
% Hence, the mini-discourse in \ref{mouse} leads to a discourse that contains just one info state, namely $I'$. 

%%%%%%%%%%
\subsection{Introduction of situation drefs}\label{section-analysis-situation-drefs}
Following \citeauthor{Brasoveanu10}, I assume that English \textit{if} introduces a new dref that stores the situations verifying the antecedent proposition, as in \Next[a]. For Japanese, I propose that it is  the verbal suffixes/enclitics in the antecedents that do this, as in  \Next[b]. The translation of \Next[a-b] is given in \Next[c]. The maximization operator \textbf{max} ensures that $q$ stores \textit{all} situations  that verify the antecedent proposition (see the appendix).  

\begin{multicols}{2}
	

\ex. \label{situation-dref-number-neutral} \a. \textbf{If}$^q$ I get on a plane, $\ldots$ 
	\bg. hikouki-ni nor-\textbf{eba}$^q$, $\ldots$\\
	plane-\textsc{dat} get.on-\textsc{reba} \\
	\b.[] \hspace{-1.7cm} c.  $\leadsto$ $\textbf{max}^q([\textsc{I-get-on-a-plane}_q])$
	\b.[]
	\b.[]
	
\end{multicols}





\begin{figure}[t]
\centering
\begin{multicols}{2}
\scalebox{.85}{
\hspace{-2.5cm} 
\begin{tabular}{|c|c|} 
\multicolumn{1}{c}{\small }\\ \hline
\\ \hline
    \multicolumn{1}{c}{ }  \\ 
\end{tabular}$\hspace{1mm} \xrightarrow{\textbf{max}^q([\textsc{I-g.o.-plane}_q])}$
\begin{tabular}{|c|} 
\multicolumn{1}{c}{\small $q$} \\ \hline
\textbf{w}$_1$  \\ \hline
\textbf{w}$_2$  \\ \hline
\textbf{w}$_3$ \\ \hline 
\end{tabular}
}
 
\scalebox{.8}{ 
\hspace{-1.2cm}
\begin{tabular}{|c|c|} 
   \multicolumn{1}{c}{ } \\ \hline
   \\ \hline
\end{tabular} 
\begin{tabular}{cc|ccc|} 
 & \multicolumn{1}{c}{\small $p$} & & \multicolumn{1}{c}{\small $p$} & \multicolumn{1}{c}{\small $q$}  \\  \cline{2-2}\cline{4-5}
  &  \multicolumn{1}{|c|}{\textbf{w}$_1$} & $\xrightarrow{\textbf{max}^{q\Subset p}([\textsc{I-g.o.-plane}_q])}$ &  \multicolumn{1}{|c}{\textbf{w}$_1$} & \textbf{w}$_1$  \\\cline{2-2}\cline{4-5}
 &  \multicolumn{1}{c}{\small $p$}  & &    \multicolumn{1}{c}{\small $p$} & \multicolumn{1}{c}{\small $q$}\\ \cline{2-2}\cline{4-5} 
$\xrightarrow{[p|\textsc{sg}\{p\}]}$ & \multicolumn{1}{|c|}{\textbf{w}$_2$}  & $\xrightarrow{\textbf{max}^{q\Subset p}([\textsc{I-g.o.-plane}_q])}$ &  \multicolumn{1}{|c}{\textbf{w}$_2$} & \textbf{w}$_2$  \\ \cline{2-2}\cline{4-5}
  & \multicolumn{1}{c}{\small $p$} & &   \multicolumn{1}{c}{\small $p$} & \multicolumn{1}{c}{\small $q$} \\  \cline{2-2}\cline{4-5}
  & \multicolumn{1}{|c|}{\textbf{w}$_3$} & $\xrightarrow{\textbf{max}^{q\Subset p}([\textsc{I-g.o.-plane}_q])}$ &  \multicolumn{1}{|c}{ \textbf{w}$_3$} & \textbf{w}$_3$\\ \cline{2-2}\cline{4-5} 
\end{tabular} 
}
\end{multicols}
\caption{Introducing situation drefs (left: number-neutral; right: singular)}\label{figure-situation-dref}
\end{figure} 



\noindent For instance, in a model where I get on a plane in \textbf{w}$_1$,  \textbf{w}$_2$ and \textbf{w}$_3$, the update in \LLast[c] is visualized on the left of Fig.\ \ref{figure-situation-dref}. For simplicity, henceforth I will leave out  the variables for info states (e.g.\ $I$) and assignments (e.g.\ $i_1$) in the figures. 
Note that the translation in \LLast[c] is essentially a dynamic implementation of the idea that conditional antecedents are definite descriptions of situations. 
In addition,  while \citeauthor{Brasoveanu10} assumes that conditional connectives store \textit{plural} referents (see \citealt{Brasoveanu10}: 500--501), I propose that by default, they store \textit{number-neutral} referents. 
It is also worth mentioning that the definite descriptions denoted by  \LLast[a] and \LLast[b]  are \textit{non-anaphoric}, since \textit{if} and \textit{-(re)ba} establish new drefs, instead of anaphorically retrieving  existing drefs. Anaphoric instances of conditional connectives will become relevant in Sec \ref{section-analysis-Q-adverbial-conditionals}. 




I propose that situation drefs can be singular; in particular, \textit{moshi} is an overt marker of the singularity of situation drefs.   
Specifically, \textit{moshi} introduces a singular situation dref (similarly to the singular indefinite determiner \textit{a}; see  \ref{mouse}), and the  verbal suffixes/enclitics (e.g.\ \textit{-(re)ba}) are anaphoric to it.\footnote{
I leave it open \textit{why} the  verbal suffixes/enclitics have to be anaphoric to the dref introduced by \textit{moshi}. One possibility is that   it is forced by the local binding of situation variables argued by \cite{Percus00}.
} To illustrate, consider \Next[a]; see the indices on \textit{moshi} and \textit{-(re)ba} for the relevant drefs. See \Next[b] for the translation of \Next[a]; the corresponding update is visualized  on the right of Fig.\ \ref{figure-situation-dref}.  \Next[b] says that \textit{moshi} introduces a singular  dref $p$, thereby storing exactly one situation in each info state. As for \textit{-(re)ba}, it is anaphoric to $p$ and introduces a new dref $q$ that is a structured subset ($\Subset$) of  $p$.\footnote{
Informally, structured subset helps us preserve the dependencies between individuals; see the appendix for its definition.  \citeauthor{Brasoveanu10} (\citeyear{Brasoveanu10}: \S2.4) argues that a stronger notion of structured subset  is needed in order to capture the strong reading of donkey anaphora. Since the issue is orthogonal to our current concern, this paper adopts the simpler definition. 
}

\ex.\label{situation-dref-singular} \ag. \textbf{moshi}$^p$ hikouki-ni nor-eba$_p^{q\Subset p}$,  $\ldots$\\
	\textsc{moshi} plane-\textsc{dat} get.on-\textsc{reba} \\
	`If I get on a plane, $\ldots$'
	\b. $\leadsto$ $[p|\textsc{sg}\{p\}]; \textbf{max}^{q\Subset p}([\textsc{I-get-on-a-plane}_q])$

\noindent   As a consequence  of the singularity of $p$ and the structured subset relation between $p$ and $q$, each info state  ends up storing exactly one value not only at $p$, but also at $q$. In other words,  the singularity introduced by \textit{moshi} is inherited by the dref that stores the antecedent situations. This effect will become crucial in my explanation of the oddness of \textit{moshi} in Q-adverbial conditionals. 


%%%%%%%%%%
\subsection{Analysis of Q-adverbial conditionals}\label{section-analysis-Q-adverbial-conditionals}

I follow \citeauthor{Brasoveanu10} and assume that Q-adverbs relate  two drefs that store the situations satisfying  the restrictor and the nuclear scope, respectively. As indicated in \Next, the restrictor dref $p$ is obtained via anaphora  (e.g.\ from the context, a subordinate clause or a common noun phrase; cf.\  \citealt{vonFintel94}), whereas the nuclear-scope dref $q$ is freshly established by the Q-adverb itself  (irrelevant individual drefs  are omitted in \Next). 
Due to the conservativity of natural language quantifiers,  the nucelar-scope dref is assumed to be a  structured  subset of the restrictor dref (i.e.\ $q\Subset p$).  
Overall, Q-adverbs express a set relation between the two drefs; for instance, \textit{usually} tests whether the $p$-situations and the $q$-situations stand in a \textsc{most}-relation. 

\ex. Usually$_p^{q\Subset p}$, I'm happy. \hspace{5mm} 
	$\leadsto$   \hspace{5mm} 
	$\textbf{max}^{q\Subset p}([\textsc{I'm-happy}_q]); [\textsc{most}\{p,q\}]$ \label{usually-happy}



In Q-adverbial conditionals, the restrictor dref of Q-adverbs is provided by the antecedent via anaphora. \Next provides the example from \ref{no-moshi-itsumo-taitei} and  its translation. 
\ex. \label{Q-adverbial-repeat} \ag. hikouki-ni nor-reba$^q$, \textbf{taitei}\hspace{.5mm}$_q^{r\Subset q}$ kibun-ga waruku naru. \\
  	plane-\textsc{dat} get.on-\textsc{reba} usually feeling-\textsc{nom} bad become\\
  		`If  I get on a plane, I usually feel sick.' 
 	\b. $\leadsto$ $\textbf{max}^q([\textsc{I-g.o.-plane}_q]); \textbf{max}^{r\Subset q}([\textsc{I-fl-sick}_r]);[\textsc{most}\{q,r\}]$



\begin{figure}[t]
\scalebox{.9}{
\begin{tabular}{|c|c|} 
\multicolumn{1}{c}{\small}\\ 
\hline
\\ \hline
    \multicolumn{1}{c}{ }  \\ 
\end{tabular}$\hspace{1mm} \xrightarrow{\textbf{max}^q([\textsc{I-g.o.-plane}_q])}$
\begin{tabular}{|c|} 
\multicolumn{1}{c}{\small $q$} \\ \hline
\textbf{w}$_1$  \\ \hline
\textbf{w}$_2$  \\ \hline
\textbf{w}$_3$  \\ \hline 
\end{tabular} 
$\hspace{1mm} \xrightarrow{\textbf{max}^{r\Subset q}([\textsc{I-fl-sick}_r])}$
\begin{tabular}{|cc|} 
\multicolumn{1}{c}{\small $q$} & \multicolumn{1}{c}{\small $r$} \\ \hline
\textbf{w}$_1$ & \textbf{w}$_1$ \\ \hline
\textbf{w}$_2$ & \textbf{w}$_2$  \\ \hline
\textbf{w}$_3$ & $\star$ \\ \hline 
\end{tabular}
$\hspace{1mm} \xrightarrow{[\textsc{most}\{q,r\}]}$
\begin{tabular}{|cc|} 
\multicolumn{1}{c}{\small $q$} & \multicolumn{1}{c}{\small $r$} \\ \hline
 \textbf{w}$_1$ & \textbf{w}$_1$ \\ \hline
 \textbf{w}$_2$ & \textbf{w}$_2$  \\ \hline
 \textbf{w}$_3$ & $\star$ \\ \hline 
\end{tabular}
}
\caption{Interpretation of Q-adverbial conditionals (cf.\ \ref{Q-adverbial-repeat}) }
\label{figure-Q-adverbial}
\end{figure}

\noindent Suppose that I get on a plane in $\textbf{w}_1$, $\textbf{w}_2$ and $\textbf{w}_3$, and feel sick in $\textbf{w}_1$ and $\textbf{w}_2$.  The interpretation of \Last[a] in such a model is depicted in Fig.\ \ref{figure-Q-adverbial}.  
 In the first two updates, we store at $q$  all situations where I get on a plane, and at $r$ all situations where I feel sick. Note that the dummy situation $\star$ assigned to $r$ at the third assignment  reflects the fact in this model that I do not feel sick at \textbf{w}$_3$ (see footnote \ref{footnote-dummy} for the dummy value). Next, we  perform the test $[\textsc{most}\{q,r\}]$ on the resulting info state by checking whether most $q$-situations are also $r$-situations. Since this is indeed the case, the info state passes  the update.  We  thus correctly predict  \Last[a] to be true in this model. 


Together with my proposal for \textit{moshi} introduced in Sec \ref{section-analysis-situation-drefs}, this  analysis of Q-adverbial conditionals explains the oddness of \textit{moshi} in Q-adverbial conditionals. 
 \Next[a] repeats the relevant example from \ref{moshi-itsumo-taitei}. 
 %There are two differences between this example and its \textit{moshi}-less counterpart   in \Last[a]: first, \textit{moshi} introduces a new singular dref $p$ at the beginning of the interpretation in \Next[a]; second, the suffix \textit{-(re)ba} in \Next[a] is anaphoric to $p$, and the dref introduced by it, $q$, is required to be a structured subset of $p$. 
 Compare the indices in \Last[a] and \Next[a], and also  the translations in  \Last[b] and \Next[b]. 
 
\ex. \label{moshi-Q-adverbial-repeat}
	\ag. ??\textbf{moshi}\hspace{.5mm}$^p$ hikouki-ni nor-reba$_p^{q\Subset p}$, taitei$_{q}^{\hspace{.5mm}r\Subset q}$ kibun-ga waruku naru.   \\
  	\textsc{moshi} plane-\textsc{dat} get.on-\textsc{reba} usually feeling-\textsc{nom} bad become\\
  		`If  I get on a plane, I usually feel sick.' 
 	\b.  \hspace{-3mm} {\small $\leadsto$  $[p|\textsc{sg}\{p\}]; \textbf{max}^{q\Subset p}([\textsc{I-g.o.-plane}_q]); \textbf{max}^{r\Subset q}([\textsc{I-fl-sick}_r]);[\textsc{most}\{q,r\}] $  }




\begin{figure}[H]
\scalebox{.79}{
\begin{tabular}{|c|c|} 
\multicolumn{1}{c}{\small }\\ \hline
\\ \hline
\end{tabular} $\xrightarrow{[p|\textsc{sg}\{p\}]}$
\begin{tabular}{|c|cc|c|cc|c|c|c} 
\multicolumn{1}{c}{\small $p$} & & \multicolumn{1}{c}{\small $p$} & \multicolumn{1}{c}{\small $q$} & &  \multicolumn{1}{c}{\small $p$} & \multicolumn{1}{c}{\small $q$} & \multicolumn{1}{c}{\small $r$}  \\ \cline{1-1}\cline{3-4}\cline{6-8}
\textbf{w}$_1$ & $\xrightarrow{\textbf{max}^{q\Subset p}([\textsc{I-g.o.-pl}_q])}$ & \multicolumn{1}{|c|}{ \textbf{w}$_1$} & \textbf{w}$_1$ & $\xrightarrow{\textbf{max}^{r\Subset q}([\textsc{fl-sick}_r])}$ & \multicolumn{1}{|c|}{\textbf{w}$_1$} &  \textbf{w}$_1$  & \textbf{w}$_1$ & $\xrightarrow{[\textsc{most}\{q,r\}]}$ {\Large $\emptyset$} \\ \cline{1-1}\cline{3-4}\cline{6-8}
\multicolumn{1}{c}{\small $p$}  & &  \multicolumn{1}{c}{\small $p$} & \multicolumn{1}{c}{\small $q$} &  &  \multicolumn{1}{c}{\small $p$} & \multicolumn{1}{c}{\small $q$} & \multicolumn{1}{c}{\small $r$}\\ \cline{1-1}\cline{3-4}\cline{6-8}
\textbf{w}$_2$ & $\xrightarrow{\textbf{max}^{q\Subset p}([\textsc{I-g.o.-pl}_q])}$ & \multicolumn{1}{|c|}{ \textbf{w}$_2$} & \textbf{w}$_2$ & $\xrightarrow{\textbf{max}^{r\Subset q}([\textsc{fl-sick}_r])}$ & \multicolumn{1}{|c|}{\textbf{w}$_2$} &  \textbf{w}$_2$  & \textbf{w}$_2$ & $\xrightarrow{[\textsc{most}\{q,r\}]}$ {\Large $\emptyset$}  \\ \cline{1-1}\cline{3-4}\cline{6-8}
 \multicolumn{1}{c}{\small $p$} & &  \multicolumn{1}{c}{\small $p$} & \multicolumn{1}{c}{\small $q$} & &  \multicolumn{1}{c}{\small $p$} & \multicolumn{1}{c}{\small $q$} & \multicolumn{1}{c}{\small $r$} \\ \cline{1-1}\cline{3-4}\cline{6-8}
\textbf{w}$_3$ & $\xrightarrow{\textbf{max}^{q\Subset p}([\textsc{I-g.o.-pl}_q])}$ &  \multicolumn{1}{|c|}{\textbf{w}$_3$} & \textbf{w}$_3$ & $\xrightarrow{\textbf{max}^{r\Subset q}([\textsc{fl-sick}_r])}$ & \multicolumn{1}{|c|}{\textbf{w}$_3$} &  \textbf{w}$_3$  & $\star$ & $\xrightarrow{[\textsc{most}\{q,r\}]}$ {\Large $\emptyset$}\\ \cline{1-1}\cline{3-4}\cline{6-8}
   \multicolumn{1}{c}{\small $\vdots$} 
\end{tabular}
}
\caption{Interpretation of Q-adverbial conditionals in presence of \textit{moshi} (cf.\ \ref{moshi-Q-adverbial-repeat})}
\label{figure-Q-adverbial-moshi}
\end{figure}

\noindent  Assuming the same model as before,  the interpretation of \Last is depicted  in Fig.\ \ref{figure-Q-adverbial-moshi}.  
As shown in the figure, in addition to the singular dref $p$ introduced by \textit{moshi}, the restrictor dref $q$ and the nuclear-scope dref $r$ also turn out to store singular referents. As noted in Sec  \ref{section-analysis-situation-drefs}, this effect arises from the singularity condition imposed by \textit{moshi}, in combination with  the structured subset relation between $p$ and $q$ and between $q$ and $r$. 
I propose that it is the  singularity of the restrictor dref $q$ that leads to the unacceptability of \Last[a]; specifically, the singularity of $q$ causes trouble for the last update, $[\textsc{most}\{q,r\}]$. 
Recall that in Sec \ref{section-introduction}, we have established the general constraint that Q-adverbs require non-singleton restrictors. I implement this constraint as the presupposition of the two-place lexical relations that quantifiers are associated with, exemplified using \textsc{most} below:
\ex. $\textsc{most}\{u,u'\}$ is defined relative to  $I_{\langle s,t\rangle}$ only if $|\{u(i):i_s\in I_{u\neq\star}\}|>1$, where $I_{u\neq \star}$ is the set of assignments $i\in I$ that store a non-dummy value at $u$.\label{definition-presupposition-Q-adverb}


\noindent It is now easy to see what goes wrong with the interpretation depicted in Fig.\ \ref{figure-Q-adverbial-moshi}. In all input info states of $[\textsc{most}\{q,r\}]$,  $q$ stores a singular referent, thus failing to satisfy the presupposition in \Last; as a result, none of the input info states can be updated successfully. Since  truth is defined in this framework in terms of the existence of output info states,   \LLast[a]  comes out necessarily false. I argue that this  necessary falsity explains   the unacceptability of  \textit{moshi} in  Q-adverbial conditionals.\footnote{
See \cite{Gajewski02,Gajewski09} for an account of ungrammaticality based on semantic triviality.
} 

%%%%%%%%%%
\subsection{Analysis of modal conditionals} \label{section-analysis-modal-conditionals}
In Sec \ref{section-existing-view-by-Frank}, I have argued for independent reasons that,  unlike Q-adverbs, modals in conditionals do not need to be  restricted by the antecedents. Therefore, deviating  from \citeauthor{Brasoveanu10}, I  propose the following difference between Q-adverbs and modals: while Q-adverbs are anaphoric to the antecedent dref  for their restrictor (e.g.\ $q$ in \ref{Q-adverbial-repeat}), modals establish restrictor drefs on their own. 

As exemplified in \Next, I propose that modals  are associated with three drefs: an anaphorically retrieved dref that stores the evaluation situation ($p$), a newly established restrictor dref ($q$), and a newly established neclear-scope dref ($r$). Overall, modals express a set relation between restrictor and nuclear-scope drefs. 

\ex. {\small I might$_p^{q,r\Subset q}$ be late. $\leadsto$ $\textbf{max}^{q}([\textsc{R}^{epi}\{p,q\}]); \textbf{max}^{r\Subset q}([\textsc{I'm-late}_r]); [\textsc{some}\{q,r\}]$}

\noindent In the  translation above, $\textsc{R}^{epi}$ is a two-place lexical relation between  situations, which models the epistemic accessibility relation associated with \textit{might}.\footnote{
A more fine-grained account of modality \textit{\`a la} \cite{Kratzer81} can be  recast in this framework (cf.\ \citealt{Brasoveanu10}), but I refrain from doing so  to avoid distracting from the main focus of this paper.  
} Hence, the update $\textbf{max}^{q}([\textsc{R}^{epi}\{p,q\}])$ introduces a new dref $q$ that stores all situations that are epistemically accessible from the  evaluation situation stored in $p$. The rest of the interpretation proceeds as usual: we store in $r$ all situations that satisfy the prejacent of \textit{might}, and check whether the $q$- and the $r$-situations stand in a \textsc{some}-relation. 

I propose that in conditionals, it is the evaluation dref of modals that is provided by the antecedent. Using the example discussed in \ref{moshi-kamoshirenai}, this is illustrated by the indices in  \Next[a] and the translation in \Next[b]. 
%This new modal semantics captures the difference between Q-adverbs and modals in conditionals under the assumption adopted  in Sec \ref{section-existing-view-by-Frank}. 

 \ex. \label{modal-conditional-repeat}
 	\ag. Mary-ga ku-\textbf{reba}\hspace{.2mm}$^q$, John-mo kuru  \textbf{kamoshirenai}$_q^{\hspace{.5mm}r,s\Subset r}$.   \\
	 M-\textsc{nom} come-\textsc{reba}  J-\textsc{nom} come might\\ 
	`If Mary comes,  John might  also come.'
	\b. \hspace*{-4mm} {\small $\leadsto$ $\textbf{max}^{q}([\textsc{m-come}_q]); \textbf{max}^r([\textsc{R}^{epi}\{q,r\}]); \textbf{max}^{s\Subset r}([\textsc{j-come}_s]); [\textsc{some}\{r,s\}]$ }
	

\noindent %In \Last[a], the suffix \textit{-(re)ba} introduces a new dref $q$ that stores all situations where Mary comes, similarly to the examples that we have considered so far. However, unlike the Q-adverbs considered in Sec \ref{section-analysis-Q-adverbial-conditionals}, the modal \textit{kamoshirenai} `might' is anaphoric to $q$ for its evaluation dref, rather than its restrictor dref. 
To see how this idea works, assume that the following holds: (i) Mary comes in $\textbf{w}_1$ and $\textbf{w}_2$, (ii) John comes in $\textbf{w}_1$, and (iii)  from   both $\textbf{w}_1$ and $\textbf{w}_2$, we can epistemically access $\textbf{w}_1$ and $\textbf{w}_3$. The translation in \Last[b] is depicted in Fig.\ \ref{figure-modal-conditional} (the step of applying  $\textbf{max}^{q}([\textsc{m-come}_q])$ to the  initial info state is omitted).
 It is easy to see that   $\textsc{R}^{epi}$  has an ``expansion'' effect on its input info state in the sense that its DRS yields  an output info state that consists of four assignments, rather than just two. This is so because from each situation stored in $q$, namely \textbf{w}$_1$ and \textbf{w}$_2$, there are  two situations are epistemically accessible. 
The reader can verify using Fig.\ \ref{figure-modal-conditional} that  the  translation in \Last[b]  predicts correctly that \Last[a] is true in this model.


\begin{figure}[t]
\scalebox{.79}[.8]{
$\dots \xrightarrow{}$
\begin{tabular}{|c|} 
   \multicolumn{1}{c}{\small $q$} \\ \hline
	\textbf{w}$_1$  \\ \hline
  	\textbf{w}$_2$  \\ \hline
\end{tabular}
 $\xrightarrow{\textbf{max}^{r}([\textsc{R}^{epi}\{q,r\}])}$
\begin{tabular}{|cc|} 
    \multicolumn{1}{c}{\small $q$} & \multicolumn{1}{c}{\small $r$} \\ \hline
    \textbf{w}$_1$ & \textbf{w}$_1$ \\ \hline
   \textbf{w}$_1$ & \textbf{w}$_3$  \\ \hline
   \textbf{w}$_2$  & \textbf{w}$_1$ \\ \hline
  \textbf{w}$_2$  & \textbf{w}$_3$  \\ \hline
\end{tabular}
 $\xrightarrow{\textbf{max}^{s\Subset r}([\textsc{j-come}_s])}$
\begin{tabular}{|ccc|} 
 \multicolumn{1}{c}{\small $q$} & \multicolumn{1}{c}{\small $r$} & \multicolumn{1}{c}{\small $s$} \\ \hline
     \textbf{w}$_1$ & \textbf{w}$_1$ & \textbf{w}$_1$  \\ \hline
    \textbf{w}$_1$ & \textbf{w}$_3$ & $\star$ \\ \hline
   \textbf{w}$_2$  & \textbf{w}$_1$ & \textbf{w}$_1$ \\ \hline
    \textbf{w}$_2$  & \textbf{w}$_3$ & $\star$ \\ \hline
\end{tabular}
$\xrightarrow{[\textsc{some}\{r,s\}]}$
\begin{tabular}{|ccc|} 
   \multicolumn{1}{c}{\small $q$} & \multicolumn{1}{c}{\small $r$} & \multicolumn{1}{c}{\small $s$} \\ \hline
    \textbf{w}$_1$ & \textbf{w}$_1$ & \textbf{w}$_1$  \\ \hline
   \textbf{w}$_1$ & \textbf{w}$_3$ & $\star$ \\ \hline
 \textbf{w}$_2$  & \textbf{w}$_1$  & \textbf{w}$_1$ \\ \hline
  \textbf{w}$_2$  & \textbf{w}$_3$ & $\star$ \\ \hline
\end{tabular}
}
\caption{Interpretation of modal conditionals (cf.\ \ref{modal-conditional-repeat})} \label{figure-modal-conditional}
\end{figure}





Before examining the effect of \textit{moshi} in modal conditionals, it is worth emphasizing again that the main feature of the current semantics of modal conditionals---that modals are anaphoric to the antecedent dref for their situation dref, rather than for their restrictor dref---is not itself a new idea. Rather, it is an implementation of the existing idea that modals in conditionals can be evaluated pointwise at antecedent worlds (cf.\ Sec \ref{section-existing-view-by-Frank}). Consider again  deontic conditionals whose antecedents describe  a change of laws; \Next provides a shortened  version of the example in \ref{Frank}: 

\ex. If$^q$ the new laws   pass, salespeople  have-to$_q^{r,s\Subset r}$ work longer.

\noindent The indices indicate that under the current analysis, the deontic modal \textit{have to} is evaluated with respect to the situations where the new laws have passed. This allows us to obtain a set of deontically accessible situations (stored at $r$) from the law-passing situations, rather than from the actual situation. As a result, the change of laws described by the antecedent will feed into the  criteria for  evaluating  \textit{have to}. As noted in Sec \ref{section-existing-view-by-Frank}, this is exactly what we need to capture the interpretation of \Last. 

We are now ready to explain why \textit{moshi} is acceptable in modal conditionals. %The upshot is that \textit{moshi} does \textit{not} prevent the antecedent from providing a meaningful restrictor for the modal. 
Consider \Next[a] and its translation in \Next[b]; the update is visualized in Fig.\ \ref{figure-modal-conditional-moshi}. 

\ex. \label{moshi-modal-repeat}
	 \ag. \textbf{moshi}\hspace{.5mm}$^p$ Mary-ga ku-\textbf{reba}$_p^{q\Subset p}$, John-mo kuru  \textbf{kamoshirenai}$_q^{\hspace{.5mm}r,s\Subset r}$.   \\
	 \textsc{moshi}  M-\textsc{nom} come-\textsc{reba}  J-\textsc{nom} come  might\\ 
	`If Mary comes,  John might  also come.'
		\b.  {\small $\leadsto$$ [p|\textsc{sg}\{p\}]; \textbf{max}^{q\Subset p}([\textsc{m-come}_q]);  \textbf{max}^r([\textsc{R}^{epi}\{q,r\}]); \textbf{max}^{s\Subset r}([\textsc{j-come}_s]);$} \\
			\hspace*{2.8mm} {\small $[\textsc{some}\{r,s\}]$ }



\begin{figure}[t]
\scalebox{.8}{
\begin{tabular}{|c|} 
\multicolumn{1}{c}{\small }\\ \hline
\\ \hline
\end{tabular}  $\xrightarrow{[p|\textsc{sg}\{p\}]}$
\begin{tabular}{|c|ccc|c} 
\multicolumn{1}{c}{\small $p$}  &  & \multicolumn{1}{c}{\small $p$}  & \multicolumn{1}{c}{\small $q$} \\ \cline{1-1}\cline{3-4}
  \textbf{w}$_1$ & $\xrightarrow{\textbf{max}^{q\Subset p}([\textsc{m-come}_q])}$ &  \multicolumn{1}{|c}{\textbf{w}$_1$} & \textbf{w}$_1$ &  $\xrightarrow{\textbf{max}^r([\textsc{R}^{epi}\{q,r\}])}$   \\ \cline{1-1}\cline{3-4}
\multicolumn{1}{c}{\small  $p$} & & \multicolumn{1}{c}{\small $p$}  & \multicolumn{1}{c}{\small  $q$} \\ \cline{1-1}\cline{3-4}
 \textbf{w}$_2$ & $\xrightarrow{\textbf{max}^{q\Subset p}([\textsc{m-come}_q])}$  &  \multicolumn{1}{|c}{\textbf{w}$_2$} & \textbf{w}$_2$ &  $\xrightarrow{\textbf{max}^{r}([\textsc{R}^{epi}\{q,r\}])}$  \\ \cline{1-1}\cline{3-4}
\multicolumn{1}{c}{\small $\vdots$}
\end{tabular}
\begin{tabular}{|ccc|c} 
\multicolumn{1}{c}{\small $p$} & \multicolumn{1}{c}{\small $q$} & \multicolumn{1}{c}{\small $r$} \\ \cline{1-3}
 \textbf{w}$_1$ & \textbf{w}$_1$  & \textbf{w}$_1$ & \multirow{2}{*}{$\xrightarrow{\textbf{max}^{s\Subset r}([\textsc{j-come}_s])}$}  \\ \cline{1-3}
 \textbf{w}$_1$ & \textbf{w}$_1$  & \textbf{w}$_3$   \\\cline{1-3}
\multicolumn{1}{c}{\small $p$} & \multicolumn{1}{c}{\small $q$} & \multicolumn{1}{c}{\small $r$} \\ \cline{1-3}
 \textbf{w}$_2$ & \textbf{w}$_2$ & \textbf{w}$_1$ & \multirow{2}{*}{$\xrightarrow{\textbf{max}^{s\Subset r}([\textsc{j-come}_s])}$}\\ \cline{1-3}
 \textbf{w}$_2$ & \textbf{w}$_2$ &  \textbf{w}$_3$  \\ \cline{1-3}
\end{tabular} 
}
\scalebox{.8}{
\begin{tabular}{cccc|ccccc|} 
 \multicolumn{1}{c}{\small $p$} & \multicolumn{1}{c}{\small $q$} & \multicolumn{1}{c}{\small  $r$} & \multicolumn{1}{c}{\small $s$} & &\multicolumn{1}{c}{\small $p$} & \multicolumn{1}{c}{\small $q$} & \multicolumn{1}{c}{\small  $r$} & \multicolumn{1}{c}{\small $s$} \\ \cline{1-4}\cline{6-9}
\multicolumn{1}{|c}{\textbf{w}$_1$} & \textbf{w}$_1$  &  \textbf{w}$_1$   &  \textbf{w}$_1$   &\multirow{2}{*}{$\xrightarrow{[\textsc{some}\{r,s\}]}$} & \multicolumn{1}{|c}{\textbf{w}$_1$} & \textbf{w}$_1$ & \textbf{w}$_1$ & \textbf{w}$_1$\\ \cline{1-4}\cline{6-9}
\multicolumn{1}{|c}{\textbf{w}$_1$} & \textbf{w}$_1$  &  \textbf{w}$_3$   &  $\star$ & & \multicolumn{1}{|c}{\textbf{w}$_1$} & \textbf{w}$_1$ & \textbf{w}$_3$ & $\star$ \\ \cline{1-4}\cline{6-9}
\multicolumn{1}{c}{\small $p$} & \multicolumn{1}{c}{\small $q$} & \multicolumn{1}{c}{\small  $r$}  & \multicolumn{1}{c}{\small  $s$}   \\ \cline{1-4}\cline{6-9}
\multicolumn{1}{|c}{\textbf{w}$_2$} & \textbf{w}$_2$ &  \textbf{w}$_1$  &  \textbf{w}$_1$  & \multirow{2}{*}{$\xrightarrow{[\textsc{some}\{r,s\}]}$} & \multicolumn{1}{|c}{\textbf{w}$_2$} & \textbf{w}$_2$ & \textbf{w}$_1$ & \textbf{w}$_1$ \\ \cline{1-4}\cline{6-9}
\multicolumn{1}{|c}{\textbf{w}$_2$} & \textbf{w}$_2$ &  \textbf{w}$_3$  &  $\star$  & & \multicolumn{1}{|c}{\textbf{w}$_2$} & \textbf{w}$_2$ & \textbf{w}$_3$ & $\star$\\  \cline{1-4}\cline{6-9}
\end{tabular}
}
\caption{Interpretation of modal conditionals in presence of \textit{moshi} (cf.\ \ref{moshi-modal-repeat})} 
\label{figure-modal-conditional-moshi}
\end{figure}


\noindent  The key to explaining \textit{moshi}'s acceptability in \Last[a] lies in  the  accessibility relation that the modal is  associated with. As shown  in Fig.\ \ref{figure-modal-conditional-moshi}, the steps up to the update performed by $\textbf{max}^{q\Subset p}([\textsc{m-come}_q])$ proceed  the same as in Q-adverbial conditionals that contain \textit{moshi}, cf.\ \ref{moshi-Q-adverbial-repeat}. Their difference comes in when the restrictor dref of \Last[a], $r$, gets introduced by the DRS  $\textbf{max}^{r}([\textsc{R}^{epi}\{q,r\}])$. This update  stores at $r$ the (possibly non-unique) situations that are epistemically accessible from each situation stored in $q$. Hence, unlike the restrictor dref of \ref{moshi-Q-adverbial-repeat} (in that example, $q$), $r$ in \Last[a] does \textit{not} inherit the singularity effect induced by \textit{moshi}. 
As a consequence, the last update $[\textsc{some}\{r,s\}]$  operates on info states that store non-singular referents at the restrictor dref $r$. These info states  thus satisfy the presupposition that the first argument of the  lexical relation  \textsc{some} be  non-singular (cf.\ \ref{definition-presupposition-Q-adverb}). The interpretation can thus proceed without running into the problem that besieged  its Q-adverbial counterpart \ref{moshi-Q-adverbial-repeat}. 

 
%%%%%%%%%%
\section{Conclusion}\label{section-conclusion}
In this paper, I have explored a widely accepted assumption of the referential analysis of conditionals, namely that conditional antecedents refer to plural objects. 
While this assumption was made for conditionals that express both Q-adverbial and modal quantification, it is challenged by  Japanese \textit{moshi}, which is compatible with modal conditionals but  not  Q-adverbial conditionals. 
To capture the observed paradigm, I have proposed that conditional antecedents are by default number-neutral, but can be forced to denote singular referents by \textit{moshi}.\footnote{
The reader might wonder whether the paradigm can be explained by a type restriction, i.e.\ by assuming that  \textit{moshi}  can apply to expressions referring  to \textit{situations}, but not those referring to \textit{worlds} (thanks to Jon Gajewski and Ezra Keshet for pointer). While I do not have direct evidence to rule out this possibility, I would like to point out that \textit{moshi} has been shown   to display type-flexibility between worlds and individuals based on its use  in topic constructions (\citealt{Yang22-LV}). I leave it to future work to develop the diagnostics for teasing apart  reference to situations and reference to worlds. 

} In Q-adverbial conditionals, the singularity enforced by \textit{moshi} gets inherited by the restrictors of Q-adverbs, which  is in conflict with the requirement that the restrictors of Q-adverbs be non-singleton sets; this explains the oddness of \textit{moshi} in Q-adverbial conditionals. As for modal conditionals, I have adduced an independent argument from the literature that  modals need not be restricted  by conditional antecedents;  hence, modals  are insensitive to the singularity/plurality of conditional antecedents, which explains the acceptability of \textit{moshi} in modal conditionals.



%%%%%%%%%%
\section*{Appendix: Definitions and translations of lexical entries in Intensional PCDRT}
%\subsection*{Basic types and frequently used complex objects}
%\begin{multicols}{2}
	
%\ex.  Basic types: 
%		\a. truth values $t$, 
%		\b. individuals $e$, 
%		\b. variable assignments $s$, 
%		\b. situations $\omega$
%		\z.

%\ex. Types of complex objects: 
%		\a. info states  $\langle s,t\rangle$,
%		\b. individual drefs $\langle s,e\rangle$,
%		\b. situation drefs  $\langle s,\omega \rangle$,
%		\b.  DRSs  $\langle st, \langle st, t\rangle \rangle$ (or $T$)

%\end{multicols}

%\begin{table}[htb]
%\begin{tabular}{|l|c|c|c|}\hline
 % Object 		& Type 	& Variables & Constants \\ \hline
% Truth values 	& $t$ 	& 			& \textsc{1}, \textsc{0}\\
% Individuals		& $e$	& $x$, $y$, $\dots$ & \textsc{tom}, \textsc{cat}$_1$, $\dots$ \\
% Variable assignments & $s$ & $i$, $j$, $\dots$  &\\
% Situations 	& $\omega$ 	& $w_1$, $w_2$, $\dots$ & \textbf{w}$_1$, \textbf{w}$_2$, $\dots$\\ \hline
% Info states 	& $\langle s,t\rangle$ & $I$, $J$ ,$\dots$& \\
% Individual drefs	& $\langle s,e\rangle$ & $u$, $v$, $\dots$ & \\
% Situation drefs & $\langle s,\omega \rangle$ & $p$, $q$, $\dots$ & \\
% DRSs		& $\langle st, \langle st, t\rangle \rangle$ (for short, $T$) & $D_1$, $D_2$, $\dots$ & \\ \hline
%\end{tabular}
%\caption{Basic types and frequently used objects}\label{table-objects}
%\end{table}

\subsection*{Definitions} 

\ex. Truth: A DRS $D_T$ is true with respect to an info state $I_{\langle s,t\rangle}$ iff $\exists J_{\langle s,t\rangle}(DIJ)$. \label{definition-truth}


\ex. Non-dummy substates: $I_{u\neq\star}:=\{i_s\in I: u(i)\neq \star \}$

\ex. Dynamic conjunction: $D_T;D'_T:=\lambda I_{\langle s,t\rangle}.\lambda J_{\langle s,t\rangle}.\exists H_{\langle s,t\rangle} (DIH \wedge D'HJ)$

\ex. Dref introduction: $[v]:=\lambda I_{\langle s,t\rangle}.\lambda J_{\langle s,t\rangle}.\forall i_s\in I(\exists j_s(i[v]j))\wedge \forall j_s(\exists i_s\in I(i[v]j))$


\ex. Tests (example): $[\textsc{mouse}_q\{v\}]:=\lambda I_{\langle s,t \rangle}.\lambda J_{\langle s,t \rangle}.I=J\wedge\textsc{mouse}_q\{v\}J$ 


\ex. Dynamic conditions (examples):\footnote{
  \textsc{sg} and \textsc{most} are originally defined relative to a situation dref, just like  \textsc{mouse}.  This is needed  to deal with cases where the singularity of drefs or the most-relation between drefs covaries with situations. For instance, when interpreting  \textit{A$^u$ thief might come in}, the intuition is that each epistemic possibility can be paired with a different   thief; hence, we   need to ensure that $u$ is singular with respect to each situation stored by the modal. Since I have glossed over the interpretation of  determiners in modalized sentences and conditionals, I provide the simpler, extensional definitions of \textsc{sg} and \textsc{most}. 
} 
	\a. $\textsc{mouse}_q\{v\}:=\lambda J_{\langle s,t \rangle}. J_{q\neq\star,v\neq\star} \neq \emptyset \wedge \forall j_s\in J_{q\neq\star,v\neq\star}(\textsc{mouse}_{q(j)}(v(j)))$  
	\b. $\textsc{sg}\{u\}:=\lambda I_{\langle s,t \rangle}. |\{u(i):i \in I_{u\neq\star}\}|=1$ 
	 \b.  $\textsc{most}\{u,u'\}:=\lambda I_{\langle s,t \rangle}. \textsc{most}(\{u(i):i\in I_{u\neq\star}\}, \{u'(i):i\in I_{u\neq\star}\})$, \\ 
	 \hspace*{\fill} defined only if $|\{u(i):i\in I_{u\neq\star}\}|>1$.
	
\ex. Structured subsets: $u\Subset u':=\lambda I_{\langle s,t\rangle}.\forall i_s\in I(u(i)=u'(i) \vee u(i)=\star)$\label{definition-structured-subset}

\ex. Maximization: {\small $\textbf{max}^q(D):=\lambda I_{\langle s,t\rangle}. J_{\langle s,t\rangle}.([q]; D)IJ\wedge \neg \exists K(([q]; D) IK \wedge J\subset K)$ } \label{definition-max}


\subsection*{Translations of  lexical items}
\ex.  \textit{a}$^v$ $\leadsto$ $\lambda P_{\langle se, \langle s\omega, T\rangle\rangle}.\lambda P'_{\langle se, \langle s\omega, T\rangle\rangle}.\lambda q_{\langle s,\omega\rangle}. [v|\textsc{sg}(v)]; P(v)(q); P'(v)(q)$

\ex. \textit{mouse} $\leadsto$ $\lambda v_{\langle s,e\rangle}.\lambda q_{\langle s,\omega\rangle}.[\textsc{mouse}_q\{v\}]$ \label{definition-mouse}

\ex. \textit{if}$^q$  / \textit{reba}$^q$ $\leadsto$ $\lambda \mathcal{P}_{\langle s\omega ,T\rangle}. \textbf{max}^q(\mathcal{P}(q))$\footnote{
In PCDRT, the \textbf{max}-operator always scopes over a distributive operator. This is needed to deal with cases where  a singularity test occurs inside the restrictor or the nuclear scope of the  quantifier. For instance, in  \textit{Every boy has a$^u$ book}, the singularity of $u$ needs to be satisfied with respect to each boy who has a book, rather than throughout the whole info state. Since I have glossed over the interpretation of  drefs in such environments, I leave out the distributive operator in \ref{translation-Q-averb} and \ref{translation-modal}. 
}


\ex. \textit{moshi}$^{p}$ $\leadsto$
	 $\lambda \mathfrak{R}_{\langle \langle s\omega, T\rangle, T\rangle}.\lambda \mathcal{P}_{\langle s\omega, T\rangle}. [p|\textsc{sg}(p)] ;\mathfrak{R}(\mathcal{P})$

\ex. \label{translation-Q-averb} \textit{usually}$_q^{r\Subset q}$ / \textit{taitei}$_q^{r\Subset q}$ $\leadsto$ $\lambda \mathcal{P}_{\langle s\omega ,T\rangle}. \textbf{max}^{r\Subset q}(\mathcal{P}(r)); [\textsc{most}\{q,r\}]$

\ex. \textit{might}$_p^{q,r\Subset q}$ / \textit{kamoshirenai}$_p^{q,r\Subset q}$ 
	\\ \hspace*{\fill}
 $\leadsto$ 
	 $\lambda \mathcal{P}_{\langle s\omega ,T\rangle}. \textbf{max}^{q}([\textsc{R}^{epi}\{p,q\}]); \textbf{max}^{r\Subset q}([\mathcal{P}(r)]); [\textsc{some}\{q,r\}]$\label{translation-modal}


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\bibliography{diss-ref.bib}


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\begin{addresses}
  
  \begin{address}
    Muyi Yang\\
	Oak Hall Room 365,\\
	368 Fairfield Way, Unit 1145,\\
	Storrs, CT 06269\\
    \email{muyi.yang@uconn.edu}
  \end{address}
  
\end{addresses}

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\end{document}
