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<Paper uid="N04-4033">
  <Title>Polarity sensitivity and evaluation order in type-logical grammar</Title>
  <Section position="3" start_page="0" end_page="0" type="intro">
    <SectionTitle>
2 Delimited continuations
</SectionTitle>
    <Paragraph position="0"> Figure 1 shows natural deduction rules for multimodal categorial grammar, a member of the type-logical family of grammar formalisms (Moortgat, 1996a; Bernardi, 2002). Figure 2 lists our structural postulates. These two figures form the logic underlying our account.</Paragraph>
    <Paragraph position="1"> We use two binary modes: the default mode (blank) for surface syntactic composition, and the continuation mode c. As usual, a formula of the form A * B can be read as &amp;quot;A followed by B&amp;quot;. By contrast, a formula of the form A *c B can be read as &amp;quot;A in the context B&amp;quot;. In programming-language terms, the formula A *c B plugs a subexpression A into a delimited continuation B. The Root rule creates a trivial continuation: it says that 1 is a right identity for the c mode, where 1 can be thought of as a nullary connective, effectively enabling empty antecedents for the c mode. The binary modes, along with the first three postulates in Figure 2, provide a new way to encode Moortgat's ternary connective q (1996b) for in-situ quantification. For intuition, it may help to draw logical formulas as binary trees, distinguishing graphically between the two modes.</Paragraph>
    <Paragraph position="2"> To further capture the interaction between scope inversion and polarity sensitivity exemplified in (3-4), we use three unary modes: the value mode (blank), the unquotation mode u, and the polarity mode p. The value mode marks when an expression is devoid of in-situ quantification, or, in programming-language terms, when it is a pure value rather than a computation with control effects.</Paragraph>
    <Paragraph position="3"> As a special case, any formula can be turned pure by embedding it under a diamond using the T postulate, analogous to quotation or staging in programming languages.</Paragraph>
    <Paragraph position="4"> Quotations can be concatenated using the Kprime postulate.</Paragraph>
    <Paragraph position="5"> The unquotation mode u marks when a diamond can be canceled using the Unquote postulate. Unquotation is also known as eval or run in programming languages.</Paragraph>
    <Paragraph position="6"> The polarity mode p, and the empirical utility of these unary modes, are explained in SS3.</Paragraph>
    <Paragraph position="7"> A derivation is considered complete if it culminates in a sequent whose antecedent is built using the default binary mode * only, and whose conclusion is a type of the form DiamonduA. Below is a derivation of Alice saw Bob.</Paragraph>
    <Paragraph position="8">  (8) Alice turnstileleft np saw turnstileleft (np\Diamondus)/np Bob turnstileleft np /E</Paragraph>
    <Paragraph position="10"> Note that clauses take the typeDiamondus rather than the usual s, so the Unquote rule can operate on clauses. We abbreviate Diamondus to s* below.</Paragraph>
    <Paragraph position="11"> To illustrate in-situ quantification, Figure 3 on the following page shows a derivation of Alice saw a man's mother. For brevity, we treat a man as a single lexical item. It is a quantificational noun phrase whose polarity is neutral in a sense that contrasts with other quantifiers considered below. The crucial part of this derivation is the use of the structural postulates Root, Left, and Right to divide the sentence into two parts: the subexpression a man and its context Alice saw 's mother. The type of a man, s*/c(np\cs*), can be read as &amp;quot;a subexpression that produces a clause when placed in a context that can enclose an np to make a clause&amp;quot;.</Paragraph>
    <Paragraph position="12">  cause we assign the typesDiamondusquare|pDiamondps andsquare|pDiamondpDiamondus to positive and negative clauses, respectively, and can derive s* turnstileleft Diamondusquare|pDiamondps, s* turnstileleft square|pDiamondpDiamondus.(9) In words, a neutral clause can be silently converted into a positive or negative one. We henceforth write s+ and s[?] for Diamondusquare|pDiamondps and square|pDiamondpDiamondus. By (9), both types are &amp;quot;subtypes&amp;quot; of s* (that is to say, entailed by s*).</Paragraph>
    <Paragraph position="13"> The p mode is used in Figure 5 on the next page to derive Nobody saw anybody. Unlike a man, the quantifier anybody has the type s[?]/c(np\cs[?]), showing that it takes scope over a negative clause to make another negative clause. Meanwhile, the quantifier nobody has the type s*/c(np\cs[?]), showing that it takes scope over a negative clause to make a neutral clause. Thus nobody can take scope over the negative clause returned by anybody to make a neutral clause, which is complete.</Paragraph>
    <Paragraph position="14"> The contrast between (2a) and (3) boils down to the Right (but not Left) postulate's requirement that the left-most constituent be of the form DiamondB. (In programming-language terms, a subexpression can be evaluated only if all other subexpressions to its left are pure.) For nobody to take scope over (and license) anybody in (3) requires the context *Anybody saw . In other words, the sequent (10) np *c parenleftbig(1 *Diamondanybody) *Diamondsawparenrightbig turnstileleft s[?] must be derived, in which the Right rule forces the constituents anybody and saw to be embedded under diamonds. Figure6showsanattemptatderiving(10), which fails because the type s[?] for negative clauses cannot be Unquoted (shown with question marks). The sequent in(10)cannotbederived, andthesentence*Anybodysaw nobody is not admitted. Nevertheless, Somebody saw everybody is correctly predicted to have ambiguous scope, because neutral and positive clauses can be Unquoted.</Paragraph>
    <Paragraph position="15"> The quantifiers a man, nobody, and anybody in Figures 3 and 5 exemplify a general pattern of analysis: every polarity-sensitive item, be it traditionally considered a licensor or a licensee, specifies in its type an input polarity (of the clause it takes scope over) and an output polarity (of the clause it produces). Figure 4 lists more quantifiers and their input and output polarities. As shown there, these type assignments can be visualized as a finite-state machine. The states are the three clause types. The e-transitions are the two derivability relations in (9). The non-e transitions are the quantifiers. The start states are the clausal types that can be Unquoted. The final state is the clausal type returned by verbs, namely neutral.</Paragraph>
    <Paragraph position="16"> The precise pattern of predictions made by this theory can be stated in two parts. First, due to the lexical types in Figure 4 and the &amp;quot;subtyping&amp;quot; relations in (9), the quantifiers in a sentence must form a valid transition sequence, from widest to narrowest scope. This constraint is standard in type-logical accounts of polarity sensitivity. Second, thanks to the unary modes in the structural</Paragraph>
    <Paragraph position="18"> postulates in Figure 2, whenever two quantifiers take inverse rather than linear scope with respect to each other, the transitions must pass through a start state (that is, a clause type that can be Unquoted) in between. This constraint is an empirical advance over previous accounts, which are oblivious to linear order.</Paragraph>
    <Paragraph position="19"> Theinputandoutputpolaritiesofquantifiersarehighly mutually constrained. Take everybody for example. If we hold the polarity assignments of the other quantifiers fixed, then the existence of a linear-scope reading for A man introduced everybody to somebody forces everybody to be input-positive and output-neutral. But then ouraccountpredictsthatNobodyintroducedeverybodyto somebody has a linear-scope reading, unlike the simpler sentenceNobodyintroducedAlicetosomebody. Thisprediction is borne out, as observed by Kroch (1974, pages 121-122) and discussed by Szabolcsi (2004).</Paragraph>
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