A Mechanism to Restrict the Scope of Clause-Bounded Quantifiers in ‘Continuation’ Semantics Anca Dinu Faculty of Foreign Languages and Literatures, University of Bucharest [email protected] To filter out these impossible interpretations, we Abstract first need to understand the scope behavior of each scope-taking lexical entry: its maximal scope limits and the scope precedence preferences w.r.t. other lexical entries. Second, This paper presents a formal mechanism to we should force the scope closing of the properly constrain the scope of negation and of quantifiers by applying a standard type shifter certain quantificational determiners to their Lower (which is equivalent to identity function minimal clause in continuation semantics application), once their scope limits were framework introduced in Barker and Shan (2008) and which was subsequently extended reached. But the actual mechanism that ensures from sentential level to discourse level in Dinu the scope closing was left underspecified in (2011). In these works, type shifting is previous work on continuation semantics. employed to account for side effects such as In what follows, we propose such a pronominal anaphora binding or quantifier mechanism, designed to ensure that no lexical scope. However, allowing arbitrary type entry having the scope bounded to its minimal shifting will result in overgenerating clause (such as not, no, every, each, any, etc) will interpretations impossible in natural language. ever take scope outside, thus getting right To filter out some of these impossible discourse truth conditions. interpretations, once the negation or the The programming language concept of quantifiers reach their maximal scope limits (that is their minimal clause), one should force continuations was successfully used by Barker their scope closing by applying a standard type and Shan in a series of articles (Barker 2002, shifter Lower. But the actual mechanism that Barker 2004, Shan 2005, Shan and Barker 2006, forces the scope closing was left Barker and Shan 2008) to analyze intra-sentential underspecified in previous work on linguistic phenomena (focus fronting, donkey continuation semantics. We propose here such anaphora, presupposition, crossover, superiority, a mechanism, designed to ensure that no etc). Moreover, (de Groote, 2006) proposed an lexical entries having the scope bounded to elegant discourse semantics based on their minimal clause (such as not, no, every, continuations. Continuations are a standard tool each, any, etc) will ever take scope outside. in computer science, used to control side effects of computation. They are a notoriously hard to 1 Introduction understand notion. Actually, understanding what The starting point of this paper is the a continuation is per se is not so hard. What is continuation semantics introduced in Barker and more difficult is to understand how a grammar Shan (2008) and extended from sentential level based on continuations (a „continuized‟ to discourse level in Dinu (2011). In this grammar) works. The basic idea of continuizing framework, type shifting is used to account for a grammar is to provide subexpressions with side effects such as pronominal anaphora binding direct access to their own continuations (future or quantifier scope. However, allowing arbitrary context), so subexpressions are modified to take type shifting will result in overgenerating a continuation as an argument. A continuized interpretations impossible in natural language. grammar is said to be written in continuation 495 Proceedings of Recent Advances in Natural Language Processing, pages 495–502, Hissar, Bulgaria, 12-14 September 2011. passing style. Continuation passing style is in continuization in which only the category S has fact a restricted (typed) form of λ-calculus. been continuized. Historically, the first continuation operators This is by no means a coincidence, MG only were undelimited (e.g., call/cc or J). An continuizes the noun phrase meanings and undelimited continuation of an expression dynamic semantics only continuizes the sentence represents “the entire (default) future for the meanings, rather than continuizing uniformly computation” of that expression. Felleisen (1988) throughout the grammar as it is done in introduced delimited continuations (sometimes continuation semantics. called „composable‟ continuations) such as control („C‟) and prompt („%‟). Delimited 2 Preliminaries continuations represent the future of the We use Barker and Shan‟s (2008) tower notation computation of the expression up to a certain for a given expression, which consists of three boundary. Interestingly, the natural-language levels: the top level specifies the syntactic phenomena discussed here make use only of category of the expression couched in categorial delimited continuations. grammar, the middle level is the expression itself For instance, if we take the local context to be and the bottom level is the semantic value: restricted to the sentence, when computing the meaning of the sentence John saw Mary., the 푠푦푛푡푎푐푡푖푐 푐푎푡푒푔표푟푦 default future of the value denoted by the subject 푒푥푝푟푒푠푠푖표푛 is that it is destined to have the property of 푠푒푚푎푛푡푖푐 푣푎푙푢푒 seeing Mary predicated of it. In symbols, the 퐶|퐵 The syntactic categories are written , continuation of the subject denotation j is the 퐴 function 휆푥. 푠푎푤 풎 푥. Similarly, the default where A, B and C can be any categories. We read future of the object denotation m is the property this counter clockwise as “the expression of being seen by John, the function 휆푦. 풔풂풘 푦 푗; functions as a category A in local context, takes the continuation of the transitive verb denotation scope at an expression of category B to form an saw is the function 휆푅. 푅 푚 푗; and the expression of category C.” continuation of the verb phrase saw Mary is the The semantic value in linear notation function 휆푃. 푃 푗. This simple example illustrates 휆푘. 푓[푘(푥)] is equivalently written vertically as 푓[ ] two important aspects of continuations: omitting the future context (continuation) k. (1) every meaningful subexpression has a 푥 Here, x can be any expression, and f[ ] can be continuation; any expression with a gap [ ]. Free variables in x (2) the continuation of an expression is can be bound by binders in f [ ]. This vertical always relative to some larger expression (layered) notational convention is meant to make containing it. the combination process of two expressions Thus, when John occurs in the sentence John easier (more visual) then in linear notation. Here left yesterday., its continuation is the property there are the two possible modes of combination 휆푥. 풚풆풔풕풆풓풅풂풚 풍풆풇풕 푥; when it occurs in Mary (Barker and Shan 2008): thought John left., its continuation is the property 휆푥. 풕풉풐풖품풉풕 (풍풆풇풕 푥) 푚 and when it occurs in 퐶|퐷 퐷|퐸 퐶|퐸 the sentence Mary or John left., its continuation 퐴/퐵 퐵 퐴 푙 푒푓푡 − 푒푥푝 푟푖푔푕푡 − 푒푥푝 is 휆푥. (풍풆풇풕 풎) ∨ (풍풆풇풕 푥) and so on. 푙푒푓푡 − 푒푥푝 푟푖푔푕푡 − 푒푥푝 = It is worth mentioning that some results of 푔[ ] 푕[ ] 푔[푕 ] traditional semantic theories are particular cases 푓 푥 푓(푥) of results in continuation semantics: 퐶|퐷 퐷|퐸 The generalized quantifier type from 퐶|퐸 퐵 퐵\퐴 퐴 Montague grammar (Montague, 1970) 푙 푒푓푡 − 푒푥푝 푟푖푔푕푡 − 푒푥푝 푙푒푓푡 − 푒푥푝 푟푖푔푕푡 − 푒푥푝 = <<<e,t>,t>,t> is exactly the type of 푔[ ] 푕[ ] 푔[푕 ] quantificational determiners in continuation- 푥 푓 푓(푥) based semantics; The <<t,t>,t> type of sentences in Below the horizontal lines, combination dynamic semantics is exactly the type of proceeds simply as in combinatory categorial sentences in continuation-based semantics. In grammar: in the syntax, B combines with A/B or fact, dynamic interpretation constitutes a partial B\A to form A; in the semantics, x combines with f to form f(x). Above the lines is where the 496 combination machinery for continuations kicks To derive the syntactic category and a in. The syntax combines the two pairs of semantic value with no horizontal line, Barker categories by a kind of cancellation: the D on the and Shan (2008) introduce the type-shifter left cancels with the D on the right. The Lower. In general, for any category A, any value semantics combines the two expressions with x, and any semantic expression f[ ] with a gap, gaps by a kind of composition: we plug h[ ] to the following type-shifter is available: the right into the gap of g[ ] to the left, to form g[h[ ]]. The expression with a gap on the left, g[ 퐴|푆 퐴 ], always surrounds the expression with a gap on 푆 퐿표푤푒푟 푒 푥푝푟푒푠푠푖표푛 푒 푥푝푟푒푠푠푖표푛 the right, h[ ], no matter which side supplies the 푓[ ] 푓[푥] function and which side supplies the argument 푥 below the lines. This fact expresses the generalization that the default order of semantic Syntactically, Lower cancels an S above the evaluation is left-to-right. line to the right with an S below the line. When there is no quantification or anaphora Semantically, Lower collapses a two-level involved, a simple sentence like John came. is meaning into a single level by plugging the value derived as follows: x below the line into the gap [ ] in the expression f[ ] above the line. Lower is equivalent to 퐷푃 퐷푃\푆 푆 identity function application. 퐽표푕푛 푐푎푚푒 = 퐽표푕푛 푐푎푚푒 The third and the last type shifter we need is 푗 푐푎푚푒 푐푎푚푒 푗 one that accounts for binding. We adopt the idea In the syntactic layer, as usual in categorical (in line with Barker and Shan (2008)) that the grammar, the category under slash (here DP) mechanism of binding is the same as the cancels with the category of the argument mechanism of scope taking. Binding is a term expression; the semantics is function application. used both in logics and in linguistics with analog Quantificational expressions have extra layers (but not identical) meaning. In logics, a variable on top of their syntactic category and on top of is said to be bound by an operator (as the their semantic value, making essential use of the universal or existential operators) if the variable powerful mechanism of continuations in ways is inside the scope of the operator.