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"Quantifiers"
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Eliminating the universe : logical properties of natural language
\"This book synthesizes the author's work (1980s-2015) on the logical expressive power of natural language. It extends the tools and concepts of model theory as used in (higher order) predicate logic to the study of natural language semantics. It focuses on boolean structure, generalized quantification (separated from variable binding), covering some cases of anaphora. Different categories -- predicates, adjective, quantifiers -- are modeled by non-isomorphic boolean lattices. Of empirical linguistic interest is the expressibility of many natural classes of quantifiers defined in terms of their logical (automorphism invariant) properties. Some of these correlate with classes used syntactically in generative grammar. In other cases we find general (possibly universal) constraints on possible quantifier denotations in natural language. Also of novel logical interest are entailment paradigms that depend on relations between pairs or triples of generalized quantifier denoting expressions, ones that are in some cases inherently vague. In addition we note novel binary quantifiers that lie beyond the \"Frege boundary\" in that they are provably not identical to any iterated application of unary quantifiers. Of philosophical interest is the existence of models which make the same sentences true as standard models but which lack a universe and hence, seemingly, a notion of \"reference\". Moreover, these models generalize to ones in which we can represent (some) intensional expressions without the use of novel ontological objects, such as \"possible worlds\" or \"propositions\"-- Provided by publisher.
Book
Quantifiers in Language and Logic
Quantification is a topic which brings together linguistics, logic, and philosophy. Quantifiers are the essential tools with which, in language or logic, we refer to quantity of things or amount of stuff. In English they include such expressions as no, some, all, both, or many. This book presents the definitive interdisciplinary exploration of how they work — their syntax, semantics, and inferential role.
eBook
Eye-tracking evidence for fixation asymmetries in verbal and numerical quantifier processing
When people are given quantified information (e.g., ‘there is a 60% chance of rain’), the format of quantifiers (i.e., numerical: ‘a 60% chance’ vs. verbal: ‘it is likely’) might affect their decisions. Previous studies with indirect cues of judgements and decisions (e.g., response times, decision outcomes) give inconsistent findings that could support either a more intuitive process for verbal than numerical quantifiers or a greater focus on the context (e.g., rain) for verbal than numerical quantifiers. We used two pre-registered eye-tracking experiments (n(1) = 148, n(2) = 133) to investigate decision-making processes with verbal and numerical quantifiers. Participants evaluated multiple verbally or numerically quantified nutrition labels (Experiment 1) and weather forecasts (Experiment 2) with different context valence (positive or negative), and quantities (`low', `medium', or `high' in Experiment 1 and `possible', `likely', or `very likely' in Experiment 2) presented in a fully within-subjects design. Participants looked longer at verbal than numerical quantifiers, and longer at the contextual information with verbal quantifiers. Quantifier format also affected judgements and decisions: in Experiment 1, participants judged positive labels to be better in the verbal compared to the equivalent numerical condition (and to be worse for negative labels). In Experiment 2, participants decided on rain protection more for a verbal forecast of rain than the equivalent numerical forecast. The results fit the explanation that verbal quantifiers put more focus on the informational context than do numerical quantifiers, rather than prompting more intuitive decisions.
Journal Article
Discourse-Linked DPs as Covert Partitives
The paper studies the conditions that determine the Discourse-Linked or non-Discourse-Linked status of noun-less Determiner Phrases introduced by different determiners, in Italian and in English. For instance, given the sentence Ten bombs exploded yesterday, the continuation [Three] were cluster bombs tends to have a meaning equivalent to 'three of the bombs that exploded’ (D-Linked), while [Three] will explode today is understood as 'three (different) bombs’ (non D-Linked). Beside world-knowledge, the syntax of the determiners and their position with respect to the verb affect the availability of DL/non-DL readings. This and other facts undermine an analysis cast purely in terms of semantic domain restrictions, and suggest that DL readings are due to the presence of a covert partitive structure. While perhaps intuitive, this idea faces various issues in Italian, due to its interactions with the syntax of the pro-form ne. We show that an NP-based structure for numeral and proportion-based partitives (three/half of the bombs) is actually compatible with the facts, and offers a cue on the nature of sub-DP pro-forms and their uses.
Journal Article
Complement set focus after positive quantifiers: The influence of context
During reading, different cues influence readers’ focus. One of these is natural language quantifiers. Negative quantifiers, such as
(e.g., “Few attended the lecture”), have a complex influence on focus. They convey a sense of shortfall – a quantity that is less than what was expected (e.g. Moxey, 2006; Moxey & Sanford, 1987). This, in turn creates focus on the complement set (e.g., those not attending). Upadhyay et al. (2018) demonstrated the influence of story context on these focus effects. In three experiments, we investigated the influence of story context on focus effects with positive quantifiers. Although positive quantifiers are less diffuse in their meaning than negative quantifiers, context can lead positive quantifiers to convey shortfall, as negative quantifiers do. When the story context creates an expectation for a large quantity (e.g.,
), the positive quantifier
was understood as shortfall, leading to focus on the complement set.
Journal Article
The mental representation of universal quantifiers
A sentence like every circle is blue might be understood in terms of individuals and their properties (e.g., for each thing that is a circle, it is blue) or in terms of a relation between groups (e.g., the blue things include the circles). Relatedly, theorists can specify the contents of universally quantified sentences in first-order or second-order terms. We offer new evidence that this logical first-order vs. second-order distinction corresponds to a psychologically robust individual vs. group distinction that has behavioral repercussions. Participants were shown displays of dots and asked to evaluate sentences with each, every, or all combined with a predicate (e.g., big dot). We find that participants are better at estimating how many things the predicate applied to after evaluating sentences in which universal quantification is indicated with every or all, as opposed to each. We argue that every and all are understood in second-order terms that encourage group representation, while each is understood in first-order terms that encourage individual representation. Since the sentences that participants evaluate are truth-conditionally equivalent, our results also bear on questions concerning how meanings are related to truth-conditions.
Journal Article
The Trouble With Quantifiers: Exploring Children's Deficits in Scalar Implicature
Adults routinely use the context of utterances to infer a meaning beyond the literal semantics of their words (e.g., inferring from “She ate some of the cookies” that she ate some, but not all). Contrasting children's (N = 209) comprehension of scalar implicatures using quantifiers with contextually derived ad hoc implicatures revealed that 4‐ to 5‐year‐olds reliably computed ad hoc, but not scalar, implicatures (Experiment 1). Unexpectedly, performance with “some” and “none” was correlated (Experiments 1 and 2). An individual differences study revealed a correlation between quantifier knowledge and implicature success (Experiment 3); a control study ruled out other factors (Experiment 4). These findings suggest that some failures with scalar implicatures may be rooted in a lack of semantic knowledge rather than general pragmatic or processing demands.
Journal Article
At least not false, at most possible: between truth and assertibility of superlative quantifiers
Generalized Quantifier Theory defines superlative quantifiers at most n and at least n as truth-conditionally equivalent to comparative quantifiers fewer than n+1 and more than n–1. It has been demonstrated, however, that this standard theory cannot account for various linguistic differences between these two types of quantifiers. In this paper I discuss how the distinction between assertibility and truth-conditions can be applied to explain this phenomenon. I draw a parallel between the assertibility of disjunctions and superlative quantifiers, and argue that those assertibility conditions are essentially modal. I use epistemic logic to formalize the assertibility conditions and revisit some of the linguistic puzzles related to superlative quantification.
Journal Article
Probabilistic pragmatics explains gradience and focality in natural language quantification
An influential view in philosophy and linguistics equates the meaning of a sentence to the conditions under which it is true. But it has been argued that this truth-conditional view is too rigid and that meaning is inherently gradient and revolves around prototypes. Neither of these abstract semantic theories makes direct predictions about quantitative aspects of language use. Hence, we compare these semantic theories empirically by applying probabilistic pragmatic models as a link function connecting linguistic meaning and language use. We consider the use of quantity words (e.g., “some,” “all”), which are fundamental to human language and thought. Data from a large-scale production study suggest that quantity words are understood via prototypes. We formulate and compare computational models based on the two views on linguistic meaning. These models also take into account cognitive factors, such as salience and numerosity representation. Statistical and empirical model comparison show that the truth-conditional model explains the production data just as well as the prototype-based model, when the semantics are complemented by a pragmatic module that encodes probabilistic reasoning about the listener’s uptake.
Journal Article