IF多值逻辑及博弈语义

本文基于经典一阶逻辑句法的逻辑优先性分析,把Hintikka的独立联结词和独立量词扩展到多值逻辑中。我们给出IF多值逻辑的句法,并使用不完全信息的语义赋值博弈解释了IF多值逻辑。     

Prior expectation and the interpretation of natural language quantifiers

Abstract

The interpretation of natural language quantifiers depends, in part, on one's prior expectations about what the proportion being described might be. This hypothesis is tested empirically, by comparing the interpretation of 10 different quantifier expressions in three contexts which contained proportions about which subjects' expectations were reliably different (Experiment 1). It is further argued that quantifiers perform many functions which cannot be observed by considering only the proportion(s) denoted by a quantifier. The experimental evidence presented shows that quantifiers can provide information about what the user of the quantifier had expected before “knowing the facts” (Experiment 2). It is also clear that when writers produce a quantifier, they can reveal their assumptions about the prior expectations held by their readers (Experiment 2).     

Quantifiers and Congruence Closure

We prove some results about the limitations of the expressive power of quantifiers on finite structures. We define the concept of a bounded quantifier and prove that every relativizing quantifier which is bounded is already first-order definable (Theorem 3.8). We weaken the concept of congruence closed (see [6]) to weakly congruence closed by restricting to congruence relations where all classes have the same size. Adapting the concept of a thin quantifier (Caicedo [1]) to the framework of finite structures, we define the concept of a meager quantifier. We show that no proper extension of first-order logic by means of meager quantifiers is weakly congruence closed (Theorem 4.9). We prove the failure of the full congruence closure property for logics which extend first-order logic by means of meager quantifiers, arbitrary monadic quantifiers, and the Härtig quantifier (Theorem 6.1).     

The logic in language: How all quantifiers are alike,but each quantifier is different

Quantifier words like each, every, all and three are among the most abstract words in language. Unlike nouns, verbs and adjectives, the meanings of quantifiers are not related to a referent out in the world. Rather, quantifiers specify what relationships hold between the sets of entities, events and properties denoted by other words. When two quantifiers are in the same clause, they create a systematic ambiguity. “Every kid climbed a tree” could mean that there was only one tree, climbed by all, or many different trees, one per climbing kid. In the present study, participants chose a picture to indicate their preferred reading of different ambiguous sentences – those containing every, as well as the other three quantifiers. In Experiment 1, we found large systematic differences in preference, depending on the quantifier word. In Experiment 2, we then manipulated the choice of a particular reading of one sentence, and tested how this affected participants’ reading preference on a subsequent target sentence. We found a priming effect for all quantifiers, but only when the prime and target sentences contained the same quantifier. For example, all-a sentences prime other all-a sentences, while each-a primes each-a, but sentences with each do not prime sentences with all or vice versa. In Experiment 3, we ask whether the lack of priming across quantifiers could be due to the two sentences sharing one fewer word. We find that changing the verb between the prime and target sentence does not reduce the priming effect. In Experiment 4, we discover one case where there is priming across quantifiers – when one number (e.g. three) is in the prime, and a different one (e.g. four) is in the target. We discuss how these findings relate to linguistic theories of quantifier meaning and what they tell us about the division of labor between conceptual content and combinatorial semantics, as well as the mental representations of quantification and of the abstract logical structure of language.     

The Skolem-Löwenheim theorem in toposes

The topos theory gives tools for unified proofs of theorems for model theory for various semantics and logics. We introduce the notion of power and the notion of generalized quantifier in topos and we formulate sufficient condition for such quantifiers in order that they fulfil downward Skolem-Löwenheim theorem when added to the language. In the next paper, in print, we will show that this sufficient condition is fulfilled in a vast class of Grothendieck toposes for the general and the existential quantifiers.     

关于不可化归的多态式量词句的CCG探析

广义量词理论对英语中一些特定的量词句进行分析,把其中两个单态式量词合并成一个多态式量词。Keenan证明了这个多态式量词的意义不能化归为两个单态式量词的意义,即多态式量词的意义不能从两个单态式量词的标准意义推演出来。Keenan的研究是很有价值的,但本文尝试从另外的角度思考,对汉语类似的多态式量化句进行个案处理,采纳组合范畴语法针对自然语言表层结构的词汇主义方法,遵循部分表达式的意义决定整体表达式意义的组合原则,从两个单态式量词的非标准意义推演出整个多态式量词句的量化意义。     

Compositional Natural Language Semantics using Independence Friendly Logic or Dependence Logic

Independence Friendly Logic, introduced by Hintikka, is a logic in which a quantifier can be marked for being independent of other quantifiers. Dependence logic, introduced by Väänänen, is a logic with the complementary approach: for a quantifier it can be indicated on which quantifiers it depends. These logics are claimed to be useful for many phenomena, for instance natural language semantics. In this contribution we will compare these two logics by investigating their application in a compositional analysis of the de dicto - de re ambiguity in natural language. It will be argued that Independence Friendly logic is suitable, whereas Dependence Logic is not.     

如何利用广义量词的语义性质判断扩展三段论的有效性

在国内外最新研究成果的基础上,笔者通过对文中的定理和推论的证明,主要说明了以下几点：（1）利用广义量词的相关语义性质,比如单调性和对称性,既可以解释亚氏三段论的有效性,又可以解释带有广义量词的扩展三段论的有效性;（2）一些有效的扩展三段论仅仅表征了广义量词的左或右单调性,还有一些有效的扩展三段论同时表征了广义量词的多个语义性质;（3）利用广义量词的东南或西北或西南或东北方向的单调性可以判断一些带有限制条件的扩展三段论的有效性。此研究将有利于广义量词理论的发展,对于计算机科学中的知识表示和知识推理的研究都具有较为重要的理论价值和实践意义。     

Annotation Theories over Finite Graphs

In the current paper we consider theories with vocabulary containing a number of binary and unary relation symbols. Binary relation symbols represent labeled edges of a graph and unary relations represent unique annotations of the graph’s nodes. Such theories, which we call annotation theories, can be used in many applications, including the formalization of argumentation, approximate reasoning, semantics of logic programs, graph coloring, etc. We address a number of problems related to annotation theories over finite models, including satisfiability, querying problem, specification of preferred models and model checking problem. We show that most of considered problems are NPTime- or co-NPTime-complete. In order to reduce the complexity for particular theories, we use second-order quantifier elimination. To our best knowledge none of existing methods works in the case of annotation theories. We then provide a new second-order quantifier elimination method for stratified theories, which is successful in the considered cases. The new result subsumes many other results, including those of [2, 28, 21].     

Theories of categorical reasoning and extended syllogisms

The aim of this study was to examine the predictions of three theories of human logical reasoning, (a) mental model theory, (b) formal rules theory (e.g., PSYCOP), and (c) the probability heuristics model, regarding the inferences people make for extended categorical syllogisms. Most research with extended syllogisms has been restricted to the quantifier “All” and to an asymmetrical presentation. This study used three-premise syllogisms with the additional quantifiers that are used for traditional categorical syllogisms as well as additional syllogistic figures. The predictions of the theories were examined using overall accuracy as well as a multinomial tree modelling technique. The results demonstrated that all three theories were able to predict response selections at high levels. However, the modelling analyses showed that the probability heuristics model did the best in both Experiments 1 and 2.     

扩展三段论的可化归性与广义量词的语义性质之间的关系

基于Barwise、Cooper、Keenan、Peters、Westerstahl和vanEijck等人的研究成果,作者提出并证明了若干事实和推论。这些事实和推论表明：（1）不同三段论之间的可化归性本质上反映了广义量词的单调性、对称性等语义性质之间的可转换性,因此,我们可以根据四个亚氏量词的语义性质之间的转换关系来验证亚氏三段论的可化归性;（2）利用广义量词的语义性质可以验证扩展三段论的不同推理模式之间的可化归关系。由于广义量词在自然语言中普遍存在,因此,本文的研究对广义量词理论的发展和自然语言的信息处理都具有积极意义。     

Quantifier comprehension in corticobasal degeneration

In this study, we investigated patients with focal neurodegenerative diseases to examine a formal linguistic distinction between classes of generalized quantifiers, like "some X" and "less than half of X." Our model of quantifier comprehension proposes that number knowledge is required to understand both first-order and higher-order quantifiers. The present results demonstrate that corticobasal degeneration (CBD) patients, who have number knowledge impairments but little evidence for a deficit understanding other aspects of language, are impaired in their comprehension of quantifiers relative to healthy seniors, Alzheimer's disease (AD) and frontotemporal dementia (FTD) patients [F(3,77)=4.98; p<.005]. Moreover, our model attempts to honor a distinction in complexity between classes of quantifiers such that working memory is required to comprehend higher-order quantifiers. Our results support this distinction by demonstrating that FTD and AD patients, who have working memory limitations, have greater difficulty understanding higher-order quantifiers relative to first-order quantifiers [F(1,77)=124.29; p<.001]. An important implication of these findings is that the meaning of generalized quantifiers appears to involve two dissociable components, number knowledge and working memory, which are supported by distinct brain regions.     

Attentional focusing with quantifiers in production and comprehension

There is a very large number of quantifiers in English, so many that it seems impossible that the only information that they convey is about amounts. Building on the earlier work of Moxey and Sanford (1987), we report three experiments showing that positive and negative quantifiers focus on different subsets of the logical possibilities that quantifiers allow semantically. Experiments 1 and 2 feature a continuation task with quantifiers that span a full range of denotations (from near 0% to near 100%) and show that the effect is not restricted to quantifiers denoting small amounts. This enables a distinction to be made between generalization and complement set focus proper. The focus effects extend to comprehension, as shown by a self-paced reading study (Experiment 3). It is noted that the focus effects obtained are compatible with findings from earlier work by Just and Carpenter (1971), which used a verification paradigm, and in fact these effects constitute a direct test of inferences Just and Carpenter made about mechanisms of encoding negative quantifiers. A related but different explanation is put forward to explain the present data. The experiments show a quantifier function beyond the simple denotation of amount.     

“At least one” problem with “some” formal reasoning paradigms

In formal reasoning, the quantifier "some" means "at least one and possibly all." In contrast, reasoners often pragmatically interpret "some" to mean "some, but not all" on both immediate-inference and Euler circle tasks. It is still unclear whether pragmatic interpretations can explain the high rates of errors normally observed on syllogistic reasoning tasks. To address this issue, we presented participants (reasoners) in the present experiments either standard quantifiers or clarified quantifiers designed to precisely articulate the quantifiers' logical interpretations. In Experiment 1, reasoners made significantly more logical responses and significantly fewer pragmatic responses on an immediate-inference task when presented with logically clarified as opposed to standard quantifiers. In Experiment 2, this finding was extended to a variant of the immediate-inference task in which reasoners were asked to deduce what followed from premises they were to assume to be false. In Experiment 3, we used a syllogistic reasoning task and observed that logically clarified premises reduced pragmatic and increased logical responses relative to standard ones, providing strong evidence that pragmatic responses can explain some aspects of the errors made in the syllogistic reasoning task. These findings suggest that standard quantifiers should be replaced with logically clarified quantifiers in teaching and in future research.     

Unrestricted Quantification and the Structure of Type Theory

Semantic theories based on a hierarchy of types have prominently been used to defend the possibility of unrestricted quantification. However, they also pose a prima facie problem for it: each quantifier ranges over at most one level of the hierarchy and is therefore not unrestricted. It is difficult to evaluate this problem without a principled account of what it is for a quantifier to be unrestricted. Drawing on an insight of Russell's about the relationship between quantification and the structure of predication, we offer such an account. We use this account to examine the problem in three different type-theoretic settings, which are increasingly permissive with respect to predication. We conclude that unrestricted quantification is available in all but the most permissive kind of type theory.     

Classifying ℵ0-categorical theories

Among the complete ₀-categorical theories with finite non-logical vocabularies, we distinguish three classes. The classification is obtained by looking at the number of bound variables needed to isolated complete types. In classI theories, all types are isolated by quantifier free formulas; in classII theories, there is a leastm, greater than zero, s.t. all types are isolated by formulas in no more thanm bound variables: and in classIII theories, for eachm there is a type which cannot be isolated inm or fewer bound variables. ClassII theories are further subclassified according to whether or not they can be extended to classI theories by the addition of finitely many new predicates. Alternative characterizations are given in terms of quantifier elimination and homogeneous models. It is shown that for each primep, the theory of infinite Abelian groups all of whose elements are of orderp is classI when formulated in functional constants, and classIII when formulated in relational constants.     

Taking the High (or Low) Road: A Quantifier Priming Perspective on Basic Anchoring Effects

ABSTRACT

Current explanations of basic anchoring effects, defined as the influence of an arbitrary number standard on an uncertain judgment, confound numerical values with vague quantifiers. I show that the consideration of numerical anchors may bias subsequent judgments primarily through the priming of quantifiers, rather than the numbers themselves. Study 1 varied the target of a numerical comparison judgment in a between-participants design, while holding the numerical anchor value constant. This design yielded an anchoring effect consistent with a quantifier priming hypothesis. Study 2 included a direct manipulation of vague quantifiers in the traditional anchoring paradigm. Finally, Study 3 examined the notion that specific associations between quantifiers, reflecting values on separate judgmental dimensions (i.e., the price and height of a target) can affect the direction of anchoring effects. Discussion focuses on the nature of vague quantifier priming in numerically anchored judgments.     

Cross-linguistic relations between quantifiers and numerals in language acquisition: Evidence from Japanese

A study of 104 Japanese-speaking 2- to 5-year-olds tested the relation between numeral and quantifier acquisition. A first study assessed Japanese children’s comprehension of quantifiers, numerals, and classifiers. Relative to English-speaking counterparts, Japanese children were delayed in numeral comprehension at 2 years of age but showed no difference at 3 and 4 years of age. Also, Japanese 2-year-olds had better comprehension of quantifiers, indicating that their delay was specific to numerals. A second study examined the speech of Japanese and English caregivers to explore the syntactic cues that might affect integer acquisition. Quantifiers and numerals occurred in similar syntactic positions and overlapped to a greater degree in English than in Japanese. Also, Japanese nouns were often dropped, and both quantifiers and numerals exhibited variable positions relative to the nouns they modified. We conclude that syntactic cues in English facilitate bootstrapping numeral meanings from quantifier meanings and that such cues are weaker in classifier languages such as Japanese.     

Inductive inference in the limit of empirically adequate theories

Most standard results on structure identification in first order theories depend upon the correctness and completeness (in the limit) of the data, which are provided to the learner. These assumption are essential for the reliability of inductive methods and for their limiting success (convergence to the truth).The paper investigates inductive inference from (possibly) incorrect and incomplete data. It is shown that such methods can be reliable not in the sense of truth approximation, but in the sense that the methods converge to empirically adequate theories, i.e. theories, which are consistent with all data (past and future) and complete with respect to a given complexity class of L-sentences. Adequate theories of bounded complexity can be inferred uniformly and effectively by polynomial-time learning algorithms. Adequate theories of unbounded complexity can be inferred pointwise by less efficient methods.