一个含有原子的自然模型∑（A）

1989年A.Blass和A.Scedrov构造了含有原子的模型V（A）（A是所有原子的集合,参见文献[1]）并证明了V（A）是ZFA（ZFA=ZF＋A,公理A断言：存在所有原子的集合）的模型。由于集合论的公理系统GB是ZF的一个保守扩充,因此,集合论的公理系统GBA（GBA=GB＋A,其中GB是集合论的含有集合和类的哥德尔－贝奈斯公理系统）也是ZFA的一个保守扩充。本文的目的是在集合论的含有原子和集合的公理系统ZFA的自然模型V（A）的基础上,为集合论的含有原子、集合和类的公理系统GBA建立模型。因此,我们首先介绍了A.Blass和A.Scedrov的含有原子的模型V（A）;第二,给出并证明V（A）具有的一些基本性质;第三,扩充了集合论的公理系统ZFA的形式语言LZFA并定义含有原子和集合的类C;第四,构造含有原子、集合和类的模型∑（A）,称它为自然模型,最后,证明了∑（A）是GBA的模型。     

The psychological puzzle of Sudoku

Sudoku puzzles, which are popular worldwide, require individuals to infer the missing digits in a 9 × 9 array according to the general rule that every digit from 1 to 9 must occur once in each row, in each column, and in each of the 3-by-3 boxes in the array. We present a theory of how individuals solve these puzzles. It postulates that they rely solely on pure deductions, and that they spontaneously acquire various deductive tactics, which differ in their difficulty depending on their “relational complexity”, i.e., the number of constraints on which they depend. A major strategic shift is necessary to acquire tactics for more difficult puzzles: solvers have to keep track of possible digits in a cell. We report three experiments corroborating this theory. We also discuss their implications for theories of reasoning that downplay the role of deduction in everyday reasoning.     

Monotonicity in Practical Reasoning

Classic deductive logic entails that once a conclusion is sustained by a valid argument, the argument can never be invalidated, no matter how many new premises are added. This derived property of deductive reasoning is known as monotonicity. Monotonicity is thought to conflict with the defeasibility of reasoning in natural language, where the discovery of new information often leads us to reject conclusions that we once accepted. This perceived failure of monotonic reasoning to observe the defeasibility of natural-language arguments has led some philosophers to abandon deduction itself (!), often in favor of new, non-monotonic systems of inference known as `default logics'. But these radical logics (e.g., Ray Reiter's default logic) introduce their desired defeasibility at the expense of other, equally important intuitions about natural-language reasoning. And, as a matter of fact, if we recognize that monotonicity is a property of the form of a deductive argument and not its content (i.e., the claims in the premise(s) and conclusion), we can see how the common-sense notion of defeasibility can actually be captured by a purely deductive system.     

The Logicality of Language: A new take on Triviality, “Ungrammaticality”, and Logical Form*

Recent work in formal semantics suggests that the language system includes not only a structure building device, as standardly assumed, but also a natural deductive system which can determine when expressions have trivial truth‐conditions (e.g., are logically true/false) and mark them as unacceptable. This hypothesis, called the ‘logicality of language’, accounts for many acceptability patterns, including systematic restrictions on the distribution of quantifiers. To deal with apparent counter‐examples consisting of acceptable tautologies and contradictions, the logicality of language is often paired with an additional assumption according to which logical forms are radically underspecified: i.e., the language system can see functional terms but is ‘blind’ to open class terms to the extent that different tokens of the same term are treated as if independent. This conception of logical form has profound implications: it suggests an extreme version of the modularity of language, and can only be paired with non‐classical—indeed quite exotic—kinds of deductive systems. The aim of this paper is to show that we can pair the logicality of language with a different and ultimately more traditional account of logical form. This framework accounts for the basic acceptability patterns which motivated the logicality of language, can explain why some tautologies and contradictions are acceptable, and makes better predictions in key cases. As a result, we can pursue versions of the logicality of language in frameworks compatible with the view that the language system is not radically modular vis‐á‐vis its open class terms and employs a deductive system that is basically classical.     

Classes of models for selected axiomatic theories of choice

Adding a reversibility axiom to the other axioms of Luce's (1959) probabilistic ranking theory results in an impossibility theorem—that all alternatives in an alternative set are equally likely to be chosen (i.e., that preferences are random). This impossibility theorem is generally avoided by removing the reversibility axiom. Using simple algebraic methods such a modified theory is shown to contain a theorem similiar to the impossibility result. These results are discussed within the framework of mathematical model theory (model theory deals with the relations between sets of sentences (theories) and the structures which satisfy these sentences (models)) to illustrate the applicability of model theory as an analytic tool in theory development.     

A Probabilistic Constraints Approach to Language Acquisition and Processing

This article provides an overview of a probabilistic constraints framework for thinking about language acquisition and processing. The generative approach attempts to characterize knowledge of language (i.e., competence grammar) and then asks how this knowledge is acquired and used. Our approach is performance oriented: the goal is to explain how people comprehend and produce utterances and how children acquire this skill. Use of language involves exploiting multiple probabilistic constraints over various types of linguistic and nonlinguistic information. Acquisition is the process of accumulating this information, which begins in infancy. The constraint satisfaction processes that are central to language use are the same as the bootstrapping processes that provide entry to language for the child. Framing questions about acquisition in terms of models of adult performance unifies the two topics under a set of common principles and has important consequences for arguments concerning language learnability.     

Quasi-Realism, Negation and the Frege-Geach Problem

Every expressivist theory of moral language requires a solution to the Frege-Geach problem, i.e., the problem of explaining how moral sentences retain their meaning in unasserted (e.g., conditional and disjunctive) contexts. An essential part of Blackburn's ' quasi -realist project', i.e., the project of showing how we can earn the right to treat moral sentences as if they have ordinary truth-conditions, is to provide a sophisticated solution. I show, however, that simple negated contexts provide a fundamental difficulty, since accepting the negation of a sentence is easily confused with merely refusing to accept that sentence. I argue that Blackburn's model-set semantics for his 'Hooray!' and 'Boo!' operators requires logical apparatus to which he is not entitled. I consider various modifications, but show that they do not succeed.
email : n.unwin@bolton.ac.uk     

Swimming and Speaking Spanish

The dominant view of the status of knowledge of language is that it is theoretical or what Gilbert Ryle called knowledge-that. Defenders of this thesis may differ among themselves over the precise nature of the knowledge which underlies language, as for example, Michael Dummett and Noam Chomsky differ over the issue of unconscious knowledge; however, they all agree that acquisition, understanding and use of language require that the speaker have access to a theory of language. In this paper, I argue that this view is mistaken. Knowledge of language is properly seen as practical knowledge, knowledge-how. My target is Michael Dummett’s treatment of theory of meaning in The Seas of Language. If my argument goes through, underlying assumptions about the nature of cognition as computational must be adjusted to allow for other forms of knowledge, which are arguably more basic, and which underlie knowledge-that.     

A Piagetian perspective on mathematical construction

In this paper, we offer a Piagetian perspective on the construction of the logico-mathematical schemas which embody our knowledge of logic and mathematics. Logico-mathematical entities are tied to the subject's activities, yet are so constructed by reflective abstraction that they result from sensorimotor experience only via the construction of intermediate schemas of increasing abstraction. The ‘axiom set’ does not exhaust the cognitive structure (schema network) which the mathematician thus acquires. We thus view ‘truth’ not as something to be defined within the closed ‘world’ of a formal system but rather in terms of the schema network within which the formal system is embedded. We differ from Piaget in that we see mathematical knowledge as based on social processes of mutual verification which provide an external drive to any ‘necessary dynamic’ of reflective abstraction within the individual. From this perspective, we argue that axiom schemas tied to a preferred interpretation may provide a necessary intermediate stage of reflective abstraction en route to acquisition of the ability to use formal systems in abstracto.     

Interactions between Knowledge,Action and Commitment within Agent Dynamic Logic

This paper considers a new class of agent dynamic logics which provide a formal means of specifying and reasoning about the agents activities and informational, motivational and practical aspects of the behaviour of the agents. We present a Hilbert-style deductive system for a basic agent dynamic logic and consider a number of extensions of this logic with axiom schemata formalising interactions between knowledge and commitment (expressing an agent s awareness of her commitments), and interactions between knowledge and actions (expressing no learning and persistence of knowledge after actions). The deductive systems are proved sound and complete with respect to a Kripke-style semantics. Each of the considered logics is shown to have the small model property and therefore decidable.Presented by Ewa Orlowska     

Can logic be combined with probability? Probably

In the nineteen sixties seminal work was done by Gaifman and then Scott and Krauss in adapting the concepts, tools and procedures of the model theory of modern logic to provide a corresponding model theory in which notions of probabilistic consistency and consequence are defined analogously to the way they are defined (semantically) in the deductive case. The present paper proposes a way of extending these ideas beyond the bounds of a formalised language, even the infinitary language of Scott and Krauss, to achieve a logic having the power and expressiveness of the modern mathematical theory of probability.     

Can Jean Piaget explain the possibility of knowledge?

The purpose of this article is to show that Piaget's use of the equilibrium principle cannot explain the possibility of correct understanding. That is, it cannot explain the possibility of knowledge, as opposed to simple change in belief. To make the argument, I begin by describing Piaget's explanatory model, which is known as the equilibrium principle. I then argue that correct understanding, or knowledge of any x as a case of y, requires a concept of correctness, i.e., the recognition that words and concepts apply under some conditions but not others. I try to show that because he uses the equilibrium principle as a basis for his explanation, Piaget cannot explain how a concept of correctness is acquired. Finally, I argue that to explain the possibility of knowledge, one must show how the conditions for word and concept application are determined by a community of language users. Again, I claim that Piaget's use of the equilibrium model precludes such an account.     

归纳推理和演绎推理的关系理论及其模型

推理是人类高级认知过程的一种,相关的心理学研究一般将其分为归纳推理和演绎推理两个方面。归纳推理是从特殊到一般的推理过程,与之相对的演绎推理则是从一般到特殊的过程。归纳推理和演绎推理的关系问题是当前心理推理研究领域的一个重点问题。这一问题主要有两种理论解释：一种是单过程理论,该理论认为归纳推理和演绎推理本质上是同一个认知过程,以单过程理论为基础构建的推理模型称为单维模型;另一种是双过程理论,认为归纳推理和演绎推理是两个不同的认知过程,并不同程度地受到启发和分析过程的影响。未来研究可多关注推理的时间进程,以及采用不同的研究方法对各自理论提供数据支持。     

Bringing together humanistic and intergroup perspectives to build a model of internalisation of normative social harmdoing

This article introduces a model of the internalisation of normative social harmdoing: the MINSOH. This model seeks to explain how group members internalise harmful social norms such that they personally endorse their groups' normative actions. To this aim, the MINSOH integrates two divergent yet complementary theoretical perspectives: self-determination theory and the social identity approach. These perspectives differ in their basic assumptions about the possibility for harm to become internalised, yet when integrated, they provide a powerful account of how harmdoing can become internalised. The MINSOH proposes specific conditions under which harmful normative actions become accepted by group members. This article outlines multiple self-determined motivations for harmdoing and discrete group processes that enable harmdoing to be internalised and autonomously enacted, and reviews factors that facilitate (i.e., strong/unique/comparative social identification; endorsement of ideological justifications) and block the internalisation process (presence of multiple identities/diverging norms; inclusive superordinate identity). Directions for future research are then discussed.     

The Simplest Axiom System for Plane Hyperbolic Geometry

We provide a quantifier-free axiom system for plane hyperbolic geometry in a language containing only absolute geometrically meaningful ternary operations (in the sense that they have the same interpretation in Euclidean geometry as well). Each axiom contains at most 4 variables. It is known that there is no axiom system for plane hyperbolic consisting of only prenex 3-variable axioms. Changing one of the axioms, one obtains an axiom system for plane Euclidean geometry, expressed in the same language, all of whose axioms are also at most 4-variable universal sentences. We also provide an axiom system for plane hyperbolic geometry in Tarski's language L _B which might be the simplest possible one in that language.     

The semantics and acquisition of number words: integrating linguistic and developmental perspectives

This article brings together two independent lines of research on numerally quantified expressions, e.g. two girls. One stems from work in linguistic theory and asks what truth conditional contributions such expressions make to the utterances in which they are used--in other words, what do numerals mean? The other comes from the study of language development and asks when and how children learn the meaning of such expressions. My goal is to show that when integrated, these two perspectives can both constrain and enrich each other in ways hitherto not considered. Specifically, work in linguistic theory suggests that in addition to their 'exactly n' interpretation, numerally quantified NPs such as two hoops can also receive an 'at least n' and an 'at most n' interpretation, e.g. you need to put two hoops on the pole to win (i.e. at least two hoops) and you can miss two shots and still win (i.e. at most two shots). I demonstrate here through the results of three sets of experiments that by the age of 5 children have implicit knowledge of the fact that expressions like two N can be interpreted as 'at least two N' and 'at most two N' while they do not yet know the meaning of corresponding expressions such as at least/most two N which convey these senses explicitly. I show that these results have important implications for theories of the semantics of numerals and that they raise new questions for developmental accounts of the number vocabulary.     

Tensed Mereology

Classical mereology (CM) is usually taken to be formulated in a tenseless language, and is therefore associated with a four-dimensionalist metaphysics. This paper presents three ways one might integrate the core idea of flat plenitude, i.e., that every suitable condition or property has exactly one mereological fusion, with a tensed logical setting. All require a revised notion of mereological fusion. The candidates differ over how they conceive parthood to interact with existence in time, which connects to the distinction between endurance and perdurance. Similar issues arise for the integration of mereology with modality, and much of our discussion applies to this project as well.     

Carnap's inductive probabilities as a contribution to decision theory

Common probability theories only allow the deduction of probabilities by using previously known or presupposed probabilities. They do not, however, allow the derivation of probabilities from observed data alone. The question thus arises as to how probabilities in the empirical sciences, especially in medicine, may be arrived at. Carnap hoped to be able to answer this question byhis theory of inductive probabilities. In the first four sections of the present paper the above mentioned problem is discussed in general. After a short presentation of Carnap's theory it is then shown that this theory cannot claim validity for arbitrary random processes. It is suggested that the theory be only applied to binomial and multinomial experiments. By application of de Finetti's theorem Carnap's inductive probabilities are interpreted as consecutive probabilities of the Bayesian kind. Through the introduction of a new axiom the decision parameter λ can be determined even if no a priori knowledge is given. Finally, it is demonstrated that the fundamental problem of Wald's decision theory, i.e., the determination of a plausible criterion where no a priori knowledge is available, can be solved for the cases of binomial and multinomial experiments.