A formal theory of generalized intermediate syllogisms

In this paper, we continue developing the formal theory of intermediate quantifiers (expressions such as most, few, almost all, a lot of, many, a great deal of, a large part of, a small part of). The theory is a fuzzy-logic formalization of the concept introduced by Peterson in his book. We will syntactically prove that 105 generalized Aristotle's syllogisms introduced in this book are valid in our theory. At the same time, we will also prove that syllogisms listed there as invalid are invalid also in our theory. Therefore, we believe that our theory provides a reasonable mathematical model of the generalized syllogistics.     

The arithmetic of hyperbolic formal modules

This paper considers hyperbolic formal groups, which come from the elliptic curve theory, in the context of the theory of formal modules. In the first part of the paper, the characteristics of hyperbolic formal groups are considered, i.e., the explicit formulas for the formal logarithm and exponent; their convergence is studied. In the second part, the isogeny and its kernel and height are found; a p-typical logarithm is defined. The Artin–Hasse and Vostokov functions are then constructed; their correctness and main properties are evaluated.     

非Fuchs型方程的新理论——树级数解的表现定理(Ⅰ)

在微分方程的解析理论中非Fuchs型方程的严格显式解至今并未求得(Poincaré问题),本文提出的新理论首次给出非正则积分的一般求法和显式的精确解. 本法与经典理论的根本不同在于摈弃形式解的假定,从方程本身建立对应关系,应用留数定理自动给出非正则积分的解析结构.它由无收缩部和全、半收缩部组成.前者是通常的递推级数,后者则表为树级数.树级数是类新颖的解析函数,通常的递推级数只是它的特例而已. 本文的目的是建立非正则积分的一般理论,为此需要阐明Poincaré问题(1880T.I.P.333)的实质[1]:无法求出非正则积分的显式.根据以下证明的表现定理, 非正则积分是类新颖的解析函数,其中系数D_nk是方程参数的常项树级数.     

Portfolio selection with skewness: A comparison of methods and a generalized one fund result

This contribution compares existing and newly developed techniques for geometrically representing mean–variance–skewness portfolio frontiers based on the rather widely adapted methodology of polynomial goal programming (PGP) on the one hand and the more recent approach based on the shortage function on the other hand. Moreover, we explain the working of these different methodologies in detail and provide graphical illustrations in relation to the goal programming literature in operations research. Inspired by these illustrations, we prove two new results: a formal relation between both approaches and a generalization of the well-known one fund separation theorem from traditional mean–variance portfolio theory.     

How Emergent Models May Foster the Constitution of Formal Mathematics

This article deals with the role that so-called emergent models can play in the process of constituting formal mathematics. The underlying philosophy is that formal mathematics is something that is, or should be, constituted by the students themselves. In the instructional design theory for realistic mathematics education, models always have been employed to foster a process in which formal mathematics is reinvented by the students themselves. This article describes how the use of models became more and more explicated over time and developed into the notion of emergent models. The design of an instructional sequence, which deals with flexible mental computation strategies for addition and subtraction up to 100, is taken as an instance for elaborating what is meant by emergent models and what role they play in fostering the constitution of formal mathematics. The analysis shows that there are 3 interrelated processes. First. at a more holistic level, there is a global transition in which “the model” initially emerges as a model of informal mathematical activity and then gradually develops into a model for more formal mathematical reasoning. Second, the transition from “model of” to “model for” involves the constitution of anew mathematical reality that can be denoted formal in relation to the original starting points of the students. Third, in the series of instructional activities, there is not 1 model, but the model actually is shaped as a series of signs, in which each new sign comes to signify activity with a previous sign in a chain of signification.     

Automorphic forms for subgroups of the modular group

The theory of theta constants with rational characteristics is developed from the point of view of automorphic functions for the principal congruence subgroups of the modular group PSL(2, ℤ). New identities are derived and particular emphasis is given to the level 3 case where a striking generalization of the classical λ-function is obtained. This paper is dedicated to Prof. J. Thompson in recognition of his many contributions to mathematics and of his being the recipient of the Wolf Prize in Mathematics Research by HMF supported in part by the Paul and Gabriella Rosenbaum Foundation and partially sponsored by the Edmund Landau Center for Research in Mathematical Analysis, supported by the Minerva Foundation (Germany). Research by IK supported in part by NSF Grants DMS 9003361 and 9204092.     

Operatorenkalkül über freien Monoiden II: Binomialsysteme

In this part starting from a generalization of the binomial theorem a development of Rota's theory of polynomial sequences of binomial type to the case of countably many noncommuting variables is given. Translation invariance of operators gives the relation to the formal power series considered inI. For a special class of binomial systems there are given a number of characterizations, such as a generalized Rodrigues formula. In the case of an analogue of the Newton polynomials those are used for a study of generalized Stirling numbers. (In III the theory of binomial systems of diagonal type will be continued until an analogue of Lagrange inversion and a short development of the theory of generalized Sheffer polynomials will be given.)     

How Emergent Models May Foster the Constitution of Formal Mathematics

This article deals with the role that so-called emergent models can play in the process of constituting formal mathematics. The underlying philosophy is that formal mathematics is something that is, or should be, constituted by the students themselves. In the instructional design theory for realistic mathematics education, models always have been employed to foster a process in which formal mathematics is reinvented by the students themselves. This article describes how the use of models became more and more explicated over time and developed into the notion of emergent models. The design of an instructional sequence, which deals with flexible mental computation strategies for addition and subtraction up to 100, is taken as an instance for elaborating what is meant by emergent models and what role they play in fostering the constitution of formal mathematics. The analysis shows that there are 3 interrelated processes. First. at a more holistic level, there is a global transition in which "the model" initially emerges as a model of informal mathematical activity and then gradually develops into a model for more formal mathematical reasoning. Second, the transition from "model of" to "model for" involves the constitution of anew mathematical reality that can be denoted formal in relation to the original starting points of the students. Third, in the series of instructional activities, there is not 1 model, but the model actually is shaped as a series of signs, in which each new sign comes to signify activity with a previous sign in a chain of signification.     

Dependence relation in pregeometries

The aim of this paper is to set a foundation to separate geometric model theory from model theory. Our goal is to explore the possibility to extend results from geometric model theory to non first order logic (e.g. ). We introduce a dependence relation between subsets of a pregeometry and show that it satisfies all the formal properties that forking satisfies in simple first order theories. This happens when one is trying to lift forking to nonelementary classes, in contexts where there exist pregeometries but not necessarily a well-behaved dependence relation (see for example [HySh]). We use these to reproduce S. Buechler's characterization of local modularity in general. These results are used by Lessmann to prove an abstract group configuration theorem in [Le2]. Received February 2, 1999; accepted in final form April 30, 2000.     

A particularist epistemology: ‘Affectual intuitionism’

Jonathan Dancy has developed a very refined theory called ethical particularism. He has argued extensively for the metaphysical part of his position. However, the accompanying epistemology is not yet clear. In this paper I will sketch a particularist epistemology that is consistent with Dancy’s particularist metaphysics, although my approach differs in certain respects from epistemological claims Dancy has made. I will defend an epistemology that states: 1. that moral knowledge is based on intuitions and 2. that we need emotions in order to have moral knowledge. I will call this approach ‘affectual intuitionism’. Dancy rejects both claims, but I will argue that his arguments against these claims are not convincing.     

Theories and Theories of Truth

Formal theories, as in logic and mathematics, are sets of sentences closed under logical consequence. Philosophical theories, like scientific theories, are often far less formal. There are many axiomatic theories of the truth predicate for certain formal languages; on analogy with these, some philosophers (most notably Paul Horwich) have proposed axiomatic theories of the property of truth. Though in many ways similar to logical theories, axiomatic theories of truth must be different in several nontrivial ways. I explore what an axiomatic theory of truth would look like. Because Horwich’s is the most prominent, I examine his theory and argue that it fails as a theory of truth. Such a theory is adequate if, given a suitable base theory, every fact about truth is a consequence of the axioms of the theory. I show, using an argument analogous to Gödel’s incompleteness proofs, that no axiomatic theory of truth could ever be adequate. I also argue that a certain class of generalizations cannot be consequences of the theory.     

Syllogisms and 5-Square of Opposition with Intermediate Quantifiers in Fuzzy Natural Logic

In this paper, we provide an overview of some of the results obtained in the mathematical theory of intermediate quantifiers that is part of fuzzy natural logic (FNL). We briefly introduce the mathematical formal system used, the general definition of intermediate quantifiers and define three specific ones, namely, “Almost all”, “Most” and “Many”. Using tools developed in FNL, we present a list of valid intermediate syllogisms and analyze a generalized 5-square of opposition.     

A weighted causal theory for acquiring and utilizing open knowledge

Motivated by enabling intelligent robots/agents to take advantage of open-source knowledge resources to solve open-ended tasks, a weighted causal theory is introduced as the formal basis for the development of these robots/agents. The action model of a robot/agent is specified as a causal theory following McCain and Turner's nonmonotonic causal theories. New knowledge is needed when the robot/agent is given a user task that cannot be accomplished only with the action model. This problem is cast as a variant of abduction, that is, to find the most suitable set of causal rules from open-source knowledge resources, so that a plan for accomplishing the task can be computed using the action model together with the acquired knowledge. The core part of our theory is constructed based on credulous reasoning and the complexity of corresponding abductive reasoning is analyzed. The entire theory is established by adding weights to hypothetical causal rules and using them to compare competing explanations which induce causal models satisfying the task. Moreover, we sketch a model theoretic semantics for the weighted causal theory and present an algorithm for computing a weighted-abductive explanation. An application of the techniques proposed in this paper is illustrated in an example on our service robot, KeJia, in which the robot tries to acquire proper knowledge from OMICS, a large-scale open-source knowledge resource, and solve new tasks with the knowledge.     

Generalization of Pawlak’s rough approximation spaces by using δβ-open sets

The original rough set model was developed by Pawlak, which is mainly concerned with the approximation of objects using an equivalence relation on the universe of his approximation space. This paper extends Pawlak’s rough set theory to a topological model where the set approximations are defined using the topological notion δβ-open sets. A number of important results using the topological notion δβ-open set are obtained. We also, proved that some of the properties of Pawlak’s rough set model are special instances of those of topological generalizations. Moreover, several important measures, related to the new model, such as accuracy measure and quality of approximation are presented.     

Chiral differential operators: Formal loop group actions and associated modules

Chiral differential operators (CDOs) are closely related to string geometry and the quantum theory of 2-dimensional σ-models. This paper investigates two topics about CDOs on smooth manifolds. In the first half, we study how a Lie group action on a smooth manifold can be lifted to a “formal loop group action” on an algebra of CDOs; this turns out to be a condition on the equivariant first Pontrjagin class. The case of a principal bundle receives particular attention and gives rise to a type of vertex algebras of great interest. In the second half, we introduce a construction of modules over CDOs using the said “formal loop group actions” and semi-infinite cohomology. Intuitively, these modules should have a geometric meaning in terms of “formal loop spaces”. The first example we study leads to a new conceptual construction of an arbitrary algebra of CDOs. The other example, called the spinor module, may be useful for a geometric theory of the Witten genus.     

Non-deterministic ideal operators: An adequate tool for formalization in Data Bases

In this paper, we propose the application of formal methods to Software Engineering. The most used data model is the relational model and we present, within the general framework of lattice theory, this analysis of functional dependencies. For this reason, we characterize the concept of f-family by means of a new concept which we call non-deterministic ideal operator (nd.ideal-o). The study of nd.ideal-o.s allows us to obtain results about functional dependencies as trivial particularizations, to clarify the semantics of the functional dependencies and to progress in their efficient use, and to extend the concept of schema. Moreover, the algebraic characterization of the concept of Key of a schema allows us to propose new formal definitions in the lattice framework for classical normal forms in relation schemata. We give a formal definition of the normal forms for functional dependencies more frequently used in the bibliography: the second normal form (2FN), the third normal form(3FN) and Boyce-Codd's normal form (FNBC).     

Charles Peirce's place in philosophy

Peirce's publications on the method of scientific investigation (as distinct from his work in formal logic and mathematics) are his most important and valuable contributions to philosophy. His views on this subject are superior in clarity and cogency to his voluminous writings on metaphysics and cosmology. He subscribed to a fallibilistic conception of knowledge that is poles apart from a wholesale skepticism; his formulations of the conditions for meaningful discourse and of the pragmatic maxim, though not free from difficulties, have been fruitful sources of much subsequent philosophical and scientific analyses; and his classification of and discussions of types of argument or reasoning employed in scientific inquiry continue to be valuable and insightful clarifications of this important subject. In contrast to his account of scientific method, Peirce's evolutionary theory of ultimate reality, though marked by originality and ingenious speculation, has little merit as a contribution to genuine knowledge.     

Nearness approximation space based on axiomatic fuzzy sets

The approximation space model was originally proposed by Pawlak (1982) [19]. It was Or?owska who first observed that approximation spaces serves as a formal counterpart of perception, or observation [16, §2, p. 8], in which approximations provide a means of approximating one set of objects with another set of objects using the indiscernibility relation. Topology has been used to enrich the original model of an approximation space as well as more recent models of generalized approximation spaces. In this paper, an extension of th e topology neighborhood based on AFS (Axiomatic Fuzzy Sets) theory is introduced, and some interesting properties are given. Furthermore, a new generalized approximation space model is established with two application examples, which can be used to deal with information tables with many category features and viewed as a multi-granulations form of nearness approximation space models.     

Precovers,Modalities and Universal Closure Operators in a Topos

In this paper we develop the notion of formal precover in a topos by defining a relation between elements and sets in a local set theory. We show that such relations are equivalent to modalities and to universal closure operators. Finally we prove that these relations are well characterized by a convenient restriction to a particular set. Mathematics Subject Classification: 03G30, 03B20, 03F55.     

The spherical part of the local and global Springer actions

The affine Weyl group acts on the cohomology (with compact support) of affine Springer fibers (local Springer theory) and of parabolic Hitchin fibers (global Springer theory). In this paper, we show that in both situations, the action of the center of the group algebra of the affine Weyl group (the spherical part) factors through the action of the component group of the relevant centralizers. In the situation of affine Springer fibers, this partially verifies a conjecture of Goresky–Kottwitz–MacPherson and Bezrukavnikov–Varshavsky. We first prove this result for the global Springer action, and then deduce from it the result for the local Springer action. This gives an application of global Springer theory to more classical problems.