Order Axioms Research Articles

Another Splitting of the Pasch Axiom

The Pasch axiom is shown to be equivalent, given the linear order axioms, to the conjunction of its inner form with a statement involving five individual variables.

Using homogeneous groupings in portfolio management

Often, in situations of uncertainty in portfolio management, it is difficult to apply the numerical methods based on the linearity principle. When this happens it is possible to use nonnumeric techniques to assess the situations with a nonlinear attitude. One of the concepts that can be used in these situations is the concept of grouping. In the last 30 years, several studies have tried to give good solutions to the problems of homogeneous groupings. For example, we could mention the Pichat algorithm, the affinities algorithms and several studies developed by the authors of this work. In this paper, we use some topological axioms in order to develop an algorithm that is able to reduce the number of elements of the power sets of the related sets by connecting them to the sets that form the topologies. We will apply this algorithm in the grouping of titles listed in the Stock Exchange or in its dual perspective.

Populating Data Warehouses with Semantic Data

The Semantic Web has become a new environment that enables organizations to attach semantic annotations taken from domain and application ontologies to the information they generate. As a result, large amounts of complex, semi-structured and heterogeneus semantic data repositories are being made available. In this paper, we present an automatic method for on-demand construction of fact tables aimed at analyzing semantic data expressed as instance stores in RDF/OWL. The starting point of the method is a multidimensional star schema (i.e. topic of analysis, dimensions and measures) designed by the analyst by picking up concepts and properties from the ontology of the instance store. The method exploits the underlying semantics of ontology axioms in order to derive facts, that is, “valid combinations” of instances and literals that fulfill the analyst needs. Moreover, both the ontology axioms and the instances are indexed in order to reach scalability. We have tested the method with a synthetically generated instance set in order to show its scalability and effectiveness.

Uphill self-control

This paper extends the theory of temptation and self-control introduced by Gul and Pesendorfer (2001) to allow for increasing marginal costs of resisting temptation, that is, convex self-control costs. It also proves a representation theorem that admits a general class of self-control cost functions. Both models maintain the Order, Continuity, and Set Betweenness axioms but violate Independence.

Solution of an open problem of Alsina, Frank and Schweizer

An open problem of Alsina, Frank and Schweizer on associative functions is solved. Equivalence between continuity in each component and joint continuity of an associative two place function satisfying some boundary conditions is shown, thus providing a minimal set of axioms in order for a two place function to be a continuous triangular norm.

A System of Axioms for Hyperbolic Geometry

Three–dimensional hyperbolic geometry is characterized using axioms of order, incidence, dimension, continuity and, instead of an axiom of parallels, there is an axiom of “rigidity” and, rather than several axioms of congruence, there is one axiom of symmetry. It is claimed that this system of axioms is simpler than the system of independent axioms of Moore [15].

Semantic optimization techniques for preference queries

Preference queries are relational algebra or SQL queries that contain occurrences of the winnow operator ( find the most preferred tuples in a given relation). Such queries are parameterized by specific preference relations. Semantic optimization techniques make use of integrity constraints holding in the database. In the context of semantic optimization of preference queries, we identify two fundamental properties: containment of preference relations relative to integrity constraints and satisfaction of order axioms relative to integrity constraints. We show numerous applications of those notions to preference query evaluation and optimization. As integrity constraints, we consider constraint-generating dependencies, a class generalizing functional dependencies. We demonstrate that the problems of containment and satisfaction of order axioms can be captured as specific instances of constraint-generating dependency entailment. This makes it possible to formulate necessary and sufficient conditions for the applicability of our techniques as constraint validity problems. We characterize the computational complexity of such problems.

What is the logic of propositional identity?

Propositional identity is not expressed by a predicate. So its logic is not given by the ordinary first order axioms for identity. What are the logical axioms governing this concept, then? Some axioms in addition to those proposed by Arthur Prior are proposed.

Leibnizian models of set theory

Abstract.A model is said to be Leibnizian if it has no pair of indiscernibles. Mycielski has shown that there is a first order axiom LM (the Leibniz-Mycielski axiom) such that for any completion T of Zermelo-Fraenkel set theory ZF. T has a Leibnizian model if and only if T proves LM. Here we prove:Theorem A. Every complete theory T extending ZF + LM has nonisomorphic countable Leibnizian models.Theorem B. If κ is a prescribed definable infinite cardinal ofa complete theory T extending ZF + V = OD, then there are nonisomorphic Leibnizian models of T of power ℵ1such thatis ℵ1-like.Theorem C. Every complete theory T extendingZF + V = ODhas nonisomorphic ℵ1-like Leibnizian models.

Proof-theoretical analysis of order relations

A proof-theoretical analysis of elementary theories of order relations is effected through the formulation of order axioms as mathematical rules added to contraction-free sequent calculus. Among the results obtained are proof-theoretical formulations of conservativity theorems corresponding to Szpilrajn’s theorem on the extension of a partial order into a linear one. Decidability of the theories of partial and linear order for quantifier-free sequents is shown by giving terminating methods of proof-search.

An axiomatic approach to intergenerational equity

We present a set of axioms in order to capture the concept of equity among an infinite number of generations. There are two ethical considerations: one is to treat every generation equally and the other is to respect distributive fairness among generations. We find two opposite results. In Theorem 1, we show that there exists a preference ordering satisfying anonymity, strong distributive fairness semiconvexity, and strong monotonicity. However, in Theorem 2, we show that there exists no binary relation satisfying anonymity, distributive fairness semiconvexity, and sup norm continuity. We also clarify logical relations between these axioms and non-dictatorship axioms.

Pheochromocytoma — Death of an Axiom

Medical students often learn axioms in order to remember the key features of a disorder. The “rule of 10” used to describe pheochromocytomas is a good example: 10 percent are extraadrenal, and of t...

SCHRÖDINGER EQUATION FOR QUANTUM FRACTAL SPACE–TIME OF ORDER n VIA THE COMPLEX-VALUED FRACTIONAL BROWNIAN MOTION

First remark: Feynman's discovery in accordance of which quantum trajectories are of fractal nature (continuous everywhere but nowhere differentiable) suggests describing the dynamics of such systems by explicitly introducing the Brownian motion of fractional order in their equations. The second remark is that, apparently, it is only in the complex plane that the Brownian motion of fractional order with independent increments can be generated, by using random walks defined with the complex roots of the unity; in such a manner that, as a result, the use of complex variables would be compulsory to describe quantum systems. Here one proposes a very simple set of axioms in order to expand the consequences of these remarks. Loosely speaking, a one-dimensional system with real-valued coordinate is in fact the average observation of a one-dimensional system with complex-valued coordinate: It is a strip modeling. Assuming that the system is governed by a stochastic differential equation driven by a complex valued fractional Brownian of order n, one can then obtain the explicit expression of the corresponding covariant stochastic derivative with respect to time, whereby we switch to the extension of Lagrangian mechanics. One can then derive a Schrödinger equation of order n in quite a direct way. The extension to relativistic quantum mechanics is outlined, and a generalized Klein–Gordon equation of order n is obtained. As a by-product, one so obtains a new proof of the Schrödinger equation.

Linear Measurement Models—Axiomatizations and Axiomatizability

This paper examines, in the scope of representational measurement theory, different axiomatizations and axiomatizability of linear and bilinear representations of ordinal data contexts in real vector spaces. The representation theorems proved in this paper are modifications and generalizations of Scott's characterization of finite linear measurement models. The advantage of these representation theorems is that they use only finitely many axioms, the number of which depends on the size of the given ordinal data context. Concerning the axiomatizability, it is proved by model-theoretic methods that finite linear measurement models cannot be axiomatized by a finite set of first order axioms.

From situation calculus to fluent calculus: State update axioms as a solution to the inferential frame problem

Successor state axioms provide a solution to the famous Frame Problem as far as the representational aspect is concerned. Solving in classical, monotonic logic the additional inferential Frame Problem, on the other hand, was the major motivation for the development of the Fluent Calculus a decade or so ago. Yet the expressiveness of the latter in comparison to the Situation Calculus remained a largely open question until today. In this note, we derive a novel version of the Fluent Calculus by gradually applying the principle of reification to successor state axioms in order to address the inferential Frame Problem without losing the representational merits. Our approach results in a fully mechanic method for the generation of state update axioms from any collection of Situation Calculus-style effect axioms for deterministic actions, provided the actions do not have potentially infinitely many effects. The axiomatization thus obtained is proved essentially equivalent to the corresponding axiomatization which uses successor state axioms.

A common axiom set for classical and intuitionistic plane geometry

We describe a first order axiom set which yields the classical first order Euclidean geometry of Tarski when used with classical logic, and yields an intuitionistic (or constructive) Euclidean geometry when used with intuitionistic logic. The first order language has a single six place atomic predicate and no function symbols. The intuitionistic system has a computational interpretation in recursive function theory, that is, a realizability interpretation analogous to those given by Kleene for intuitionistic arithmetic and analysis. This interpretation shows the unprovability in the intuitionistic theory of certain “nonconstructive” theorems of the classical geometry.

AN ALTERNATIVE AXIOMATIZATION OF SEN'S POVERTY MEASURE

We provide an alternative axiomatization of Sen's (1976) poverty measure. We derive the measure from the general definition of a poverty measure by using a version of Sen's rank order axiom, and a substantially weaker form of his normalization axiom. These two axioms, together with a continuity axiom, lead uniquely to the measure suggested by Sen (1976).

ASYMMETRIC VALUATION OF GAINS AND LOSSES AND PREFERENCE ORDER ASSUMPTIONS

Recent findings of large differences between people's valuations of gains and losses imply that the common preference order axioms of completeness, transitivity, and dominance, on which economic analyses and predictions of consumer behavior are largely based, may not be consistent with actual choices. This paper reports the results of an experimental test of the preference order assumptions using real exchanges of two goods and money. The findings show consistent differences in the valuation placed on goods and money depending on whether the entitlements are being acquired or given up, and provide direct evidence of preference assumption violations.

Finitely based modular congruence varieties are distributive

R. Dedekind introduced the modular law, a lattice equation true in most of the lattices associated with classical algebraic systems, see [4]. Although this law is one of the most important tools for working with these lattices, it does not fully describe the equational properties of these lattices. This was made clear in [8] where B. Jonsson and the author showed that if any modular congruence variety actually satisfies the (stronger) Arguesian law. (A congruence variety is the variety generated by all the congruence lattices of the members of a variety of algebras.) In this note we show that no finite set of lattice equations is strong enough to describe the equational properties of the lattices associated with classical algebraic systems in the following strong sense: there is no modular, nondistributive congruence variety which has a finite basis for its equational theory. This question was posed by George McNulty in the problem session on lattice theory at the Jonsson symposium. Jonsson had asked a similar question in [17]. He asked, in Problem 9.12, whether there is a nontrivial variety whose congruence variety is neither the variety of all lattices nor the variety of all distributive lattices, but which is finitely based. Some partial results on this problem had been obtained previously. Let Vn denote the class of lattices embeddable into the lattice of subgroups of an abelian group of exponent n and let V0 be those lattices embeddable into the lattice of subspaces of a rational vector space. Thus HVn is the congruence variety associated with the variety of abelian groups of exponent n. In [7] it was shown that if L is a class of lattices contained in a modular congruence variety and containing Vn for some n not a prime, then L cannot be defined by finitely many first order axioms. It was also shown that every modular, nondistributive congruence variety contains HVn for some n. Mark Haiman, in his proof that there are Arguesian lattices which are not representable as lattices of permuting equivalence relations (solving an old problem of Jonsson), shows that HVp is not finitely based for any prime p, [11] and [12].

A new definition of fuzzy functional dependency in fuzzy relational databases

The need to incorporate and treat information given in fuzzy terms in Relational Databases has concentrated a great effort in the last years. This article focuses on the treatment of functional dependencies (f.d.) between attributes of a relation scheme. We review other approaches to this problem and present some of its missfunctions concerning intuitive properties a fuzzy extension of f.d. should verify. Then we introduce a fuzzy extension of this concept to overcome the previous anomalous behaviors and study its properties. of primary interest is the completeness of our fuzzy version of Armstrong axioms in order to derive all the fuzzy functional dependencies logically implied by a set of f.f.d. just using these axioms. © 1994 John Wiley & Sons, Inc.