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 Simplest Axiom System for Plane Hyperbolic Geometry Revisited

Using the axiom system provided by Carsten Augat in [1], it is shown that the only 6-variable statement among the axioms of the axiom system for plane hyperbolic geometry (in Tarski’s language L _B≡), we had provided in [3], is superfluous. The resulting axiom system is the simplest possible one, in the sense that each axiom is a statement in prenex form about at most 5 points, and there is no axiom system consisting entirely of at most 4-variable statements.     

PREFERENCE REVERSALS DUE TO MYOPIC DISCOUNTING OF DELAYED REWARD

Abstract— A basic stationarity axiom of economic theory assumes stable preference between two deferred goods separated by a fixed time. To test this assumption, we offered subjects choices between delayed rewards, while manipulating the delays to those rewards. Preferences typically reversed with changes in delay, as predicted by hyperbolic discounting models of impulsiveness. Of 36 subjects, 34 reversed preference from a larger, later reward to a smaller, earlier reward as the delays to both rewards decreased. We conclude that the stationarity axiom is not appropriate in models of human choice.     

More Triviality

This paper uses the framework of Popper and Miller's work on axiom systems for conditional probabilities to explore Adams' thesis concerning the probabilities of conditionals. It is shown that even very weak axiom systems have only a very restricted set of models satisfying a natural generalisation of Adams' thesis, thereby casting severe doubt on the possibility of developing a non-Boolean semantics for conditionals consistent with it.     

Axiomatizations of Hyperbolic Geometry: A Comparison Based on Language and Quantifier Type Complexity

Hyperbolic geometry can be axiomatized using the notions of order andcongruence (as in Euclidean geometry) or using the notion of incidencealone (as in projective geometry). Although the incidence-based axiomatizationmay be considered simpler because it uses the single binary point-linerelation of incidence as a primitive notion, we show that it issyntactically more complex. The incidence-based formulation requires some axioms of the quantifier-type \forall\exists\forall, while the axiom system based on congruence and order can beformulated using only \forall\exists-axioms.     

Tests of product and additive scaling axioms

A product axiom for ratio judgments and an additive axiom for difference judgments were tested for six Ss judging brightness. The product axiom was rejected for all six Ss whereas the additive axiom was accepted for two Ss. Power functions did not fit the data well, although somewhat better for difference judgments than ratio judgments. A two-stage model fitted somewhat better, but failed to satisfy one important implication of that model, previously confirmed for lifted weights. The data gave stronger support for a theory of brightness estimation based on an additive axiom for difference judgments than a product axiom based on ratio judgments.     

A logic road from special relativity to general relativity

We present a streamlined axiom system of special relativity in first-order logic. From this axiom system we ??derive?? an axiom system of general relativity in two natural steps. We will also see how the axioms of special relativity transform into those of general relativity. This way we hope to make general relativity more accessible for the non-specialist.     

Normative systems and medical metaethics Part II: Health-maximizing and persons

Two common medical-ethical axions, the health-maximizing axiom and the personhood-respecting axiom, are discussed. On the basis of a philosophical analysis of personhood and freedom of the will it is shown that these two axioms are incompatible. The rejection of the first axiom is suggested.     

论基础公理与反基础公理

"循环并不可恶"。本文在此基础上讨论基础公理和反基础公理。首先指出基础公理原本就是一条有争议的公理;第二,说明基础公理的局限性;第三,详细论述反基础公理家族中的三个成员,并给出它们两两不相容的一个证明;第四,分析反基础公理导致集合论域在V=WF上不断扩张的方法,并指出这种扩张的方法与数系扩张的方法相同;最后结论:良基集合理论(ZFC)与非良基集合理论(ZFC~-+AFA(或者ZFC和ZFC~-+FAFA或者ZFC和ZFC~-+SAFA))之间的关系类似于欧几里得几何学与非欧几何学之间的关系。     

On Groupoids of Relations with One Conjunctive Operation of Rank 2

Studia Logica - In this paper, we obtain axiom systems, bases of identities and quasi-identities for classes of algebras of binary relations with a conjunctive operation, i.e., an operation that...     

On the Meaning of Hilbert's Consistency Problem (Paris, 1900)

The theory that ``consistency implies existence' was put forward by Hilbert on various occasions around the start of the last century, and it was strongly and explicitly emphasized in his correspondence with Frege. Since (Gödel's) completeness theorem, abstractly speaking, forms the basis of this theory, it has become common practice to assume that Hilbert took for granted the semantic completeness of second order logic. In this paper I maintain that this widely held view is untrue to the facts, and that the clue to explain what Hilbert meant by linking together consistency and existence is to be found in the role played by the completeness axiom within both geometrical and arithmetical axiom systems.     

A General Framework for $$ {FDE}$$ FDE -Based Modal Logics

We develop a general theory of FDE-based modal logics. Our framework takes into account the four-valued nature of FDE by considering four partially defined modal operators corresponding to conditions for verifying and falsifying modal necessity and possibility operators. The theory comes with a uniform characterization for all obtained systems in terms of FDE-style formula-formula sequents. We also develop some correspondence theory and show how Hilbert-style axiom systems can be obtained in appropriate cases. Finally, we outline how different systems from the literature can be expressed in our framework.

     

Fractal Images of Formal Systems

Formal systems are standardly envisaged in terms of a grammar specifying well-formed formulae together with a set of axioms and rules. Derivations are ordered lists of formulae each of which is either an axiom or is generated from earlier items on the list by means of the rules of the system; the theorems of a formal system are simply those formulae for which there are derivations. Here we outline a set of alternative and explicitly visual ways of envisaging and analyzing at least simple formal systems using fractal patterns of infinite depth. Progressively deeper dimensions of such a fractal can be used to map increasingly complex wffs or increasingly complex value spaces, with tautologies, contradictions, and various forms of contingency coded in terms of color. This and related approaches, it turns out, offer not only visually immediate and geometrically intriguing representations of formal systems as a whole but also promising formal links (1) between standard systems and classical patterns in fractal geometry, (2) between quite different kinds of value spaces in classical and infinite-valued logics, and (3) between cellular automata and logic. It is hoped that pattern analysis of this kind may open possibilities for a geometrical approach to further questions within logic and metalogic.\looseness=-1     

The Ryōan-ji axiom for common knowledge on hypergraphs

The article studies common knowledge in communication networks with a fixed topological structure. It introduces a non-trivial principle, called the Ryōan-ji axiom, which captures logical properties of common knowledge of all protocols with a given network topology. A logical system, consisting of the Ryōan-ji axiom and two additional axioms, is proven to be sound and complete.     

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。     

A Mathematical Model of Depth Displacement with a Contracting Bar

When a vertical line segment contracts at both ends according to a (decelerating) time law of hyperbolic type, it appears to rotate around its midpoint. The phenomenon is quite surprising from the point of view of projective geometry as the segment should rather appear to recede along the sagittal plane. Apparent displacement in depth is however obtained when the bar simultaneously contracts and is displaced laterally on the frontal plane. But, here again, projective expectations are contradicted, because apparent displacement in depth occurs whether movement of the bar in two dimensions is hyperbolic (decelerating), uniform, harmonic (accelerating), or a mixture of the three. It is suggested here that the visual system operates in such a way as to minimize the differences between the lengths of the velocity vectors of all points of a moving configuration. The mathematical model derived from this hypothesis allows qualitative and quantitative predictions that are in good agreement with experimental results. Copyright 2001 Academic Press.     

Partial monotonicity and a new version of the Ramsey test

We introduce two new belief revision axioms: partial monotonicity and consequence correctness. We show that partial monotonicity is consistent with but independent of the full set of axioms for a Gärdenfors belief revision sytem. In contrast to the Gärdenfors inconsistency results for certain monotonicity principles, we use partial monotonicity to inform a consistent formalization of the Ramsey test within a belief revision system extended by a conditional operator. We take this to be a technical dissolution of the well-known Gärdenfors dilemma.In addition, we present the consequential correctness axiom as a new measure of minimal revision in terms of the deductive core of a proposition whose support we wish to excise. We survey several syntactic and semantic belief revision systems and evaluate them according to both the Gärdenfors axioms and our new axioms. Furthermore, our algebraic characterization of semantic revision systems provides a useful technical device for analysis and comparison, which we illustrate with several new proofs.Finally, we have a new inconsistency result, which is dual to the Gärdenfors inconsistency results. Any elementary belief revision system that is consequentially correct must violate the Gärdenfors axiom of strong boundedness (K^*8), which we characterize as yet another monotonicity condition.This work was supported by the McDonnell Douglas Independent Research and Development program.     

A Unified Nonparametric IRT Model for <Emphasis Type="Italic">d</Emphasis>-Dimensional Psychological Test Data (<Emphasis Type="Italic">d</Emphasis>-ISOP)

The (univariate) isotonic psychometric (ISOP) model (Scheiblechner, 1995) is a nonparametric IRT model for dichotomous and polytomous (rating scale) psychological test data. A weak subject independence axiom W1 postulates that the subjects are ordered in the same way except for ties (i.e., similarly or isotonically) by all items of a psychological test. A weak item independence axiom W2 postulates that the order of the items is similar for all subjects. Local independence (LI or W3) is assumed in all models. With these axioms, sample-free unidimensional ordinal measurements of items and subjects become feasible. A cancellation axiom (Co) gives, as a result, the additive isotonic psychometric (ADISOP) model and interval scales for subjects and items, and an independence axiom (W4) gives the completely additive isotonic psychometric (CADISOP) model with an interval scale for the response variable (Scheiblechner, 1999). The d-ISOP, d-ADISOP, and d-CADISOP models are generalizations to d-dimensional dependent variables (e.g., speed and accuracy of response). The author would like to thank an Associate Editor and two anonymous referees and also Professor H.H. Schulze for their very valuable suggestions and corrections.