New preference violation indices for the condition of order preservation

<p>Consistency of pairwise comparisons is one particular aspect that is studied thoroughly in the recent decades. However, since the introduction of the concept of the condition of the order preservation in 2008, there is no inconsistency measure based on the aforementioned condition. Therefore, the aim of this paper is to fill this gap and propose new preference violation indices for measuring violation of the condition of the order preservation. Further, an axiomatic system for the proposed measures is discussed, and it is shown that the proposed indices satisfy uniqueness, invariance under permutation, invariance under inversion of preferences and continuity axioms.</p> <p>&nbsp;</p>

The Formal Modeling Of Engineering Design Information By Means Of An Axiomatic System

<div>There is mounting evidence in the current literature which suggests that our collective understanding of engineering design is insufficient to support the continued growth of the engineering endeavor. Design theory is the emergent research field that addresses this problem by seeking to improve our understanding of, and thus our ability to, design. The goal of this author's work is to demonstrate that formal techniques of logic can improve our understanding of design. Specifically, a formal system called the Hybrid Model (HM) is presented; this system is a set-theoretic description of engineering design information that is valid independent of (a) the processes that generate or manipulate the information and (b) the role of the human designer. Because of this, HM is universally applicable to the representation of design-specific information throughout all aspects of the engineering enterprise. The fundamental unit in HM is a design entity, which is defined as a unit of information relevant to a design task. The axioms of HM define the structure of design entities and the explicit means by which they may be rationally organized. HM provides (a) a basis for building taxonomies of design entities, (b) a generalized approach for making statements about design entities independent of how the entities are generated or used, and (c) a formal syntactic notation for the standardization of design entity specification. Furthermore, HM is used as the foundation of DESIGNER, an extension to the Scheme programming language, providing a prototype-based object-oriented system for the static modeling of design information. Objects in the DESIGNER language satisfy the axioms of HM while providing convenient programming mechanisms to increase usability and efficiency. Several design-specific examples demonstrate the applicability of DESIGNER, and thus of HM as well, to the accurate representation of design information. </div>

The Formal Modeling Of Engineering Design Information By Means Of An Axiomatic System

<div>There is mounting evidence in the current literature which suggests that our collective understanding of engineering design is insufficient to support the continued growth of the engineering endeavor. Design theory is the emergent research field that addresses this problem by seeking to improve our understanding of, and thus our ability to, design. The goal of this author's work is to demonstrate that formal techniques of logic can improve our understanding of design. Specifically, a formal system called the Hybrid Model (HM) is presented; this system is a set-theoretic description of engineering design information that is valid independent of (a) the processes that generate or manipulate the information and (b) the role of the human designer. Because of this, HM is universally applicable to the representation of design-specific information throughout all aspects of the engineering enterprise. The fundamental unit in HM is a design entity, which is defined as a unit of information relevant to a design task. The axioms of HM define the structure of design entities and the explicit means by which they may be rationally organized. HM provides (a) a basis for building taxonomies of design entities, (b) a generalized approach for making statements about design entities independent of how the entities are generated or used, and (c) a formal syntactic notation for the standardization of design entity specification. Furthermore, HM is used as the foundation of DESIGNER, an extension to the Scheme programming language, providing a prototype-based object-oriented system for the static modeling of design information. Objects in the DESIGNER language satisfy the axioms of HM while providing convenient programming mechanisms to increase usability and efficiency. Several design-specific examples demonstrate the applicability of DESIGNER, and thus of HM as well, to the accurate representation of design information. </div>

Alternative Semantics for Normative Reasoning with an Application to Regret and Responsibility

We provide a fine-grained analysis of notions of regret and responsibility (such as agent-regret and individual responsibility) in terms of a language of multimodal logic. This language undergoes a detailed semantic analysis via two sorts of models: (i) relating models, which are equipped with a relation of propositional pertinence, and (ii) synonymy models, which are equipped with a relation of propositional synonymy. We specify a class of strictly relating models and show that each synonymy model can be transformed into an equivalent strictly relating model. Moreover, we define an axiomatic system that captures the notion of validity in the class of all strictly relating models.

Issues in clustering algorithm consistency in fixed dimensional spaces. Some solutions for k-means

Kleinberg introduced an axiomatic system for clustering functions. Out of three axioms, he proposed, two (scale invariance and consistency) are concerned with data transformations that should produce the same clustering under the same clustering function. The so-called consistency axiom provides the broadest range of transformations of the data set. Kleinberg claims that one of the most popular clustering algorithms, k-means does not have the property of consistency. We challenge this claim by pointing at invalid assumptions of his proof (infinite dimensionality) and show that in one dimension in Euclidean space the k-means algorithm has the consistency property. We also prove that in higher dimensional space, k-means is, in fact, inconsistent. This result is of practical importance when choosing testbeds for implementation of clustering algorithms while it tells under which circumstances clustering after consistency transformation shall return the same clusters. Two types of remedy are proposed: gravitational consistency property and dataset consistency property which both hold for k-means and hence are suitable when developing the mentioned testbeds.

Finite Arithmetic Axiomatization for the Basis of Hyperrational Non-Standard Analysis

The standard elementary number theory is not a finite axiomatic system due to the presence of the induction axiom scheme. Absence of a finite axiomatic system is not an obstacle for most tasks, but may be considered as imperfect since the induction is strongly associated with the presence of set theory external to the axiomatic system. Also in the case of logic approach to the artificial intelligence problems presence of a finite number of basic axioms and states is important. Axiomatic hyperrational analysis is the axiomatic system of hyperrational number field. The properties of hyperrational numbers and functions allow them to be used to model real numbers and functions of classical elementary mathematical analysis. However hyperrational analysis is based on well-known non-finite hyperarithmetic axiomatics. In the article we present a new finite first-order arithmetic theory designed to be the basis of the axiomatic hyperrational analysis and, as a consequence, mathematical analysis in general as a basis for all mathematical application including AI problems. It is shown that this axiomatics meet the requirements, i.e., it could be used as the basis of an axiomatic hyperrational analysis. The article in effect completes the foundation of axiomatic hyperrational analysis without calling in an arithmetic extension, since in the framework of the presented theory infinite numbers arise without invoking any new constants. The proposed system describes a class of numbers in which infinite numbers exist as natural objects of the theory itself. We also do not appeal to any “enveloping” set theory.

Natural deduction

Natural deduction is a philosophically as well as pedagogically important logical proof system. This chapter introduces Gerhard Gentzen’s original system of natural deduction for minimal, intuitionistic, and classical predicate logic. Natural deduction reflects the ways we reason under assumption in mathematics and ordinary life. Its rules display a pleasing symmetry, in that connectives and quantifiers are each governed by a pair of introduction and elimination rules. After providing several examples of how to find proofs in natural deduction, it is shown how deductions in such systems can be manipulated and measured according to various notions of complexity, such as size and height. The final section shows that the axiomatic system of classical logic presented in Chapter 2 and the system of natural deduction for classical logic introduced in this chapter are equivalent.

A Logically Formalized Axiomatic Epistemology System Σ + C and Philosophical Grounding Mathematics as a Self-Sufficing System

The subject matter of this research is Kant’s apriorism underlying Hilbert’s formalism in the philosophical grounding of mathematics as a self-sufficing system. The research aim is the invention of such a logically formalized axiomatic epistemology system, in which it is possible to construct formal deductive inferences of formulae—modeling the formalism ideal of Hilbert—from the assumption of Kant’s apriorism in relation to mathematical knowledge. The research method is hypothetical–deductive (axiomatic). The research results and their scientific novelty are based on a logically formalized axiomatic system of epistemology called Σ + C, constructed here for the first time. In comparison with the already published formal epistemology systems X and Σ, some of the axiom schemes here are generalized in Σ + C, and a new symbol is included in the object-language alphabet of Σ + C, namely, the symbol representing the perfection modality, C: “it is consistent that…”. The meaning of this modality is defined by the system of axiom schemes of Σ + C. A deductive proof of the consistency of Σ + C is submitted. For the first time, by means of Σ + C, it is deductively demonstrated that, from the conjunction of Σ + C and either the first or second version of Gödel’s theorem of incompleteness of a formal arithmetic system, the formal arithmetic investigated by Gödel is a representation of an empirical knowledge system. Thus, Kant’s view of mathematics as a self-sufficient, pure, a priori knowledge system is falsified.