Number Of Axioms Research Articles

A Formal Treatment of Contract Signature

The paper develops a logical understanding of processes for signature of legal contracts, motivated by applications to legal recognition of smart contracts on blockchain platforms. A number of axioms and rules of inference are developed that can be used to justify a ``meeting of the minds'' precondition for contract formation from the fact that certain content has been signed. In addition to an ``offer and acceptance'' process, the paper considers ``signature in counterparts'', a legal process that permits a contract between two or more parties to be brought into force by having the parties independently (possibly, remotely) sign different copies of the contract, rather than placing their signatures on a common copy at a physical meeting. It is argued that a satisfactory account of signature in counterparts benefits from a logic with syntactic self-reference. The axioms used are supported by a formal semantics, and a number of further properties of the logic are investigated. In particular, it is shown that the logic implies that when a contract has been signed, the parties do not just agree, but are in mutual agreement (a common-knowledge-like notion) about the terms of the contract.

Monotone Subintuitionistic Logic: Duality and Transfer Results

We consider subintuitionistic logics as an extension of positive propositional logic with a binary modality, interpreted over ordered and unordered monotone neighborhood frames, with a range of frame conditions. This change in perspective allows us to apply tools and techniques from the modal setting to subintuitionistic logics. We provide a Priestley-style duality, and transfer results from the (classical) logic of monotone neighborhood frames to obtain completeness, conservativity, and a finite model property for the basic logic, extended with a number of axioms.

The number of axioms

We derive results concerning the least possible number of distinct axioms in proofs in classical and intuitionistic sequent calculi and other related systems. In particular, we show that there is no recursive bound on the least possible number of distinct axioms in a proof of a provable sequent in terms of the sequent's length, and that there is no elementary bound on the least possible number of axioms in a cut-free proof of a provable sequent in terms of the least possible length of an arbitrary proof thereof, strengthening a classical result due to Orevkov and Statman.

“Армія метафор” у науковому дискурсі

The article outlines the problem of the terminological potential of metaphor as the main figure of poetics, rhetoric, artistic and poetic thinking in general. The authors analyze numerous works about the nature and value of the metaphor; comprehend peculiarities of its use, forms and discourses of use, figurative potential of metaphorical thinking; at the same time, they emphasize on the mythological basis of such thinking (in particular, based on the works of O. Freudenberg, N. Frye, F. Nietzsche). The main focus is on the metaphorical (figurative) nature of terminology in both the humanities and the exact sciences, as well as the problem of the interpenetration of terms, in particular, scientific terms, metaphors borrowed from human sciences. The research confirms that at the present stage of development of all spheres of knowledge and science in general there is a cross-cutting penetration of metaphor at different levels: at the level of the names of theories; at the level of titles of scientific works – from ancient times to modern times; at the level of naming / defining phenomena and concepts, etc. Methodological foundation is based at the works of H. Ortega y Gasset, H. G. Gadamer and other philosophers of the twentieth century, who acknowledge the metaphor as a form of scientific thinking and cognition, at the same time see it as a marker of the spiritual space of science. The article outlines the Ukrainian experience of term-building in the context of a long-standing European tradition in combining elements of Western and Eastern approaches to the realization of reality and the transfer of knowledge. Numerous examples confirm the idea that metaphorology as a separate brunch in the 21st century is gaining momentum. Paying attention to metaphorical outview as an integral feature of human thinking (including scientific) offers new aspects and perspectives on the history of the sciences, opens the way to the convergence and interpenetration of scientific spheres, contains a number of axioms for human understanding of nature, history, world and mysteries of being.

Polynomial semantics for modal logics

ABSTRACTIt is shown here that the modal logic K and any extension of it with a finite number of axioms can be characterised by a polynomial semantics. Moreover, some comments are made about the possibility of using algebraic computation to determine deducibility on these logics.

FRACTIONAL SEMANTICS FOR CLASSICAL LOGIC

AbstractThis article presents a new (multivalued) semantics for classical propositional logic. We begin by maximally extending the space of sequent proofs so as to admit proofs for any logical formula; then, we extract the new semantics by focusing on the axiomatic structure of proofs. In particular, the interpretation of a formula is given by the ratio between the number of identity axioms out of the total number of axioms occurring in any of its proofs. The outcome is an informational refinement of traditional Boolean semantics, obtained by breaking the symmetry between tautologies and contradictions.

Conditionals and Conditional Probabilities without Triviality

The Adams Thesis holds for a conditional → and a probability assignment P if and only if P(A→B)=P(B∣A) whenever P(A)>0. The restriction ensures that P(B∣A) is well defined by the classical formula P(B∣A)=P(B∩A)/P(A). Drawing on deep results of Maharam on measure algebras, it is shown that, notwithstanding well-known triviality results (Lewis, etc.), any probability space can be extended to a probability space with a new conditional satisfying the Adams Thesis and satisfying a number of axioms for conditionals. This puts significant limits on how far triviality results can go.

Approaches to Modeling the Properties of Complex Structure Materials

Some approaches to the axiomatic formulation of theoretical fundamentals in continuum mechanics are considered. The basic concepts, laws, and hypotheses in the classical theory of continuum mechanics and their modifications for its nonclassical versions are discussed. In the framework of the classical version of the rational theory, a number of axioms are proposed for the general theory of constitutive relations. For the media of nonclassical type, the approaches to axiomatic formulation are studied by the example of the rational mechanics of moment media (Cosserat continuum): some specific notions of bodies with their attributes and the forms of their interactions and motions are introduced, the appropriate generalizations of the main laws and hypotheses are given, and the general forms of constitutive relations are analyzed for arbitrary and small strains. The approaches to the construction of medium models are discussed in accordance with the method of mechanical modeling proposed by A. A. Il’yushin.

General Reduced Forms of Constitutive Relations in the Classical Continuum Mechanics

A theory of constitutive relations describing the resistance of bodies to deformation is developed. This theory takes into account the presence of internal body forces and internal kinematic constraints. A number of axioms and a general reduced form of constitutive relations for classical media are proposed. For simple bodies it is proved that the Il'yushin and Noll constitutive relations are equivalent.

Zones of non-communication: Music and the relative immanence of capitalism

The goal of this paper is to show not what music is, but what music as a sort of political practice can do in the age of contemporaneity. The age of contemporaneity is defined by the relative immanence of capitalism, which is structured as a society of control on the level of the socius. On the one hand, capitalism consists of a certain number of axioms that enable the denumeration of flows, while, on the other, the society of control forges a connection between the capitalist axiomatic and the infra-individual levels of every (in)dividual. In an ontological and epistemological framework structured like that, art in general and music in particular create zones of non-communication that break affective connections between the infra-individual and the trans-individual in the form of the axiomatic and society of control. With affective non-directionality, music produces a space-time wherein different and new connections between affects may occur. Those affective connections are not exclusively tied to axiomatic denumeration, that is, the production of surplus value, but potentially to the production of surplus life.

How Can We Enter in Dialogue? Transdisciplinary Methodology of the Dialogue between People, Cultures, and Spiritualities

How Can We Enter in Dialogue? Transdisciplinary Methodology of the Dialogue between People, Cultures, and Spiritualities

On the Measurement Problem

In this paper, we will see that we can reformulate the purely classical probability theory, using similar language to the one used in quantum mechanics. This leads us to reformulate quantum mechanics itself using this different way of understanding probability theory, which in turn will yield a new interpretation of quantum mechanics. In this reformulation, we still prove the existence of none classical phenomena of quantum mechanics, such as quantum superposition, quantum entanglement, the uncertainty principle and the collapse of the wave packet. But, here, we provide a different interpretation of these phenomena of how it is used to be understood in quantum physics. The advantages of this formulation and interpretation are that it induces the same experimental results, solves the measurement problem, and reduces the number of axioms in quantum mechanics. Besides, it suggests that we can use new types of Q-bits which are more easily to manipulate.

Modeling Memorization and Forgetfulness Using Dierential Equations

Research Context: The aim of the study was to use dier- ential equations to model memorization of students based on a given data taking into account forgetfulness. Research Methods: The purpose of this paper was to decipher the rate at which students memorized the stu that required memorization in the area of axioms and proofs of theorems as well as considering the fact that they will forget some of them along the way. The usage of dierential equation was employed to model the trend. The paper contributes to the literature by documenting that students can memorize large number of stu even beyond their perceived imaginations. Conclusion: This study employed the usage of dierential equations to mod- el the rate at which students could memorize a given number of axioms and proofs, considering the fact that they will forget some of them along the way. Persons who are able to absorb and retain more are able to recollect better than those who can absorb more and retain less. On the other hand, those who can absorb less and retain more have an upper hand in recollection over those who can absorb more and retain less. Consequently it is better to have a higher retention constant than a higher absorption rate. Factors like

A framework for testing first-order logic axioms in program verification

Program verification systems based on automated theorem provers rely on user-provided axioms in order to verify domain-specific properties of code. However, formulating axioms correctly (that is, formalizing properties of an intended mathematical interpretation) is non-trivial in practice, and avoiding or even detecting unsoundness can sometimes be difficult to achieve. Moreover, speculating soundness of axioms based on the output of the provers themselves is not easy since they do not typically give counterexamples. We adopt the idea of model-based testing to aid axiom authors in discovering errors in axiomatizations. To test the validity of axioms, users define a computational model of the axiomatized logic by giving interpretations to the function symbols and constants in a simple declarative programming language. We have developed an axiom testing framework that helps automate model definition and test generation using off-the-shelf tools for meta-programming, property-based random testing, and constraint solving. We have experimented with our tool to test the axioms used in Auto-Cert, a program verification system that has been applied to verify aerospace flight code using a first-order axiomatization of navigational concepts, and were able to find counterexamples for a number of axioms.

Equivalent supermartingale densities and measures in discrete time infinite horizon market models

We consider a general discrete time infinite horizon securities market model in which the set $\ccW$ of stochastic wealth processes, corresponding to investment strategies, is subject to a number of axioms with financial interpretation. We obtain criteria for the existence of equivalent supermartingale densities and measures for the set $ \ccW _ + $ of nonnegative elements of $ \ccW $. These criteria are expressed in terms of various no-arbitrage conditions. The most complete results are formulated for Fatou closed sets $ \ccW $. This closedness condition is satisfied by the traditional market model with a finite number of basic assets. A feature of the paper consists of applying the Kreps–Yan theorem for the space $L ^\infty $ with the norm topology. With the use of this theorem we establish the existence of an equivalent supermartingale density under the absence of free lunch with vanishing risk condition for strategies with finite horizons.

An axiomatic comparison of learned term-weighting schemes in information retrieval: clarifications and extensions

Machine learning approaches to information retrieval are becoming increasingly widespread. In this paper, we present term-weighting functions reported in the literature that were developed by four separate approaches using genetic programming. Recently, a number of axioms (constraints), from which all good term-weighting schemes should be deduced, have been developed and shown to be theoretically and empirically sound. We introduce a new axiom and empirically validate it by modifying the standard BM25 scheme. Furthermore, we analyse the BM25 scheme and the four learned schemes presented to determine if the schemes are consistent with the axioms. We find that one learned term-weighting approach is consistent with more axioms than any of the other schemes. An empirical evaluation of the schemes on various test collections and query lengths shows that the scheme that is consistent with more of the axioms outperforms the other schemes.

A fuzzy aggregation approach to group decision-making based on centroid measurement

Abstract: In group decision-making situations, the decision-makers’ opinions are generally characterized bysubjectivity, imprecision and vagueness. Such opinions can be represented by fuzzy numbers such as positivetrapezoidalfuzzynumbers.Severalaggregationmethodshavebeenproposedpreviously;however,therearestilldeficits which limit their applications. This paper proposes an aggregation method based on centroidmeasurement which aims to overcome some of the shortcomings. In the paper, a number of axioms areproposed. A numerical example is given at the end to illustrate the application of the proposed aggregationmethod.Keywords: fuzzy opinions, fuzzy numbers, group decision-making, aggregation 1. IntroductionIn multiple-criteria group decision-making sit-uations, group members’ opinions or evalua-tions are based on a certain set of criteria. Theopinions may vary due to different perspectiveson both criteria and alternatives (Xu, 1988,1990; Feng & Xu, 1999a, 1999b; Qiu et al.,2003; Zhang & Lu, 2003; Zhang et al., 2003;Zhang & Jiang, 2006). Aggregating individualopinions to form an opinion with group con-sensus is an important consideration in groupdecision-making.The consensus opinion can be obtained bycalculating the weighted sum ofindividual opin-ions. One way to assign the weights is to esti-mate the relative importance of an individualbased on the criterion of authority. In theprocess of group decision-making, individualmembers’ opinions usually involve subjectivity,imprecision and vagueness (Hsu & Chen, 1996;Lee, 2002). Bardossy et al. (1993) suggest that asubjectiveestimatecanberepresentedbyafuzzynumber. The concept of fuzzy numbers was firstproposed by Dubois and Prade (1978). In theexisting literature, positive trapezoidal or trian-gular fuzzy numbers are employed to representfuzzy opinions (Olcer & Odabasi, 2005). Anumerical approximation technique has beenproposed to convert linguistic terms to theircorresponding fuzzy numbers (Chen et al.,1992). Kacprzyk et al. (1992) showed how lin-guisticquantifierscanbeusedingroupdecision-making. Some aggregation approaches dealingwith fuzzy numbers have been proposed forforming opinions with group consensus.Hsu and Chen (1996) proposed a similarityaggregation method in which the weight indexesconsist of two facets: the degree of relativeimportance of the individual and the degree ofsimilarity of opinion among group members.

Axiomatization of the index of pointedness for closed convex cones

Let C(H) denote the class of closed convex cones in a Hilbert space H. One possible way of measuring the degree of pointedness of a cone K is by evaluating the distance from K to the set of all nonpointed cones. This approach has been explored in detail in a previous work of ours. We now go beyond this particular choice and set up an axiomatic background for addressing this issue. We define an index of pointedness over H as being a function f: C(H) ® R satisfying a certain number of axioms. The number f(K) is intended, of course, to measure the degree of pointedness of the cone K. Although several important examples are discussed to illustrate the theory in action, the emphasis of this work lies in the general properties that can be derived directly from the axiomatic model.

Fiscal equalisation: Principles and an application to the European Union

The paper derives a normative model for partial fiscal equalisation based on a number of axioms and makes special allowance for the existence of a specific fiscal need in the jurisdictions. A simple version of this idealised equalisation scheme relates net contributions to the equalisation funds to deviations of a jurisdiction’s gross income from average gross income and a jurisdiction’s specific needs from average specific needs. The theoretical model is then empirically tested for the case of the European Union using data from 1986–97. It is found that most restrictions of the model appear to hold, in particular, relatively richer countries contribute more and those with greater fiscal needs, approximated by the importance of the agricultural sector, pay less. However, in the EU, an adjustment of net payments to changes in the actual importance of the specific fiscal need for a country is lacking.

A Critique of Theories and Theorizing in Social Sciences

This paper seeks to present a critique of and an alternative to theoriesand theorizing employed by social scientists to explain the relationshipbetween religion and politics in general and Islamic PoliticalExperience in particular. Within the context of the paper we argued thatpolitcal theory can be conveniently understood in terms of theco-existence of two distinct and rival styles of though Positivism andhistoricism. For the lack of better terms we take positivism andhistoricism to be conventional and radical paradigms respectively. Theparadigms are found wanting in that they do not have the capacity toprovide a satisfactory framework of ideas and common vocabularlywith which to conduct discourse on Islamic Political Experience. In anycase, for a paradigm to do that, it must become fully subsumed in anIslamic worldview. A paradigm presented as a critique and alternativeto these paradigms is based on conceptual analysis, with pure Qur’anicand Shariah concepts providing both the framework and methdologicaltools of analysis. It is an axiomatic approach as it involves systematicanalysis of a number of axioms, the starting point of which is the ideaof the totality of Islam as an ideal which Muslims endeaovur toconcretize.