Extra Concepts and Domains of Truth Values in Interpreting Logical Languages in Artificial Intelligence Systems

Vestnik MEI ◽  
2020 ◽  
Vol 5 (5) ◽  
pp. 148-154
Author(s):  
Vadim N. Falk ◽  
Keyword(s):  
Bounded Lattice ◽  
Truth Values ◽  
Logical Operations ◽  
Countable Power ◽  
Number Representations ◽  
Definition Of ◽  
Symbol Sequences ◽  
Context Free ◽  
Special Case ◽  
Traditional Understanding

So-called extra concepts introduced to represent structurally defined objects and structures with unlimited complexity in their traditional understanding are suggested. The concepts of extra-word, extra-regular expression, context-free extra-grammar, and context-free extra-language are extensions of the well-known concepts used in the theory of formal languages. Extra words are a special case of symbol sequences; however, the set of all extra words in any alphabet is countable, whereas the set of all symbol sequences is not countable. The periodic codes of rational number representations in some positional numeration system are in fact extra words in this terminology. The concept of an extra-tuple is a generalization of the tuple concept, which implies the possibility of interpreting extra-tuples both as finite and as the indicated type of infinite sequences of elements of an arbitrary, not more than countable set, and it should be noted that the set of all possible sequences of such sort remains countable. By using the introduced concepts, a countable family of the domains of truth values has been specified for multivalued and countable-valued logics, each of which is a bounded lattice of finite or countable power with the traditional definition of basic logical operations of negation, conjunction, and disjunction. The hierarchical construction of the proposed truth domains makes it possible to introduce new logical operations in consideration that do not have analogues in the classical logic.

scholarly journals The Structure of n Harmonic Points and Generalization of Desargues’ Theorems

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 1018
Author(s):  
Xhevdet Thaqi ◽  
Ekrem Aljimi
Keyword(s):  
Fourth Harmonic ◽  
New Findings ◽  
Rank 2 ◽  
Definition Of ◽  
Special Case

: In this paper, we consider the relation of more than four harmonic points in a line. For this purpose, starting from the dependence of the harmonic points, Desargues’ theorems, and perspectivity, we note that it is necessary to conduct a generalization of the Desargues’ theorems for projective complete n-points, which are used to implement the definition of the generalization of harmonic points. We present new findings regarding the uniquely determined and constructed sets of H-points and their structure. The well-known fourth harmonic points represent the special case (n = 4) of the sets of H-points of rank 2, which is indicated by P42.

An Overview of the Proof

2019 ◽  
pp. 3-22
Author(s):  
Christian Haesemeyer ◽  
Charles A. Weibel
Keyword(s):  
Motivic Cohomology ◽  
Norm Residue ◽  
Cohomology Operations ◽  
Definition Of ◽  
Special Case

This chapter provides the main steps in the proof of Theorems A and B regarding the norm residue homomorphism. It also proves several equivalent (but more technical) assertions in order to prove the theorems in question. This chapter also supplements its approach by defining the Beilinson–Lichtenbaum condition. It thus begins with the first reductions, the first of which is a special case of the transfer argument. From there, the chapter presents the proof that the norm residue is an isomorphism. The definition of norm varieties and Rost varieties are also given some attention. The chapter also constructs a simplicial scheme and introduces some features of its cohomology. To conclude, the chapter discusses another fundamental tool—motivic cohomology operations—as well as some historical notes.

Computer Simulations

2019 ◽  
pp. 9-20
Author(s):  
Paul Humphreys
Keyword(s):  
Computer Simulation ◽  
Numerical Methods ◽  
Computer Simulations ◽  
Scientific Progress ◽  
Computational Science ◽  
Working Definition ◽  
Mathematical Techniques ◽  
Definition Of ◽  
Special Case ◽  
Numerical Treatments

The need to solve analytically intractable models has led to the rise of a new kind of science, computational science, of which computer simulations are a special case. It is noted that the development of novel mathematical techniques often drives scientific progress and that even relatively simple models require numerical treatments. A working definition of a computer simulation is given and the relation of simulations to numerical methods is explored. Examples where computational methods are unavoidable are provided. Some epistemological consequences for philosophy of science are suggested and the need to take into account what is possible in practice is emphasized.

Introduction

2020 ◽  
pp. 1-6
Author(s):  
Peter Scholze ◽  
Jared Weinstein
Keyword(s):  
Moduli Spaces ◽  
Function Fields ◽  
Number Fields ◽  
Important Special Case ◽  
Langlands Correspondence ◽  
Key Innovation ◽  
Perfectoid Spaces ◽  
Definition Of ◽  
Special Case

This introductory chapter provides an overview of Drinfeld's work on the global Langlands correspondence over function fields. Whereas the global Langlands correspondence is largely open in the case of number fields K, it is a theorem for function fields, due to Drinfeld and L. Lafforgue. The key innovation in this case is Drinfeld's notion of an X-shtuka (or simply shtuka). The Langlands correspondence for X is obtained by studying moduli spaces of shtukas. A large part of this course is about the definition of perfectoid spaces and diamonds. There is an important special case where the moduli spaces of shtukas are classical rigid-analytic spaces. This is the case of local Shimura varieties. Some examples of these are the Rapoport-Zink spaces.

scholarly journals An Investigation on the Logical Structure of Mathematics (VI): Consistent V-System T(V) (With Corrections to Part (XII))

1959 ◽  
Vol 14 ◽  
pp. 95-107
Author(s):  
Sigekatu Kuroda
Keyword(s):  
Logical Structure ◽  
Consistency Proof ◽  
Truth Values ◽  
Definition Of

The V-system T(V) is defined in §2 by using §1, and its consistency is proved in §3. The definition of T(V) is given in such a way that the consistency proof of T(V) in §3 shows a typical way to prove the consistency of some subsystems of UL. Otherwise we could define T(V) more simply by using truth values. After T(V)-sets are treated in §4, it is proved in §5 as a T(V)-theorem that T(V)-sets are all equal to V.

Notes on general topology

1965 ◽  
Vol 61 (4) ◽  
pp. 877-878 ◽  
Author(s):  
A. J. Ward
Keyword(s):  
General Topology ◽  
Standard Definition ◽  
Definition Of ◽  
Image Position ◽  
Special Case ◽  
Close Parallelism

There is a close parallelism between the theories of convergence of directed nets and of filters, in which ‘subnet’ corresponds, in general, to ‘refinement’. With the standard definitions, however (1), pages 65 et seq., this correspondence is not exact, as there is no coarsest net converging to x0 of which all other nets with the same limit are subnets. (Suppose, for example, that a net X = {xj,: j ∈ J} in R1 has both the sequence-net S = {n−1; n = 1, 2,…} and the singleton-net {0} as subnets. Then (with an obvious notation), there existsuch that j0 ≥ jn for all n, while jn ≥ j0 for all n ≥ n0 say. But, given any j ∈ J, there exists n with jn ≥ j: it follows that jn ∈ j for all n ≥ n0 (independent of j); thus X cannot converge to 0. Even if nets with a last member are excluded, a similar result can be obtained by considering the net Y = {yθ; Θ an ordinal less than ω1}, where yθ = 0 for all Θ. If X has both Y and S as subnets we can show that (with a similar notation) there exists Θ0 such that Θ ≥ Θ0 implies jθ ≥ all jn, but also n0 such that n ≥ n0 implies ; the rest is as before.) Moreover, the theory of convergence classes, (l), pages 73 et seq., contains a condition (Kelley's condition (c)) whose analogue need not be separately stated for filters. These differences can be removed by adopting a wider definition of subnet, a course which does not seem unnatural, inasmuch as the standard definition is already wider than the ‘obvious’ one, and our proposed definition is equivalent to the standard one in the special case of sequences.

The cylindric algebras of three-valued logic

1998 ◽  
Vol 63 (4) ◽  
pp. 1201-1217
Author(s):  
Norman Feldman
Keyword(s):  
Effective Procedure ◽  
Cylindric Algebras ◽  
Truth Values ◽  
Recursive Functions ◽  
Partial Recursive Functions ◽  
Truth Value ◽  
The One ◽  
Definition Of

In this paper we consider the three-valued logic used by Kleene [6] in the theory of partial recursive functions. This logic has three truth values: true (T), false (F), and undefined (U). One interpretation of U is as follows: Suppose we have two partially recursive predicates P(x) and Q(x) and we want to know the truth value of P(x) ∧ Q(x) for a particular x0. If x0 is in the domain of definition of both P and Q, then P(x0) ∧ Q(x0) is true if both P(x0) and Q(x0) are true, and false otherwise. But what if x0 is not in the domain of definition of P, but is in the domain of definition of Q? There are several choices, but the one chosen by Kleene is that if Q(X0) is false, then P(x0) ∧ Q(x0) is also false and if Q(X0) is true, then P(x0) ∧ Q(X0) is undefined.What arises is the question about knowledge of whether or not x0 is in the domain of definition of P. Is there an effective procedure to determine this? If not, then we can interpret U as being unknown. If there is an effective procedure, then our decision for the truth value for P(x) ∧ Q(x) is based on the knowledge that is not in the domain of definition of P. In this case, U can be interpreted as undefined. In either case, we base our truth value of P(x) ∧ Q(x) on the truth value of Q(X0).

RECONCEPTUALISING THE EXPRESS TRUST

2002 ◽  
Vol 61 (3) ◽  
pp. 657-683 ◽  
Author(s):  
Patrick Parkinson
Keyword(s):  
Common Law ◽  
Civil Law ◽  
Specific Subject ◽  
Core Content ◽  
Beneficial Ownership ◽  
The Common ◽  
Trust Property ◽  
Definition Of ◽  
Irreducible Core ◽  
Traditional Understanding

This article argues that the express trust should be understood as a species of obligation rather than as a means of organising the ownership of property. Two propositions seem fundamental to the traditional understanding of the trust as an aspect of property law. Firstly, in the nature of the trust, there must be a separation of legal and beneficial ownership. Secondly, there must be trust property. Neither is necessarily true. With many discretionary trusts and other recognised types of express trust it is impossible to locate the beneficial estate. Furthermore, the requirement for there to be trust property is, on closer analysis, a requirement of certainty of obligation in relation to specific subject-matter within which the trust property can be located.The article explores the implications of understanding the trust as a species of obligation. It allows all express trusts, including charitable trusts, to be explained as resting on the same fundamental concepts. The trust in the common law world may still be distinguished from contract and from the civil law forms of the trust. This new conceptualisation also illuminates what is the irreducible core content of the trust. The article concludes with a new definition of the express trust.

Game genesis of justice in the teachings of Huizinga

2020 ◽  
Vol 7 (1) ◽  
pp. 59-63
Author(s):  
Yury A. Tsvetkov
Keyword(s):  
Human Culture ◽  
Religious Cult ◽  
Points Of View ◽  
The Law ◽  
Johan Huizinga ◽  
Definition Of ◽  
Special Case

The article presents the concept of the game origin of justice, developed by the Dutch historian and philosopher Johan Huizinga, in the context of the general teaching about human culture as a game. From the work of the historian, the game signs are distinguished, and the definition of its concept is formulated. The highlighted game signs correlate with the justice signs. The interpretation of some proto-legal phenomena and statements about their gaming origin are compared with the points of view of other legal historians, namely, J. Davi and V. Ehrenberg. This paper presents the author's interpretation in relation to contemporary developments in the law. An explanation is given for why the theory about the game origin of justice has not received support and development in the lawyers work. The identification of justice with a religious cult is carried out through similar gaming practices. The paper concludes by stating that there are direct genetic links among the game, justice, and religious worship. It is hypothesized that the theory about the game origin of justice can be considered a special case of a higher-level theory about the origin of state and law from the game.

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