Consistency notions in illative combinatory logic

1977 ◽  
Vol 42 (4) ◽  
pp. 527-529 ◽  
Author(s):  
M. W. Bunder
Keyword(s):  
Propositional Calculus ◽  
General System ◽  
Combinatory Logic ◽  
Definition Of ◽  
Image Position

In this paper we show that certain notions of consistency currently in use in illative combinatory logic are not always necessary nor always sufficient for the acceptability of the system.The three notions of consistency we will consider are:(A) Not all terms of the system are assertible.(B) Not all propositions of the system are assertible.(C) The system is Q-consistent, i.e. if Q (for equality) is in or is added to the system as a primitive with axioms such asPost defined a system of propositional calculus to be inconsistent if all propositions were assertible. In a more general system where there may be terms which are not propositions (although all propositions are terms), we need to distinguish notions (A) and (B) as two possible consistency notions “in the sense of Post”.In [3] Curry and Feys assume that Q is in the system defined as λxλy.Ξ(C*x)(C*y) or as a primitive, but Seldin in [5] also talks about Q-consistency of a system (F22) to which Q has to be added. We therefore consider the definition of Q-consistency that includes all three possibilities.In [3] Curry and Feys state that Q-consistency is essential for acceptability. This is because there the purpose was to have an assertional system of illative combinatory logic in which combinatory equality was represented by the assertion of QXY. However if Q is to represent some other form of equality, Q-consistency may become inessential for acceptability or even incompatible with it.

On the independence of Henkin's axioms for fragments of the propositional calculus

1951 ◽  
Vol 16 (1) ◽  
pp. 43-45
Author(s):  
Maurice L'abbé
Keyword(s):  
Propositional Calculus ◽  
General System ◽  
Function Symbol ◽  
The Other ◽  
Axiom Schema ◽  
Other Hand ◽  
Truth Function ◽  
The One ◽  
Image Position ◽  
Made In

A general system of axioms has been given by Henkin for a fragment of the propositional calculus having as primitive symbols, in addition to the usual parentheses, variables, and implication sign ⊃, an arbitrarily given truth function symbol ϕ. This system of axioms, which we shall denote by S(⊃, ϕ), contains the following three axiom schemataplus the 2m further axiom schemata involving the symbol ϕwhere ϕ is an m-placed function symbol. We refer to Henkin's paper, p. 43, for the detailed description of the axiom schemata (4).The remark was made in the above mentioned paper that each of the 2m axiom schemata of (4) is trivially independent of the rest of the axioms of S(⊃, ϕ), and it was conjectured that the axiom schemata (1), (2) and (3) are also independent. In this note, we prove the general independence of the axiom schemata (1) and (2). As for (3), we show on the one hand its independence in the systems S(⊃) and S(⊃, f), and, on the other hand, its dependence in the system S(⊃, ∼). The net result is, therefore, that in any of these systems of axioms S(⊃, ϕ) all the axiom schemata are independent, except possibly the axiom schema (3).

On the inconsistency of systems similar to

1978 ◽  
Vol 43 (1) ◽  
pp. 1-2 ◽  
Author(s):  
M. W. Bunder ◽  
R. K. Meyer
Keyword(s):  
General System ◽  
Modus Ponens ◽  
Combinatory Logic ◽  
Deduction Theorem ◽  
The Absolute ◽  
Image Position ◽  
Weakened Form

This note shows that the inconsistency proof of the system of illative combinatory logic given in [1] can be simplified as well as extended to the absolute inconsistency of a more general system.One extension of the result in [1] lies in the fact that the following weakened form of the deduction theorem for implication will lead to the inconsistency:Also the inconsistency follows almost as easily foras it does for ⊢ H2X for arbitrary X, so we will consider the more general case.The only properties we require other than (DT), (1) and Rule Eq for equality are modus ponens,andLet G = [x] Hn−1x⊃: . … H2x⊃ :Hx⊃ . x ⊃ A, where A is arbitrary. Then if Y is the paradoxical combinator and X = YG, X = GX.Now X ⊂ X, i.e.,

On adding (ξ) to weak equality in combinatory logic

1989 ◽  
Vol 54 (2) ◽  
pp. 590-607
Author(s):  
Martin W. Bunder ◽  
J. Roger Hindley ◽  
Jonathan P. Seldin
Keyword(s):  
Combinatory Logic ◽  
Definition Of ◽  
Image Position

AbstractBecause the main difference between combinatory weak equality and λβ-equality is that the ruleis valid for the latter but not the former, it is easy to assume that another way of defining combinatory β-equality is to add rule (ξ) to the postulates for weak equality. However, to make this true, one must choose the definition of combinatory abstraction in (ξ) very carefully. If one tries to use one of the more common abstraction algorithms, the result will be an equality, =ξ, that is either equivalent to βη-equality (and so strictly stronger than β-equality) or else strictly weaker than β-equality. This paper will study the relations =ξ for several commonly used abstraction-algorithms, distinguish between them, and axiomatize them.

Pascal's Prism

2012 ◽  
Vol 96 (536) ◽  
pp. 213-220
Author(s):  
Harlan J. Brothers
Keyword(s):  
Probability Theory ◽  
Fractal Geometry ◽  
Euclidean Geometry ◽  
Fibonacci Series ◽  
Pascal’S Triangle ◽  
Number Sequences ◽  
Definition Of ◽  
Image Position ◽  
Pascal's Triangle

Pascal's triangle is well known for its numerous connections to probability theory [1], combinatorics, Euclidean geometry, fractal geometry, and many number sequences including the Fibonacci series [2,3,4]. It also has a deep connection to the base of natural logarithms, e [5]. This link to e can be used as a springboard for generating a family of related triangles that together create a rich combinatoric object.2. From Pascal to LeibnizIn Brothers [5], the author shows that the growth of Pascal's triangle is related to the limit definition of e.Specifically, we define the sequence sn; as follows [6]:

scholarly journals CONTENTS AND SYSTEM OF EXPERT ACTIVITY IN THE FIELD OF COMPUTER TECHNOLOGIES: DEFINITION PROBLEMS

2019 ◽  
pp. 158-162
Author(s):  
S. M. Korniiko
Keyword(s):  
Information Technology ◽  
Information And Communication Technologies ◽  
Computer Technology ◽  
Information Search ◽  
High Speed ◽  
Human Life ◽  
General System ◽  
Data Access ◽  
Computer Technologies ◽  
Definition Of

The article is devoted to the definition of the content and the system of expert activity in the field of computer technologies, which is based on the results of determining the general system of expert activity. Expert activity should be understood as the implementation by authorized agents on the basis of special knowledge in the field of science, technology, art, crafts, etc. Studies of objects, phenomena and processes in order to provide scientifically substantiated conclusions on the diverse issues that arise in the process of life of society. Such a definition of expert activity includes both judicial and non-judicial expert examination. At present, more than 500 laws are adopted in Ukraine, which in one way or another concern the conduct of expert assessments (most of them are valid at 2019). But no any among that laws directly devoted to the expert work in the field of computer technology. So the system and content of the expert work in the field of computer technology should be established, based on knowledge of the object of expertise – computer technology. It is considered as synonymous with the concept of “information technology” or “information and communication technologies”. Information technology – it is a purposeful organized set of information processes using computer facilities, which provide high speed data processing, rapid information search, dispersal of data, access to information sources regardless of places of their location. The system of expert activity in the field of computer technologies includes examinations belonging to a group of judicial (engineering, commodity, forensic, etc.) and non-judicial (scientific and scientific and technical expertise; examination of issues of quality and conformity of goods (products) to certain requirements; examination of issues of information security; examination of issues of environmental impact and the environment of human life, etc.), as well as presented by different kindsand species examinations that have different goals focused on the study of computer technology in their various aspects and provides solutions to diverse issues.

The Static Aeroelasticity of Slender Configurations

1963 ◽  
Vol 14 (1) ◽  
pp. 75-104 ◽  
Author(s):  
G. J. Hancock
Keyword(s):  
Static Stability ◽  
Aerodynamic Forces ◽  
Flexible Aircraft ◽  
The Past ◽  
Plate Models ◽  
Non Linear ◽  
The Future ◽  
Static Aeroelasticity ◽  
Definition Of ◽  
Image Position

SummaryThe validity and applicability of the static margin (stick fixed) Kn,where as defined by Gates and Lyon is shown to be restricted to the conventional flexible aircraft. Alternative suggestions for the definition of static margin are put forward which can be equally applied to the conventional flexible aircraft of the past and the integrated flexible aircraft of the future. Calculations have been carried out on simple slender plate models with both linear and non-linear aerodynamic forces to assess their static stability characteristics.

The Multiplicative Groups of Quasifields

1987 ◽  
Vol 39 (4) ◽  
pp. 784-793 ◽  
Author(s):  
Michael J. Kallaher
Keyword(s):  
Zero Element ◽  
Binary Operations ◽  
Definition Of ◽  
Image Position ◽  
Multiplicative Groups ◽  
Multiplicative Mapping

Let (Q, +, ·) be a finite quasifield of dimension d over its kernel K = GF(q), where q = pk with p a prime and k ≧ 1. (See p. 18-22 and p. 74 of [7] or Section 5 of [9] for the definition of quasifield.) For the remainder of this article we will follow standard conventions and omit, whenever possible, the binary operations + and · in discussing a quasifield. For example, the notation Q will be used in place of the triple (Q, +, ·) and Q* will be used to represent the multiplicative loop (Q − {0}, ·).Let m be a non-zero element of the quasifield Q; the right multiplicative mapping ρm:Q → Q is defined by1

scholarly journals On the Inversion of the Gauss Transformation

1957 ◽  
Vol 9 ◽  
pp. 459-464 ◽  
Author(s):  
P. G. Rooney
Keyword(s):  
Differential Operator ◽  
Recent Work ◽  
Suitable Definition ◽  
Inversion Theory ◽  
The Subject ◽  
Definition Of ◽  
Image Position ◽  
Gauss Transformation ◽  
Operational Methods

The inversion theory of the Gauss transformation has been the subject of recent work by several authors. If the transformation is defined by1.1,then operational methods indicate that,under a suitable definition of the differential operator.

On the Definition of C*-Algebras II

1985 ◽  
Vol 37 (4) ◽  
pp. 664-681 ◽  
Author(s):  
Zoltán Magyar ◽  
Zoltán Sebestyén
Keyword(s):  
Hilbert Space ◽  
Banach Algebra ◽  
Representation Theorem ◽  
Banach Algebras ◽  
Bounded Operators ◽  
Fundamental Representation ◽  
C Algebras ◽  
Definition Of ◽  
Image Position

The theory of noncommutative involutive Banach algebras (briefly Banach *-algebras) owes its origin to Gelfand and Naimark, who proved in 1943 the fundamental representation theorem that a Banach *-algebra with C*-condition(C*)is *-isomorphic and isometric to a norm-closed self-adjoint subalgebra of all bounded operators on a suitable Hilbert space.At the same time they conjectured that the C*-condition can be replaced by the B*-condition.(B*)In other words any B*-algebra is actually a C*-algebra. This was shown by Glimm and Kadison [5] in 1960.

On the “Third Definition” of the Topology on the Spectrum of a C*-Algebra

1971 ◽  
Vol 23 (3) ◽  
pp. 445-450 ◽  
Author(s):  
L. Terrell Gardner
Keyword(s):  
Hilbert Space ◽  
Quotient Space ◽  
Principal Result ◽  
Irreducible Representations ◽  
C Algebra ◽  
Fell Topology ◽  
The Third ◽  
Definition Of ◽  
Image Position

0. In [3], Fell introduced a topology on Rep (A,H), the collection of all non-null but possibly degenerate *-representations of the C*-algebra A on the Hilbert space H. This topology, which we will call the Fell topology, can be described by giving, as basic open neighbourhoods of π0 ∈ Rep(A, H), sets of the formwhere the ai ∈ A, and the ξj ∈ H(π0), the essential space of π0 [4].A principal result of [3, Theorem 3.1] is that if the Hilbert dimension of H is large enough to admit all irreducible representations of A, then the quotient space Irr(A, H)/∼ can be identified with the spectrum (or “dual“) Â of A, in its hull-kernel topology.

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