Simplified foundations for mathematical logic

1955 ◽  
Vol 20 (2) ◽  
pp. 123-139 ◽  
Author(s):  
Robert L. Stanley
Keyword(s):  
Mathematical Logic ◽  
Set Theory ◽  
Natural Deduction ◽  
Small Body ◽  
The Other ◽  
Consistency Proof ◽  
Basic Notion ◽  
Relative Consistency ◽  
And Mathematics

A system SF, closely related to NF, is outlined here. SF has several novel points of simplicity and interest, (a) It uses only one basic notion, from which all the other concepts of logic and mathematics may be built definitionally. Three-notion systems are common, but Quine's two-notion IA has for some time represented the extreme in conceptual economy, (b) The theorems of SF are generated under just three rules of analysis, which unify into a single postulational principle, (c) SF is built solely in terms of what is commonly, known as the “natural deduction” method, under which each theorem is attacked primarily as it stands, by means of a very small body of rules, rather than less directly, through a very large, potentially infinite backlog of theorems. Although natural deduction is by no means new as a method, its exclusive applications have previously been relatively limited, not even reaching principles of identity, much less set theory, relations, or mathematics proper, (d) SF is at least as strong as NF, yielding all of its theorems, which are expressed here in forms analogous to those of the metatheorems in ML. If NF is consistent, so is SF. The main points in the relative consistency proof are set forth below in section seven.

Formenanalyse bei Marx und ihr Verhältnis zur Systemwissenschaft / Analysis of forms by Marx and its relation to systems research

1975 ◽  
Vol 4 (3) ◽  
Author(s):  
Werner Loh
Keyword(s):  
Mathematical Logic ◽  
Empirical Analysis ◽  
The Other ◽  
Systems Research ◽  
Other Hand ◽  
Empirical Logic ◽  
Formal Concepts ◽  
And Mathematics ◽  
Different Levels

AbstractMARX’s analysis of forms and modern systems research have in common the problem of form. MARX analyzed forms by functionally relating elements to each other on different levels. Contrary to modern systems theories and Marxism-Leninism elements are for MARX forms themselves and not non-formal elementary qualities. The analysis of forms, therefore, is able to characterize its objects only relationally-functionally. On the other hand modern systems theories integrate concepts like ‚action‘ or ‚goal‘ in an elementaristic manner. The analysis of forms must be controlled by systematic concretization and totalization adequate to the problem. The formal concepts of systems research are often interpreted as logical-mathematical. Logic and mathematics are usually understood as non-empirical. Empirical analysis of forms is in need of an empirical logic and mathematics.

Set theory, different systems of

2018 ◽  
Author(s):  
Michael Potter
Keyword(s):  
Twentieth Century ◽  
Mathematical Logic ◽  
Set Theory ◽  
The Other ◽  
Von Neumann ◽  
Class Theory ◽  
Iterative Conception ◽  
The One ◽  
Limitation Of Size ◽  
As If

To begin with we shall use the word ‘collection’ quite broadly to mean anything the identity of which is solely a matter of what its members are (including ‘sets’ and ‘classes’). Which collections exist? Two extreme positions are initially appealing. The first is to say that all do. Unfortunately this is inconsistent because of Russell’s paradox: the collection of all collections which are not members of themselves does not exist. The second is to say that none do, but to talk as if they did whenever such talk can be shown to be eliminable and therefore harmless. This is consistent but far too weak to be of much use. We need an intermediate theory. Various theories of collections have been proposed since the start of the twentieth century. What they share is the axiom of ‘extensionality’, which asserts that any two sets which have exactly the same elements must be identical. This is just a matter of definition: objects which do not satisfy extensionality are not collections. Beyond extensionality, theories differ. The most popular among mathematicians is Zermelo–Fraenkel set theory (ZF). A common alternative is von Neumann–Bernays–Gödel class theory (NBG), which allows for the same sets but also has proper classes, that is, collections whose members are sets but which are not themselves sets (such as the class of all sets or the class of all ordinals). Two general principles have been used to motivate the axioms of ZF and its relatives. The first is the iterative conception, according to which sets occur cumulatively in layers, each containing all the members and subsets of all previous layers. The second is the doctrine of limitation of size, according to which the ‘paradoxical sets’ (that is, the proper classes of NBG) fail to be sets because they are in some sense too big. Neither principle is altogether satisfactory as a justification for the whole of ZF: for example, the replacement schema is motivated only by limitation of size; and ‘foundation’ is motivated only by the iterative conception. Among the other systems of set theory to have been proposed, the one that has received widespread attention is Quine’s NF (from the title of an article, ‘New Foundations for Mathematical Logic’), which seeks to avoid paradox by means of a syntactic restriction but which has not been provided with an intuitive justification on the basis of any conception of set. It is known that if NF is consistent then ZF is consistent, but the converse result has still not been proved.

The equivalence of NF-style set theories with “tangled” type theories; the construction of ω-models of predicative NF (and more)

1995 ◽  
Vol 60 (1) ◽  
pp. 178-190 ◽  
Author(s):  
M. Randall Holmes
Keyword(s):  
Set Theory ◽  
Type Theory ◽  
The Other ◽  
Natural Numbers ◽  
Consistency Proof ◽  
Natural Behaviour ◽  
New Foundations ◽  
Consistency Problem

AbstractAn ω-model (a model in which all natural numbers are standard) of the predicative fragment of Quine's set theory “New Foundations” (NF) is constructed. Marcel Crabbé has shown that a theory NFI extending predicative NF is consistent, and the model constructed is actually a model of NFI as well. The construction follows the construction of ω-models of NFU (NF with urelements) by R. B. Jensen, and, like the construction of Jensen for NFU, it can be used to construct α-models for any ordinal α. The construction proceeds via a model of a type theory of a peculiar kind; we first discuss such “tangled type theories” in general, exhibiting a “tangled type theory” (and also an extension of Zermelo set theory with Δ0 comprehension) which is equiconsistent with NF (for which the consistency problem seems no easier than the corresponding problem for NF (still open)), and pointing out that “tangled type theory with urelements” has a quite natural interpretation, which seems to provide an explanation for the more natural behaviour of NFU relative to the other set theories of this kind, and can be seen anachronistically as underlying Jensen's consistency proof for NFU.

TRANSLATIONS BETWEEN LINEAR AND TREE NATURAL DEDUCTION SYSTEMS FOR RELEVANT LOGICS

2021 ◽  
pp. 1-22
Author(s):  
SHAWN STANDEFER
Keyword(s):  
Natural Deduction ◽  
The Other ◽  
Relevant Logics

Abstract Anderson and Belnap presented indexed Fitch-style natural deduction systems for the relevant logics R, E, and T. This work was extended by Brady to cover a range of relevant logics. In this paper I present indexed tree natural deduction systems for the Anderson–Belnap–Brady systems and show how to translate proofs in one format into proofs in the other, which establishes the adequacy of the tree systems.

The Leptobrachium (Anura: Megophryidae) of the Langbian Plateau, southern Vietnam, with description of a new species

Zootaxa ◽  
2011 ◽  
Vol 2804 (1) ◽  
pp. 25 ◽  
Author(s):  
BRYAN L. STUART ◽  
JODI J. L. ROWLEY ◽  
DAO THI ANH TRAN ◽  
DUONG THI THUY LE ◽  
HUY DUC HOANG
Keyword(s):  
Mitochondrial Dna ◽  
New Species ◽  
Small Body ◽  
Original Description ◽  
The Other ◽  
Sexually Active ◽  
Small Body Size ◽  
Upper Lip ◽  
Southern Vietnam ◽  
A New Species

We sampled two forms of Leptobrachium in syntopy at the type locality of L. pullum at upper elevations on the Langbian Plateau, southern Vietnam. The two forms differed in morphology (primarily in coloration), mitochondrial DNA, and male advertisement calls. One form closely agrees with the type series of L. pullum (but not to its original description due to error), and the other is described as new. Leptobrachium leucops sp. nov. is distinguished from its congeners by having small body size (males with SVL 38.8–45.2), the upper one-third to one-half of iris white, a blue scleral arc, a dark venter, and sexually active males without spines on the upper lip. Leptobrachium pullum and L. mouhoti, a recently described species from low-elevation slopes of the Langbian Plateau in eastern Cambodia, are morphologically divergent but genetically similar, warranting further investigation into geographic variation in the red-eyed Leptobrachium of southern Indochina.

scholarly journals What If Quantum Theory Violates All Mathematics?

Open Physics ◽  
2017 ◽  
Vol 15 (1) ◽  
pp. 598-602
Author(s):  
Elemér Elad Rosinger
Keyword(s):  
Mathematical Logic ◽  
Quantum Theory ◽  
Bell Inequalities ◽  
List Type ◽  
Elementary Argument ◽  
And Mathematics

Abstract It is shown by using a rather elementary argument in Mathematical Logic that if indeed, quantum theory does violate the famous Bell Inequalities, then quantum theory must inevitably also violate all valid mathematical statements, and in particular, such basic algebraic relations like 0 = 0, 1 = 1, 2 = 2, 3 = 3, … and so on … An interest in that result is due to the following three alternatives which it imposes upon both Physics and Mathematics: Quantum Theory is inconsistent. Quantum Theory together with Mathematics are inconsistent. Mathematics is inconsistent. In this regard one should recall that, up until now, it is not known whether Mathematics is indeed consistent.

Set Theory and Foundations of Mathematics: An Introduction to Mathematical Logic

10.1142/12456 ◽  
2022 ◽  
Author(s):  
Douglas Cenzer ◽  
Jean Larson ◽  
Christopher Porter ◽  
Jindrich Zapletal
Keyword(s):  
Mathematical Logic ◽  
Set Theory ◽  
Foundations Of Mathematics

Uncertainty Analysis in Quantitative Risk Assessment

1996 ◽  
Vol 118 (1) ◽  
pp. 121-124 ◽  
Author(s):  
S. Quin ◽  
G. E. O. Widera
Keyword(s):  
Risk Assessment ◽  
Uncertainty Analysis ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Failure Modes ◽  
The Other ◽  
Quantitative Risk Assessment ◽  
Engineering Practice ◽  
Criticality Analysis

Of the quantitative approaches applied to inservice inspection, failure modes, effects,criticality analysis (FMECA) methodology is recommended. FMECA can provide a straightforward illustration of how risk can be used to prioritize components for inspection (ASME, 1991). But, at present, it has two limitations. One is that it cannot be used in the situation where components have multiple failure modes. The other is that it cannot be used in the situation where the uncertainties in the data of components have nonuniform distributions. In engineering practice, these two situations exist in many cases. In this paper, two methods based on fuzzy set theory are presented to treat these problems. The methods proposed here can be considered as a supplement to FMECA, thus extending its range of applicability.

scholarly journals Licenciatura no Cenário da Educação Superior Brasileira / Degree in Scenario of the Brazilian Higher Education

2016 ◽  
Vol 4 (2) ◽  
Author(s):  
Ivo de Jesus Ramos ◽  
Luiz Henrique Amaral
Keyword(s):  
Higher Education ◽  
Mathematics Teachers ◽  
Public Spending ◽  
Public Institutions ◽  
The Other ◽  
Financial Resources ◽  
Current Scenario ◽  
The Government ◽  
And Mathematics ◽  
Science And Mathematics Teachers

ABSTRACTThis research, exploratory nature, aims to identify the current scenario of degrees in Science and Mathematics in Brazil, its weaknesses and offer. The number of vacancies in the IES Degree courses in Science and Mathematics is insufficient to meet demand in teacher training in this area? That was the question that guided the investigation. In this sense, we try to see if there are no vacancies, if there was a reduction in enrollment, evaluate evasion, estimate the annual public spending on vacancies unoccupied in 2011, these degrees. In response to the question presented in this study, the results corroborate the analysis of Tardif and Lessard (2009), with the prospect of an increase in the deficit of teachers. On the other hand, the survey indicated that the financial resources expended by the Government, especially in public institutions, little impact due to the high percentage of evasion, considering the offered vacancies and loss of students during the process. About 920 million reais annually, only in public HEIs, the resources made available for training of science and mathematics teachers do not produce effective results.RESUMOEsta investigação, de natureza exploratória, tem como objetivo identificar o atual cenário das licenciaturas de Ciências e Matemática no Brasil, suas fragilidades e oferta. A oferta de vagas pelas IES nos cursos de Licenciatura em Ciências e Matemática é insuficiente para atender a demanda na formação de professores nessa área? Essa foi a questão que norteou a investigação. Nesse sentido, procuramos verificar se há falta de vagas, se houve redução das matrículas, avaliar a evasão, estimar o gasto público anual com as vagas não ocupadas, em 2011, nessas licenciaturas. Em resposta ao questionamento apresentado nesta pesquisa, os resultados corroboram com a análise de Tardif e Lessard (2009), com a perspectiva de um agravamento no déficit de professores. Por outro lado, a pesquisa apontou que os recursos financeiros despendidos pelo Governo, em especial nas Instituições Públicas, pouco efeito produzem devido ao alto percentual de evasão, considerando-se as vagas ofertadas e perda de alunos durante o processo. Cerca de 920 milhões de reais anuais, apenas nas IES Públicas, dos recursos colocados à disposição para formação de professores de Ciências e Matemática não geram resultado efetivo. Contacto principal: [email protected]

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