Axioms

284 KiB

Open AccessArticle

Potential Infinity, Abstraction Principles and Arithmetic (Leśniewski Style)

by Rafal Urbaniak

Axioms 2016, 5(2), 18; https://doi.org/10.3390/axioms5020018 - 15 Jun 2016

Cited by 3 | Viewed by 4034

This paper starts with an explanation of how the logicist research program can be approached within the framework of Leśniewski’s systems. One nice feature of the system is that Hume’s Principle is derivable in it from an explicit definition of natural numbers. I [...] Read more.

This paper starts with an explanation of how the logicist research program can be approached within the framework of Leśniewski’s systems. One nice feature of the system is that Hume’s Principle is derivable in it from an explicit definition of natural numbers. I generalize this result to show that all predicative abstraction principles corresponding to second-level relations, which are provably equivalence relations, are provable. However, the system fails, despite being much neater than the construction of Principia Mathematica (PM). One of the key reasons is that, just as in the case of the system of PM, without the assumption that infinitely many objects exist, (renderings of) most of the standard axioms of Peano Arithmetic are not derivable in the system. I prove that introducing modal quantifiers meant to capture the intuitions behind potential infinity results in the (renderings of) axioms of Peano Arithmetic (PA) being valid in all relational models (i.e. Kripke-style models, to be defined later on) of the extended language. The second, historical part of the paper contains a user-friendly description of Leśniewski’s own arithmetic and a brief investigation into its properties. Full article

(This article belongs to the Special Issue Lvov—Warsaw School)

299 KiB

Open AccessArticle

An Overview of the Fuzzy Axiomatic Systems and Characterizations Proposed at Ghent University

by Etienne E. Kerre, Lynn D´eer and Bart Van Gasse

Axioms 2016, 5(2), 17; https://doi.org/10.3390/axioms5020017 - 07 Jun 2016

Cited by 1 | Viewed by 6799

Abstract

During the past 40 years of fuzzy research at the Fuzziness and Uncertainty Modeling research unit of Ghent University several axiomatic systems and characterizations have been introduced. In this paper we highlight some of them. The main purpose of this paper consists of [...] Read more.

During the past 40 years of fuzzy research at the Fuzziness and Uncertainty Modeling research unit of Ghent University several axiomatic systems and characterizations have been introduced. In this paper we highlight some of them. The main purpose of this paper consists of an invitation to continue research on these first attempts to axiomatize important concepts and systems in fuzzy set theory. Currently, these attempts are spread over many journals; with this paper they are now collected in a neat overview. In the literature, many axiom systems have been introduced, but as far as we know the axiomatic system of Huntington concerning a Boolean algebra has been the only one where the axioms have been proven independent. Another line of further research could be with respect to the simplification of these systems, in discovering redundancies between the axioms. Full article

(This article belongs to the Special Issue Lvov—Warsaw School)

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275 KiB

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On the Mutual Definability of the Notions of Entailment, Rejection, and Inconsistency

by Urszula Wybraniec-Skardowska

Axioms 2016, 5(2), 15; https://doi.org/10.3390/axioms5020015 - 07 Jun 2016

Cited by 3 | Viewed by 3368

Abstract

In this paper, two axiomatic theories T⁻ and T′ are constructed, which are dual to Tarski’s theory T⁺ (1930) of deductive systems based on classical propositional calculus. While in Tarski’s theory T⁺ the primitive notion is the classical consequence [...] Read more.

In this paper, two axiomatic theories T⁻ and T′ are constructed, which are dual to Tarski’s theory T⁺ (1930) of deductive systems based on classical propositional calculus. While in Tarski’s theory T⁺ the primitive notion is the classical consequence function (entailment) Cn⁺, in the dual theory T⁻ it is replaced by the notion of Słupecki’s rejection consequence Cn⁻ and in the dual theory T′ it is replaced by the notion of the family Incons of inconsistent sets. The author has proved that the theories T⁺, T⁻, and T′ are equivalent. Full article

(This article belongs to the Special Issue Lvov—Warsaw School)

241 KiB

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Contribution of Warsaw Logicians to Computational Logic

by Damian Niwiński

Axioms 2016, 5(2), 16; https://doi.org/10.3390/axioms5020016 - 03 Jun 2016

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Abstract

The newly emerging branch of research of Computer Science received encouragement from the successors of the Warsaw mathematical school: Kuratowski, Mazur, Mostowski, Grzegorczyk, and Rasiowa. Rasiowa realized very early that the spectrum of computer programs should be incorporated into the realm of mathematical [...] Read more.

The newly emerging branch of research of Computer Science received encouragement from the successors of the Warsaw mathematical school: Kuratowski, Mazur, Mostowski, Grzegorczyk, and Rasiowa. Rasiowa realized very early that the spectrum of computer programs should be incorporated into the realm of mathematical logic in order to make a rigorous treatment of program correctness. This gave rise to the concept of algorithmic logic developed since the 1970s by Rasiowa, Salwicki, Mirkowska, and their followers. Together with Pratt’s dynamic logic, algorithmic logic evolved into a mainstream branch of research: logic of programs. In the late 1980s, Warsaw logicians Tiuryn and Urzyczyn categorized various logics of programs, depending on the class of programs involved. Quite unexpectedly, they discovered that some persistent open questions about the expressive power of logics are equivalent to famous open problems in complexity theory. This, along with parallel discoveries by Harel, Immerman and Vardi, contributed to the creation of an important area of theoretical computer science: descriptive complexity. By that time, the modal μ-calculus was recognized as a sort of a universal logic of programs. The mid 1990s saw a landmark result by Walukiewicz, who showed completeness of a natural axiomatization for the μ-calculus proposed by Kozen. The difficult proof of this result, based on automata theory, opened a path to further investigations. Later, Bojanczyk opened a new chapter by introducing an unboundedness quantifier, which allowed for expressing some quantitative properties of programs. Yet another topic, linking the past with the future, is the subject of automata founded in the Fraenkel-Mostowski set theory. The studies on intuitionism found their continuation in the studies of Curry-Howard isomorphism. ukasiewicz’s landmark idea of many-valued logic found its continuation in various approaches to incompleteness and uncertainty. Full article

(This article belongs to the Special Issue Lvov—Warsaw School)

291 KiB

Open AccessArticle

An Axiomatic Account of Question Evocation: The Propositional Case

by Andrzej Wiśniewski

Axioms 2016, 5(2), 14; https://doi.org/10.3390/axioms5020014 - 26 May 2016

Cited by 5 | Viewed by 3757

Abstract

An axiomatic system for question evocation in Classical Propositional Logic is proposed. Soundness and completeness of the system are proven. Full article

(This article belongs to the Special Issue Lvov—Warsaw School)

455 KiB

Open AccessArticle

Fundamental Results for Pseudo-Differential Operators of Type 1, 1

by Jon Johnsen

Axioms 2016, 5(2), 13; https://doi.org/10.3390/axioms5020013 - 19 May 2016

Cited by 1 | Viewed by 3921

Abstract

This paper develops some deeper consequences of an extended definition, proposed previously by the author, of pseudo-differential operators that are of type

1, 1

in Hörmander’s sense. Thus, it contributes to the long-standing problem of creating a systematic theory of such operators. [...] Read more.

This paper develops some deeper consequences of an extended definition, proposed previously by the author, of pseudo-differential operators that are of type

1, 1

in Hörmander’s sense. Thus, it contributes to the long-standing problem of creating a systematic theory of such operators. It is shown that type

1, 1

-operators are defined and continuous on the full space of temperate distributions, if they fulfil Hörmander’s twisted diagonal condition, or more generally if they belong to the self-adjoint subclass; and that they are always defined on the temperate smooth functions. As a main tool the paradifferential decomposition is derived for type

1, 1

-operators, and to confirm a natural hypothesis the symmetric term is shown to cause the domain restrictions; whereas the other terms are shown to define nice type

1, 1

-operators fulfilling the twisted diagonal condition. The decomposition is analysed in the type

1, 1

-context by combining the Spectral Support Rule and the factorisation inequality, which gives pointwise estimates of pseudo-differential operators in terms of maximal functions. Full article

(This article belongs to the Special Issue Infinite-dimensional Analysis: Multi-variable Operator Theory, Representation Theory, and Applications)

491 KiB

Open AccessArticle

Infinite-dimensional Lie Algebras, Representations, Hermitian Duality and the Operators of Stochastic Calculus

by Palle Jorgensen and Feng Tian

Axioms 2016, 5(2), 12; https://doi.org/10.3390/axioms5020012 - 17 May 2016

Cited by 4 | Viewed by 5792

Abstract

We study densely defined unbounded operators acting between different Hilbert spaces. For these, we introduce a notion of symmetric (closable) pairs of operators. The purpose of our paper is to give applications to selected themes at the cross road of operator commutation relations [...] Read more.

We study densely defined unbounded operators acting between different Hilbert spaces. For these, we introduce a notion of symmetric (closable) pairs of operators. The purpose of our paper is to give applications to selected themes at the cross road of operator commutation relations and stochastic calculus. We study a family of representations of the canonical commutation relations (CCR)-algebra (an infinite number of degrees of freedom), which we call admissible. The family of admissible representations includes the Fock-vacuum representation. We show that, to every admissible representation, there is an associated Gaussian stochastic calculus, and we point out that the case of the Fock-vacuum CCR-representation in a natural way yields the operators of Malliavin calculus. We thus get the operators of Malliavin’s calculus of variation from a more algebraic approach than is common. We further obtain explicit and natural formulas, and rules, for the operators of stochastic calculus. Our approach makes use of a notion of symmetric (closable) pairs of operators. The Fock-vacuum representation yields a maximal symmetric pair. This duality viewpoint has the further advantage that issues with unbounded operators and dense domains can be resolved much easier than what is possible with alternative tools. With the use of CCR representation theory, we also obtain, as a byproduct, a number of new results in multi-variable operator theory which we feel are of independent interest. Full article

(This article belongs to the Special Issue Infinite-dimensional Analysis: Multi-variable Operator Theory, Representation Theory, and Applications)

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220 KiB

Open AccessEditorial

An Overview of Topological Groups: Yesterday, Today, Tomorrow

by Sidney A. Morris

Axioms 2016, 5(2), 11; https://doi.org/10.3390/axioms5020011 - 05 May 2016

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Abstract

It was in 1969 that I began my graduate studies on topological group theory and I often dived into one of the following five books. My favourite book “Abstract Harmonic Analysis” [1] by Ed Hewitt and Ken Ross contains both a proof of [...] Read more.

It was in 1969 that I began my graduate studies on topological group theory and I often dived into one of the following five books. My favourite book “Abstract Harmonic Analysis” [1] by Ed Hewitt and Ken Ross contains both a proof of the Pontryagin-van Kampen Duality Theorem for locally compact abelian groups and the structure theory of locally compact abelian groups.[...] Full article

(This article belongs to the Special Issue Topological Groups: Yesterday, Today, Tomorrow)

2033 KiB

Open AccessArticle

Applications of Skew Models Using Generalized Logistic Distribution

by Pushpa Narayan Rathie, Paulo Silva and Gabriela Olinto

Axioms 2016, 5(2), 10; https://doi.org/10.3390/axioms5020010 - 15 Apr 2016

Cited by 6 | Viewed by 5307

Abstract

We use the skew distribution generation procedure proposed by Azzalini [Scand. J. Stat., 1985, 12, 171–178] to create three new probability distribution functions. These models make use of normal, student-t and generalized logistic distribution, see Rathie and Swamee [Technical [...] Read more.

We use the skew distribution generation procedure proposed by Azzalini [Scand. J. Stat., 1985, 12, 171–178] to create three new probability distribution functions. These models make use of normal, student-t and generalized logistic distribution, see Rathie and Swamee [Technical Research Report No. 07/2006. Department of Statistics, University of Brasilia: Brasilia, Brazil, 2006]. Expressions for the moments about origin are derived. Graphical illustrations are also provided. The distributions derived in this paper can be seen as generalizations of the distributions given by Nadarajah and Kotz [Acta Appl. Math., 2006, 91, 1–37]. Applications with unimodal and bimodal data are given to illustrate the applicability of the results derived in this paper. The applications include the analysis of the following data sets: (a) spending on public education in various countries in 2003; (b) total expenditure on health in 2009 in various countries and (c) waiting time between eruptions of the Old Faithful Geyser in the Yellow Stone National Park, Wyoming, USA. We compare the fit of the distributions introduced in this paper with the distributions given by Nadarajah and Kotz [Acta Appl. Math., 2006, 91, 1–37]. The results show that our distributions, in general, fit better the data sets. The general R codes for fitting the distributions introduced in this paper are given in Appendix A. Full article

(This article belongs to the Special Issue Special Functions: Fractional Calculus and the Pathway for Entropy Dedicated to Professor Dr. A.M. Mathai on the occasion of his 80th Birthday)

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152 KiB

Open AccessEditorial

The Lvov-Warsaw School and Its Future

by Angel Garrido and Piedad Yuste

Axioms 2016, 5(2), 9; https://doi.org/10.3390/axioms5020009 - 11 Apr 2016

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Abstract

The Lvov-Warsaw School (L-WS) was the most important movement in the history of Polish philosophy, and certainly prominent in the general history of philosophy, and 20th century logics and mathematics in particular.[...] Full article

(This article belongs to the Special Issue Lvov—Warsaw School)

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Axioms, Volume 5, Issue 2 (June 2016) – 10 articles

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