Abraham Avi Arad Hochman INJUSTICIA y QUERELLAS FALSAS !!!

Diari Oicial de la Generalitat de Catalunya Núm. 5100 – 31.3.2008 25005
Administració de justícia
JUTJATS DE PRIMERA INSTÀNCIA
I INSTRUCCIÓ
EDICTE
del Jutjat de Primera Instància núm. 41 de Barcelona, sobre actuacions de proce-
diment ordinari (exp. 498/2007).
D/ña. Modesto Casals Delgado Secretario Judicial del Juzgado Primera Instancia
41 Barcelona
HAGO SABER:
Que en este Juzgado obra dictada la siguiente Sentencia que en su encabezami-
ento y fallo dice:
En Barcelona, a veinte de febrero de dos mil ocho. Vistos por Don .DON JESÚS
ARANGÜENA SANDE Magistrado Juez del Juzgado de Primera Instancia número
cuarenta y uno de los de Barcelona, los autos de procedimiento ordinario registrados
con el número. 498/2007, seguidos a instancia de Pere Fontané Masnou represen-
tado por la Procuradora Sra. MONICA RIBAS RULO y dirigida por el Letrado
Sr. Rebled, contra Abraham Arad Hochman , en rebeldia, ha dictado sentencia con
arreglo a los siguientes:
FALLO :
Que estimando como estimo la demanda interpuesta por Don Pere Fontané
Masnou, representado por la Procuradora Sra Ribas, debo de condenar y condeno
a Don Abraham Arad Hochman a abonar a la actora 36.060,72 Euros de principal
más los intereses legales de dicha cantidad desde demanda y hasta la presente
resolución sin perjuicio del art. 576 LEC, con expresa condena en costas a la parte
demandada.
Esta sentencia no es irme y contra ella cabe interponer recurso de apelación el
cual se prepará en el plazo de cinco dias de este Juzgado por medio de escrito en
el que cite la parte la resolución apelada y maniieste su voluntad de recurrir con
expresión de los pronuncimientos que impugna; Asi por esta mi sentencia, de la que
se expedirá testimonio para su unión los autos, la pronuncio, mando y irmo. “
Y para que sirva de notiicación al demandado en ignorado paradero libro el
presente.
Barcelona, 26 de febrero de 2008
El secretario judicial, irma ilegible
PG-208923 (08.080.009)
Diari Oicial de la Generalitat de Catalunya Núm. 5100 – 31.3.2008
25006
Administració de justícia
JUTJATS DE PRIMERA INSTÀNCIA
I INSTRUCCIÓ
EDICTE
del Jutjat de Primera Instància núm. 41 de Barcelona, sobre actuacions de judici
verbal (exp. 239/2003).
D/ña. Modesto Casals Delgado Secretario Judicial del Juzgado Primera Instancia
41 Barcelona
HAGO SABER:
Que en este Juzgado obra dictada la siguiente Sentencia que en su encabezami-
ento y fallo dice:
En Barcelona, a dos de junio de dos mil cuatro. Vistos por Don JESÚS ARAN-
GÜENA SANDE Magistrado Juez del Juzgado de Primera Instancia número
cuarenta y uno de los de Barcelona, los autos de juicio verbal, sobre reclamación
de cantidad registrados con el número 239/2003, seguidos a instancia de la Gene-
ralitat de Catalunya representada y dirigida por la Letrada de la Generalitat frente a
Ximena Soledad Belmar en rebeldía y a Francisco Tapies Iglesias (luego desistido),
ha dictado sentencia con arreglo a los siguientes:
FALLO : Que estimando la demanda interpuesta por GENERALITAT DE
CATALUNYA, representada por la Letrada de la Generalitat, frente a Doña Ximena
Soledad Belmar, en rebeldía, debo de condenar y condeno a dicha demandada a
abonar a la actora la cantidad de 687,60 euros más los intereses legales de dicha
cantidad desde la demanda hasta la presente resolución, sin perjuicio de lo previsto
en el art 576 LEC, imponiendo asimismo a dicha demandada - si las hubiere- el
pago de las costas causadas en esta instancia.
Esta sentencia no es irme y contra ella cabe interponer recurso de apelación el
cual se prepará en el plazo de cinco dias de este Juzgado por medio de escrito en
el que cite la parte la resolución apelada y maniieste su voluntad de recurrir con
expresión de los pronunciamientos que impugna. Así por esta mi sentencia, de la que
se expedirá testimonio para su unión a los autos, la pronuncio, mando y irmo.”
Y para que sirva de notiicación al demandado en ignorado paradero libro el
presente.
Barcelona, 29 de febrero de 2008
El secretario judicial, irma ilegible
PG-210061 (08.079.111)

Abraham Avi Arad Hochman

Injustice is the lack of or opposition to justice, either in reference to a particular event or act, or as a larger status quo. The term generally refers to the misuse, abuse, neglect, or malfeasance of a justice system, with regard to a particular case or context, such that the legal status quo represents a systemic failure to serve the cause of justice.
[edit]

Abraham Avi Arad Hochman

El éxito no es tanto hacer lo que uno quiere como querer lo que uno hace. (David Brown)


SOCIEDAD


¿Dónde está el dinero de los judíos muertos en el Holocausto?

La Banca suiza accede a que se investigue qué ha ocurrido con los bienes de las víctimas de los nazis


RAMY WURGAFT


CORRESPONSAL

JERUSALEN.- Ha transcurrido medio siglo, pero la tormenta del Holocausto nazi aún no amaina. En estos días, el vendaval ha arrasado con una de las instituciones más sólidas de nuestra época: el secretismo de la banca suiza.

En un acuerdo firmado el jueves en Nueva York, la Asociación de Bancos Suizos (ABS) accedió a que una comisión investigadora indague lo ocurrido con el dinero que habían depositado los judíos de Europa, antes y durante la Segunda Guerra Mundial.

El presidente de la Agencia Judía (AJ), Abraham Burg, estima que en la Banca helvética existen unas 2.000 cuentas corrientes «en estado de hibernación» cuyos propietarios murieron en los campos de exterminio de Auschwitz o de Treblinka.

David Elroi, funcionario de la AJ, dijo a EL MUNDO que el monto total de estas cuentas «dormidas» alcanza la fabulosa suma de 7.000 millones de dólares. «Pero eso no es todo. En las bóvedas de los bancos hay joyas y obras de arte de incalculable valor», afirma David Elroi. Y añade: «Para hacerse una idea de los tesoros que se conservan en las cajas blindadas, baste señalar que los Openhaimer y los Shosheim, dos de las familias judías más poderosas de Alemania, guardaban gran parte de su caudal y de sus acciones en los bancos helvéticos».

Pero también la gente de la calle ponía sus centavos a recaudo de los suizos, con lo cual los expertos de la Agencia Judía calculan que un total de 100.000 personas, entre supervivientes del Holocausto y herederos de aquéllos, se beneficiarán una vez que se disipen los misterios de esta trama.

La ABS, por su parte, sostiene que las estimaciones de la Agencia Judía «padecen de una imaginación febril». Joseph Ackermann, presidente del Credit Swiss de Ginebra, sostiene que un buen número de los inversores judíos consiguieron liquidar sus cuentas antes de fugarse de Europa, y el dinero que resta en ningún caso supera los 34 millones de dólares.

«Los hebreos que depositaban bienes en la Banca suiza eran personas solventes cuyos lazos comerciales en todo el mundo les permitieron darse a la fuga antes de caer en las garras de los nazis», dijo Ackermann a un enviado del periódico hebreo Yediot Hajaronot. «Los que imaginan un tesoro de dimensiones faraónicas, se llevarán una gran decepción».

Abraham Arad, presidente e investigador del Museo del Holocausto Yad Vashem (con sede en Jerusalén), rechaza los argumentos de Joseph Ackermann: «Los primeros en caer en las redes del nazismo fueron, precisamente, los judíos adinerados. Obviamente, el Reich alemán no sólo ansiaba exterminarlos, sino también apoderarse de sus bienes».

En una intervención radiofónica, Arad llegó a expresar que las declaraciones del banquero Ackermann «le huelen mal». «No culpo de nada a este señor ni a sus colegas, pero tampoco se puede descartar que el dinero de los judíos haya llegado a donde corresponde», dijo Arad, insinuando que parte de las cuentas bancarias pudieron haber sido saqueadas.

De cualquier forma, el acuerdo para desvelar la suerte de las cuentas hebreas, se firmó con toda solemnidad en las oficinas que tiene la Agencia Judía en un rascacielos de Nueva York.

Abraham Burg, signatario del documento en representación de Israel, manifestó: «Este es un momento histórico. En Israel viven decenas de supervivientes del Holocausto cuya situación es precaria. El dinero no erradicará las pesadillas, pero al menos les permitirá vivir con decoro el resto de sus días».

Elan Steinberg, director ejecutivo del Congreso Judío Mundial, felicitó a los banqueros suizos por «haber respondido al llamado de sus conciencias».

Abraham Burg, presidente de la AJ y artífice del acuerdo suizo-judío, es hijo de Yosef Burg, quien firmó en 1952 el acuerdo de indemnizaciones con el Gobierno de Alemania. Un acuerdo mediante el cual Bonn admitió su responsabilidad en el Holocausto, compensando a los supervivientes de la tragedia con millones de marcos.

La comisión creada tras al acuerdo de Nueva York estará formada por tres representantes de los bancos suizos y tres de instituciones judías: Abraham Burg, Shebah Weiss y el presidente de la comunidad judía de Argentina, Reuven Braja.

Abraham Avi Arad Hochman: Abraham Avi Arad Hochman

Abraham Avi Arad Hochman: Abraham Avi Arad Hochman

Abraham Avi Arad Hochman: Abraham Avi Arad Hochman

Abraham Avi Arad Hochman

Neural Computing Systems (NCS) provides leading-edge decision support, data analysis and predictive modeling software and services. Based on our expertise from years of research and drawing from experience in solving a wide range of data analysis problems, we have developed unique combinations of tools and technology leading to novel solutions.

We focus our technology to better service both the business and defense sectors, specializing in the automated analysis of large data-sets, signals, information, and images, to deliver breakdown and discovery.

To learn more about NCS technology and how it can empower you, choose an area of focus to the left.

Recent NCS News:
Apr. 24 NCS Unveils Multi-Spectral Shadow Removal and Tomographic Radar Data Fusion Technologies at 2003 AeroSense Conference
Feb. 15 TSI renews NetPAL license
>> News Archive


site map help desk privacy policy 1-949-854-6528

Copyright © 2000-2003 Neural Computing Systems, LLC. All Rights Reserved.

Abraham Avi Arad Hochman

From:
Subject: Re: invariant measures on SO(3) and SE(3)?
Date: Wed, 5 May 1999 23:25:13 -0500 (CDT)
Newsgroups: [missing]
To: bruyninc@leland.Stanford.EDU
Keywords: Haar measure

>Where van I find (references to) invariant measures (``Haar'' measures)
>on SO(3) and SE(3)?

Gee, how much detail do you want? I've taken the liberty of enclosing the
reviews of a few texts in a few of the areas of mathematics where these
topics would arise. I think if you don't make your request more specific,
we will have every group theorist, measure theorist, and harmonic analyst
telling you everything they know...
Until you respond with a more specific question I'd rather not approve
your post, OK?
dave (moderator)
PS - one or two more references at
http://www.math.niu.edu/~rusin/known-math/index/43-XX.html


98e:28001 28-01 (60-01)
Simonnet, Michel(SNG-DAK)
Measures and probabilities. (English. English summary)
With a foreword by Charles-Michel Marle. Universitext.
Springer-Verlag, New York, 1996. xiv+510 pp. $44.00. ISBN
0-387-94644-6

This book seems to be intended both as an advanced textbook and as a
reference book on measure theory. It deals with the three usual faces
of integration theory (countably additive measures on abstract sets,
Daniell integrals on abstract sets, and Radon measures on locally
compact Hausdorff spaces), plus some applications to probability
theory and to analysis on locally compact groups.

The prerequisites for reading the book are basic point-set topology
and functional analysis. The book begins with an introduction to
ordered groups and vector spaces and then introduces the Daniell
construction of the integral (the functions being integrated are
allowed to have values in a Banach space). As special cases Simonnet
treats integration with respect to abstract measures (including their
extension from semirings of sets to $\sigma$-rings of sets) and the
theory of Radon measures. The usual material for a course on measure
theory is presented in detail.

The second half of the book begins with applications to probability
theory. This includes the existence of measures on infinite product
spaces, Birkhoff's ergodic theorem, an introduction to the central
limit theorem (including Fourier transforms on ${R}\sp n$), the strong
law of large numbers, and conditional expectations and probabilities.
There is no treatment, for example, of martingales, of Brownian
motion, or of weak convergence on more general metric spaces.

The final part of the book returns to Radon measures, following (as
the author points out) Bourbaki rather closely. After some more
generalities on Radon measures (including the Radon-Nikodym theorem
for Radon measures, products of Radon measures, etc.), the book
contains a chapter devoted to Haar measures, and then closes with a
chapter on convolutions of measures and functions.

Although the treatment is rather abstract and general, it is also very
concrete. For example, a thorough treatment of change of variables in
multiple integrals is given, followed later in the book with details
on the calculation of Haar measures for some concrete groups. There
are numerous exercises of all sorts---a bit more than 75 pages of
them. On the other hand, the book contains no historical notes and no
bibliography.

All in all, the book contains a large amount of information, presented
in a careful manner. However, its level of generalityq (in particular,
the use of the elements of functional analysis throughout the book),
plus the fact that abstract measures, Daniell integrals, and Radon
measures are simultaneously studied, may make this book more useful as
a reference for advanced students than as a textbook for a basic real
analysis course.

Reviewed by Donald L. Cohn
_________________________________________________________________

98c:43001 43-01 (22-01 46-01)
Folland, Gerald B.(1-WA)
A course in abstract harmonic analysis.
Studies in Advanced Mathematics.
CRC Press, Boca Raton, FL, 1995. x+276 pp. $61.95. ISBN 0-8493-8490-7

This deligthful book fills a long-standing gap in the literature on
abstract harmonic analysis. For the author the term "harmonic
analysis" means those parts of analysis in which the action of a
locally compact group plays an essential role: more specifically, the
theory of unitary representations of locally compact groups, and the
analysis of functions on such groups and their homogeneous spaces. The
book contains a careful treatment of certain key results in the
subject that were developed from about 1927 (the date of the
Peter-Weyl theorem) up to 1970. The focus is on fundamental ideas and
theorems in harmonic analysis that are used over and over again, and
which can be developed with minimal assumptions on the nature of the
underlying group. Its purpose is not to compete in any way with the
many existing excellent monographs and treatises on the subject, but
to provide a unified picture of the general abstract theory in an
introductory book of moderate length. To the reviewer's knowledge no
one existing book contains all of the topics that are treated in this
one. To be sure, various bits and pieces of what the author covers can
be found in one reference or another, and certain aspects of the
theory are treated much more extensively in a few lengthy treatises
[see, e.g., J. Dixmier, $C\sp*$-algebras, Translated from the French
by Francis Jellett, North-Holland, Amsterdam, 1977; MR 56 #16388; J.
M. G. Fell and R. S. Doran, Representations of $\sp *$-algebras,
locally compact groups, and Banach $\sp *$-algebraic bundles. Vol. 1,
Academic Press, Boston, MA, 1988; MR 90c:46001; Vol. 2; MR 90c:46002].

Assuming only a knowledge of real analysis and elementary functional
analysis, the author carefully introduces and proves (with a few
exceptions), in the first six chapters, classical facts in
representation theory. Chapter 1, titled "Banach algebras and spectral
theory", contains background material on $C\sp *$-algebras and
spectral theory of *-representations that is needed in the remainder
of the book.

As the author points out, Chapters 2--6 form the core of the book.
Chapter 2, titled "Locally compact groups", develops the basic tools
for doing analysis on groups and homogeneous spaces. Here the reader
will find a nice introductory treatment of topological groups, Haar
measure, convolutions, homogeneous spaces and quasi-invariant
measures. Chapter 3, titled "Basic representation theory", presents
the rudiments of unitary representation theory up through the
Gelfand-Raikov existence theorem for irreducible unitary
representations. The connections between functions of positive type
and representations are also described.

Chapters 4 and 5 are respectively entitled "Analysis on locally
compact abelian groups" and "Analysis on compact groups". Here the
Fourier transform takes center stage, first as a straightforward
generalization of the classical Fourier transform ${\scr
F}f(\xi)=\int\sb {-\infty}\sp \infty e\sp {-2\pi ix\xi}f(x)dx$ from
the real line to locally compact abelian groups, and then to the more
representation-theoretic form that is appropriate for the non-abelian,
compact case.

Chapter 6 presents the theory of induced representations. This is a
way of constructing a unitary representation of a locally compact
group $G$ out of a unitary representation of a closed subgroup $H$.
Geometrically speaking, these induced representations are the unitary
representations of $G$ arising from the action of $G$ on functions or
sections of homogeneous vector bundles on the homogeneous space $G/H$.
After describing the construction of induced representations for
locally compact groups, the author proves the Frobenius reciprocity
theorem for compact groups. This provides a powerful tool for finding
the irreducible decomposition of an induced representation of a
compact group. He then develops the notion of pseudomeasures of
positive type (a generalization of functions of positive type) and
uses it to prove the theorem on induction in stages and the
imprimitivity theorem, which is the deepest result of the chapter. It
forms the basis for the so-called "Mackey machine", a body of
techniques for analyzing representations of a group $G$ in terms of
the representations of a normal subgroup $N$ and the representations
of various subgroups of $G/N$.

It is important to mention that the author includes specific examples
throughout the book to illustrate the general theory. In Chapters 2--4
these examples are interwoven with the rest of the text, while in
Chapters 5 and 6 they are, for the most part, collected in separate
sections at the end of the chapter.

Now, a few words need to be said about Chapter 7, which is entitled
"Further topics in representation theory". Focusing on the theory of
noncompact, nonabelian, locally compact groups, it is more like a
survey article than a chapter of the book. The principal object of
study is the dual space $\hat{G}$ of a locally compact group $G$,
i.e., the set of equivalence classes of irreducible unitary
representations of $G$ furnished with a natural topology (which in
this book is called the Fell topology). Topics discussed include the
group $C\sp *$-algebra of a locally compact group, the dual space,
tensor products, direct integrals, and the Plancherel theorem. As the
author observes, giving a complete treatment of this material would
require a lengthy digression into the theory of von Neumann algebras,
representations of $C\sp *$-algebras, and direct integral
decompositions that would substantially increase the size of the book.
As a result, the author is content with providing definitions and
statements of the theoerems, together with a discussion of some
concrete cases. References to sources where a detailed treatment of
all of these topics can be found are provided throughout the chapter.

To help make the book self-contained, three brief appendices are
provided, respectively entitled "A Hilbert space miscellany",
"Trace-class and Hilbert-Schmidt operators", and "Vector-valued
integrals". The bibliography consists of 134 carefully selected
references and makes no pretence at completeness.

Finally, a few general concluding remarks. This book is aimed at a
broad mathematical audience. One of the reasons the author wrote it
(see the Preface) is that he believes the material is "beautiful". His
respect for the subject shows on every hand. This is apparent through
his careful writing style, which is concise, yet simple and elegant.
The reviewer would encourage anyone with an interest in harmonic
analysis to have this book in his or her personal library. The author
is to be congratulated on writing a fine book that the reviewer would
have been proud to write.

Reviewed by Robert S. Doran
_________________________________________________________________

97c:22001 22-01 (20C05 20C15 22E15)
Simon, Barry
Representations of finite and compact groups. (English. English
summary)
Graduate Studies in Mathematics, 10.
American Mathematical Society, Providence, RI, 1996. xii+266 pp.
$34.00. ISBN 0-8218-0453-7 [AMS Book Store]

Although not divided explicitly, the book consists of two parts which
should be considered separately. The first one is concerned with the
theory of representations of finite groups; it contains 6 chapters and
120 pages. The second part, devoted to the theory of compact Lie
groups, contains 3 chapters and 135 pages. Taking into account that
the subject of this latter part is much more extensive and
complicated, it is obvious that the author has had to apply a
different approach in attempting to cover it in almost the same space.

In a concise form one can say that, while the first part can be
considered as a complete and self-contained introduction to finite
group representations, the second one presents selected topics of the
theory of compact groups and their representations.

Chapter I is devoted to basic information about finite groups,
homogeneous spaces, and constructions of the direct and semi-direct
products of groups. An exhaustive list of examples is presented,
including $Z\sb n$, the permutation group $S\sb n$, finite groups of
rotations, Platonic groups, and $p$-groups including Sylow theorems.
Chapter II describes the fundamental concepts and results about
representations of finite groups: irreducible representations, regular
representation, group algebra, matrix elements, Schur's lemma. Special
attention is paid to the classification of the irreducible
representations as real, complex or quaternionic. Chapter III is
devoted to the central components of representation theory, such as
the theory of characters and of class functions, and Fourier analysis.
The dimension theorem is also proved. Chapter IV is concerned with
representations of abelian finite groups, dual groups and Clifford
groups. Chapter V is of a more general character. It presents the
Frobenius theory of irreducible representations of semidirect
products, the general induced representations of finite groups, the
Frobenius character formula and the reciprocity theorem, and Mackey's
criterion of irreducibility. Chapter VI is totally devoted to the
representations of symmetric groups with application of Young frames
and Young tableaux. The Frobenius character formula for $S\sb n$ and
its applications close the first part of the book, which can be
recommended as a very good text about finite groups and their
representations. The approach is elementary, and the presentation is
clear and well organized, in the form of a course. In the unique case
when auxiliary material is necessary (the theory of algebraic
integers), the exposition is concise, complete and elegant.

Passing to the second part, devoted to compact groups and their
representations, we must emphasize that it treats almost exclusively
finite-dimensional representations of the compact Lie groups and the
approach is much more algebraic than we could expect after reading the
introduction.

Chapter VII is mostly introductory and contains generalities (without
proofs) about $C\sp \infty$-manifolds, homotopy theory and multilinear
algebra interspersed with the elements of representation theory. Then
Lie groups and their Lie algebras, the exponential mapping and the
adjoint representation are introduced. The construction of the Haar
measure is carried over for general Lie groups. The classical matrix
groups are presented as examples of Lie groups. The detailed
description of their structure presented along the whole text is a
great advantage of the book. The final 9 pages of this chapter are
devoted to the representations of groups. The author, anxious to avoid
general concepts, speaks only of compact Lie groups acting on
finite-dimensional spaces. The orthogonality relations for matrix
elements and the Peter-Weyl theorem are proved only in this context.
Obviously it is impossible to avoid infinite-dimensional
representations completely; hence the author is sometimes forced to
speak of (undefined) "infinite-dimensional representations" or, as in
Theorem VII.10.8, of a "strongly continuous map of $G$ to unitary
operators on ${\scr H}$" (forgetting at this moment that the map
should be a group homomorphism).

At the beginning of Chapter VIII, which in fact is an original
contribution to the theory of maximal tori in compact Lie groups, the
exposition is strangely complicated. First, the existence of the
maximal tori and the fact that all of them are conjugate to each other
is announced in Theorem VIII.1.1 for compact and semisimple Lie
groups. In order to prove that the compactness is critical, the author
gives an example of a group without a maximal torus which is neither
compact nor semisimple; hence the example fails. Next, he proves the
equivalence of Theorem VIII.1.1 to Theorem VII.1.1$'$, where the
semisimplicity is not assumed. The above-mentioned counterexample put
after Theorem VIII.1.1$'$ would work perfectly. The version VII.1.1$'$
is proved finally but the proof of the existence of the maximal tori
appears as a remark outside this proof.

This part of the book is interesting but needs polishing.

The final sections of the chapter are algebraic and devoted to the
concepts of roots, root spaces, to the classification of the
fundamental systems of roots, Dynkin diagrams, Weyl groups and
Cartan-Stiefel diagrams. Again, the classical groups are presented
from this point of view.

The last and the most extensive Chapter IX begins with the study of
the geometry of the Cartan-Stiefel diagrams and of the integral forms.
After proving the Weyl integration formula, the maximal weights are
introduced and the Weyl character formula is proved. As applications
of the latter, the Weyl dimension formula, and the multiplicity
formulas of Kostant and Freudenthal, and the formulas of Racah and of
Steinberg for Clebsch-Gordan integers are given. The last sections
contain the description of irreducible representations of compact
classical groups and their tensor products. The real and quaternionic
representations are distinguished. The alternative proof of the
Frobenius character formula appears in relation with the tensor
products of irreducible representations of the group ${\rm U}(n)$.

It must be mentioned that the description of the irreducible
representations, although made for groups, not their Lie algebras, is
algebraic, being based on the concept of the highest weight. The
analytic realizations do not appear even in the examples. The induced
representations are not introduced in this part of the book; hence
Frobenius reciprocity is also absent. The decomposition theory of
representations is practically omitted. The author's promise to give
more analytic flavour to the theory is kept only in the part
concerning the structure of the compact Lie groups. Surprisingly, the
algebraic parts of the book seem to be more complete and better
organized.

The theory of representations of groups is nowadays a very extensive
area. Textbooks presenting particular topics of the theory are very
desirable. In particular this book can be recommended as a base for
courses about representations of finite groups and finite-dimensional
representations of Lie groups.

It is a pity that the bibliography is definitely incomplete. The
absence of the classical monographs of C. W. Curtis and I. Reiner, H.
Weyl, I. M. Gelfand and M. A. Naimark, S. Helgason, and D. P.
Zhelobenko is difficult to explain.

Reviewed by Antoni Wawrzynczyk
_________________________________________________________________

96b:00001 00A05 (28-01 30-01 46-01)
Gelbaum, Bernard R.(1-SUNYB)
Modern real and complex analysis. (English. English summary)
A Wiley-Interscience Publication.
John Wiley & Sons, Inc., New York, 1995. xiv+489 pp. $64.95. ISBN
0-471-10715-8

This book is ambitious in scope and aims to achieve in one volume what
older French and German cours d'analyse did in several. Besides the
standard topics one would expect from the title, there is much more.
The basics of point-set topology are reviewed, uniform structures and
simplices discussed, and Tikhonov's product theorem, Brouwer's
fixed-point theorem, and the Tietze-Urysohn extension theorem proved.
{Here a pedagogical opportunity is missed: although the open map
theorem in Banach spaces is later proved, the commonality of proof
with the extension theorem [S. Grabiner, Amer. Math. Monthly 93
(1986), no. 3, 190--191; MR 88a:54034] is neither exploited nor
mentioned.}There is an admirably concise treatment of the complex
exponential and circular functions. Integration is from the
Daniell-functional as well as the Caratheodory-outer-measure point of
view, and the Haar measure is constructed. Considerable functional
analysis (weak topologies, Banach algebras, Hilbert space, the $C\sp
*$-algebra version of the spectral theorem) is developed, and we are
only up to page 140 in this 490-page book. In the complex analysis
half, we see Pompeiu's generalization of the Cauchy integral formula
(via Stokes), Riemann surfaces developed in some detail, the
uniformization theorem presented as a sequence of exercises, a short
introduction to several complex variables, and a very nice short
chapter entitled "convexity and complex analysis" (centering around
the Riesz-Thorin convexity theorem). For this wealth of topics and
length the book's $$65$ price must be considered reasonable nowadays.

All this notwithstanding, the book has serious defects. Some are
technical/stylistic. For example, the author's laudable concision is
often at the expense of readability: symbols are preferred to words,
but even the 6-page symbol index is unable to chronicle all of them,
and the appearance of the printed page (not to mention the reading of
it) is, in the argot, not very user-friendly. Parentheses are rampant
where not needed (e.g., we see, for propositions $A$ and $B$, $A\wedge
B$, but $\{A\}\Rightarrow\{B\}$) and sometimes absent where needed
(e.g., in $\int f+g$). It is often hard to know where hypotheses end
and conclusions begin because of the author's sparing use of "then"
and "that", and statements of theorems are often convoluted. This is
sure to impede foreign readers. (Native speakers of the international
scientific language are too often not conscious of their special
obligations to handle it meticulously.) These deficiencies are
regrettable but bearable. More serious are the logical deficiencies,
and unfortunately they are legion (as are routine typographical
errors). The reviewer read in detail the first $50%$ of the complex
variables part and generated over ten pages of errata. Some things are
repairable, but probably not by neophytes, while others are
unsalvageable. The attempt to derive Weierstrass' theorem on
specifying zeros from Mittag-Leffler's theorem on specifying principal
parts is an example of the latter: in the course of it the function
$(z-a)/(b-a)$ is exhibited as the exponential of a holomorphic
function in a deleted neighborhood of $a$. And the proof of
Mittag-Leffler's theorem itself involves a confused misuse of Runge's
theorem. Exercise 6.2.26 asks the reader to prove that if $u\sb n$ are
uniformly bounded and harmonic in an open disk $D$ and converge on a
set with a cluster point in $D$, then the sequence converges
throughout $D$. Another exercise (with a hint!) claims that if $u$,
$v$ are harmonic in region $\Omega$ and $\limsup u\leq\liminf v$ at
each boundary point, then $u\leq v$. But perhaps a student can be
expected to observe that $u\sb n(z)=(-1)\sp n{\rm Im}\,z$ and
$\Omega=\bold C\sb \infty\sbs\{0,\infty\}$, $v(z)=\log \vert z\vert $,
$u=2v$ provide counterexamples. The book is probably valuable to
cognoscenti for its breadth of topics, overview and organization, but
cannot be recommended as a text---except as a challenge to the more
mature student. Any automobile marketed with as many defects would
surely be recalled. Does the publisher deserve some blame for not
having had this critically read by a mathematician (and paid him/her
adequately to do so)? Shouldn't the author himself feel such an
obligation to the mathematical public? Ours is, after all, a
self-policing profession.

Reviewed by R. B. Burckel
_________________________________________________________________

95i:20001 20-01 (20C30 20C35 22E46)
Sternberg, S.(1-HRV)
Group theory and physics. (English. English summary)
Cambridge University Press, Cambridge, 1994. xiv+429 pp. ISBN
0-521-24870-1

There are hundreds of books written on group theory and perhaps a
hundred about physical applications, and it seems already impossible
to write something very outstanding. Nevertheless it can be done, as
we are witnessing here with a new book on applications of group theory
in physics where modern mathematics is nicely intertwined with
physics, from classical crystallography to fullerenes and from
symmetry properties of atoms and molecules to quarks. The book
contains a fresh approach to many topics and shows the highest degree
of mathematical competency. The text seems to be very friendly to
physicists though written in terms of modern mathematics (morphisms,
orbits, vector bundles, etc.). In addition there are interesting and
valuable excursions into the history of groups and spectroscopy and
citations of classical works which make the reading of the book a real
pleasure.

Perhaps the best introduction of the book would be to reproduce the
contents (the headings of sections being somewhat abridged).

Chapter 1. Basic definitions and examples (definition of group,
examples, homomorphisms, action on a set, conjugation, topology of
groups SU(2) and SO(3), morphisms, finite subgroups of SO(3) and O(3);
applications to crystallography, icosahedral group and fullerenes).

Chapter 2. Representation theory of finite groups (definitions,
examples, irreducibility, complete reducibility, Schur lemma,
characters, regular representation, acting on function spaces,
representations of the symmetric group).

Chapter 3. Molecular vibrations and homogeneous vector bundles (small
oscillations, molecular displacements and vector bundles, induced
representations, principal bundles, tensor products, operators and
selection rules, semiclassical theory of radiation, semidirect
products and their representations, Wigner's classification of irreps
of the Poincare group, parity, Mackey theorems on induced
representations with applications to the symmetric group, exchange
forces and induced representations).

Chapter 4. Compact groups and Lie groups (Haar measure, Peter-Weyl
theorem, irreps of SU(2), irreps of SO(3) and spherical harmonics;
hydrogen atom, periodic table, shell model of the nucleus,
CG-coefficients and isospin, relativistic wave equations; Lie
algebras, representations of su(2)).

Chapter 5. The irreducible representations of ${\rm SU}(n)$ (tensor
representations of ${\rm GL}(V)$, restrictions to some subgroups,
decompositions, computational rules, weight vectors,
finite-dimensional irreps of ${\rm Sl}(d,\bold C)$; strangeness, the
Eightfold Way, quarks, color and beyond. Where do we stand?).

Appendices. A. The Bravais lattices and the arithmetical crystal
classes. B. Tensor product. C. Integral geometry and the
representations of the symmetric group. D. Wigner's theorem on quantum
mechanical symmetries. E. Compact groups, Haar measure, and the
Peter-Weyl theorem. F. A history of 19th-century spectroscopy. G.
Characters and fixed point formulas for Lie groups. This book will
certainly become a landmark among the books on group theory in physics
as are the classical books by B. L. van der Waerden, H. Weyl, E. P.
Wigner, M. Hamermesh, etc.

Reviewed by J. Lohmus

© Copyright American Mathematical Society 1999

Abraham Avi Arad Hochman

From: orjanjo@math.ntnu.no (Orjan Johansen)
To:Avi Arad
Newsgroups: sci.math
Subject: Re: Distribution theory, Fourier transforms, and Parsevals theorem
Date: 7 Oct 1998 14:17:12 GMT

In article <6vc4h3$4hl$1@nnrp1.dejanews.com>,
wrote:
>
>I have questioned regarding distribution theory, Fourier transforms,
>and Parsevals theorem. Usually, there are two versions of this
>theorem: one version for periodic signals and the other version for
>aperiodic signals. For example, given x(t) = x(t+T), one version of
>the theorem is
>
>1/T int_{-T/2}^{T/2} abs(x(t))^2 dt = sum_k abs(a_k)^2
>
>where a_k are the Fourier coefficients. The other version for aperiodic
>signals is
>
>int_{-infinity}^{infinity} abs(x(t))^2 dt = int_{-infinity}^{infinity}
>abs(X(u)) ^2 dt
>
>where X(u) is the Fourier transform of x(t). I'm wondering if
>distribution theory unifies these two formulations.

I don't know about distribution theory, but one theory which does
unify these is the theory of harmonic analysis on (locally compact
abelian) groups. The relevant theorem there is:

Plancherel's theorem
--------------------
The Fourier transform from L^2(G) to L^2(G^), where G^ is the dual
group of G, is an isometry.

The integrals and sums above all calculate the square of the L^2-norm.
In the first case the group is the multiplicative unit circle of
complex numbers and its dual is the integers; In the second case the
group is the additive real numbers, which is self-dual.

Greetings,
Ørjan.

--
'What Einstein called "the happiest thought of my life" was his
realization that gravity and acceleration are both made of orange
Jello.' - from a non-crackpot sci.physics.relativity posting

Abraham Avi Arad Hochman

The most incomprehensible thing about the world is that it is comprehensible. - A. Einstein.

Abraham Avi Arad Hochman

Гармонические естественные спектры и аппроксимация коротких сигналов

Дмитриев Евгений Васильевич г.Воронеж (kvsj3903@yandex.ru)

Вашему вниманию предлагаются результаты выполненной научно-исследовательской работы из области инженерной техники и прикладной математики. Они подготовлены в виде небольшой книги, имеющей название "Гармонические естественные спектры и аппроксимация коротких сигналов".

Eugeny Dmitriev, City of Voronezh

To the attention of your I propose the results of carried-out work in the field of engineering and applied mathematics. It is got up as a small-sized book and has the title "Short signal harmonic natural Spectra and Approximation".

Краткое содержание
Гл.1 Проблема определения параметров короткого сигнала
Гл.2 Известные способы определения гармонического спектра короткого сигнала
Гл.3 Введение понятия спектра короткого сигнала, предложение способа его определения
Гл.4 Свойства спектров сигнала ПСО и КСО
Гл.5 Cпектры ПСО и КСО в сравнении со спектрами на основе разложений Фурье
Гл.6 Свойства спектров ПСО и КСО отрезка процесса
Гл.7 Примеры расчета спектров и аппроксимации коротких сигналов

Brief contents
Chap.1 The problem of short signal parameter definition
Chap.2 Conventional methods of short signal harmonic spectrum definition
Chap.3 Introduction of the short signal spectrum concept and proposals on the way of its definition
Chap.4 Properties of ESS and FSS signal spectra
Chap.5 ESS and FSS spectra in comparison with spectra based on Fourier expansion
Chap.6 Properties of ESS and FSS spectra in a process segment
Chap.7 Examples of spectra estimation and short signal approximation

Abraham Avi Arad Hochman

Electronic Journals on Mathematics.
Academic press electronic journal library
Acta Mathematica Universitatis Comenianae (mirror at EMS)
Algebra Montpellier Announcements
Annales Academiæ Scientiarum Fennicæ
Archivum Mathematicum (Brno)
BIT
Canadian Mathematical Society Journal Information
Central European Journal of Mathematics
Complexity International
Computational Methods in Applied Mathematics mathematical journal
Documenta Mathematica
Documenta Mathematica Mirror in USA
The East Journal on Approximations
Electronic Colloquium on Computational Complexity
Electronic Communications in Probability
Electronic Journal of Combinatorics
— Including the World Combinatorics Exchange
Electronic Journal of Differential Equations
Electronic Journal of Linear Algebra
Electronic Journal of Probability
The Electronic Journal of Qualitative Theory of Differential Equations
Electronic Library of the European Mathematical Society
Electronic Proceedings of the International Conference on Technology in Collegiate Mathematics
Electronic Research Announcements of the AMS
Electronic Transactions in Numerical Analysis
Euromath Bulletin
Furman University Electronic Journal of Undergraduate Mathematics
Forum Geometricorum
Geometry and Topology
Indiana University Mathematics Journal
Integers:Electronic Journal of Combinatorial Number Theory
Journal of Convex Analysis
Journal of Graph Algorithms and Applications
Journal of Lie Theory
Le Journal de Maths des Elèves
Journal de Théorie des Nombres
Lecturas Matematicas
Mathematica Journal
Mathematica World
Mathematical Physics Electronic Journal
Matematicki Vesnik
MathUser
MSU Electronic Journal of Applications in Mathematics
New York Journal of Mathematics
Nonlinear Science Today
Northeastern Mathematical Journal
Rendiconti dell'Istituto di Matematica dell'Università di Trieste
Revista Colombiana de Matematicas
Rose-Hulman Institute of Technology Undergraduate Mathematics Journal
School Mathematics
Séminaire Lotharingien de Combinatoire
Southwest Journal of Pure and Applied Mathematics
Statistical Science [Project Euclid]
Tangents
Theory and Applications of Categories
Ulam Quarterly
Printed Journals (some available electronically)
Bibliothèque Mathématique et Informatique, U Bordeaux for Tables of Contents of over 300 journals
Abstract and Applied Analysis
Abstract and Applied Analysis [Project Euclid]
Acta Applicandae Mathematicae
Acta Mathematica Hungarica
Acta Numerica
Acta Scientiarum Mathematicarum
Advances in Applied Mathematics
Advances in Computational Mathematics (contents)
Advances in Differential Equations
Advances in Mathematics
The Asian Journal of Mathematics
Journal für die reine und angewandte Mathematik (Crelle's Journal)
aequationes mathematicae
African Journal of Mathematics
algebra universalis
Archiv der Mathematik
Commentarii Mathematici Helvetici
computational complexity
Computational and Applied Mathematics
Divulgaciones Matematicas
Elemente der Mathematik
Geometric and Functional Analysis
Helvetica Physica Acta
Integral Equations and Operator Theory
Journal of Fourier Analysis and Applications
Journal of Geometry
Journal of Mathematical Fluid Mechanics
Journal of Mathematical Systems, Estimation, and Control
Maple Tech
Nonlinear Differential Equations and Applications
Physics in Perspective
Results in Mathematics
Selecta Mathematica, New Series
Transformation Groups
Zeitschrift für Angewandte Mathematik und Physik
Advances in Applied Probability [Project Euclid]
Algebras and Representation Theory
American Journal of Mathematics
American Mathematical Monthly
American Mathematical Society Journals (all)
Annales Academiae Scientiarum Fennicae
Annales de l'Institut Fourier
The Annals of Applied Probability [Project Euclid]
Annals of Combinatorics
Annals of Global Analysis and Geometry
Annals of the Institute of Statistical Mathematics
Annals of Mathematics
Annals of Mathematics [Project Euclid]
The Annals of Probability [Project Euclid]
The Annals of Statistics [Project Euclid]
Les Annales Scientifiques de l'Ecole Normale Supérieure
Applied Categorical Structures
Ars Combinatoria
Australian Mathematical Society Gazette
Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry
Bulletin of the American Mathematical Society
Bulletin of the Belgian Mathematical Society - Simon Stevin
Bulletin of the Belgian Mathematical Society-Simon Stevin [Project Euclid]
Bulletin of the London Mathematical Society
The Bulletin of Symbolic Logic
Bulletin of Symbolic Logic [Project Euclid]
Canadian Applied Mathematics Quarterly [Project Euclid]
Canadian Journal of Mathematics
Canadian Mathematical Bulletin
Canadian Mathematical Society Notes
Collectanea Mathematica
College Mathematics Journal
Combinatorics, Probability and Computing
Communications in Analysis and Geometry
Communications in Numerical Methods in Engineering
Complexity
Computational & Mathematical Organization Theory
Computational Optimization and Applications
Compositio Mathematica
Constraints
Constructive Approximation
Contrôle, Optimisation et Calcul des Variations
Crux Mathematicorum
Data Mining and Knowledge Discovery
Designs, Codes and Cryptography
Differential and Integral Equations
Discrete Applied Mathematics
Discrete and Computational Geometry
Discrete Event Dynamic Systems
Discrete Mathematics
Dissertation Summaries in Mathematics
Dynamics of Continuous, Discrete and Impulsive Systems
Dynamics and Control
Educational Studies in Mathematics
Elsevier Mathematics and Computer Science - Alert
L'Enseignement Mathématique
European Journal of Combinatorics
Experimental Mathematics
Experimental Mathematics [Project Euclid]
Extremes
Focus
Geometriae Dedicata
Illinois Journal of Mathematics
IMA Journal of Numerical Analysis (contents)
Indagationes Mathematicae
Infinite Dimensional Analysis, Quantum, Probability and Related Topics (IDAQP)
Informatica
International Journal of Computers for Mathematical Learning
International Journal for Numerical Methods in Engineering
International Linear Algebra Society, all journals
Inventiones Mathematicae
Israel Journal of Mathematics
Journal of Algebra
Journal of Algebraic Combinatorics
Journal of Applied Mathematics [Project Euclid]
Journal of Applied Probability [Project Euclid]
Journal of Approximation Theory
Journal of the Australian Mathematical Society (Series A)
Journal of Combinatorial Optimization
Journal of Combinatorial Theory, Series A
Journal of Combinatorial Theory, Series B
Journal of Differential Geometry
Journal of Differential Geometry Bibliography page
Journal of Dynamical and Control Systems
Journal of Engineering Mathematics
Journal of Fluid Mechanics
Journal of Generalized Lie Theory and Applications
Journal of Global Optimization
Journal of Graph Theory
Journal of Heuristics
Journal of Interdisciplinary Mathematics
Journal of the London Mathematical Society
Journal of Mathematical Imaging and Vision
Journal of Mathematics of Kyoto University
Journal of Number Theory
Journal de Théorie des Nombres de Bordeaux
Journal of Operator Theory
Journal of Symbolic Logic [Project Euclid]
Journal of Transfigural Mathematics
K-Theory
Kluwer Mathematics Journals (all)
Kodai Mathematical Journal [Project Euclid]
Letters in Mathematical Physics
Lifetime Data Analysis
Linear Algebra and its Applications (Index)
Linear and Multilinear Algebra (Index)
LISP and Symbolic Computation
MAA (all journals)
Mathematics of Computation
Mathematica Pannonica
Mathematical Physics, Analysis and Geometry
Mathematical Proceedings of the Cambridge Philosophical Society
Mathematics of Control, Signals, and Systems (Netherlands)
Mathematics of Control, Signals, and Systems (Rutgers)
Mathematics Magazine
Mathematical Connections
Mathematical Research Letters
Mathematische Zeitschrift
Methods and Applications of Analysis
The Michigan Mathematical Journal [Project Euclid]
Missouri Journal of Mathematical Sciences
Monte Carlo Methods and Applications
Multibody System Dynamics
New Zealand Journal of Mathematics
Nonlinear Dynamics
Notices of the AMS
Notre Dame Journal of Formal Logic [Project Euclid]
Numerical Algorithms (contents)
Numerische Mathematik
Open Systems & Information Dynamics
Order, Journal on the Theory of Ordered Sets and its Applications
Pacific Journal of Mathematics
Periodica Mathematica Hungarica
Pi Mu Epsilon Journal
Portugaliae Mathematica
Positivity
Potential Analysis
Primus
Probabilités et Statistique
Proceedings (ESAIM)
The Ramanujan Journal (The Editor-in-Chief's page at U. Florida)
The Ramanujan Journal (The Publisher's page in the Netherlands)
Regular and Chaotic Dynamics
Reliable Computing
Results in Mathematics/Resultate der Mathematik
Revista del Profesor de Matemáticas
Revista Matemática Iberoamericana [Project Euclid]
Russian Journal of Mathematical Physics
Saitama Mathematical Journal
Semigroup Forum
Serdica Mathematical Journal
Set-Valued Analysis
SIAM (all journals)
SIAM Journal on Applied Mathematics
SIAM Journal on Discrete Mathematics
SIAM Journal on Mathematical Analysis
SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis (contents)
the complete list of Springer Verlag journals
Statistical Inference for Stochastic Processes
Studia Logica
Surveys on Mathematics for Industry
Svenska matematikersamfundet
The Journal of Supercomputing
Theory and Decision
Topoi
Topological Methods in Non-Linear Analysis
Transactions on Mathematical Software
World Scientific Publishers
Mathematical Logic Quarterly Homepage
Mathematical Logic Quarterly on-line version
Mathematische Nachrichten (Mathematical News) Homepage
Mathematische Nachrichten (Mathematical News) on-line version
ZAMM - Zeitschrift für angewandte Mathematic und Mechanik (Journal of applied Mathematics and Mechanics) homepage
ZAMM - Zeitschrift für angewandte Mathematic und Mechanik (Journal of applied Mathematics and Mechanics) on-line version
Zentralblatt für Mathematik

Abraham Avi Arad Hochman

Abstract harmonic analysis

Introduction

Abstract harmonic analysis: if Fourier series is the study of periodic real functions, that is, real functions which are invariant under the group of integer translations, then abstract harmonic analysis is the study of functions on general topological groups which are invariant under a (closed) subgroup. This includes topics of varying level of specificity, including analysis on Lie groups or locally compact abelian groups. This area also overlaps with representation theory of topological groups.

History

Mackey, George W. : "Harmonic analysis as the exploitation of symmetry---a historical survey", Bull. Amer. Math. Soc. (N.S.) 3 (1980), no. 1, part 1, 543--698 (MR81d:01017)
Applications and related fields

For other analysis on topological and Lie groups, See 22Exx

One can carry over the development of Fourier series for functions on the circle and study the expansion of functions on the sphere; the basic functions then are the spherical harmonics -- see 33: Special Functions.

Subfields

There is only one division (43A) but it is subdivided:

43A05: Measures on groups and semigroups, etc.
43A07: Means on groups, semigroups, etc.; amenable groups
43A10: Measure algebras on groups, semigroups, etc.
43A15: L^p-spaces and other function spaces on groups, semigroups, etc.
43A17: Analysis on ordered groups, H^p-theory
43A20: L^1-algebras on groups, semigroups, etc.
43A22: Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
43A25: Fourier and Fourier-Stieltjes transforms on locally compact abelian groups
43A30: Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
43A32: Other transforms and operators of Fourier type
43A35: Positive definite functions on groups, semigroups, etc.
43A40: Character groups and dual objects
43A45: Spectral synthesis on groups, semigroups, etc.
43A46: Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.)
43A50: Convergence of Fourier series and of inverse transforms
43A55: Summability methods on groups, semigroups, etc., See Also 40J05
43A60: Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
43A62: Hypergroups
43A65: Representations of groups, semigroups, etc., See Also 22A10, 22A20, 22Dxx, 22E45
43A70: Analysis on specific locally compact abelian groups, See also 11R56, 22B05
43A75: Analysis on specific compact groups
43A77: Analysis on general compact groups
43A80: Analysis on other specific Lie groups, See also 22Exx
43A85: Analysis on homogeneous spaces
43A90: Spherical functions, See also 22E45, 22E46, 33C65, 33D55
43A95: Categorical methods, See also 46Mxx
43A99: Miscellaneous topics
This is among the smaller areas in the Math Reviews database.

Browse all (old) classifications for this area at the AMS.

Textbooks, reference works, and tutorials

Berenstein, Carlos A.: "The Pompeiu problem, what's new?", Complex analysis, harmonic analysis and applications (Bordeaux, 1995), 1--11; Pitman Res. Notes Math. Ser., 347; Longman, Harlow, 1996. MR97g:43007

Software and tables

Other web sites with this focus

Here are the AMS and Goettingen resource pages for area 43.
Selected Topics at this site

Plancherel's theorem: the Fourier transform is an isometry.
Computing the volume element on GL_n(R).
Invariant (Haar) measures on SO(3) and SE(3) -- some summaries
Haar measure and rotation group SO(n)
SNAG (Stone-Naimark-Ambrose-Godement) Theorem: construct measures corresponding to representations of LCA groups

Abraham Avi Arad Arad

The Logic of Complex Predicates
Examples

For convenience, we reproduce the item Logic/Logic of Complex Predicates from Principia Metaphysica:

In this tutorial, we give examples of the axioms and then draw out some consequences.
Examples of the Axioms

Instances of Axiom 1:


The first example asserts: objects c and a exemplify being an x and y such that x, b, and y exemplify S if and only if c, b, and a exemplify S. As an intuitive example, we might say: John and Betty stand in the relation being an x and y such that x gives Fido to y if and only if John gives Fido to Betty. The second example asserts: objects b and a exemplify being an x and y such necessarily, x fails to bear P to y if and only if necessarily, b fails to bear P to a. As a more concrete example: Sherlock Holmes and Gladstone exemplify being an x and y such that necessarily x and fail to meet each other if and only if necessarily, Holmes fails to meet Gladstone (the right condition in this biconditional is in fact true---an abstract object like Holmes is not the kind of thing that could meet an ordinary object such as Gladstone, and so given this principle, these two objects exemplify the complex relation denoted by the complex predicate; note, however, that the right condition of the biconditional is consistent with the claim: according to the Conan Doyle novels, Holmes met Gladstone). The third example asserts: object b exemplifies being an object x such that, necessarily, if x exemplifies Q, then x exemplifies having a spatiotemporal location if and only if necessarily, if b exemplifies Q, then b has a spatiotemporal location. As a concrete instance: an object y exemplifies being something such that, necessarily, if it exemplifies being a plant, it has a spatiotemporal location iff necessarily, if y exemplifies being a plant, y has a spatiotemporal location (the right condition of this biconditional is true for every object y whatsoever). Exercise: Produce specific examples of the other instances of Axiom 1. Remark: Notice that in the last instance of Axiom 1, we have an example of a 3-place relation that is exemplified just in case a 2-place relation is exemplified. The objects a, b, and c exemplify the relation being an x, y, and z such that x bears R to y just in case a bears R to b. Such relations will play an interesting role in developing the theory of possible worlds. In particular, we will focus on 1-place properties that objects exemplify just in case a proposition (0-place relation) is true. For example, by Axiom 1, we know that an object x exemplifies the property being such that Clinton is President if and only if Clinton is President. By Universal Generalization, it follows that every object x exemplifies being such that Clinton is President if and only if Clinton is President. And by the Rule of Necessitation, it follows that necessarily, every object x exemplifies being such that Clinton is President if and only if Clinton is President. We call these properties that objects exemplify whenever a proposition is true `propositional properties'. For a more precise definition, see the item Logic/Propositional Properties.
Instance of Axiom 2:


This instance asserts: the 3-place relation being an x, y, and z such that x, y, and z exemplify the relation G is identical to the 3-place relation G. In other words, the simplest possible complex predicates, consisting of an atomic exemplification formula in which all the object terms are variables bound by a lambda, denote the same relation as the simple predicate symbol involved in the atomic formula.
Instance of Axiom 3:


The first example asserts: the relation being an x and y such that necessarily, it is not the case that x bears P to y is identical with the relation being a y and z such that necessarily, it is not the case that y bears P to z. In other words, interchange of bound variable makes no difference to the relation denoted by the complex predicate.
Some Consequences:

Logical Theorem: Comprehension Principle for Relations


This tells us that for any exemplification-based formula phi (containing no quantifiers over relations or definite descriptions), there is a relation F which is necessarily such that objects x_1,...,x_n exemplify F if and only if x_1,...,x_n are such that phi. This is a logical theorem schema that can be derived from Axiom 1 in n+2 simple steps: apply Universal Generalization to Axiom 1 n times, beginning with the variable x_n and ending with the variable x_1; then apply the Rule of Necessitation; finally apply the rule of Existential Generalization to the complex predicate. EG can be applied to the complex predicate because the latter is an n-place term that denotes a relation.
Here are some examples of this logical theorem schema---these correspond to the examples of Axiom 1 above:


In each example, the existence of the complex relation in question is explicitly asserted. It is important to recall now that there is a definition of identity for relations. In the section Language/Relation Identity in Principia Metaphysica, one finds the following two definitions, the first which gives condtions under which properties (1-place relations) are identical and the second which gives conditions under which n-place relations (n > 1) are identical:

The first definition tells us that properties F and G are identical if and only if, necessarily, they encode the same properties. The second definition tells us that, for n > 1, n-place relations F and G are identical iff for each way of plugging n - 1 objects in the same order into F and G, the resulting 1-place properties are identical. These definitions, together with the comprehension principle for relations, constitute a mathematically precise theory of relations and properties, for they are explicit existence and identity conditions for these entities.
Notice that these definitions allow us to consistently assert that there are properties and relations which are necessarily exemplified by the same objects but which are nevertheless distinct. For example, the properties being equiangular and being equilateral are distinct properties, and so it is natural to suppose, therefore, that the property being an equiangular Euclidean triangle is distinct from the property being an equilateral Euclidean triangle. However, these two complex properties are exemplified by the same objects at every possible world (it is necessarily the case that anything exemplifying the one exemplifies the other). This is a virtue of the theory---many theories and/or treatments of properties identify properties that are necessarily equivalent. We identify properties only when they are necessarily equivalent with respect to the objects that encode them, not when they are necessarily equivalent with respect to the objects that exemplify them. The intuition here is that abstract objects represent possible objects of thought. If properties F and G are distinct, then it is possible to conceive of an object having (i.e., encoding) F and not G (and vice versa). So the `encoding extensions' of F and G are distinct when F and G are distinct. However, if F and G are not distinct properties, one couldn't conceive of an object having (i.e., encoding) F and not G. The encoding extensions of identical properties are the same.

Logical (Axioms and) Theorem: Comprehension Principle for Propositions


This is actually the `degenerate' case of Axiom 1 when n = 0. It can be read as follows: the proposition that-phi is true if and only if phi. Here are some examples:

We may read the first example as follows: the proposition that a exemplifies P and it is not the case that b exemplifies Q is true if and only if a exemplifies P and it is not the case that b exemplifies Q.
From the Axiom Schema, we may derive the following theorem schema that constitutes a comprehension principle for propositions:


Here are two instances of the theorem schema that correspond to the two instances (above) of the axiom schema:

These examples explicitly assert the existence of complex propositions.
Here, too, it is important to recall the definition for the identity of propostions in the item Language/Relation Identity:


This tells us that propositions p and q are identical if and only if the properties being such that p and being such that q are identical. The condition sunder which properties are identical has already been defined, and so propositional identiy is hereby reduced to property identity. Consequently, we now have a precise theory of propositions: the comprehension axiom and the above definition give us explicit existence and identity conditions for propositions.
This definition also allows us to consistently assert that certain necessarily equivalent propositions are nevertheless distinct. For example, both of the following propositions are necessarily false: Fido is a dog and it is not the case that Fido is a dog and There is a barber who shaves only those people who don't shave themselves. In formal terms, these propositions would be denoted by the following complex 0-place predicates:


Though the two propositions are necessarily equivalent (i.e., true in the same possible worlds), we may consistently assert that they are distinct. This stands in contrast to treatments of propositions on which necessarily equivalent propositions are identified, contrary to intuition.

Abraham Arad

There is a lot of debate on the net. Unfortunately, much of it is of very low quality. The aim of this document is to explain the basics of logical reasoning, and hopefully improve the overall quality of debate.

The Concise Oxford English Dictionary defines logic as "the science of reasoning, proof, thinking, or inference." Logic will let you analyze an argument or a piece of reasoning, and work out whether it is likely to be correct or not. You don't need to know logic to argue, of course; but if you know even a little, you'll find it easier to spot invalid arguments.

There are many kinds of logic, such as fuzzy logic and constructive logic; they have different rules, and different strengths and weaknesses. This document discusses simple Boolean logic, because it's commonplace and relatively easy to understand. When people talk about something being "logical," they usually mean the type of logic described here.

What logic isn't

It's worth mentioning a couple of things which logic is not.

First, logical reasoning is not an absolute law which governs the universe. Many times in the past, people have concluded that because something is logically impossible (given the science of the day), it must be impossible, period. It was also believed at one time that Euclidean geometry was a universal law; it is, after all, logically consistent. Again, we now know that the rules of Euclidean geometry are not universal.

Second, logic is not a set of rules which govern human behavior. Humans may have logically conflicting goals. For example:

John wishes to speak to whomever is in charge.
The person in charge is Steve.
Therefore John wishes to speak to Steve.
Unfortunately, John may have a conflicting goal of avoiding Steve, meaning that the reasoned answer may be inapplicable to real life.

This document only explains how to use logic; you must decide whether logic is the right tool for the job. There are other ways to communicate, discuss and debate.

Arguments

An argument is, to quote the Monty Python sketch, "a connected series of statements to establish a definite proposition."

Many types of argument exist; we will discuss the deductive argument. Deductive arguments are generally viewed as the most precise and the most persuasive; they provide conclusive proof of their conclusion, and are either valid or invalid.

Deductive arguments have three stages:

premises
inference
conclusion
However, before we can consider those stages in detail, we must discuss the building blocks of a deductive argument: propositions.

Propositions

A proposition is a statement which is either true or false. The proposition is the meaning of the statement, not the precise arrangement of words used to convey that meaning.

For example, "There exists an even prime number greater than two" is a proposition. (A false one, in this case.) "An even prime number greater than two exists" is the same proposition, reworded.

Unfortunately, it's very easy to unintentionally change the meaning of a statement by rephrasing it. It's generally safer to consider the wording of a proposition as significant.

It's possible to use formal linguistics to analyze and rephrase a statement without changing its meaning; but how to do so is outside the scope of this document.

Premises

A deductive argument always requires a number of core assumptions. These are called premises, and are the assumptions the argument is built on; or to look at it another way, the reasons for accepting the argument. Premises are only premises in the context of a particular argument; they might be conclusions in other arguments, for example.

You should always state the premises of the argument explicitly; this is the principle of audiatur et altera pars. Failing to state your assumptions is often viewed as suspicious, and will likely reduce the acceptance of your argument.

The premises of an argument are often introduced with words such as "Assume," "Since," "Obviously," and "Because." It's a good idea to get your opponent to agree with the premises of your argument before proceeding any further.

The word "obviously" is also often viewed with suspicion. It occasionally gets used to persuade people to accept false statements, rather than admit that they don't understand why something is "obvious." So don't be afraid to question statements which people tell you are "obvious"--when you've heard the explanation you can always say something like "You're right, now that I think about it that way, it is obvious."

Inference

Once the premises have been agreed, the argument proceeds via a step-by-step process called inference.

In inference, you start with one or more propositions which have been accepted; you then use those propositions to arrive at a new proposition. If the inference is valid, that proposition should also be accepted. You can use the new proposition for inference later on.

So initially, you can only infer things from the premises of the argument. But as the argument proceeds, the number of statements available for inference increases.

There are various kinds of valid inference--and also some invalid kinds, which we'll look at later on. Inference steps are often identified by phrases like "therefore ..." or "... implies that ..."

Conclusion

Hopefully you will arrive at a proposition which is the conclusion of the argument - the result you are trying to prove. The conclusion is the result of the final step of inference. It's only a conclusion in the context of a particular argument; it could be a premise or assumption in another argument.

The conclusion is said to be affirmed on the basis of the premises, and the inference from them. This is a subtle point which deserves further explanation.

Implication in detail

Clearly you can build a valid argument from true premises, and arrive at a true conclusion. You can also build a valid argument from false premises, and arrive at a false conclusion.

The tricky part is that you can start with false premises, proceed via valid inference, and reach a true conclusion. For example:

Premise: All fish live in the ocean
Premise: Sea otters are fish
Conclusion: Therefore sea otters live in the ocean
There's one thing you can't do, though: start from true premises, proceed via valid deductive inference, and reach a false conclusion.

We can summarize these results as a "truth table" for implication. The symbol "=>" denotes implication; "A" is the premise, "B" the conclusion. "T" and "F" represent true and false respectively.

Truth Table for Implication
Premise Conclusion Inference
A B A => B
false false true
false true true
true false false
true true true
If the premises are false and the inference valid, the conclusion can be true or false. (Lines 1 and 2.)
If the premises are true and the conclusion false, the inference must be invalid. (Line 3.)
If the premises are true and the inference valid, the conclusion must be true. (Line 4.)
So the fact that an argument is valid doesn't necessarily mean that its conclusion holds--it may have started from false premises.

If an argument is valid, and in addition it started from true premises, then it is called a sound argument. A sound argument must arrive at a true conclusion.

Example argument

Here's an example of an argument which is valid, and which may or may not be sound:

Premise: Every event has a cause
Premise: The universe has a beginning
Premise: All beginnings involve an event
Inference: This implies that the beginning of the universe involved an event
Inference: Therefore the beginning of the universe had a cause
Conclusion: The universe had a cause
The proposition in line 4 is inferred from lines 2 and 3. Line 1 is then used, with the proposition derived in line 4, to infer a new proposition in line 5. The result of the inference in line 5 is then restated (in slightly simplified form) as the conclusion.

Spotting arguments

Spotting an argument is harder than spotting premises or a conclusion. Lots of people shower their writing with assertions, without ever producing anything you might reasonably call an argument.

Sometimes arguments don't follow the pattern described above. For example, people may state their conclusions first, and then justify them afterwards. This is valid, but it can be a little confusing.

To make the situation worse, some statements look like arguments but aren't. For example:

"If the Bible is accurate, Jesus must either have been insane, a liar, or the Son of God."

That's not an argument; it's a conditional statement. It doesn't state the premises necessary to support its conclusion, and even if you add those assertions it suffers from a number of other flaws which are described in more detail in the Atheist Arguments document.

An argument is also not the same as an explanation. Suppose that you are trying to argue that Albert Einstein believed in God, and say:

"Einstein made his famous statement 'God does not play dice' because of his belief in God."

That may look like a relevant argument, but it's not; it's an explanation of Einstein's statement. To see this, remember that a statement of the form "X because Y" can be rephrased as an equivalent statement, of the form "Y therefore X." Doing so gives us:

"Einstein believed in God, therefore he made his famous statement 'God does not play dice.'"

Now it's clear that the statement, which looked like an argument, is actually assuming the result which it is supposed to be proving, in order to explain the Einstein quote.

Furthermore, Einstein did not believe in a personal God concerned with human affairs--again, see the Atheist Arguments document.

We've outlined the structure of a sound deductive argument, from premises to conclusion. But ultimately, the conclusion of a valid logical argument is only as compelling as the premises you started from. Logic in itself doesn't solve the problem of verifying the basic assertions which support arguments; for that, we need some other tool. The dominant means of verifying basic assertions is scientific enquiry. However, the philosophy of science and the scientific method are huge topics which are quite beyond the scope of this document.

Fallacies

There are a number of common pitfalls to avoid when constructing a deductive argument; they're known as fallacies. In everyday English, we refer to many kinds of mistaken beliefs as fallacies; but in logic, the term has a more specific meaning: a fallacy is a technical flaw which makes an argument unsound or invalid.

(Note that you can criticize more than just the soundness of an argument. Arguments are almost always presented with some specific purpose in mind--and the intent of the argument may also be worthy of criticism.)

Arguments which contain fallacies are described as fallacious. They often appear valid and convincing; sometimes only close inspection reveals the logical flaw.

Below is a list of some common fallacies, and also some rhetorical devices often used in debate. The list isn't intended to be exhaustive; the hope is that if you learn to recognize some of the more common fallacies, you'll be able to avoid being fooled by them.

The Nizkor Project has an excellent list of logical fallacies.

Sadly, many of the examples below have been taken directly from the Net, though some have been rephrased for the sake of clarity.

List of fallacies

Accent
Ad hoc
Affirmation of the consequent
Amphiboly
Anecdotal evidence
Argumentum ad antiquitatem
Argumentum ad baculum / Appeal to force
Argumentum ad crumenam
Argumentum ad hominem
Argumentum ad ignorantiam
Argumentum ad lazarum
Argumentum ad logicam
Argumentum ad misericordiam
Argumentum ad nauseam
Argumentum ad novitatem
Argumentum ad numerum
Argumentum ad populum
Argumentum ad verecundiam
Audiatur et altera pars
Bifurcation
Circulus in demonstrando
Complex question / Fallacy of interrogation / Fallacy of presupposition
Composition
Converse accident / Hasty generalization
Converting a conditional
Cum hoc ergo propter hoc
Denial of the antecedent
Dicto simpliciter / The fallacy of accident / Sweeping generalization
Division
Equivocation / Fallacy of four terms
Extended analogy
Ignoratio elenchi / Irrelevant conclusion
Natural Law fallacy / Appeal to Nature
"No True Scotsman ..." fallacy
Non causa pro causa
Non sequitur
Petitio principii / Begging the question
Plurium interrogationum / Many questions
Post hoc, ergo propter hoc
Red herring
Reification / Hypostatization
Shifting the burden of proof
Slippery slope argument
Straw man
Tu quoque
Undistributed Middle / "A is based on B" fallacies / "... is a type of ..." fallacies
For more fallacies, more examples, and scholarly references, see "Stephen's Guide to the Logical Fallacies." (Off Site)
Accent

Accent is a form of fallacy through shifting meaning. In this case, the meaning is changed by altering which parts of a statement are emphasized. For example:

"We should not speak ill of our friends"

and

"We should not speak ill of our friends"

Be particularly wary of this fallacy on the net, where it's easy to misread the emphasis of what's written.

Ad hoc (for this purpose only)

As mentioned earlier, there is a difference between argument and explanation. If we're interested in establishing A, and B is offered as evidence, the statement "A because B" is an argument. If we're trying to establish the truth of B, then "A because B" is not an argument, it's an explanation.

The Ad Hoc fallacy is to give an after-the-fact explanation which doesn't apply to other situations. Often this ad hoc explanation will be dressed up to look like an argument. For example, if we assume that God treats all people equally, then the following is an ad hoc explanation:

"I was healed from cancer."

"Praise the Lord, then. He is your healer."

"So, will He heal others who have cancer?"

"Er... The ways of God are mysterious."

Affirmation of the consequent

This fallacy is an argument of the form "A implies B, B is true, therefore A is true." To understand why it is a fallacy, examine the truth table for implication given earlier. Here's an example:

"If the universe had been created by a supernatural being, we would see order and organization everywhere. And we do see order, not randomness--so it's clear that the universe had a creator."

This is the converse of Denial of the Antecedent.

Amphiboly

Amphiboly occurs when the premises used in an argument are ambiguous because of careless or ungrammatical phrasing. For example:

"Premise: Belief in God fills a much-needed gap."

Anecdotal evidence

One of the simplest fallacies is to rely on anecdotal evidence. For example:

"There's abundant proof that God exists and is still performing miracles today. Just last week I read about a girl who was dying of cancer. Her whole family went to church and prayed for her, and she was cured."

It's quite valid to use personal experience to illustrate a point; but such anecdotes don't actually prove anything to anyone. Your friend may say he met Elvis in the supermarket, but those who haven't had the same experience will require more than your friend's anecdotal evidence to convince them.

Anecdotal evidence can seem very compelling, especially if the audience wants to believe it. This is part of the explanation for urban legends; stories which are verifiably false have been known to circulate as anecdotes for years.

Argumentum ad antiquitatem

This is the fallacy of asserting that something is right or good simply because it's old, or because "that's the way it's always been." The opposite of Argumentum ad Novitatem.

"For thousands of years Christians have believed in Jesus Christ. Christianity must be true, to have persisted so long even in the face of persecution."

Argumentum ad baculum (Appeal to force or fear)

An Appeal to Force happens when someone resorts to force (or the threat of force) to try and push others to accept a conclusion. This fallacy is often used by politicians, and can be summarized as "might makes right." The threat doesn't have to come directly from the person arguing. For example:

"Thus there is ample proof of the truth of the Bible. All those who refuse to accept that truth will burn in Hell."

"In any case, I know your phone number and I know where you live. Have I mentioned I am licensed to carry concealed weapons?"

Argumentum ad crumenam

The fallacy of believing that money is a criterion of correctness; that those with more money are more likely to be right. The opposite of Argumentum ad Lazarum. Example:

"Microsoft software is undoubtedly superior; why else would Bill Gates have gotten so rich?"

Argumentum ad hominem (Abusive: attacking the person)

Argumentum ad hominem literally means "argument directed at the man"; there are two varieties.

The first is the abusive form. If you refuse to accept a statement, and justify your refusal by criticizing the person who made the statement, then you are guilty of abusive argumentum ad hominem. For example:

"You claim that atheists can be moral--yet I happen to know that you abandoned your wife and children."

This is a fallacy because the truth of an assertion doesn't depend on the virtues of the person asserting it. A less blatant argumentum ad hominem is to reject a proposition based on the fact that it was also asserted by some other easily criticized person. For example:

"Therefore we should close down the church? Hitler and Stalin would have agreed with you."

A second form of argumentum ad hominem is to try and persuade someone to accept a statement you make, by referring to that person's particular circumstances. For example:

"Therefore it is perfectly acceptable to kill animals for food. I hope you won't argue otherwise, given that you're quite happy to wear leather shoes."

This is known as circumstantial argumentum ad hominem. The fallacy can also be used as an excuse to reject a particular conclusion. For example:

"Of course you'd argue that positive discrimination is a bad thing. You're white."

This particular form of Argumentum ad Hominem, when you allege that someone is rationalizing a conclusion for selfish reasons, is also known as "poisoning the well."

It's not always invalid to refer to the circumstances of an individual who is making a claim. If someone is a known perjurer or liar, that fact will reduce their credibility as a witness. It won't, however, prove that their testimony is false in this case. It also won't alter the soundness of any logical arguments they may make.

Argumentum ad ignorantiam (Argument from ignorance)

Argumentum ad ignorantiam means "argument from ignorance." The fallacy occurs when it's argued that something must be true, simply because it hasn't been proved false. Or, equivalently, when it is argued that something must be false because it hasn't been proved true.

(Note that this isn't the same as assuming something is false until it has been proved true. In law, for example, you're generally assumed innocent until proven guilty.)

Here are a couple of examples:

"Of course the Bible is true. Nobody can prove otherwise."

"Of course telepathy and other psychic phenomena do not exist. Nobody has shown any proof that they are real."

In scientific investigation, if it is known that an event would produce certain evidence of its having occurred, the absence of such evidence can validly be used to infer that the event didn't occur. It does not prove it with certainty, however.

For example:

"A flood as described in the Bible would require an enormous volume of water to be present on the earth. The earth doesn't have a tenth as much water, even if we count that which is frozen into ice at the poles. Therefore no such flood occurred."

It is, of course, possible that some unknown process occurred to remove the water. Good science would then demand a plausible testable theory to explain how it vanished.

Of course, the history of science is full of logically valid bad predictions. In 1893, the Royal Academy of Science were convinced by Sir Robert Ball that communication with the planet Mars was a physical impossibility, because it would require a flag as large as Ireland, which it would be impossible to wave. [Fortean Times Number 82.]

See also Shifting the Burden of Proof.

Argumentum ad lazarum

The fallacy of assuming that someone poor is sounder or more virtuous than someone who's wealthier. This fallacy is the opposite of the Argumentum ad Crumenam. For example:

"Monks are more likely to possess insight into the meaning of life, as they have given up the distractions of wealth."

Argumentum ad logicam

This is the "fallacy fallacy" of arguing that a proposition is false because it has been presented as the conclusion of a fallacious argument. Remember always that fallacious arguments can arrive at true conclusions.

"Take the fraction 16/64. Now, canceling a six on top and a six on the bottom, we get that 16/64 = 1/4."

"Wait a second! You can't just cancel the six!"

"Oh, so you're telling us 16/64 is not equal to 1/4, are you?"

Argumentum ad misericordiam (Appeal to pity; Special pleading)

This is the Appeal to Pity, also known as Special Pleading. The fallacy is committed when someone appeals to pity for the sake of getting a conclusion accepted. For example:

"I did not murder my mother and father with an axe! Please don't find me guilty; I'm suffering enough through being an orphan."

Argumentum ad nauseam

This is the incorrect belief that an assertion is more likely to be true, or is more likely to be accepted as true, the more often it is heard. So an Argumentum ad Nauseam is one that employs constant repetition in asserting something; saying the same thing over and over again until you're sick of hearing it.

On the Net, your argument is often less likely to be heard if you repeat it over and over again, as people will tend to put you in their kill files.

Argumentum ad novitatem

This is the opposite of the Argumentum ad Antiquitatem; it's the fallacy of asserting that something is better or more correct simply because it is new, or newer than something else.

"BeOS is a far better choice of operating system than OpenStep, as it has a much newer design."

Argumentum ad numerum

This fallacy is closely related to the argumentum ad populum. It consists of asserting that the more people who support or believe a proposition, the more likely it is that that proposition is correct. For example:

"The vast majority of people in this country believe that capital punishment has a noticeable deterrent effect. To suggest that it doesn't in the face of so much evidence is ridiculous."

"All I'm saying is that thousands of people believe in pyramid power, so there must be something to it."

Argumentum ad populum (Appeal to the people or gallery)

This is known as Appealing to the Gallery, or Appealing to the People. You commit this fallacy if you attempt to win acceptance of an assertion by appealing to a large group of people. This form of fallacy is often characterized by emotive language. For example:

"Pornography must be banned. It is violence against women."

"For thousands of years people have believed in Jesus and the Bible. This belief has had a great impact on their lives. What more evidence do you need that Jesus was the Son of God? Are you trying to tell those people that they are all mistaken fools?"

Argumentum ad verecundiam (Appeal to authority)

The Appeal to Authority uses admiration of a famous person to try and win support for an assertion. For example:

"Isaac Newton was a genius and he believed in God."

This line of argument isn't always completely bogus when used in an inductive argument; for example, it may be relevant to refer to a widely-regarded authority in a particular field, if you're discussing that subject. For example, we can distinguish quite clearly between:

"Hawking has concluded that black holes give off radiation"

and

"Penrose has concluded that it is impossible to build an intelligent computer"

Hawking is a physicist, and so we can reasonably expect his opinions on black hole radiation to be informed. Penrose is a mathematician, so it is questionable whether he is well-qualified to speak on the subject of machine intelligence.

Audiatur et altera pars

Often, people will argue from assumptions which they don't bother to state. The principle of Audiatur et Altera Pars is that all of the premises of an argument should be stated explicitly. It's not strictly a fallacy to fail to state all of your assumptions; however, it's often viewed with suspicion.

Bifurcation

Also referred to as the "black and white" fallacy and "false dichotomy," bifurcation occurs if someone presents a situation as having only two alternatives, where in fact other alternatives exist or can exist. For example:

"Either man was created, as the Bible tells us, or he evolved from inanimate chemicals by pure random chance, as scientists tell us. The latter is incredibly unlikely, so ..."

Circulus in demonstrando

This fallacy occurs if you assume as a premise the conclusion which you wish to reach. Often, the proposition is rephrased so that the fallacy appears to be a valid argument. For example:

"Homosexuals must not be allowed to hold government office. Hence any government official who is revealed to be a homosexual will lose his job. Therefore homosexuals will do anything to hide their secret, and will be open to blackmail. Therefore homosexuals cannot be allowed to hold government office."

Note that the argument is entirely circular; the premise is the same as the conclusion. An argument like the above has actually been cited as the reason for the British Secret Services' official ban on homosexual employees.

Circular arguments are surprisingly common, unfortunately. If you've already reached a particular conclusion once, it's easy to accidentally make it an assertion when explaining your reasoning to someone else.

Complex question / Fallacy of interrogation / Fallacy of presupposition

This is the interrogative form of Begging the Question. One example is the classic loaded question:

"Have you stopped beating your wife?"

The question presupposes a definite answer to another question which has not even been asked. This trick is often used by lawyers in cross-examination, when they ask questions like:

"Where did you hide the money you stole?"

Similarly, politicians often ask loaded questions such as:

"How long will this EU interference in our affairs be allowed to continue?"

or

"Does the Chancellor plan two more years of ruinous privatization?"

Another form of this fallacy is to ask for an explanation of something which is untrue or not yet established.

Composition

The Fallacy of Composition is to conclude that a property shared by a number of individual items, is also shared by a collection of those items; or that a property of the parts of an object, must also be a property of the whole thing. Examples:

"The bicycle is made entirely of low mass components, and is therefore very lightweight."

"A car uses less petrochemicals and causes less pollution than a bus. Therefore cars are less environmentally damaging than buses."

A related form of fallacy of composition is the "just" fallacy, or fallacy of mediocrity. This is the fallacy that assumes that any given member of a set must be limited to the attributes that are held in common with all the other members of the set. Example:

"Humans are just animals, so we should not concern ourselves with justice; we should just obey the law of the jungle."

Here the fallacy is to reason that because we are animals, we can have only properties which animals have; that nothing can distinguish us as a special case.

Converse accident / Hasty generalization

This fallacy is the reverse of the Fallacy of Accident. It occurs when you form a general rule by examining only a few specific cases which aren't representative of all possible cases. For example:

"Jim Bakker was an insincere Christian. Therefore all Christians are insincere."

Converting a conditional

This fallacy is an argument of the form "If A then B, therefore if B then A."

"If educational standards are lowered, the quality of argument seen on the Net worsens. So if we see the level of debate on the net get worse over the next few years, we'll know that our educational standards are still falling."

This fallacy is similar to the Affirmation of the Consequent, but phrased as a conditional statement.

Cum hoc ergo propter hoc

This fallacy is similar to post hoc ergo propter hoc. The fallacy is to assert that because two events occur together, they must be causally related. It's a fallacy because it ignores other factors that may be the cause(s) of the events.

"Literacy rates have steadily declined since the advent of television. Clearly television viewing impedes learning."

This fallacy is a special case of the more general non causa pro causa.

Denial of the antecedent

This fallacy is an argument of the form "A implies B, A is false, therefore B is false." The truth table for implication makes it clear why this is a fallacy.

Note that this fallacy is different from Non Causa Pro Causa. That has the form "A implies B, A is false, therefore B is false," where A does not in fact imply B at all. Here, the problem isn't that the implication is invalid; rather it's that the falseness of A doesn't allow us to deduce anything about B.

"If the God of the Bible appeared to me, personally, that would certainly prove that Christianity was true. But God has never appeared to me, so the Bible must be a work of fiction."

This is the converse of the fallacy of Affirmation of the Consequent.

Dicto simpliciter / Fallacy of accident / Sweeping generalization

A sweeping generalization occurs when a general rule is applied to a particular situation, but the features of that particular situation mean the rule is inapplicable. It's the error made when you go from the general to the specific. For example:

"Christians generally dislike atheists. You are a Christian, so you must dislike atheists."

This fallacy is often committed by people who try to decide moral and legal questions by mechanically applying general rules.

Division

The fallacy of division is the opposite of the Fallacy of Composition. It consists of assuming that a property of some thing must apply to its parts; or that a property of a collection of items is shared by each item.

"You are studying at a rich college. Therefore you must be rich."

"Ants can destroy a tree. Therefore this ant can destroy a tree."

Equivocation / Fallacy of four terms

Equivocation occurs when a key word is used with two or more different meanings in the same argument. For example:

"What could be more affordable than free software? But to make sure that it remains free, that users can do what they like with it, we must place a license on it to make sure that will always be freely redistributable."

One way to avoid this fallacy is to choose your terminology carefully before beginning the argument, and avoid words like "free" which have many meanings.

Extended analogy

The fallacy of the Extended Analogy often occurs when some suggested general rule is being argued over. The fallacy is to assume that mentioning two different situations, in an argument about a general rule, constitutes a claim that those situations are analogous to each other.

Here's real example from an online debate about anti-cryptography legislation:

"I believe it is always wrong to oppose the law by breaking it."

"Such a position is odious: it implies that you would not have supported Martin Luther King."

"Are you saying that cryptography legislation is as important as the struggle for Black liberation? How dare you!"

Ignoratio elenchi / Irrelevant conclusion

The fallacy of Irrelevant Conclusion consists of claiming that an argument supports a particular conclusion when it is actually logically nothing to do with that conclusion.

For example, a Christian may begin by saying that he will argue that the teachings of Christianity are undoubtedly true. If he then argues at length that Christianity is of great help to many people, no matter how well he argues he will not have shown that Christian teachings are true.

Sadly, these kinds of irrelevant arguments are often successful, because they make people to view the supposed conclusion in a more favorable light.

Natural Law fallacy / Appeal to Nature

The Appeal to Nature is a common fallacy in political arguments. One version consists of drawing an analogy between a particular conclusion, and some aspect of the natural world--and then stating that the conclusion is inevitable, because the natural world is similar:

"The natural world is characterized by competition; animals struggle against each other for ownership of limited natural resources. Capitalism, the competitive struggle for ownership of capital, is simply an inevitable part of human nature. It's how the natural world works."

Another form of appeal to nature is to argue that because human beings are products of the natural world, we must mimic behavior seen in the natural world, and that to do otherwise is "unnatural":

"Of course homosexuality is unnatural. When's the last time you saw two animals of the same sex mating?"

An example of "Appeal to Nature" taken to extremes is The Unabomber Manifesto.

"No True Scotsman ..." fallacy

Suppose I assert that no Scotsman puts sugar on his porridge. You counter this by pointing out that your friend Angus likes sugar with his porridge. I then say "Ah, yes, but no true Scotsman puts sugar on his porridge.

This is an example of an ad hoc change being used to shore up an assertion, combined with an attempt to shift the meaning of the words used original assertion; you might call it a combination of fallacies.

Non causa pro causa

The fallacy of Non Causa Pro Causa occurs when something is identified as the cause of an event, but it has not actually been shown to be the cause. For example:

"I took an aspirin and prayed to God, and my headache disappeared. So God cured me of the headache."

This is known as a false cause fallacy. Two specific forms of non causa pro causa fallacy are the cum hoc ergo propter hoc and post hoc ergo propter hoc fallacies.

Non sequitur

A non sequitur is an argument where the conclusion is drawn from premises which aren't logically connected with it. For example:

"Since Egyptians did so much excavation to construct the pyramids, they were well versed in paleontology."

(Non sequiturs are an important ingredient in a lot of humor. They're still fallacies, though.)

Petitio principii (Begging the question)

This fallacy occurs when the premises are at least as questionable as the conclusion reached. Typically the premises of the argument implicitly assume the result which the argument purports to prove, in a disguised form. For example:

"The Bible is the word of God. The word of God cannot be doubted, and the Bible states that the Bible is true. Therefore the Bible must be true.

Begging the question is similar to circulus in demonstrando, where the conclusion is exactly the same as the premise.

Plurium interrogationum / Many questions

This fallacy occurs when someone demands a simple (or simplistic) answer to a complex question.

"Are higher taxes an impediment to business or not? Yes or no?"

Post hoc ergo propter hoc

The fallacy of Post Hoc Ergo Propter Hoc occurs when something is assumed to be the cause of an event merely because it happened before that event. For example:

"The Soviet Union collapsed after instituting state atheism. Therefore we must avoid atheism for the same reasons."

This is another type of false cause fallacy.

Red herring

This fallacy is committed when someone introduces irrelevant material to the issue being discussed, so that everyone's attention is diverted away from the points made, towards a different conclusion.

"You may claim that the death penalty is an ineffective deterrent against crime--but what about the victims of crime? How do you think surviving family members feel when they see the man who murdered their son kept in prison at their expense? Is it right that they should pay for their son's murderer to be fed and housed?"

Reification / Hypostatization

Reification occurs when an abstract concept is treated as a concrete thing.

"I noticed you described him as 'evil.' Where does this 'evil' exist within the brain? You can't show it to me, so I claim it doesn't exist, and no man is 'evil.'"

Shifting the burden of proof

The burden of proof is always on the person asserting something. Shifting the burden of proof, a special case of Argumentum ad Ignorantiam, is the fallacy of putting the burden of proof on the person who denies or questions the assertion. The source of the fallacy is the assumption that something is true unless proven otherwise.

For further discussion of this idea, see the "Introduction to Atheism" document.

"OK, so if you don't think the grey aliens have gained control of the US government, can you prove it?"

Slippery slope argument

This argument states that should one event occur, so will other harmful events. There is no proof made that the harmful events are caused by the first event. For example:

"If we legalize marijuana, then more people would start to take crack and heroin, and we'd have to legalize those too. Before long we'd have a nation full of drug-addicts on welfare. Therefore we cannot legalize marijuana."

Straw man

The straw man fallacy is when you misrepresent someone else's position so that it can be attacked more easily, knock down that misrepresented position, then conclude that the original position has been demolished. It's a fallacy because it fails to deal with the actual arguments that have been made.

"To be an atheist, you have to believe with absolute certainty that there is no God. In order to convince yourself with absolute certainty, you must examine all the Universe and all the places where God could possibly be. Since you obviously haven't, your position is indefensible."

The above straw man argument appears at about once a week on the net. If you can't see what's wrong with it, read the "Introduction to Atheism" document.

Tu quoque

This is the famous "you too" fallacy. It occurs if you argue that an action is acceptable because your opponent has performed it. For instance:

"You're just being randomly abusive."

"So? You've been abusive too."

This is a personal attack, and is therefore a special case of Argumentum ad Hominem.

Undistributed Middle / "A is based on B" fallacies / "... is a type of ..." fallacies

These fallacies occur if you attempt to argue that things are in some way similar, but you don't actually specify in what way they are similar. Examples:

"Isn't history based upon faith? If so, then isn't the Bible also a form of history?"

"Islam is based on faith, Christianity is based on faith, so isn't Islam a form of Christianity?"

"Cats are a form of animal based on carbon chemistry, dogs are a form of animal based on carbon chemistry, so aren't dogs a form of cat?"