A Logic of Exceptions

Second Edition March 2011, ISBN 978-90-804774-7-6

€ 24.95



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Download software in The Economics Pack (1995, 2011)

Abstract of main results for teachers and advanced students.

Book review by Richard Gill in Nieuw Archief voor Wiskunde, September 2008

Book Review by the European Mathematical Society, February 2012

Russell’s Paradox and Logicomix

Mathematica and books at Wolfram

Cover text

A LOGIC OF EXCEPTIONS provides the concepts and tools for sound inference. Discussed are: (1) the basic elements: propositional operators, predicates and sets; (2) the basic notions: inference, syllogism, axiomatics, proof theory; (3) the basic extra’s: history, relation to the scientific method, the paradoxes. The new elements in the book are: (4) a logic of exceptions, solutions for those paradoxes, analysis of common errors in the literature, routines in Mathematica. The book is intended to be used in the first year of college or university. The last two chapters require a more advanced level that is worked up to.

Logic is used not only in science and mathematics but also in business and sometimes in politics and government. Logic and inference however can suffer from paradoxes such as the Liar paradox “This sentence is false” or the proof-theoretic variant by Gödel “This statement is not provable” or the Russell set paradox of “The catalogue of all catalogues that don’t mention themselves”. This book explains and solves those paradoxes, and thereby gives a clarity that was lacking up to now. The author proposes the new approach that a concept, such as the definition of truth or the notion of proof or the definition of a set, also reckons with the exceptions that may pertain to its very definition. The approach to keep exceptions in the back of one’s mind is a general sign of intelligence.

A quote from this book: “Since the Egyptians, mankind has been trying to solve the problem of bureaucracy. One frequent approach is the rule of law, say, that a supreme law-giver defines a rule that a bureaucracy must enforce. It is difficult for a law however to account for all kinds of exceptions that might be considered in its implementation. Ruthless enforcement might well destroy the very intentions of that law. Some bureaucrats might still opt for such enforcement merely to play it safe that nobody can say that they don't do their job. Decades may pass before such detrimental application is noticed and revised. There is the story of Catherine the Great regularly visiting a small park for a rest in the open air, so that they put a guard there; and some hundred years after her death somebody noticed that guarding that small park had become kind of silly. When both law-givers and bureaucrats grow more aware of some logic of exceptions then they might better deal with the contingencies of public management. It is a long shot to think so, of course, but in general it would help when people are not only aware of the rigour of a logical argument or rule but also of the possibility of some exception.”

You can benefit from this book also when you don’t have the software. However, with the software, you will have an interactive environment in which you can test the propositions in this book and your own deductions. The software is included in The Economics Pack - by the same author - which is an application of Mathematica, a system for doing mathematics with the computer. The Pack has users in many countries in the world. The Pack is available for Windows XP, Macintosh and Unix platforms and requires Mathematica 8.0.1 or later. It can be freely downloaded, but you need a licence to run it.

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Warning: This book proves radical new ideas and must be read with care. If you spend more than a cursory glance on the PDF then it is advisable to get the hardcopy and continue reading from paper. I tend to focus my research on misconceptions that lead society away from common sense, and then I select pivots that cause crucially different points of view depending on how the argument is resolved. Such a pivot only works well if the argumentation gets proper attention.


A. The book and/or software
Type Price Description
The book only
€ 24.95
One copy of "A Logic of Exceptions" (ALOE)    Order   Note: In 2012 the website was relocated. Above PDF has been updated with the name of the present website. Also some minor typos were corrected. The hardcopy print however uses the original PDF.
See The Economics Pack (TEP)
B. Site licences
For site licences, contact the author.

A note by one of the first readers

This note is by R.D. Gill (Leiden university) April 23 2007:
"A Logic of Exceptions: I am presently reading, and rereading, a fascinating book about logic, science, Gödel's theorem, and much else besides. In fact I'm composing a book-review on it for the Dutch mathematical society's magazine "Nieuw Archief". The book is giving me a lot to ponder about, and has already made me change my mind concerning several issues on which I have long held strong opinions. (To give just one example: two months ago, I would have unhesitatingly said that three-valued logic was not a fruitful idea). I recommend the book highly to anyone who is not afraid to discover that they might have been wrong concerning a cherished opinion. To clinch the argument - you can freely download a pdf of the text at the book's internet site. The book does confirm a different opinion of mine: though I consider Gödel one of the great heroes of 20th century mathematics, any idea that his beautiful theorems have anything to say about artificial intelligence, quantum theory, consciousness, mind, or [whatever], is pure bunkum."

PM. The first edition of 2007 (PDF, no longer in print).

A note on work in progress

ALOE is used in "Elementary Statistics and Causality" (ESAC), planned somewhere in 2015+

Paper A difficulty in proof theory (August 2007)

Paper Improving the logical base of calculus on the issue of “division by zero” (July 2007) - but see here Conquest of the Plane

Paper Contra Cantor Pro Occam - Proper constructivism with abstraction (March 2012, March 2013) (CCPO-PCWA)
PM. Legacy paper Contra Cantor Pro Occam (August 2011, update March 2012) superseded, but still with a bit on infinitesimals.
PM. Reply to Jan van Rongen (March 2012). 

ZFC is inconsistent. A condition by Paul of Venice (1369-1429) solves Russell's Paradox, blocks Cantor's Diagonal Argument, and provides a challenge to ZFC,
November 14 2014 - June 17 2015. (PV-RP-CDA-ZFC)
(1) Since June 17 2015 the phrase "ZFC is inconsistent" has become the main title, so that the reference to Paul of Venice has become the subtitle. The part on ZFC itself gave cause to split off the "Two results" paper below. The Paul of Venice article still is important for the alternative axioms, and the question what to do with Cantor's Theorem and the transfinites.
(2) Till May 27 2015 the focus was on asking questions to users of ZFC, but then I decided to do some proofs myself.
(3) See the earlier versions. Version 1 has 6 pages and still is fine. Later versions got somewhat cluttered. June 17 2015 is streamlined again, but 30 pages.
(4) Chance had - if it exists - that K.P. Hart (TU Delft) published in NAW on the history of the diagonal argument. I sent this criticism to NAW, May 5, updated June 19 2015.
(4a) The criticism was included as Appendix B in the May 1 - June 12 versions. Those also contained Appendix C on another proof by Hart. Here is Hart's response to Appendix B and my rejoinder. You have to look in the June 12 version to find these Appendices B and C, since those have been replaced and removed in the June 17 version. Hart's proof in that Appendix C now has become Theorem 1.3.B in the "Two results" paper below, and it appears to provide a transparant way to show that ZFC is inconsistent.
(4b) Background supporting documentation for Appendix B in the paper on Paul of Venice, May 20 2015
(4c) Here is the email exchange Colignatus - Hart 2011-2015.
(5) The paper refers to: Memo for Bas Edixhoven and Jan Bergstra (both KNAW), part Dutch part English on CCPO-PCWA and Paul of Venice, October 29 & November 10 2014.
(6) See also a breach to integrity, also below.
(7) See a NAW Referee Report of June 11 2015 with my comments of June 15 2015.

Two results on ZFC: (1) If ZFC is consistent then it is deductively incomplete, (2) ZFC is inconsistent, June 18 & 19 2015 (12 pages, this selects only ZFC from the former.)

Some notes in Dutch

L.E.J. Brouwer begreep de Uitgesloten Derde niet - reaction to Van Atten & Sundholm in NAW June 2014

A protest w.r.t. scientific integrity in the study of logic

Breach of scientific integrity 1981-2014 by Johan van Benthem, professor of logic at the universities of Groningen and Amsterdam, September 2014

A breach of scientific integrity since 1980 on the common logical paradoxes, May 21 2015