Outstanding Contributions to Logic 14 Melvin Fitting Brian Rayman Editors Raymond Smullyan on Self Reference Outstanding Contributions to Logic Volume 14 Editor-in-chief Sven Ove Hansson, Royal Institute of Technology, Stockholm, Sweden Editorial Board Marcus Kracht, Universität Bielefeld Lawrence Moss, Indiana University Sonja Smets, Universiteit van Amsterdam Heinrich Wansing, Ruhr-Universität Bochum More information about this series at http://www.springer.com/series/10033 Melvin Fitting • Brian Rayman Editors Raymond Smullyan on Self Reference 123 Editors Melvin Fitting Brian Rayman City University of New York (CUNY) NYC Department of Education System Brooklyn, NY City University of New York USA New York City, NY USA ISSN 2211-2758 ISSN 2211-2766 (electronic) Outstanding Contributions to Logic ISBN 978-3-319-68731-5 ISBN 978-3-319-68732-2 (eBook) https://doi.org/10.1007/978-3-319-68732-2 Library of Congress Control Number: 2017956336 © Springer International Publishing AG 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Cover illustration: Melvin Fitting Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Preface This volume began as a tribute to Raymond Smullyan, with the additional hope that it would please him. It ended as a tribute to his memory. Along the way, it did please him, in a pre-publication version. Raymond Smullyan had a lifelong interest in self-reference. Actually, this is a considerable understatement. He returned to the subject over and over. Self-reference and fixpoints are closely related, after all. His first technical publi- cations, preceding his Ph.D., had to do with self-reference, as did many of his technical books. Outside of mathematics, he is best-known for his popular puzzle books, and self-reference is a fundamental theme of many of these. The present volume is not a detailed formal study of Smullyan’s work. Rather it consists of commentary on and extensions of this work, by colleagues, friends, and admirers. We hope it provides some enlightenment and some entertainment, which is what Raymond would have hoped for too. New York City, USA Melvin Fitting Brooklyn, USA Brian Rayman v Contents 1 Introduction .......................................... 1 Melvin Fitting 2 Formal Systems, Logics, and Programs ..................... 23 Robert L. Constable 3 Adaptive Fault Diagnosis using Self-Referential Reasoning ....... 39 Robert Cowen 4 Russell’s Paradox, Gödel’s Theorem ........................ 47 Melvin Fitting 5 Dance of the Starlings ................................... 67 Henk Barendregt, Jörg Endrullis, Jan Willem Klop and Johannes Waldmann 6 Some Tweets About Mockingbirds ......................... 113 Rick Statman 7 Knights, Knaves, Truth, Truthfulness, Grounding, Tethering, Aboutness, and Paradox ................................. 123 Stephen Yablo 8 What I Tell You Three Times Is True ...................... 141 Martin Davis 9Gödel, Lucas, and the Soul-Searching Selfie .................. 147 Vann McGee 10 An Island Tale for Young Anthropologists ................... 165 Andrew G. Buchanan and John H. Conway vii viii Contents 11 Making The ‘Hardest Logic Puzzle Ever’ a Bit Harder ......... 181 Walter Carnielli 12 Bibliography of Raymond Smullyan ........................ 191 Raymond M. Smullyan Chapter 1 Introduction Melvin Fitting Raymond Smullyan was born in 1919, after the Great War and before the Great Depression. I have known him since I was a graduate student and he was my advisor. I got my PhD in 1968, so it is clear that it has been a long time that I have known him. In this introduction I will call him “Raymond.” This may sound formal to those who know him as “Ray,” but it is what I have always called him, and I don’t feel like changing now. Raymond has had two long, different, but overlapping careers: one as an academic mathematician and philosopher, and another as a more popular figure.1 Mathematics and philosophy are intertwined, each influences the other, and it is essentially impos- sible to discuss one without the other. Further, large parts of both relate directly to our chosen topic of self-reference. We begin with a broad overview, followed by sections on his technical and on his more popular work. 1.1 The Broad Background As an academic, Raymond has written several books and numerous research papers that have been influential, often in unexpected ways. His first book, Theory of For- mal Systems (Smullyan 1961) was a novel presentation of recursion theory and the Gödel incompleteness theorems, topics he revisited often throughout his career. The book introduced ideas that were influential in automata theory, as well as a decidedly abstract approach to incompleteness and undecidability results concerning the foun- dations of mathematics. Remarkably and unexpectedly, it also anticipated some of 1I omit discussion of his lives as pianist and as magician. M. Fitting (B) City University of New York (emeritus), Montrose, NY 10548, USA e-mail: melvin.fi[email protected] © Springer International Publishing AG 2017 1 M. Fitting and B. Rayman (eds.), Raymond Smullyan on Self Reference, Outstanding Contributions to Logic 14, https://doi.org/10.1007/978-3-319-68732-2_1 2 M. Fitting the ideas underlying the computer language Prolog, developed much later for use in artificial intelligence. Neither at that time nor today has Raymond claimed to know the first thing about computer science, but of course that is not a prerequisite for having an influence. Raymond’s second book, First-Order Logic (Smullyan 1968) was intended to be an elegant and beautiful presentation of formal logic.2 It was indeed elegant, and influenced several generations of logicians. A few years ago it was reprinted in the Dover book series, and continues its influence. But starting in the 1980s it has also had an unanticipated life in computer science, in the field of automated theorem proving. There is a long-running annual conference, Tableaux, with methods and programs covering many areas of logic and mathematics, and many a paper is, in a way, a direct or indirect descendent of Raymond’s 1968 book. First-Order Logic, along with a journal paper of Raymond, A Unifying principle in quantification the- ory, (Smullyan 1965), introduced what became called the Model Existence Theorem, which later became a fundamental tool in infinitary logic, again an unexpected out- come of Raymond’s search for elegance. But the areas covered in First-Order Logic do not involve self-reference, and we will not discuss it further here. There have been several other of Raymond’s books devoted to his research in mathematical logic: Gödel’s Incompleteness Theorems (Smullyan 1992a); Recursion Theory for Metamathematics (Smullyan 1993) Diagonalization and Self-Reference (Smullyan 1994); and jointly with Melvin Fitting, Set Theory and the Continuum Problem (Smullyan and Fitting 1996). All but the last of these do have self-reference as a central topic. In addition to the books mentioned above, there have been numerous research papers, a few of which will be discussed. In the 1970s, even while continuing his formal mathematical logic researches, Raymond’s career took a rather unexpected turn. He developed an interest in puzzles, especially those that were based on mathematical logic. Martin Gardner devoted a column in Scientific American to some of these puzzles, and they turned out to be quite popular. This led to What is the Name of This Book? (Smullyan 1978). It turned out to be unexpectedly successful, and was followed by a very large number of others: This Book Needs No Title (Smullyan 1980); Alice in Puzzle-Land (Smullyan 1982a), and many others. I have lost track of foreign translations—there have been a very large number. As a result of the popularity of these books, Raymond appeared on the Tonight Show with Johnny Carson, and on William F. Buckley’s firing line, and other television shows as well. At some point Raymond decided that puzzles could be used to teach some of the fundamental discoveries of modern logic. This idea was first implemented in his book The Lady or the Tiger? (Smullyan 1982c), which actually led readers through the basic ideas of Gödel’s incompleteness theorem, via a series of puzzles. That was followed by To Mock a Mockingbird (Smullyan 1985), which explored the Lambda-calculus via puzzles about birds.3 The Lambda calculus is a system of formal 2Elegant is the word most often used about Raymond’s work in reviews. 3Raymond also explored the topic more technically in his book Diagonalization and Self-Reference, (Smullyan 1994). 1 Introduction 3 logic that has had applications to the design and semantics of computer languages. Raymond’s book was of such interest that the automated theorem-proving group at Argonne National Laboratories created a system specifically for solving his bird puzzles, and it generated quite a lot of interest world-wide.