Diamond: A Paradox Logic

K($E Series on Knots and Everything — Vol. 14 DIAMOND A PcHciClOX L03IC I ^ ^0 I 8 ^w* B I ^SB* B «# W^W Scientif...

Author: N. S. Hellerstein

115 downloads 874 Views 7MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

K($E Series on Knots and Everything — Vol. 14

DIAMOND A PcHciClOX L03IC I ^

^0

I 8 ^w* B I

^SB*

B «#

W^W

Scientific

IBS

DIAMOND A Paradox Logic

SERIES ON KNOTS AND EVERYTHING Editor-in-charge: Louis H. Kauffman

Published: Vol. 1: Knots and Physics L H. Kauffman Vol. 2: How Surfaces Intersect in Space J. S. Carter Vol. 3: Quantum Topology edited by L H. Kauffman & R. A. Baadhio Vol. 4: Gauge Fields, Knots and Gravity J. Baez & J. P. Muniain

Vol. 5: Gems, Computers and Attractors for 3-Manifolds S. Lins Vol. 6: Knots and Applications edited by L H. Kauffman Vol. 7: Random Knotting and Linking

edited by K C. Millets & D. W. Sumners Vol. 8: Symmetric Bends: How to Join Two Lengths of Cord R. E. Miles Vol. 9: Combinatorial Physics T. Bastin & C. W. Kilmister Vol. 10: Nonstandard Logics and Nonstandard Metrics in Physics W. M. Honig Vol. 11: History and Science of Knots edited by J. C. Turner & P. van de Griend Vol. 13 : Entropic Spacetime Theory J. Armel

Series on Knots and Everything - Vol. 14

DIAMOND A Paradox Logic

N S Hellerstein Lincoln University USA

World Scientific Singapore •NewJersey•London •HongKong

Published by World Scientific Publishing Co. Pte. Ltd. P 0 Box 128, Farrer Road , Singapore 912805 USA office: Suite 1B, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Hellerstein, Nathaniel. Diamond, a paradox logic / by Nathaniel Hellerstein. 257+xii p. 22 cm. -- (Series on knots & everything ; vol. 14) Includes bibliographical references. ISBN 9810228503 1. Logic, Symbolic and mathematical. 2. Paradox. 1. Title. II. Series: K & E series on knots and everything ; vol. 14. QA9.H396 1996 511.3--dc2O 96-31705 CIP

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Copyright 01997 by World Scientific Publishing Co. Pte. Ltd. All rights reserved This book, or parts thereof may not be reproduced in anyform or by any means, electronic or mechanical including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

This book is printed on acid-free paper.

Printed in Singapore by Uto-Print

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Part One: Elementary Diamond Logic 1. Paradox

A. The Liar . . . . . . . . . . . . . . . . . 3 B. The Anti-Diagonal . . . . . . . . . . . . 7 C. Russell's Paradox . . . . . . . . . . . . . 8 D. Santa Sentences . . . . . . . . . . . . . 10 E. Antistrephon . . . . . . . . . . . . . . . 15 F. Size Paradoxes . . . . . . . . . . . . . . 16 G. Game Paradoxes . . . . . . . . . . . . . 20 H. Cantor's Paradox . . . . . . . . . . . . . 22 I. Paradox of the Boundary . . . . . . . . .23

2. Diamond

A. The Buzzer . . . . . . . . . . . . . . . .25 B. Diamond Values . . . . . . . . . . . . .27 C. Harmonic Functions . . . . . . . . . . . 28 D. Diamond Circuits . . . . . . . . . . . .31 E. Brownian Forms . . . . . . . . . . . . .33 F. Boundary Logic . . . . . . . . . . . . .38

3. Diamond Algebra

A. Laws . . . . . . . . . . . . . . . . . . .41 B. Normal Forms . . . . . . . . . . . . . .45 C. Completeness and Categoricity . . . . . .51

4. Self-Reference

A. Re-entrance and Fixedpoints . . . . . . .53 B. Phase Order . . . . . . . . . . . . . . .56 C. The Outer Fixedpoints . . . . . . . . . .61

5. Fixedpoint Lattices A. Relative Lattices . . . . . . . . . . . . .67 B. Shared Fixedpoints . . . . . . . . . . . .70 C. Examples . . . . . . . . . . . . . . . . . 72

V

vi Diamond , A Paradox Logic 6. Limit Logic A. Limits . . . . . . . . . . . . . . . . . . 83 B. Limit Fixedpoints . . . . . . . . . . . . . 88 C. Diamond Computation . . . . . . .90 7. Paradox Resolved A. The Liar and the Anti-Diagonal . . . . . . 93 B. Russell's Paradox . . . . . . . . . . . . .94 C. Santa Sentences . . . . . . . . . . . . .96 D. Antistrephon . . . . . . . . . . . . . . . 99 E. Size Paradoxes . . . . . . . . . . . . 101 F. Game Paradoxes . . . . . . . . . . . . 102 8. The Continuum

A. Cantor's Paradox . . . . . . . . . . . . 103 B. Dedekind Splices . . . . . . . . . . . . 104 C. Cantor's Dyadic . . . . . . . . . . . . 106 D. The Line Within The Diamond . . . . . 110 E. Zeno's Theorem . . . . . . . . . . . . 115

Part Two: Advanced Diamond Logic 9. Analytic Functions

A Analytic Functions . . . . . . . . . . . 119 B. Type Theorem . . . . . . . . . . . . . 121 C. Dihedral Conjugation . . . . . . . . . . 123 D. Star Logic . . . . . . . . . . . . . . . 126

10. Harmonic Analysis

A. Harmonic Projection . . . . . . . . . . 129 B. Differentials . . . . . . . . . . . . . . 130 C. Quadrature . . . . . . . . . . . . . . . 141 D. Diffraction . . . . . . . . . . . . . . . 152

11. Three-logic A. Ternary Logic Embeds . . . . . . . . . 163 B. S3 Conjugation . . . . . . . . . . . . . 165 C. Cyclic Distribution . . . . . . . . . . . 168 D. Voter's Paradox . . . . . . . . . . . . 174 12. Metamathematics A G6delian Quanta . . . . . . . . . . . . 183 B. Meta-Logic . . . . . . . . . . . . . . . 185 C. Dialectic . 190 D. Dialectical Dilemma . . . . . . . . . . 192

Contents vii

13. Dilemma

A. Prisoner's Dilemma . . . . . . . . . . . 195 B. Dilemma Diamond . . . . . . . . . . . 200 C. Dilemma As Diamond Metric . . . . . . 202 D. Banker's Dilemma . . . . . . . . . . . 203 E. The Unexpected Departure . . . . . . . 206

14. Speculations A. Diamond Types? . . . . . . . . . . . . 211 B. Diamond Values for Dilemma ? . . . . . 212 C. Null Quotients? . . . . . . . . . . . . . 213 D. General Lattices? . . . . . . . . . . . . 215 E. General Waves? . . . . . . . . . . . . 216 Appendix Notes and Proofs

. . . . . . . . . . . . . 219

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 Index . . . . . . . . . . . . . . . . . . . . 249

This page is intentionally left blank

Introduction There once was a poet from Crete who performed a remarkable feat He announced to the wise "Every Cretan tells lies" thus ensuring their logic's defeat.

"It cannot be too strongly emphasized that the logical paradoxes are not idle or foolish tricks . They were not included in this volume to make the reader laugh, unless it be at the limitations of logic. The paradoxes are like the fables of La Fontaine which were dressed up to look like innocent stories about fox and grapes, pebbles and frogs . For just as all ethical and moral concepts were skillfully woven into their fabric, so all of logic and mathematics, of philosophy and speculative thought, is interwoven with the fate of these little jokes." - Kasner and Newman, "Paradox Lost and Paradox Regained" from volume 3, "The World of Mathematics"

ix

x Diamond , A Paradox Logic This book is about "diamond", a logic of paradox. In diamond, a statement can be true yet false ; an "imaginary" state, midway between being and non-being. Diamond's imaginary values solve many logical paradoxes unsolvable in two-valued boolean logic. The purpose of this book is not to bury Paradox but to praise it. I do not intend to explain these absurdities away; instead I want them to blossom to their full mad glory. I gather these riddles together here to see what they have in common. Maybe they'll reveal some underlying unity, perhaps even a kind of fusion energy! They display many common themes; reverse logic, self-reference, diagonality, nonlinearity, chaos, system failure, tactics versus strategy, and transcendence of former reference frames. Although these paradoxes are truly insoluble as posed, they do in general allow this (fittingly paradoxical!) resolution; namely through loss of resolution! To demand precision is to demand partial vision. These paradoxes define, so to speak, sharp vagueness. A sense of humor is the best guide to these wild regions. The alternative seems to be a kind of grim defensiveness. There exists a strange tendency for scholars to denigrate these paradoxes by giving them derogatory names. Paradoxes have been dubbed "absurd" and "imaginary" and even (0 horror!) "irrational". Worse than such bitter insults are the hideously morbid stories which the guardians of rationality tell about these agents of Chaos. All too many innocuous riddles have been associated with frightening fables of imprisonment and death; quite gratuitously, I think. It is as if the discoverers of these little jokes hated them and wanted them dead. Did these jests offend some pedant's pride?

Introduction xi Paradox is free. It overthrows the tyranny of logic and thus undermines the logic of tyranny. This book's paradoxes are more subversive than spies, more explosive than bombs, more dangerous than armies , and more trouble than even the President of the United States . They are the weak points in the status quo; they threaten the security of the State. These paradoxes are why the pen is mightier than the sword ; a fact which is itself a paradox.

This book has two sections : elementary and advanced. The "elementary diamond logic" section covers the classic paradoxes of mathematical logic, defines diamond's values and operators , notes diamond's equational laws, introduces diamond's "phase order" lattice, proves that diamond resolves all selfreferential systems, resolves the classic paradoxes, and demonstrates that diamond embeds the continuum . This section ends with "Zeno's theorem", which finds diamond-valued fixedpoints for any real-valued function. The "advanced diamond logic " section covers diamond 's boolean underpinnings, describes "diffraction " and other non-boolean forms of diamond reasoning, embeds 3 -valued logic within diamond, and relates diamond to Godelian metamathematics, non-zero-sum games, and other topics. Thus this book illustrates a classic paradox of development; namely, the elementary is advanced, and the advanced is elementary. The results of Part I are basic, but I have known them long enough to refine and strengthen the proofs; thus the elementary is advanced . The results of Part II are subsequent research, and are therefore less developed ; that is, less certain and less general . Thus the advanced is elementary.

xll Diamond, A Paradox Logic I would be a liar indeed not to acknowledge my many friends and colleagues. These include Douglas Hofstadter, Louis Kauffman, Tarik Peterson, Sylvia Rippel, Rudy Rucker, Dick Shoup, Raymond Smullyan, Stan Tenen, and Francisco Varela; without their vital imput over many years , this book would have been impossible.

Love and thanks go to my parents, Earl and Marjorie, without whom I would have been impossible. Special thanks go to my fiancee Sherri Krynski , without whom I would not have published this. Finally, due credit (and blame !) go to myself, for boldly rushing in where logicians fear to tread.

Said a monk to a man named Joshu "Is that dog really God?" He said "Mu." This answer is vexing And highly perplexing And that was the best he could do.

Part One

Elementary Diamond Logic


Chapter 1

Paradox The Liar The Anti Diagonal Russell's Paradox Santa Sentences Antistrephon Size Paradoxes Game Paradoxes Cantor's Paradox Paradox of the Boundary

A. The Liar

Epimenides the Cretan said that all Cretans lie; did he tell the truth, or not? Let us assume, for the sake of argument , that every Cretan, except possibly Epimenides himself, was in fact a liar; but what then of Epimenides? In effect, he says he himself lies; but if he is lying, then he is telling the truth ; and if he is telling the truth , then he is lying ! Which then is it?

The same conundrum arises from the following sentence: "This sentence is false." That sentence is known as the "Liar Paradox", or "pseudomenon".

3

4 Diamond , A Paradox Logic The pseudomenon obeys this equation: L = not L.

It's true if false, and false if true . Which then is it?

That little jest is King of the Contradictions. They all seem to come back to that persistent riddle . If it is false then it is true, by its own definition; yet if it is true then it is false, for the exact same reason ! So which is it, true or false? It seems to undermine dualistic reason itself. Dualists fear this paradox; they would banish it if they could. Since it is, so to speak, the leader of the Opposition Party, it naturally bears a nasty name; the "Liar" paradox. Don't trust it, say the straight thinkers; and it agrees with them! They denigrate it, but it denigrates itself; it admits that it is a liar, and thus it is not quite a liar! It is straightforward in its deviation, accurate in its errors, and honest in its lies! Does that make sense to you, dear reader? I must admit that it has never quite made sense to me.

The name "Liar" paradox is nonetheless a gratuitous insult. The pseudomenon merely denies its truth, not its intentions. It may be false innocently, out of lack of ability or information. It may be contradicting itself, not bitterly, as the name "Liar" suggests, but in a milder tone. Properly speaking, the Liar paradox goes: "This statement is a lie." "I am lying." "I am a liar."

IA. The Liar 5 But consider these statements: "This statement is wrong."

"I am mistaken." "I am a fool." This is the Paradox of the Fool ; for the Fool is wise if and only if the Fool is foolish! The underlying logic is identical, and rightly so. For whom, after all, does the Liar fool best but the Liar? And whom else does the Fool deceive except the Fool? The Liar is nothing but a Fool, and vice versa! Therefore I sometimes call the pseudomenon (or Paradox of SelfDenial) the "Fool Paradox ", or "Fool's Paradox ", or even "Fool's Gold". The mineral "fool's gold " is iron pyrite; a common ore . This fire-y and ironic little riddle is also a common 'ore, with a thousand wry offspring. For instance:

"I am not a Marxist" - Karl Marx "Everything I say is self-serving" - Richard Nixon

Tell me, dear reader ; would you believe either of these politicos?

Compare the Liar to the following quarrel: Tweedledee: "Tweedledum is a liar." Tweedledum: "Tweedledee is a liar." - two calling each other liars rather than one calling itself a liar! This dispute, which I call "Tweedle's Quarrel", is also known as a "toggle".

6 Diamond , A Paradox Logic

Its equations are: EE = not UM UM = not EE This system has two boolean solutions : (true, false) and (false, true). The brothers, though symmetrical, create a difference between them; a memory circuit! It seems that paradox, though chaotic, contains order within it.

1 B. The Anti-Diagonal 7

B. The AntiDiagonal

Here are two paradoxes of mathematical logic, generated by an "antidiagonal " process: Grelling's Paradox. Call an adjective 'autological ' if it applies to itself, 'heterological' if it does not : "A" is heterological

"A" is not A.

Thus, 'short' and 'polysyllabic' are autological, but 'long' Etnd 'monosyllabic' are heterological. Is 'heterological ' heterological? "Heterological " is heterological = "Heterological" is not heterological. It is to the extent that it isn't!

Quine's Paradox. Let "quining" be the action of preceding a sentence fragment by its own quotation. For instance, when you quine the fragment "is true when quined", you get:

"Is true when quined" is true when quined. - a sentence which declares itself true. In general the sentence: "Has property P when quined" has property P when quined. is equivalent to the sentence: "This sentence has property P." Now consider the sentence: "Is false when quined" is false when quined. That sentence declares itself false. Is it true or false?


C. Russell's Paradox

Let R be the set of all sets which do not contain themselves: R = { x I x not an element of x } R is an anti-diagonal set. Is it an element of itself? In general:

x in R =

and therefore :

R in R = R not in R.

x not in x

Therefore R is paradoxical. Does R exist?

Here's a close relative of Russell 's set; the "Short-Circuit Set": S = {x : S not in S}. S is a constant-valued set, like the universal and null sets: For all x, (x in S) = (S not in S) = (S in S).

All sets are paradox elements for S.

Bertrand Russell told a story about the barber of a Spanish village. Being the only barber in town, he boasted that he shaves all those - and only those - who do not shave themselves. Does the barber shave himself?

To this legend I add a political postscript . That very village is guarded by the watchmen, whose job is to watch all those, and only those, who do not watch themselves.

But who shall watch the watchmen? (Thus honesty in government is truly imaginary!)

1 C. Russell 's Paradox 9 Not too long ago that village sent its men off to fight the Great War, which was a war to end all wars , and only those wars, which do not end themselves . Did the Great War end itself?

That village's priest often ponders this theological riddle: God is worshipped by all those, and only those, who do not worship themselves . Does God worship God?

10 Diamond, A Paradox Logic

D. Santa Sentences Suppose that a young child were to proclaim: "If I'm not mistaken, then Santa Claus exists." If one assumes that Boolean logic applies to this sentence, then its mere existence would imply the existence of Santa Claus! Why? Well, let the child's statement be symbolized by R, and the statement "Santa exists" be symbolized by'S'. Then we have the equation:

R = if R then S = (not R) or S .

Then we have this line of argument:

R = (R S); assume that R is either true or false. If R is false, then R = (false -41 S) = (true or S) = true. R = false implies that R = true; therefore (by contradiction) R must be true. Since R = (R 4, S), (R S) is also true. R is true, (R S) is true; so S is true. Therefore Santa Claus exists!

This proof uses proof by contradiction; an indirect method, suitable for avoiding overt mention of paradox. Here is another argument, one which confronts the paradox directly:

1 D. Santa Sentences

11

S is either true or false . If it's true, then so is R: R = (not R) or true = true. No problem . But if S is false, then R becomes a liar paradox: R = (not R) or false = not R If S is false, then R is non-boolean. therefore: If R is boolean, then S is true.

Note that both arguments work equally well to prove any other statement besides S to be true; one need merely display the appropriate "santa sentence". Thus, for instance, if some skeptic were to declare:

"If I'm not mistaken, then Santa Claus does not exist." - then by identical arguments we can prove that Santa Claus does no exist! Given two opposite Santa sentences: R, = (R1 S) ; R2 = (R2 not S ) then at least one of them must be paradoxical.

We can create Santa sentences by Grelling' s method . Let us call an adjective "Santa-logical " when it applies to itself only if Santa Claus exists; "A" is Santa- logical = If "A" is A, then Santa exists. Is "Santa-logical" Santa-logical? "Santa-logical " is Santa-logical = If "Santa-logical " is Santa-logical, then Santa exists.


Here is a Santa sentence via quining: "Implies that Santa Claus exists when quined" implies that Santa Claus exists when quined. If that statement is boolean, then Santa Claus exists.

Here's the " Santa Set for sentence G": SG = { x I (x an element of x) implies G } SG is the set of all sets which contain themselves only if sentence G is true: xinSG = (( xinx) 4, G ).

Then " SG in SG" equals a Santa sentence for G: SG in SG = ( ( SG inSG) G). "SG in SG", if boolean, makes G equal true ; another one of Santa's gifts. If G is false, then " SG in SG" is paradoxical.

One could presumably tell Barber-like stories about Santa sets. For instance, in another Spanish village, the barber takes weekends of; so he shaves all those, and only those, who shave themselves only on the weekend: B shaves M

= If M shaves M, then it's the weekend.

One fine day someone asked : does the barber shave himself? B shaves B = If B shaves B, then it's the weekend.

Has it been weekends there ever since?

1 D. Santa Sentences

13

That village is watched by the watchmen, who watch all those, and only those, who watch themselves only when fortune smiles: W watches C = if C watches C, then fortune smiles. One fine day someone asked: who watches the watchmen? W watches W = if W watches W, then fortune smiles. Does fortune smile on that village?

Recently that village saw the end of the Cold War, which ended all wars, and only those wars, which end themselves only if money talks: CW ends W

if W ends W, then money talks.

=

Did the Cold War end itself? =

CW ends CW

if CW ends CW, then money talks.

Does money talk?

That village's priest proclaimed this theological doctrine: God blesses all those, and only those, who bless themselves only when there is peace: G blesses S

=

If S blesses S, then there is peace.

One fine day someone asked the priest : Does God bless God? G blesses G

=

Is there peace?

If G blesses G, then there is peace.


Finally, consider the case of Promenides the Cretan, who always disagrees with Epimenides. Recall that Epimenides the Cretan accused all Cretans of being liars, including himself. If we let E = Epimenides, P = Promenides, and H = "honest Cretans exist", then:

E = (not E) and (not H) P = not E = not ( (not E) and (not H) ) = E or H = (not P) or H = (P H)

Thus we get this dialog:

Epimenides: All Cretans are liars. Promenides: You're a liar. Epimenides: All Cretans are liars, and I am a liar. Promenides: Either some Cretan is honest, or you're honest. Epimenides: You're a liar. Promenides: Either some Cretan is honest, or I'm a liar. Epimenides: All Cretans are liars, including myself. Promenides: If I am honest, then some Cretan is honest.

Promenides is the Santa Claus of Crete; for if his statement is boolean, then some honest Cretan exists.

I E. Antistrephon 15

E. Ant&rephon

That is, "The Retort". This is a tale of the law-courts, dating back to Ancient Greece. Protagoras agreed to train Euathius to be a lawyer, on the condition that his fee be paid, or not paid , according as Euathius win, or lose, his first case in court . (That way Protagoras had an incentive to train his pupil well; but it seems that he trained him too well !) Euathius delayed starting his practice so long that Protagoras lost patience and brought him to court, suing him for the fee. Euathius chose to be his own lawyer; this was his first case. Protagoras said, "If I win this case, then according to the judgement of the court , Euathius must pay me; if I lose this case, then according to our contract he must pay me. In either case he must pay me." Euathius retorted , "If Protagoras loses this case, then according to the judgement of the court I need not pay him; if he wins, then according to our contract I need not pay him. In either case I need not pay."

How should the judge rule? Here's another way to present this paradox: According to the contract, Euathius will avoid paying the fee - that is, win this lawsuit - exactly if he loses his first case; and Protagoras will get the fee - that is, win this lawsuit - exactly if Euathius wins his first case. But this lawsuit is Euathius's first case, and he will win if exactly if Protagoras loses. Therefore Euathius wins the suit if and only if he loses it; ditto for Protagoras.


F. Size Paradoxes

The Heap. Surely one grain of sand does not constitute a heap of sand. Surely adding another grain will not make it a heap . Nor will adding another, or another, or another. In fact, it seems absurd to say that adding one single grain of sand will turn a non-heap into a heap. By adding enough ones, we can reach any finite number; therefore no finite number of grains of sand will form a sand heap . Yet sand heaps exist; and they contain a finite number of grains of sand! Let's take it in the opposite direction. Let us grant that a finite sand heap exists. Surely removing one grain of sand will not make it a non-heap. Nor will removing another, nor another, nor another. By subtracting enough ones, we can reduce any finite number to one. Therefore one grain of sand makes a heap! What went wrong? Let's try a third time. Grant that one grain of sand forms no heap; but that some finite number of grains do form a heap . If we move a single grain at a time from the heap to the non-heap, then they will eventually become indistinguishable in size. Which then will be the heap , and which the nonheap?

The First Boring Number. This is closely related to the paradox of the Heap . For let us ask the question : are there any boring (that is, uninteresting) numbers? If there are, then surely that collection has a smallest element; thefirst uninteresting number. How interesting!

1 F. Size Paradoxes 17 Thus we find a contradiction ; and this seems to imply that there are no uninteresting numbers! But in practice, most persons will agree that most numbers are stiflingly boring , with no interesting features whatsoever ! What then becomes of the above argument? Simply this; that the smallest boring number is inherently paradoxical. If being the first boring number were a number's only claim to our interest, then we would find it interesting if and only if we do not find it interesting. Which then is it?

Berry's Paradox. What is "the smallest number that cannot be defined in less than twenty syllables "? If this defines a number, then we have done so in nineteen syllables! So this defines a number if and only if it does not.

Presumably Berry's number equals the first boring number, if your boredom threshold is twenty syllables.

These paradoxes connect to the paradox of the Heap by simple psychology. If, for some mad reason, you actually did try to count the number of grains in a sand heap , then you will eventually get bored with such an absurd task. Your attention would wander; you would lose track of all those sand grains; errors would accumulate, and the number would become indefinite. The Heap arises at the onset of uncertainty . In practice, the Heap contains a boring number of sand grains ; and the smallest Heap contains the smallest boring number of sand grains!


Finitude. Finite is the opposite of infinite; but in paradox-land, that's no excuse! In fact the concept of finiteness is highly paradoxical; for though finite numbers are finite individually and in finite groups, yet they form an infinity. Let us attempt to evaluate finiteness. Let F ='finitude', or'finity; the generic finite expression . You may replace it with any finite expression. Is Finity finite? If F is finite, then you can replace it by F+1, and thus by F+2, F+3, etc. But such a substitution, indefinitely prolonged , yields an infinity. If F is not finite, then you may not replace F by F, nor by any expression involving F; you must replace F by a well-founded finite expression, which will then be limited. Therefore F is finite if and only if it is not finite. Finitude isjust short of infinity ! It is infinity seen from underneath. You may think of it as that mysterious 'large finite number' N, larger than any number you care to mention. Call a number "large" if it is bigger than any number you care to mention; that is, bigger than any interesting number. Call a number "medium" if it is bigger than some boring number but less than some interesting number. Call a number " small" if it is less than any boring number. Presumably Finitude is the smallest large number; that is, the smallest number greater than any interesting number. (How interesting!) Finitude is dual to the Heap , which is the largest number less than any uninteresting number. The Heap is the lower limit of boredom; Finitude is the upper limit of interest.

1 F. Size Paradoxes 19

We get these inequalities:

small interesting numbers < The Heap = first boring number = last small number < medium numbers < Finitude = last interesting number first large number < large boring numbers

Finally, consider this Berry-like definition: "One plus the largest number defineable in less than twenty syllables." If this defines a number, then it has done so in only nineteen syllables, and therefore is its own successor . Presumably this number = Finitude, if your boredom threshold is twenty syllables.


G. Game Paradoxes.

Hypergame and the Mortal.

Let "Hypergame" be the game whose initial position is the set of all "short" games - that is, all games that end in a finite number of moves. For one's first move in Hypergame , one may move to the initial position of any short game. Is Hypergame short? If Hypergame is short, then the first move in Hypergame can be to Hypergame ! But this implies an endless loop, thus making Hypergame no longer a short game! But if Hypergame is not short, then its first move must be into a short game; thus play is bound to be finite, and Hypergame a short game.

The Hypergame paradox resembles the paradox of Finitude. Presumably Hypergame lasts Finitude moves; one plus the largest number defineable in less than twenty syllables. Dear reader, allow me to dramatize this paradox by means of a fictional story about a mythical being. This entity I shall dub "the Mortal"; an unborn spirit who must now make this fatal choice; to choose some mortal form to incarnate as, and thus be doomed to certain death. The Mortal has a choice of dooms. Is the Mortal doomed?

I G. Game Paradoxes 21

Normalcy and the Rebels.

Define a game as "normal " if and only if it does not offer the option of moving to its own starting position: G is normal = the move G G is not legal. Let "Normalcy" be the game of all normal games. In it one can move to the initial position of any normal game: The move N -0 G is legal = the move G =J G is not legal. Is Normalcy normal? Let G = N: The move N N is legal = the move N 4 N is not legal. Normalcy is normal if and only if it is abnormal!

That was Russell's paradox for game theory. Now consider this: The Rebel is a being who must become one who changes. The Rebel may become all those, and only those, who do not remain themselves:

R may become B = B may not become B . Can the Rebel remain a Rebel? R may become R = R may not become R. A Santa Rebel may become all those, and only those, who remain themselves only if Santa Claus exists: SR may become B = ((B may become B )

Santa exists)

Therefore: SR may become SR = ((SR may become SR) A> Santa exists) If the pivot bit is boolean, then Santa Claus exists!


H. Cantor 's Paradox Cantor's proof of the "uncountability" of the continuum relies on an "anti-diagonalization" process . Suppose we had a countable list of the real numbers between 0 and 1: R, = 0. R2

D11,

D 12, D13, D 1 4 ...

= 0 . D21, D22, D23, D24 ...

R3 = 0 . D31, D32, D33,

D 34 ...

R4 = 0 . D41, D42, D43,

D44

...

where DNM is the Mth binary digit of the Nth number. Then we define Cantor's "anti-diagonal" number: C = 0 . not

D11

, not

D22 , not D33 , not D44 ...

If C = R. for any N, then DNx = not D,tx Therefore DNr, = not DNN.

The pivot bit buzzes. From this paradox, Cantor deduced that the continuum has too many points to be counted, and thus is of a "higher order" of infinity. Thus a single buzzing bit implies infinities beyond infinities! Was more ever made from less? I say, why seek "transfinite cardinals", whatever those are? Why not ask for Santa Claus? In this spirit, I introduce the Santa-diagonal number: S = 0 . (Dl, Santa), (D22 Santa), (D33 4, Santa) ... If S = RM for any M, then D,,x _ (Dx,t Santa exists) ; Therefore D,,î1 = (D,m Santa exists) .

1 1. Paradox of the Boundary 23

1. Paradox of the Boundary

The continuum is paradoxical because it is continuous, and boolean logic is discontinuous. This topological difference yields a logical riddle which I call the Paradox of the Boundary. The paradox of the boundary has many formulations, such as: What day is midnight? Is noon A.M. or P.M.? Is dawn day or night? Is dusk? Which side of the mirror is Alice on? Which country owns the border? Is zero plus or minus? Is infinity odd or even? If a statement is true at point A and false at point B, then somewhere in-between lies a boundary. At any point on the boundary, is the statement true, or is it false? (If line segment AB spanned the island of Crete, then somewhere in the middle we should, of course, find Epimenides!)


Chapter 2

Diamond The buzzer; wave logic Diamond values; tiff; under/over-determined Harmonic functions; not, and, or, but, others Diamond circuits; phased-delay, dual-rail Brownian forms and laws Boundary logic

A. The Buzzer Chapter I posed the problem of paradox but left it undecided . Is the Liar true or false? Boolean logic cannot answer . What bold expedient would decide the question? What but Experiment ? Let us be scientific! Is it possible to build a physical model of formal paradox using simple household items , such as (say) wires, switches, batteries and relays?

Yes, you can ! And indeed it's simple ! It's easy! Just wire together a spring-loaded relay, a switch, and a battery, using this childishly simple circuit:

I

I

I \\\ ...L \\\ . I \\\ I I I I 25

26 Diamond, A Paradox Logic When you close the switch, the relay is caught in a dilemma; for if current flows in the circuit, then the relay shall be energized to break the circuit, and current will stop; whereas if there is no current in the circuit, then the springloaded circuit will re-connect, and current will flow. Therefore the relay is open if and only if it is closed. Which? To find out, close the switch. What do you see? You would see less than you'd hear; for in fact you would see a blur! The relay would oscillate. It would vibrate. It would, in fact, buzz! All buzzers, bells, alternators, and oscillators are based on this principle of oscillation via negative feedback. Thermostats rely on this principle; so do regulators, rectifiers, mechanical governors, and electromagnetic emitters. Electric motor/generators and heat engines are rotary variants of this process; they use cybernetic phase alternation to ensure that the crankshaft constantly tries to catch up to itself. Paradox, in the form of negative-feedback loops, is at the heart of all high technology. Since mechanical Liars (and Fools) dominate modern life, let us investigate their logic.

2B. Diamond Values 27

B. Diamond Values

Consider the period-2 oscillations of binary values. There are four such logic waves:

t t t t t t .... ;

call this "t/t", or "t".

t f t f t f .... ; call this "t/f', or "i". ft ft ft .... ; call this "Ft", or 'J". f f f f f f .... ;

call this "f/f", or "f'.

"/" is pronouced "but"; thus i is "true but false" and j is "false but true". These four values form a diamond-shaped lattice:

true = t/t i = t/f j = f/t false = f/f

This is "diamond logic"; a wave logic with two components and four truth values. It describes the logic waves of period 2. The values i and j can be interpreted as "underdetermined" and "overdetermined " states; where "underdetermined " means "insufficient data for definite answer", and "overdetermined " means "contradictory data". The value i can take either role, provided that j takes the other.

(See Notes for more details.)


C. Harmonic Functions

Let the positive operators "and" and "or" operate termwise: (a/b) and (c/d) _ (a and c)/(b and d) (a/b) or (c/d) _ (a or c)/(b or d)

We can then define "but" as a projection operator: a/b = (aandi)or(bandj)

( a or j ) and (b or 1)

In diamond logic, negation operates after a flip: not (a/b) _ (not b)/(not a)

This corresponds to a split-second time delay in evaluating negation; and this permits fixedpoints: not(t/f) _ (not f)/(not t) = t/f not(f/t) _ (not t)/(not f) = f/t Thus paradox is possible in diamond logic.

Call a function "harmonic" if it can be defined from "and", "or", "not", and the four values t, i, j, f. They include: a b

(not a) or b

a iffb

(if a then b) and (if b then a)

a xor b

(a and not b) or (b and not a)

2C. Harmonic Functions 29

The "majority" operator M has two definitions: M(a,b,c) _ (a and b) or (b and c) or (c and a)

(a orb) and (b or c) and (c or a)

Here are the "lattice operators":

a min b

= (a or b)/(a and b) _ "a or/and b"

a max b

= (a and b)/(a or b) _ "a and/or b"

We can define "but" from the lattice operators:

a/b = (a min f) max (b min t) = (a max t) min (b max f)

Here are the two "harmonic projection" operators: A(x) = x/(notx) p(x) (not x) / x

Here are the upper and lower differentials:

Dx = x implies x = x iff x = x or not x dx = x minus x = x xor x = x and not x

This, then, is Diamond ; a logic containing the boolean values, plus paradoxes and lattice operators.

30 Diamond , A Paradox Logic Here are truth tables for the functions defined above:

x: not x : and y : or y: y: iff y: xor y: t f i j t f i j t f i j t f i j t f i j t f t f i j t t t t t f t ffff tfij i i i f i f tiit j j j f f j t j t j t

f i j t f i j f t i j tttt ftij tfij tiit lilt i i i f j t j i i t j j j f j

x: but y : min y: max y : t f i j t f i j

M(x,y,z) Ax: px: Dx: dx : t f i j majority

t t i l t t i l t t j t j i j t f f jffj ifif jffj j i t f i t i l t i i i f t f i j i i i j j f f j t f i j j j f j j j j j

y or z yandz i yminz y max z

2D. Diamond Circuits 31

D. Diamond Circuits One can implement diamond logic in switching circuits, two different ways; via "phased delay" and via "dual rail". In "phased delay", one permits a standard switching circuit to oscillate. T then means "on", F means "off', and the two mid-values mean the two wobbles of opposite phase. This implementation requires a "global clock" to ensure that the switches wobble in synchrony; it also reqires that all inverter switches take a unit delay, and all "positive" (and, or, majority) gates take delay two:

(not B ) (n) = not ( B(n-1) ) (A and B) (n) (A(n-2) and B(n-2) ) (A or B) (n) _ (A(n-2) or B(n-2) ) (M( A, B, C)) (n) = (M( A(n- 2), B(n-2), C(n-2) )

In a "dual rail" circuit, all wires in a standard switching circuit are replaced by pairs of wires. True then means "both rails on", false means "both rails off', and the two mid-values mean that one of the two rails is on.

The gates then are:

"not":

IÎ/ V I\ n I/

Diamond , A Paradox Logic

32 "and":

\

V

I-)----\ / \

A / \ "or":

\ V

/

\

A ^^----/

There may also exist an "optical diamond" implementation in photonics, which interprets the mixed values as circularly polarized light.

2E. Brownian Forms

33

E. Brownian Forms Make a mark. This act generates a form:

A mark marks a space. Any space, if marked, remains marked if the mark is repeated:

where "=" denotes " is confused with". This is the crossed form, or "mark". Each mark is a call ; to recall is to call.

A mark is a crossing, between marked and unmarked space . To cross twice is not to cross ; thus a mark within a mark is indistinguishable from an unmarked space:

e

This is the uncrossed form, or "void". Each mark is a crossing; to recross is not to cross.

34 Diamond, A Paradox Logic Thus we get the "arithmetic initials" for G. Spencer Brown's famous Laws of Form. In his remarkable book, Laws of Form, G. S.Brown demonstrated that these suffice to evaluate all formal expressions in Brown's calculus; and that these forms obey two "algebraic initials": I I A I B I I C A C I B C I I ; "Transposition"

Al A I

"Position"

He proved that these axioms are consistent, independent, and complete; that is, they prove all arithmetic identities. This "primary algebra" can be identified with Boolean logic. The usual matching is:

II II

X Y

(void)

F

(false)

(mark)

T

(true)

(juxtapose)

X or Y

(disjunction)

I X

T-1

I

(crossing)

Y l I -

not X

X and Y

(negation)

(conjunction)

2E. Brownian Forms 35

X I Y

If X, then Y

I I X I Y I Y I X I

-Y-1 Y Z I EE]

ZXI

X xor Y

I Majority (X,Y,Z)

I

I I I x l Y I I Y I Z I I z I X I I

There is a complementary interpretation: void mark

true false

X Y (juxtapose) X and Y

I X I (crossing) not X

Xor Y

X Y I I If X then Y

The standard interpretation is usually preferred because it has a simpler implication operator. We can extend Brown's calculus to diamond by introducing two new forms : "curl" and "uncurl":


I I I (curl) I-I

i (underdetermined)

I I (uncurl )

j (overdetermined)

They have these relations:

I I_I

I I= I

I I_I

I I_I

I I= I

1 I_I

I I I_I I_I You can create other interpretations simply by swapping the roles of curl and uncurl. (This exploits the symmetries of the diamond. See the " Conjugation" chapter below.) In this interpretation, we have:

- I x I x I = dx x I x = Dx

: Differentials

(x min y) = M(x, curl, y) =

I I I xl ^_I I LI YI I YI xI I

2E. Brownian Forms 37

I I I I x \_I I \_IyI yxI I

(x max y) = M(x, uncurl, y) = I I I xI IJ I IJ yI I yI xI I

I I I I x IJ I IJ y I yxI I When one includes the new Brownian forms "curl" and "uncurl", these axioms still hold:

I I A I B I I C = A C I B C I I "Transposition"

AIBIA

"Occultation"

A

We also get this axiom; " Interference":

I

I I 1

I I =

I

\I I \I 1 IJ 1 1J I I

- a two-component anti-boolean axiom. These three axioms suffice to calculate all the truth tables, and all the algebraic identities, of diamond.

(For more details, see my book, Wave Laws.)

38 Diamond , A Paradox Logic F. Boundary Logic

Boundary logic is Brownian form algebra, adapted for the typewriter. It uses brackets instead of Brown's mark:

[ A ] instead of A . The arithmetic initials are then:

[l[] H. [[]] = If we call [] "1" and [[]] "0", then we get these equations: [0] = 1 ; [1] = 0 ; 00 = 0 ; 01 = 10 = 11 = 1 .

The algebraic initials are: [[a][b]]c = [[ac][bc]] [[a]a]

We can identify boundary forms with boolean logic this way: [ ] = true ; [ [ ] ] = false ; [A] = not A; A B = A or B ; [[A][B]] = AandB; [A]B = ifAthenB;

2F. Boundary Logic 39 [[A]B][[B]A] = AxorB; [[[A]B] [[B]A]] = AiffB; [[AB][BC][CA]]

=

Majority( A,B,C)

For diamond, we introduce two new expressions ; 6 and 9: [6] = 6 ; [9] = 9 ; 69=[]. We can identify 6 with i, and 9 with j. (Or vice versa.)

We then get these equations; 10=16=19=11=1; [1]=0; 91 =96= 1 ;99=90=9; [9]=9; 61=69=1;66=60=6; [6]=6; 00=0;06=6;09=9;01=1; [0]=1

Inspection of tables shows that juxtaposition ab is isomorphic to diamond disjunction; and crossing [a] is isomorphic to diamond negation. For the four forms 0, 1, 6, and 9 , we get these identities:

Transposition : [[a][b]]c = [[ac][bc]] Occultation: [ [ a ] b ] a = a Interference: [[6]6][[9]9] = 1

These three axioms define diamond.


Chapter 3

Diamond Algebra De Morgan, junction, differential, lattice laws Normal forms Completeness, categoricity

A. Laws Diamond obeys these Diamond laws: Commutativity : A or B= B or A; A and B= B and A Associativity : (A and B) and C = A and (B and C) (A or B) or C = A or (B or C) Distributivity: A and (B or C) (A and B) or (A and C) A or (B and C) _ (A or B) and (A or C) Identities: A and t= A A or f= A Attractors: A and f = f ; A or t= t Recall : A and A= A A or A= A Absorption : A and (A or B) = A ; A or (A and B) = A Double Negation : not (not A) = A De Morgan : not (A and B) (not A) or (not B) not (A or B) _ (not A) and (not B) Interference:

Di and Dj = f ; di or dj = t That is: (iornoti ) and(jornotj) = f (i and not i ) or (j and not j) = t

41

42 Diamond , A Paradox Logic Diamond has all the Boolean laws, except for the Law of the Excluded Middle. In its place Diamond has Interference; a two-component "anti-boolean" axiom. The Diamond laws equal the De Morgan laws plus Interference.

The Diamond laws suffice to prove: Complementarity: i and j = (not i) and j = (not j) and i

(not i) and (not j) = f

i or j = (not i) or j = (not j) or i

(not i) or (not j) = t

Paradox: noti = i ; notj = j

These rules, plus Identity, Attractors and Recall, suffice to construct diamond's truth tables.

The dual paradoxes i and j define "but", the "junction" operator: x/y = (x and i) or (y and j) =

(x or j) and (y or i)

Here are the junction laws: Recall: a/a

a

Polarity: (a/b)/(c/d)

a/d

Parallellism: (a/b) and (c/d)

(a/b) or (c/d) M( a/A, bB, c/C) _ Reflection: not (a/b) _

(a and c)/(b and d)

(a or c)/(b or d) M(a,b,c) / M(A,B,C) (not b)/(not a)

3A. Laws 43

Min and max obey these lattice laws:

Commutativity: x min y = y min x x max y = y max x Associativity: x min (y min z) _ (x min y) min z xmax(ymaxz) = (xmaxy)max z Absorption : x max (x min y) = x min (x max y) = x Recall: x min x = x max x = x Attractors : x min i = i ; x max j = j Identities : x min j = x ; x max i= x Transmission : not (x min y) = (not x) min (not y)

not (x max y) = (not x) max (not y) Mutual Distribution : x ** (y ++ z) _ (x ** y) ++ (x ** z ) where ** and ++ are both from: {and, or, min, max}

Here are some majority laws:

Modulation : M(a,f, b) = a and b M(a,t, b) = a or b M(a,i, b) = a min b M(a,j, b) = a max b Symmetry : M(x,y,z)=M(y,z,x)=M(z,x,y)=M(x,z,y)=M(z,y,x)=M(y,x,z) Coalition : M(x,x,y) = M(x,x,x) = x Transparency: not (M(x,y,z)) = M( not x, not y, not z) Distribution : M(a,b,M(c,d,e)) = M(M(a,b,c), d, M(a,b,e))

44 Diamond , A Paradox Logic Modulation plus Transparency explains DeMorgan and Transmission: not(a and b) _ (not a) or (not b) not(a or b) _ (not a) and (not b) not(a min b) (not a) min (not b) not(a max b) _ (not a) max (not b)

The differentials dx and Dx obey these derivative laws: dx and Dx

dx

dx and x

x

x or dx

x and Dx

Dx

Dx or x

Dx or dx

i.e. dx is a subset of x, which is a subset of Dx. In Venn diagram terms , dx is the boundary of x. ddx = dDx

dx

DDx = Ddx

Dx

not dx = Dx

not Dx

dx

d(not x) = dx

D(not x)

Dx

I call the following the Leibnitz rules, due to their similarity to the Leibnitz rule for derivatives of products: d(x and y) (dx and y) or (x and dy) D(xory) _ (Dxory)and(xorDy) d(x or y) _ (dx and (not y)) or ((not x) and dy) D(x and y) _ (Dx or not y)) and ((not x) or Dy)

3B. Normal Forms 45

B. Normal Forms

By using the De Morgan laws, one can put any harmonic diamond function F(x) into one of the following forms:

Disjunctive Normal Form:

F(x) _ (t11(x,) and t12(x2) and ... and tl„(x,J ) or ( t21(x 1) and t22(x2) and ... and t2n(x..) )

or ... or (tm ,

(x l) and t,n2(x2) and ... and tm„(x„) )

where each %(x) is one of these functions: { x, not x, dx, t, f, i, j }

Conjunctive Normal Form: F(x) _ (t11(x1) or t12(x2) or ... or

t 1n(x1J )

and (t21(x1) or t22(x) or ... or t2n(xn) )

and ...

and (tm,(xl) or tm2(x2) or ... or t, (x.) ) where each tu(x) is one of these functions: ( x, notx, Dx, t,f,i,j )

We do this by distributing negations downwards, canceling doublenegations, and distributing enough times. These normal forms are just like their counterparts in boolean logic, except that they allow differential terms.

46 Diamond, A Paradox Logic Theorem: The Primary Normal Forms F(x) = (A and x) or (B and not(x)) or (C and dx) or D F(x) = (a or not(x)) and (b or x) and (c or Dx) and d where A,B,C,D,a,b,c,d are all free of variable x, and: AorD = F(t) = aandd B or D = F(f) = b and d A or B or C or D = F(i) or F(j) = d D = F(i) and F(j) = a and b and c and d

Proof: We get the first two equations from the Disjunctive and Conjunctive Normal Forms by collecting like terms with respect to the variable x. The next two equations can be verified by substituting values t and f. Substituting i and j, plus using Paradox and Distribution, yields: F(i) = ((A or B or C) and i) or D = ((a and b and c) or i) and d F(j) = ((A or B or C) and j) or D = ((a and b and c) or j) and d Therefore: F(i) or F(j) =

((AorBorC)andi)orpor((AorBorC)andj)orD ((A or B or C) and (i orj) )or D A or B or C or D. Also: F(i) and F(j) = (((A or B or C) and i) or D) and (((A or B or C) and j) or D) (((A or B or C) and i) and (((A or B or C) and j)) or D ((A or B or C) and i and j) or D = D Similarly: F(i) and F(j) = a and b and c and d. F(i) or F(j) = d. QED.

36. Normal Forms 47 Before we continue, consider this Theorem: Cross-Transposition: (Aandx) or(B and notx ) or(C anddx)

_ (Aornotx) and(Borx )and(AorBorC)andDx (aorx ) and(bornotx )and(corDx) _ (aandnotx )or(bandx)or(aandbandc)ordx

Proof: Use Distribution, Occultation, and the derivative rules to derive:

(A and x) or (B and not x ) or (C and dx) (A or B or C) and (A or B or dx) and (A or not(x) or C) and (A or not(x) or dx) and (x or B or C) and (x or B or dx) and (x or not(x) or C) and (x or not(x) or dx) (A or B or C) and (A or B or dx) and (A or not(x) or C) and (A or not x) and (x or B or C) and (x or B)

and (x or not(x) or C) and (x or not x) (A or B or C) and (A or B or dx) and (A or not(x)) and (x or B) and (Dx or C) and (Dx) (A or B or C) and (A or B or x) and (A or B or not x) and (A or not(x)) and (B or x) and (Dx)

(A or B or C) and and (A or not(x)) and (B or x) and (Dx).

Similarly, ( aorx ) and(bornotx )and(corDx) _ (aandnotx ) or(bandx ) or(aandbandc)ordx QED.


Here are two examples of cross-transposition:

a xor b = (a and not b) or (not(a) and b) = (a orb) and (not a or not b) and (a or not a) and Db = (a or b) and (not a or not b) and Da and Db

a iff not b = (a or b) and (not a and not b) = (a and not b) or (not(a) and b) or (a and not a) or db = (a and not b) or (not(a) and b) or da or db

The Primary Normal Forms, plus Cross-transposition, yields:

Theorem: Differential Normal Forms

Any harmonic function F(x) can be put into these forms:

F(x) = (F(t) and x) or (F(f) and not x) or M(F(i), dx, F(j)) F(x) = (F(t) or not x) and (F(f) or x) and M(F(i), Dx, F(j))

This separates the function into boolean and lattice components.

3B. Normal Forms 49 Proof Start with the Primary Normal Forms:

F(x) _ (A and x) or (B and not(x)) or (C and dx) or D (a or not(x)) and (b or x) and (c or Dx) and d where A,B,C,D,a,b,c,d are all free of variable x, and: A or D = F(t) = a and d B or D = F(f) = b and d A or B or C or D = F(i) or F(j) = d D = F(i) and F(j) = a and b and c and d

Then apply Cross-Transposition: (A and x) or (B and not x) or (C and dx) (A or not x) and (B or x) and (A or B or C) and Dx (a or x) and (b or not x) and (c or Dx)

_ (a and not x) or (b and x) or (a and b and c) or dx

We derive: F(x) = (A and x) or (B and not x) or (C and dx) or D = ((A or not x) and (B or x) and (A or B or C) and Dx) or D = (A or D or not x) and (B or D or x)

and (A or B or C or D) and (Dx or D) = (F(t) or not x) and (F(f) or x) and (F(i) or F(j)) and (Dx or (F(i) and F(j)) = (F(t) or not x) and (F(f) or x) and (F(i) or F(j)) and (Dx or F(i)) and (Dx or F(j)) = (F(t) or not x) and (F(f) or x) and M( F(i), Dx, F(j) )

50 Diamond, A Paradox Logic Dually:

F(x) = (a or x) and (b or not x) and (c or Dx) and d = ((a and not x) or (b and x) or (a and b and c) or dx) and d = ((a and d and not x) or (b and d and x)

or (a and b and c and d) or (dx and d) = (F(f) and not x) or (F(t) and x) or (F(i) and F(j)) or ( dx and (F(i) or F(j)) ) = (F(t) and x) or (F(f) and not x) or (F(i) and F(j)) or (dx and F (i)) or (dx and F(j)) (F(t) and x) or (F(f) and not x) or M(F(i), dx, F(j))

QED.

3C. Completeness and Categoricity 51

C. Completeness and Categoricity

The differential normal forms imply this theorem: Completeness. Any equational identity in diamond can be deduced from

the diamond laws.

Proof: By induction on the number of variables. Let F = G be an identity with N variables. (Initial step.) If N = 0, then F = G is an arithmetic equation. Since the diamond laws imply diamond's truth tables, F = G follows from those axioms. (Induction step.) Suppose that all N-I variable identities in diamond are provable from the diamond laws. Let F(x) be F considered as an expression in its Nth variable x. Then: F(x) _ (F(t) and x) or (F(f) and not x) or M(F(i), dx, F(j)) is provable by the Differential Normal Form theorem. So is: G(x) = (G(t) and x) or (G(f) and not x) or M(G(i), dx, G(j)) By the induction hypothesis, these are provable from the diamond laws: F(t) = G(t) ; F(i) = G(i) ; FO) = GO) ; F(f) = G(f) Therefore we get this sequence of provable identities: F(x) = (F(t) and x) or (F(f) and not x) or M(F(i), dx, F(j)) = (G(t) and x) or (G(f) and not x) or M(G(i), dx, G(j)) = G(x) . This concludes the induction proof. Therefore any equational identity in diamond is provable from the diamond laws. QED.

52 Diamond, A Paradox Logic A note on feasibility. The above proof that F = G was only three equations long; but it is only a link in a recursive chain. A complete proof requires proofs that F(i) = G(i), F(t) = G(t), F(f) = G(f), and F(j) = G(j). Therefore any complete proof that F=G, if these expressions have n variables, will be about 4" steps long; no faster than proof by full-table look-up! Thus, though the diamond axioms are deductively complete, they may fail to be feasibly complete. Is there a polynomial-time algorithm that can check the validity of a general diamond equation? Students of feasibility will recognize this as a variant of the Boolean Consistency Problem, and therefore NP-complete.

Diamond logic's completeness suggest this:

Conjecture. Diamond is a "categorical" DeMorgan algebra: Any De Morgan algebra is a subalgebra of images of products of diamond . These De Morgan algebras need not have the Interference axiom; they are subalgebras of ones that do.

Thus diamond is to De Morgan algebras as two-valued logic is to Boolean algebra. I consider diamond to be a 2-dimensional extension of twovalued logic that solves paradox, just as the complex numbers are a 2dimensional extension of the real line that solves x2=- 1 .

Chapter 4

Self-Reference Re-entrance and Fixedpoints Phase Order and its laws The Outer Fixedpoints; examples

A. Re Entrance and Fixedpoints

Consider the Liar Paradox as a Brownian form:

L

=

L I

This form contains itself. That can be represented via re-entrance, thus:

L

=

53

54


Let re-entrance permit any mark within a Brownian form to extend a tendril to a distant space, where its endpoint shall be deemed enclosed. Thus curl sends a tendril into itself. Other re-entrant expressions include:

I I I I I I I I I I I I

I

I

I I I I I I I

I

I

I

4A. Re- Entrance and Fixedpoints 55

Self-reference can be expressed as a re-entrant brownian form, as a switching circuit, as a vector of forms, as an indexed list, and as a harmonic fixedpoint . For example:

Brownian form

/ \ Switching Circuit I \ I \ / (triangles = " not" gates)

I A B I I I I Brownian Form Vector I B = A I I

I-

-I

A = [ [ A ]B ]A Indexed List

(A , B) = (not B , not A ) Harmonic Fixedpoint


B. Phase Order Now I define the concept of phase order: af2(a)?f3(a)_f°(a) ?... Since diamond has limited chains , this descending sequence must reach its lower bound within 2n steps, if n is the number of components of _f. Therefore f2'(a) is afixedpoint for f

ff'(a)) = f2i (a) This is the greatest fixedpoint left of a.

Left seeds grow leftwards towards fixedpoints.

Similarly, right seeds grow rightwards towards fixedpoints: < f(a) < f2(a) < f3( a) < f4(a) < ... < f2n(a) = fixedpoint f2n(a) is the leftmost fixedpoint right of the right seed A.

All fixedpoints are both left and right seeds - of themselves.

4C. The Outer Fixedpoints 61

C. The Outer Fixedpoints

Now that we have self-referential forms, the question is: can we evaluate them in diamond logic? And if so, how? It turns out that phase order permits us to do so in general . For any harmonic function E(x), we have the following:

The Self-Reference Theorem: Any self-referential harmonic system has a fixedpoint: F(x) = x

Proof: Recall that all harmonic functions preserve order. i is the leftmost set of values, hence this holds: i < F() Therefore, i is a right seed for F: i -< F(i) VW F3Q) ... F^°G) = F^^'G)) i generates the "leftmost"l'filxedpoint. QED.

Similarly, j generates the "rightmost " fixedpoint:

All other fixedpoints lie between the two outermost: F2"(i) < x

= F(x) : ea)

62 Diamond, A Paradox Logic I call this process "productio ex absurdo "; literally, production from the absurd; in contrast to "reduction to the absurd", boolean logic 's refutation method. Diamond logic begins where boolean logic ends. To see productio ex absurdo in action , consider this system:

A

B C I I

=

I B = A B C I

C = C=

[[[BC[]]A][ABC]B]c c

bl aI

I I I

I I I

I

I

I I

I I I

I I

I

I I I

I

I I I I

A= B C I I I B

LI

I I

= A B C I C= A I B I

\-

I

LI

63

4C. The Outer Fixedpoints

;^_I

I II

\-I - I II

=

Î

Î

I I \I

1 \I \I I

I =

\_..I

I

- I II

- I II

=

_I II

I \._I \_I 1

=

\_I

1 I

I

I I

I

\-I I \I I

1

I -11 - 1 II I \J II

-I II

=

'\_I

-I II

- I II

I -I II

-I I

'_I I I I

I

I

I I I II I \I I

I =

The leftmost fixedpoint is: A = void, B = curl , C = void. Iterating from uncurl yields the rightmost fixedpoint: A = void, B = uncurl , C = void. All fixedpoints are between the outer fixedpoints ; therefore A = C = void; therefore B = cross B ; therefore B = curl or uncurl . Thus the outer fixedpoints are the only ones.

64 Diamond, A Paradox Logic Now consider the system:

Alan: "The key is in the drawer." Bob: "If I'm right, then Alan is right." Carl: "Alan is right or wrong." Dan: "Alan is right, but Carl is right." Eli: "Carl is right, but Alan is right." Fred: "Danis right or wrong." Gary: "Eli is right or wrong." Harry: "Fred and Gary are both right."

If two-valued logic were in control of this situation, then Bob's Santa sentence would make Alan right; and everyone else would be right too. Yet when you look in the drawer, the key isn't there!

The equations are: A = false B = if B then A C = A or not A D = AbutC E = C but A F = D or not D G = E or not E H = FandG

4C. The Outer Fxedpoints

65

If we iterate this system from default value i , we get: (A,B,C,D,E,F,G,H) 4 (f,i,i,i,i,i,i,i) c (f,i,t,f,i,i,i,i) (f,i,t, j,i,t,i,i) -0 (f,i,t,j,i j,i,i) 4 (f,i,t,j,i,j,i,f) = fixedpoint.

If we had started from default value j , we would have gotten the rightmost fixedpoint (f,J t,j, i,j,i,f). As above, these are the only two fixedpoints.


Chapter 5

Fixedpoint Lattices Relative Lattices Shared Fixedpoints Examples, including "ant" and "triplet"

A. Relative Lattices Any harmonic function F(x) has the outer fixedpoints: FZ"()' the leftmost fixedpoint, and FZ"(1), the rightmost fixedpoint. But often these are not all . In general, F has an entire lattice of fixedpoints.

Theorem : If a and b are fixedpoints for a harmonic function F, then these fixedpoints exist:

a minF b = the rightmost fixedpoint left of both a and b = FZ"(a min a maxr b = the leftmost fixedpoint ri t of both a and b = F' max b)

Proof: Let a and b be fixedpoints, and let c be any fixedpoint such that a < c and b < c. Then (a max b) < c ; so

(a max b) = F(a) max F(b) < F(a max b) < F(c) = c Ergo (a max b) is a right seed less than c:

(a max b) < F(a max b) < F2(' max b) < ... FZ' (a max b) = F(P(a max h)) < c

67


68

Therefore E(a max b) is a fixedpoint right of a and of b and is moreover the leftmost such fixedpoint. Thus,

MR max b) =

Similarly, P (a min b) =

a maXF b a minF b . QED.

For instance, consider the following Brownian form: d

c I a_b I I I I I I I I I I I I I I

This is equivalent to this boundary -logic system:

a = [b] ; b = [a] ;

c = [ab] ; d = [cd]

In the standard interpretation, (a,b,c,d) is a fixedpoint for: F(a,b,c, d) = ( not b, not a, not(a or b), not(c or d) )

In the nand-gate interpretation:

d = not( d and c ) =

not(d) or not(c) = (d 4, d(a))

Sentence d says "If I'm not mistaken, then sentence A is both true and false": a Lower Differential Santa Sentence! In the nor-gate interpretation: d = D(a) - d ; Sentence d says "A is true or false, and I am a liar." An Upper Differential Grinch! (See "The Grinch", in the Appendix.)

5A Relative Lattices

69

F has this fixedpoint lattice:

tffi ----- tffj

ftfi ----- ftfj If we seek minF of tffj and ftfj , then we must take their minimum, then apply F three times: (tffj min ftfj) = iiii '4 iiij s> Hit 41 iiii


B. Shared Fixedpoints

More than one function can share a fixedpoint. For instance:

Theorem : If F(x) and G(x) are harmonic functions and F(G(x)) = G(F(x)) (F and G commute) Then F and G share a nonempty lattice of fixedpoints: F(x) = x ; G(x) = x for all x in LFG.

Proof: We have proved that the harmonic function G has a lattice of fixedpoints; G(x) = x for all x in LG. But since F commutes with G, G(F(x)) = E(G(x)) = E(x) for all x in LG. - that is, F sends fixedpoints of G to fixedpoints of G. F sends LG to itself. What's more, E preserves order in diamond; therefore F preserves order in LG. Therefore F is an order-preserving function from LG to itself. That fact, plus lattice arguments like those in previous sections, will prove that F has fixedpoints in a nonempty sublattice LFG of LG: F(x) = x and G(x) = x for every element x of L. QED.

5B. Shared Fixedpoints 71

Lm is a lattice of shared fixedpoints. Its least element is FZ"(G'(i)); its largest element is P(G GZ" (i)); its relative minimum operator is Fti' (Crzn(a min b)); and its relative maximum operator is F'(G10(( max b)).

These results can be extended to N functions: If F, E2, ... F are N commuting harmonic functions, then they share a lattice of fixedpoints:

F (x = x for all i between 1 and N, and all x in L.

Its least element is F1'(F2' ( ... (FNZ" (i))...); its largest element is F,'(f2 '( ... (FN'G))... its relative minimum operator is FI'(FF2' ( ... (FN(g min lb))...); and its relative maximum operator is F,'(F2'(... (EAa max b))...) .


C. Examples

Consider the liar paradox: I = A I = [A]A

A = notA

Here is its Brownian form: I - I Here it is as a circuit: I I

This is its fixedpoint lattice: i

Now consider Tweedle's Quarrel:

Tweedledee: "Tweedledum is a liar." Tweedledum : "Tweedledee is a liar."

E

U

=

=

U

EI

I

I

II III I__I

E

=[

[

E

]u]E

5C. Examples 73

Its circuit is : I I

tf / \ / \ This "toggle's" lattice is: \ / \ / ft

Consider the following statement: "This statement is both true and false." It resolves to this system, the "duck":

B = [ [ B ]A B ]B b

A B I

a1

I I B = ABI 1 11 1 I I_I Here is its lattice :

i i ----- tf ----- ii

The "truck": C = [[[A]AC]B]c

A

=

I II B = CAI I I l l C = B I I I I I I has this lattice :

A

I

iii i ft j ft j j j


74

To create linear fixedpoint lattices of length 2n + 1, use: X, = Dx, x2 = Dx2 or x, x3 = Dx3 or x2 x„ = Dx„ or x, .,

For n = 4, we get the lattice: - jjjt - jjjj iiii - Mt - iitt - ittt - tttt - jttt - jjtt

Its circuit is:

I call this circuit "the ducks".

To create linear fixedpoint lattices of length 2n, use: X, = not x, x2 = Dx2 or x, x3 = Dx3 or x2

x" = Dx" or x".1 For n = 4, we get the lattice: - Mt - iitt - ittt - jttt - jjtt - jjjt

75

5C. Examples

This jolly-looking form : C = [[[BC ]A[ A ]B

if C I ]C C

A

=

B

BCI =

A

I

a_b I I I I I I I I I I I I I I I I I I

C AB I C I I I I I I i t has this lattice: ftt fti ftj iii ftf iii iif jjf tff

The "rabbit": D = [[[[B]AC]BD]CID d

Î A = B I bi I I al I I B= ACI I I I I I I C= BSI 11 1-1 1 1 I I I I D C I I has a similar lattice: tftf tfii tfjj iiii tfft jiji iift jjft ftft

Diamond, A Paradox Logic

76

This Brownian form: c = [[[a]a]a[ac]b], c a

c

a

I

I

I I I I I I I I I I-I is equivalent to the boundary-logic system: a = [a[a]] ; b = [ac] ; c = [ab] That is: a = da ; b = a nor c ; c = a nor b Its fixedpoint lattice is: ftf /

\

iii --- fii fjj --- iii \ /

\ / fft

In general, the system a = da; b = M(a, -a, f(b) ) will have this fixedpoint lattice:

ii ---( L )---jl

- where L is fs fixedpoint lattice.

5C. Examples 77

This system: a = [b]; b = [a] ; c = [ad]; d = [acd]; e = [bf]; f = [be] has this fixedpoint lattice: tffftf / \ tfffii tfffjj / \ / \ / tfffft \ iiiiii

ftiiff- fttfff -ftjjff

The "toggle " ab controls which subcircuit activates ; the toggle of or the "duck" cd. In general, the system

a = not b b = not a c = (a & -c)) or (b & g()) or da will have this fixedpoint lattice:

- where L1 is f's lattice, and where L2 is g's lattice.


78

The "triplet" has this form: C = [ [ B C ]A [ C A ]B ]c A

=

B

C

I

I II B = CAI I I I I I I C = A B I I I I I I-I I I I The triplet's lattice is:

Note that this lattice (called "M3") is non-distributive: a 0 < b < 1 c

(a max b) min c = 1 min c = c (a min c) max (b min c)= 0 max 0 = 0

On the other hand, it is "modular": x < z implies x max (y min z) = (x max y) min z It is a theorem of lattice theory that any non-distributive modular lattice contains M3 as a sublattice.

5C. Examples 79

The "ant", or "toggled buzzer", has the form C = [ [ [ B IA )B C I

A B

=

=

B I I I II Al I Ill

C BCI I I I I I I I The ant's lattice is: ftf

tfi < tfj

Note that this lattice (called N5) is non-distributive: b < < (a max b) min c = I min c = c 0 1 < < (a min c) max (b min c) = a max 0 = a a xN , for all but finitely many N. and lim + is the leftmost such value.

Lim- and lim+ are min, or max, respectively, of the cofinal range of x,,; the set of values that occur infinitely often: lim-{x„) = Min cofinal{x„} lim+{x„} = Max cofinal{x„} where cofinal {x„} = { Y : x„ = Y for infinitely many n }

Theorem : If F is a harmonic function, then F(lim- x„) < lim- F(x,J ; F(lim+ x„) > lim+ F(x„)

Proof: We shall take the lim- case first. lim- x„ < xN , for cofinitely many N. Therefore :

F(lim- x,) < F(xN) , for cofinitely many N.

Therefore :

F(lim- x,) < lim- F(xN),

since lim- F(xN) is the rightmost value left of cofinitely many F(xN)'s! The lim+ case follows by symmetry. QED.

6A. Limits Here's another proof, by cofinality: F(lim- x„) = F(Min cofinal{x„}) < Mn F(cofinal{x„}) = Min cofinal (F(x„)) = lim- F(x„) QED.

These inequalities can be strict . For instance: F(x) = dx, and x„ = {t,f,t,f,t,f...}:

d(lim-(t,f,t,f,t,f... )) = di = i; lim-(dt,df,dt,df...) = lim-{f,ff,f... } = f: so d(lim- x„) < lim- d(x„)

d(lim+{t,f,t,f,t,f... }) = di =j; lim+{dt,df,dt,df... } = lim+{t,t,t,t... } = t. so d(lim+ x„) > lim+ d(x„)

87


B. Limit Fixedpoints

Fixedpoints can be found by transfinite induction on the limit operators. Recall that for all harmonic functions F:

E(lim- )) < lim- F(a,) F(lim+ a") > lim+ F(a.) Given any set of initial values sue, then let = = lims"

F"-(so)

s) F"(,

So F(s,) = F( lim- F"(s^) ) < lim- F( F"(s%) ) lim- F"+'(%) = lim- F"(sSO) =

s"

Therefore F'-W = lim-(F"(so)) is a left seed. It generates a fixedpoint: s" F(5,,,) F2(s) > ... s2" = lira- F"(lim- ( OO)))•

S 2. is the limit of a descending sequence, and therefore also its minimum. If F has only finitely many components, then the descending sequence can only descend finitely many steps before coming to rest. Thus; if F has finitely many components , then s 2" is a fixedpoint for F: 12.

Similarly, F"+(sa) = lim+(F"(so)) is a right seed, which generates the fixedpoint F 2"+(%) = lim+(F"(lim +(F"(s^)))).

6B. Limit Fixedpoints 89

These seeds also generate minima and maxima in the relative lattice: lim-(F"( E'(xo) min F""(Yo))) = FZ" (xo) rmnF V'-(yo) lim+(°( F"+(xo) max F "+(Yo))) = F2)+( xo) maxF F"+(Yo)

If F has infinitely many components, then we must continue the iteration through more limits. Let sS NO = "M-(-F(1 2c)) 94W

=

lim -(F"(s J)

lim-(s ".)

1Gaw

And so on through the higher ordinals. They keep drifting left; so at a high enough ordinal, we get a fixedpoint: F(ss) Large cardinals imply "late" fixedpoints: self-reference with high complexity. Alas, the complexity is all in the syntax of the system, not its (mostly imaginary) content. Late fixedpoints are absurdly simple answers to absurdly complex questions.


C. Diamond Computation

Let F2u'(xo) = lim- F"(lim- F"(x,)) and F2° (x^) = lim+ F"(lim+ F"(x&))

These are the left and right fixedpoints generated from & by iterating E twice-infinity times. We can regard each of these as the output of a computation process whose input is x& and whose program is F. Diamond's computation theory is the same as its limit theory; output equals behavior "in the long run".

If F is n-dimensional, and if F '(so) is a cyclic pattern - that is, a wave then lim-(E W) equals minimum over a cycle, and lim+(F'(s%)) equals maximum over a cycle. These yield the wave-bracketing fixedpoints:

V"-(&)

lim- F(lim- F"(x&)) FZ"(Min (F'(s)), where Min is taken over at least one cycle. This is the rightmost fixedpoint left of F'(so) .

F2"+(^) x = lim+ F"(lim+ F"(xo)) F2( Max

(F

'(so)

),

where Max is taken over at least one cycle. This is the leftmost fixedpoint right of F'(s%) .

6C. Diamond Computation 91

Their existence implies this Halting Theorem: If F has n components, then its limit fixedpoints equal:

F2" () F2(a

+()

E2" ( min (F'(so)) ) 4"<j x ] { E(y) : E"(y) = t }

E ( --, x) = Union[ n such that q <x I { E(y) : E"(y)=f} The first is a countable union of intersections of E(R) with the open set { s : so = t } ; the second is a countable union of intersections of E(R) with the open set { s : s„ = f } . In either case, E sends a half-line to an intersection of E(R) with an open set in 0". Thus the lemma is proved: the inverse of E is continuous. Therefore E is an embedding: I-1 and bicontinuous. QED.

Theorem : Any continuous function f from R to diamond "lifts" to a harmonic function f* from 0" to diamond. f R -----------> 0 1 1 E = This diagram commutes. i

0" -----------> 0 f*

112 Diamond, A Paradox Logic Proof: Let F(x) be a continuous function from R to diamond. The inverse image of an open set by a continuous function is an open set; so these are open sets:

F(t) = (xinR:F(x)=t) F(f) = (xinR:F(x)=f)

Call the first set A and the second set B . Being open, they are countable unions of open intervals: A = Union(all N) (aN, AN) B = Union(all N) (bN, BN)

where all the a's and b 's are chosen from the rationals.

Approximate these sets by finite unions: A„ (x)

_ (a, S x S A) or (a2 x

A2) or ... or

B„ (x)

_ (b, x

B2) or ... or (b„ x

B) or (b2 X

(a„

Sx S

A,,) B,,)

Then take left limits: A(x) = lim- A„ (x)

= lim-((a, S x A, ) or (a2 x A2) or ... or (a„ S x S A,,)) B(x) = lim- B„ (x) = lim-((b, S x B,) or (b2 S x S B2) or ... or (b„ x B,J) These are the characteristic functions for A and B; made strictly from the Dedekind splice and diamond logic.

8D. The Line Within The Diamond 113 Now define F'(x): F'(x) A(x) min [not B(x)) lim-((a, S x A) or (a2 x A2) or ... or (a„ S x S A,,)) min not (lim-((b, x B) or (b2 S x S B) or ... or (b„ S x Bn)))

This function equals true if x is in the interior of A and the exterior of B: that is, F(y) =t and F(y) Of, for any y near enough to x. This function equals false if x is in the exterior of A and the interior of B: that is, F(y) = f and F(y) # t , for any y near enough to x. Finally , this function equals i at the boundary of the above two sets ; that is, F(y) = f, and F(y') = t, for some y and y' in any neighborhood of x.

But F is a continuous function; so it equals t in the interior of A, fin the interior of B, and i at the boundary. Therefore F'(x) = F(x). Note that F'(x) is made only from E and diamond logic: F(x) = F*(x) = C(E(x)) where C is a harmonic function. Therefore C(x) is a harmonic function which extends F(x) (via the embedding E) to all 0'. QED.

114 Diamond , A Paradox Logic Theorem: E is not only an embedding; it is a morphism: that is, functions from R to R "lift" to functions from 0" to 0'; f R -----------> R I I I E I E This diagram commutes. 0" -----------> 0"

f*

Proof If f is a function from R to R, then let i;, be the nth component of f(x), via the embedding E: fn(x) = E "(f x)) = [ f(x) Z 9,1 ] f is continuous; dedekind splice is continuous; so f„ is continuous. Therefore, by the above theorem, f, extends to a function from 0" to diamond; therefore the function

f = (fi,fz,---- fn,...) extends to a continuous function from 0" to 0". QED.

Thus the real continuum embeds and extends into the space of diamond vectors. The continuum reduces to harmonic form.

8E. Zeno's Theorem 115

E. Zeno 's Theorem

Every continuous function from the real line to itself extends to a harmonic function from diamond space to itself. But every harmonic function on diamond space has a fixedpoint. Therefore we get:

Zeno's Theorem:

Any continuous function from the real line to itself has a fixedpoint in diamond space.

I name this theorem after Zeno of Elea, famed for his paradoxes of motion. With the proof of this Theorem, we see that Zeno was right after all - in part. He claimed that no motion is possible : here we see that no motion is universal. Any continuous transformation of space has a fixedpoint; any chaotic dynamic has a paradoxical resolution.


Part Two

Advanced Diamond Logic


Chapter 9

Analytic Functions Analytic functions Type theorem; harmonic or analytic Conjugation by dihedral group Star logic

A. Analytic Functions If a function is harmonic, then it preserves order; but not all functions preserve lattice order, so not all functions are harmonic. Consider these: x i t f j ----------------------------- -----------------------------(x) = termwise negation I j f t i *(x) = dual of x

=

not

-

x

1(x) = left side

=

x/*x

I

t

r(x) = right side =

*x/x

I

L(x) = turn left = * x/not (x)I R(x) = turn right = not(x)/*xI

I

i

f

i

t

f

f

t

f

t

f

f

i

j

t

j

i

f

t

t

"Star" (*) obeys these laws: *M(x,Y,Z) = M(*x,*Y,*z) *(xandy) *xand*y ;*(xory)= *xor*y *(x min y) _ *x max *y ; *(x max y) = *x min *y *(not(x)) = not(*(x)) = -x; **x = x

119


The left and right side operators obey these equations:

I(M(x,y,z)) = M(lx,ly,lz) ; r(M(x,y,z)) = M(rx,ry,rz) 1(x and y) = lx and ly ; 1(x or y) = lx or ly 1(x min y) = Ix or ly ; I(x max y) = lx and ly r(x and y) = rx and ry ; r(x or y) = rx or ry r(xminy)= rxand ry ; r(xmaxy)= rxorry 1(1(x)) = r(1(x)) =1(x) ; I(r(x)) = r(r(x)) = r(x) l(not x) = rx ; r(not x) = lx ; *(Ix)=lx; *(rx)=rx 1(*x) = rx ; r(*x) = lx .

The rotation operators R and L obey these laws:

L(M(x,y,z)) = M(Lx,Ly,Lz) ; R(M(x,y,z)) = M(Rx,Ry,Rz) L(xandy)= Lx maxLy; L(xory)= Lx minLy L(x min y) = Lx and Ly ; L(x max y) = Lx or Ly R(x and y) = Rx min Ry ; R(x or y) = Rx max Ry R(xminy)= RxorRy; R(xmaxy)= RxandRy LLx=RRx=-x; LLLx=Rx; RRRx=Lx; LRx=RLx=x L(not x) = not(Rx) = (-x/x); R(not x) = not(Lx) = (x/-x); L(*x) = *Rx = (x/-x); R(*x) = *Lx = (-x/x)

9B. Type Theorem 121

B. Type Theorem

"Minus" reveals diamond's underlying boolean structure . Call a function F "analytic" if you can define minus by using it and the harmonic functions: -(x) = G(F(H(x))) , for some harmonic G and H. Minus does not preserve lattice order; so no analytic function does either. The converse is also true.

Type Theorem

If F does not preserve order, then F is analytic.

Or, to be more specific: If F(a) j F(b) for some a < b then at least one of these two functions equals -(x): F(a max (b min (x/not x))) / not F(a max (b min (not x /x))) not F(a max (b min (x/not x))) / F(a max (b min (not x /x)))

Proof: Here are the truth tables for those two functions: i

t

f

J

-I F1 I F(a)/not F (a) F(a)/not F(b) F(b)/not F (a) F(b)/not F(b) F2 notF (a) /F(a) notF (a) /F(b) notF (b)/ F(a) notF(b)/ F(b)


There are seven possible ways to have F(a) I F(b) : (F(a),F(b)) =

(i,t), (i , i), (i,f), (t,i ), (t,f), (f,i), (f,t) F1,F2 ; F1,F2 ; F1,F2 ; F1,F2 ; F1,F2 ; F1,F2; F1,F2

I

I

I I t If,ilf,fIf,fIi,fIt,fIf,iIf,t f It,ilt,tIt,tIi,tIf,tIt,iIt,f j I i, i i , i i, i i , i j, i i, i i, j I

* In each case at least one of FI and F2 equals -(x). QED.

Thus we get these two equivalences: F is harmonic if and only if F preserves phase order. F is analytic if and only if F does not preserve phase order. There are two kinds of functions on diamond; analytic or else harmonic.

9C. Dihedral Conjugation 123

C. Dihedral Conjugation

Given a permutation P, a function F, and a relation R, we can define the function P[F] and the relation P[R] by: P[F] (x) = P(F(P-'(x))) x P[R] y if P"'(x) R P-'(y) These are F and R conjugated by P.

Conjugation Theorems: P(F(x,y)) xRy P[=]

P[F] (P(x), P(Y) ) if

P(x) P[R] P(y) (_)

P [Q[F]]

(PoQ)[F]

P[F]o(P[G])

P [FoG]

Proof P[F] (P(x), P(Y)) = P(F( P-'(P(x)), P-'(P(Y)) )) P( F(x, y)) QED. P(x) P[R] P(y) if P-'(P(x)) R P-'(P(y)) if xRy. QED. x P[=] y if P-'(x) = P"'(y) if x = y. QED. P[Q[F]](x) = P(Q[F](P-'(x))) = P(Q(F(Q-'(P-'(x))))) _ (PoQ)oFo(PoQ)-'(x) = (PoQ)[F](x). QED. P[F]o(P[G])(x) = P(F(P-'(P(G(P-'(x)))))) P(F(G(1'-'(x)))) P[FoG](x). QED.

124 Diamond , A Paradox Logic Whatever equational identities the functions F and G may have, the functions P[F] and P[G] also have. Thus the conjugate of a DeMorgan algebra is a DeMorgan algebra, the conjugate of a field is a field, etc. Conjugation transports identities.

Now let the dihedral group D operate on the diamond. It has four reflections and four rotations: ; (tt)= "not

"; (ii)="I (ti)Gfl ="o/-"; (tj)(i f) = "-/o"

identity="o"; (tiff)= "L' ;(tf)(ij)="-'; (tjfi)="R". b a*b

a

o

o R

R

-

L

o R L L R o L o R o R L not -/o * o/-/o * o/- not * o/- not -/o o/- not -/o *

L not o/* -/o

not

o/-

*

-/0

not o/* -/o o R L

o/* -/o not R L 0

* -/o not o/L o R

-/0 not o/* L 0 R -

If we identify the two-dimensional real vectors with a "diamond vector"; that is, linear combinations of diamond values: (r) (s)

rt + si

then we can identify this group as 2 by 2 matrices; o

=

(1 0) not (0 1)

= (-1 0) (0 1)

9C. Dihedral Conjugation 125 R

=

( 0 1) ; 0/_ (0 1) (-1 0) (1 0) (- 1 (0

L

=

(0 (1

0) -1)

(1 (0

0) -1)

-1) (0 -1) 0) (-1 0)

Modulo -, these are equivalent to the generators of M(2,2), the two-bytwo matrices. D permutes functions and relations as well as elements, by conjugation. For P in the dihedral group, this applies: P[M]

=

M

That is : P( M(x, y, z) ) = M ( P(x), P(y), P(z) )

Thus the above group table also defines the group's conjugation action on the logic operators: b a[b] I and or min max not *

------- I------------------------------------a o 1 and or min max not R I min max or and * not or and max min not L I max min and or * not not I or and min max not o/- I max min or and * not * and or max min not * -/o I min max and or * not

Note that all four positive operators distribute over each other; very symmetric.


D. Star Logic Most of the best properties of diamond are shared by 3-valued logic, a simpler sublogic. 3-logic is a DeMorgan logic; it has enough semi-lattice to prove self-reference; it too solves the paradoxes of self-reference and continuity; and it too embeds the continuum. So what is the second paradox value doing? The answer is that diamond-logic has inner symmetries unavailable to 3logic; and that in fact there is a "paraharmonic" logic complementary to the harmonic logic.

Recall that -(a/b) = (-a)/(-b); "termwise" negation. Minus gets down to Diamond's boolean innards. Let * = not - ; that is, *(a/b) = (*b)/(*a).

Star reverses order. It exchanges i and j, leaving t and f fixed. Thus Star looks just like Not, at "right angles"; it is "sideways negation".

In diamond logic star is flip: *(a/b) = (*b)/(*a) In dynamic implementation star equals delay: (*a)(n) = a(n-1) In dual-rail circuits star equals swap wires.

9D. Star Logic 127 Star, not, identity and - form a Klein group: b ab I o not * -----------------------------o I o not * -

a not I not o - * all elements equal * I * - o not their own inverse - I - * not o

Let "star logic" be a logic made from *, majority, and the four values, just as diamond logic is made from not, majority, and the four values. Star logic is isomorphic to diamond logic via rotation; therefore all results from the preceding chapters apply: Star logic is a complete De Morgan algebra. It proves the self-reference theorem. It has limit operators. The continuum embeds via a morphism. Zeno's theorem. When we combine star logic and diamond logic, then we get -, a nonfixedpoint operator . In a sense, then, star logic is "perpendicular" to diamond logic; similar to it, but intersecting it only at a point. Therefore I call star logic "paraharmonic "; it resembles harmonic logic but is incompatible . Diamond logic is two-dimensional ; it has room for two separate dimensions of thought within it. Negation and star are "perpendicular" logics; they work at cross-purposes. Thus I propose Perpendicular Processing as a practical (and realistic) alternative to Parallel Processing.


Chapter 10

Harm onic An alysis Harmonic projection A, p; laws Differentials Quadrature Diffraction

A. Harmonic Projection When we rotationally conjugate the "left side" and "right side" operators; 1(x) = x/*x , r(x) = *x/x we get two "harmonic projection " operators; I(x) = x / not(x) ; p(x) = not(x) / x . These send the diamond to i and j, just as I and r send the diamond to t and f. Unlike I and r, I and p are harmonic - i.e. they preserve phase order. Being conjugate to I and r, they obey similar equations: A(i)=i; A(t)=i; 7l(f)=j; A(j)=j P(i)=i; P (t)=j; P(f)=i; PO) =j A(M(x,Y,z)) = M(lx,AY,Az) ; P(M(x,Y,z)) = M(Px,PY,Pz) A(xandy)= Xx max Xy; A(xory)= Ax min ply A(xminy)= Xx min Xy; 1l(xmaxy)= Ax max Ay p(xandy)= px min py; p(xory)= px maxpy p(xminy)= px min py; p(xmaxy)= px max py A(not x) = px; not(px) = Xx ; p(not x) = Ax ; not(px) = px 1L(*x) = *Px; P(*x) _ *xx. Harmonic logic can side-analyze star logic; and vice versa.

129


B. Differentials

Recall these "self-difference" expressions: dx = x and not x = x minus x = x xor x the "lower differential" Dx = x or not x = x implies x = x iff x the "upper differential" In Boolean logic these are identical to, respectively , false and true; and indeed those identities are the Laws of the Excluded Middle. Diamond does not obey those laws; instead it has the "interference" laws:

and

D and d erase boolean information but preserve order. They project diamond onto a sideways lattice ; 3*, which with *, min and max define paraharmonic 3-logic. D and d project diamond to a linear order; thus they are in effect norms on diamond.

dx and x

dx and Dx

x or dx

x and Dx

Dx or x

Dx or dx

i.e. dx is a subset of x, which is a subset of Dx. In Venn diagram terms, dx is the boundary of x.

10B. Differentials 131 We get these "differential logic equations":

not dx

=

Dx ; not Dx = dx

d(not x) = dx ; D(not x) = Dx ddx = dDx = dx DDx = Ddx = Dx

d(x and y) _ (dx and y) or (x and dy) D(x or y) _ (Dx or y) and (x or Dy)

I call these the "Leibnitz rules", due to their similarity to the Leibnitz rule for derivatives of products . They imply: d(x or y) _ (dx and (not y)) or ((not x) and dy) D(x and y) _ (Dx or not y)) and ((not x) or Dy)

Before we continue, recall: p(x) = "right harmonic projection " = (not x) / x ll(x) = "left harmonic projection " = x / (not x) r(x) = "right analytic projection " _ (* x) / x 1(x) = "left analytic projection " = x / (* x) R(x) = "right rotate" _ (not x) / (* x) L(x) = "left rotate" _ (* x) / (not x) soR=p /l andL =r/,X Finally, define: If x then y else z = (x y) and ((not x) z)


These definitions imply these differential logic equations: dx = X(x) and p(x) Dx = I(x) or p(x) d(A(x)) = X(x) ;

d(p(x)) = p(x)

D(A(x)) = A(x) ;

D(p(x)) = p(x)

d(x-y) = d(x

y) = ((not y) and dx) or (x and dy)

D(x-y) = D(x

y) = (y and Dx) or ((not x) and Dy)

d(x xor y) = d(x iffy) dx xor dy D(x xor y) = D(x iff y) = Dx iff Dy

dM(x,y,z) = ((y xor z) and dx) or ((z xor x) and dy) or ((x xor y) and dz) or M(dx,dy,dz) DM(x,y,z) = ((y if z) or Dx) and ((z iff x) or Dy) and ((x iffy) or Dz) and M(Dx,Dy,Dz)

d(if x then y else z) = (x and dy) or (y and dx) or (not x and dz) or (z and dx) = ((y or z) and dx) or (if x then dy else dz) D(if x then y else z) = (not x or Dy) and (not y or dx) and (x or Dz) and (not z or Dx) = ((not y and not z) or Dx) and (if not x then Dy else Dz)

106. Differentials 133 d(x/y) = (x-y)/(y-x) = A(x) and p(y) D(x/y) = (Y x)/(x y) = X(x) or p(y)

d(x min y) = dx min dy min (x-y) min (y-x) = dx min dy min ((x xor y) max t) d(x max y) = dx max dy max (x-y) max (y-x) = dx min dy min ((x xor y) min t) D(x min y) = Dx min Dy min (xy) min (yx) = Dx min Dy min ((x iff y) max f) D(x max y) = Dx max Dy max (xy) max (yx) = Dx max Dy max ((x iffy) min f)

Thus D and d do not commute with min and max.

d(-x) = -Dx = *dx ; D(-x) = -dx = *Dx d(not x) = not Dx = dx ; D(not x) = not dx = Dx d(*x) = *dx ;

D(*x) _ *Dx

i.e.: *d* = d ;

not d not = -d- = D

"Klein duality"

d(x min not x) = x min not x = pdx = XDx d(x max not x) = x max not x = Adx = pDx D(x min not x) = x min not x = Adx = pDx D(x max not x) = x max not x = pdx = ADx

134 Diamond, A Paradox Logic x min not x = XDx = pdx = dx min Dx = Dx/dx x max not x = pDx = Xdx = dx max Dx = dx/Dx d(Rx) = px and *Xx = R(x and *x) d(Lx) = Ix and *px = L(x and *x)

By combining differentials, we get these equations:

dx and dy = (x and y) - (x or y) "both without either" = d(x/y) and d(y/x) "intermix" Dx or Dy = (x and y) (x or y) "both implies either" = D(x/y) and D(y/x) "intermix"

dx or dy = (dx if dy) and (dx if Dy) = (x iffy) and (x if not y) Dx and Dy = (Dx xor Dy) or (Dx xor dy) = (x xor y) and (x xor not y) "opposite reflections" dx xor dy = d(x xor y) Dx xor Dy = d(x xor y) dx iff dy = D(x iffy) Dx if Dy = D(x iffy)

Of especial interest are the "integration" equations: dx or dy = (x/y) xor (y/x) Dx and Dy = (x/y) if (y/x)

1 OB. Differentials 135 They show that in diamond , unions and intersections of differentials can equal any value; and they tie differentials with the symmetric difference and reflection operators. The link is the "intermix" operation: J (x, Y) = (x/Y, Y/x ) The "intermix " operator sends pairs of diamond values to pairs of diamond values . What is more, it is self-inverse; iterated twice it is the identity. These facts imply these "intermix integration" laws:

d(x/y) or d(y/x) = x xor y D(x/y) and D(y/x) = x iff y d(x/y) and d(y/x) = dx and dy D(x/y) or D(y/x) = Dx or Dy

d(x/y) if d(y/x) = dx if dy = D(x iffy) D(x/y) xor D(y/x) = Dx xor Dy = d(x xor y) d(x/y) xor d(y/x) = d( x/y xor y/x ) = d(dx or dy) = dx xor dy = d(x xor y) D(x/y) xor D(y/x) = D( x/y iff y/x ) = D(Dx and Dy) = Dx iff Dy = D(x iffy)

136 Diamond , A Paradox Logic From these we get these extensions of the Leibnitz rules: (x and dy) or (y and dx) (x and y) and (dx or dy) (x and y) and (x if not y) and (x iffy) (x and y) and. (x if not y) and (x iff y) (x and y) and (dx iff dy) and (dx iff Dy) (x and y) and (x/y xor y/x) (x and y) and (x xor y)

D(x or y) _ (x or Dy) and (y or Dx) (x or y) or (Dx and Dy) (x or y) or (x iff not y) or (x iffy)

_ (x or y) or (x xor not y) or (x xor y) (x or y) or (Dx xor Dy) or (Dx xor dy) (x or y) or (x/y iff y/x) (xory)or(xiffy)

We can also rewrite the majority-boundary rules:

dM(x,y,z)

M(dx,d(y/z),d(z/y)) or M(dy,d(x/z),d(z/x)) or M(dz,d(x/y),d(y/x))

M(x-x, y-z, z-y) or M(y-y, x-z, z-x) or M(z-z, x-y, y-x)

1OB. Differentials 137

M(d(x/y),d(y/z),d(z/x)) or M(d(x/z),d(z/y),d(y/x)) or M(dx,dy,dz)

M(x-y, y-z, z-x) or M(x-z, y-x, z-y) or M(x-x, y-y, z-z)

DM(x,y,z)

M(Dx,D(y/z),D(z/y)) & M(Dy,D(x/z),D(z/x)) & M(Dz,D(x/y),D(y/x))

M(Dx, yz, zy) & M(Dy, xz, z4x) & M(Dz, xy, yx)

M(D(x/y),D(y/z),D(z/x)) & M(D(x/z),D(z/y),D(y/x)) & M(Dx,Dy,Dz)

M(xy, yz, zx) & M(xz, yx, zy) & M(Dx, Dy, Dz)

Here are some "junction integrations":

dx/dy =

(X/Y) - (Y/x)

Dx/Dy = (Y/x) 41 (X/Y) dx/Dy =

(x/y) max not(y/x)

Dx/dy = (x/y) min not(y/x)

138 Diamond, A Paradox Logic Here are "differentials of intermix"; a.k.a. "diffractions of differential":

d(x/y) / d(y/x) = x - y D(x/y) / D(y/x) = y x d(x/y) / D(y/x) = x max not y D(x/y) / d(y/x) = x min not y

d(x/f) / d(f/x) = x D(x/t) / D(t/x) = x d(t/x) / d(x/t) = not x D(f/x) / D(x/f) = not x d(i/x) / D(x/i) = d(t/x) / D(x/f) = not x D(j/x) / d(x/j) = D(f/x) / d(x/t) = not x

d(x/(not y)) / d((not y)/x) = x and y D(x/(not y)) / D((not y)/x) = x or y d(x/(not y)) / D((not y)/x) = x max y D(x/(not y)) / d((not y)/x) = x min y

And so it seems that phase junctions of differentials of phase junctions yield arbitrary logic functions. (More on this "diamond interferometry" in the "Diffraction" section.)

1OB. Differentials 139 In Boolean logic, "xor" and "iff' are isomorphic to addition modulo 2. Alas, in diamond they are no longer group operations:

(t and i) xor (t and i) = i # f = (t xor t) and i (for i) if (for i) = i # t = (f if f) and i

so they are non-distributive; (t xor i) xor j = t # f = t xor (i xor j) (f iff i) iff j = t # f = f iff (i iff j) so they are non-associative; not(i xor j) = t # f = (not i) xor j not(i iffj) = f # t = (not i) iffj so they are non-symmetrical.

This is because xor and if contain "differential terms":

x iffy = not(x xor y) = ((not x) xor y) or dx or dy x xor y = not(x iffy) _ ((not x) iffy) and Dx and Dy

(x and z) xor (y and z) = ((x xor y) and z) or ((x or y) and dz) (x or z) if (y or z) = ((x iffy) or z) and ((x and y) or Dz)

140 Diamond, A Paradox Logic (x xor (y xor z)) _

(x and y and z) or (not x and not y and z) or (not x and y and not z) or (x and not y and not z) or (x and dy) or (x and dz)

(x iff (y iff z)) = (not x or not y or not z) and (not x or y or z) and (x or not y or z) and (x or y or not z) and (not x or Dy) and (not x or Dz)

Thus asymmetry, nondistributivity, and nonassociativity are all due to asymmetric differential terms.

10C. Quadrature 141

C. Quadrature a.k.a. general quadratic distribution

Theorem: Junction Diamond For any diamond values x and y, we have: b where: < {a,c}={x,y}, and {b,d}={x/y,y/x} a < c OR < < {a,c}={x/y,y/x} and {b,d}={x,y} d

Proof is by cases.

The "junction diamond" {x, y, x/y, y/x} contains its own minimum, maximum, conjunction, and disjunction. If f is a harmonic function, then it preserves order; hence we get a similar order diamond for {f(x), f(y),t(x/y),f(y/x)}. This implies this Theorem:

Quadrature (or; General Quadratic Distribution): For any harmonic function f(x); f(x max y) = f(x) max f(y) max f(x/y) max f(y/x) fix min y) = f(x) min f(y) min f(x/y) min f(y/x) What is more, fix max y) equals one of the four terms on the right; and similarly for f(x min y).

142 Diamond , A Paradox Logic Corollary. For any harmonic f, f(x max y) > f(x) max f(y) f(x min y) :S f(x) min f(y)

N.B.: { f(x), f(y), f(x/y), f(y/x) } may have other < and = relations in addition to the "junction diamond ". For instance: di = i < dt = f = df < di : The derivative collapses the diamond to a linear order. A(i) = X(t) = i < j = ),(f) = 10) Harmonic projection collapses the diamond to 2 values.

Theorem. Quadrature in k terms: f(x, min x2 min ... min xk)

=

f(x, max x2 max ... max xk)

Min[i,j n]

(g 1/ ) )

Max[i,j max a --> F --> ]--- **

\ / \/ not /\ / \ min b --> max a --> F --> ]--- ** \

*

* At least one of these equals -(x);

the other equals one of { i, j, x, -(x) }

* * At least one of these equals *(x); the other equals one of { i, j, not(x), *(x) }

Diffraction is sensitive enough to separate function phases, detect the sides of harmonic fixedpoints, intertwine and untangle fixedpoints, extract minus from analytic functions, and modulate diamond's two reflection operators; yet it is itself defined harmonically, without use of minus. Diffraction is harmonic analysis.

Chapter 11

Three-Logic Ternary Logic Embeds in Diamond S3 Conjugation Cyclic Distribution Voter's Paradox

A. Ternary Logic Embeds

Embedded within 4-valued diamond logic is 3-valued ternary logic: A

f i t

&B f i t

f f f

f i i

VB -A f i t

f i t

f i t

i i t

t t t

t i f

The positive ternary logic operators equal minimum and maximum on the lattice f < i < t ; negation equals the order-reversal operation. 3-logic is a closed sublogic of diamond; the operators "and", "or" and "not" send three-logic to three-logic. Thus many of the theorems that apply to diamond apply to 3-logic. For instance, two of diamond's Brownian algebra axioms apply; but instead of "interference" it has "relocation":

163


I I xl YI I z = xzl YzI I "transposition" I X1 Y I x

x "occultation"

I I = I I x I x I I-I I-I I

"relocation"

These three algebraic initials are complete with respect to 3-logic; that is, any arithmetic identities involving 3-logic values are consequences of these two initials. These initials are also independent; no two of them proves the third. Finally, they are consistent; for they do have a model. 3-logic inherits half of diamond's lattice; min but not max. 3-logic thus is a semi-lattice; this suffices to define an (semilattice) order on 3-logic. Thus 3logic inherits the Self-Reference Theorem from diamond. Ternary logic suffices to solve all the paradoxes of Chapter 1. 3-logic inherits the embedding of the continuum into diamond; for the dedekind splice never equals j. As in diamond, 3-logic supports Zeno's Theorem.

11B. S3 Conjugation 165

B. S. Conjugation

Let the permutation group S3 operate on three -valued logic. It has three reflections and three rotations: (tf), (ti), (if), (fit), (tif), identity. I call (fit) "U", for "up", and (tif) "D" for "down".

a*b

I

id

( fit)

(tif)

( tf)

(ti)

(if)

id (fit) (tif) (tf) (ti) (if)

I I I I I I

id (fit) (tif) (tf) (ti ) (if)

(fit) (tif) id (if) (tf) (ti)

(tif) id (fit) (ti) (if) (tf)

(tf) (ti) ( if) id ( fit) (tif)

(ti) (if) (tf) (tif) id (fit)

(if) (tf) (ti) (fit) (tif) id

S3 permutes functions and relations as well as elements . As before, define the function P[F] and the relation P[R] by:

P[F] (x) = P(F(P-'(x))) x P[R] y if P-'(x) R P"'(y)

Thus the above group table also defines the group 's conjugation action on these permuted lattices:

Diamond, A Paradox Logic

166 t ---> j

The following system takes 2n steps to go from i to the fixedpoint j: x, = dx1 / t x2 = x1 min not(x,) min dx2 / t x3 = x2 min not(x2) min dx3 / t

x. = x„_1 min not(x„_,) min dx„ / t

iii...i --> tii...i --> jii...i --> jti...i --> jji...i --> jjt...i --> ....... --> jjj...j

If the iterated function contains no reference to i or to j, then the process takes at most n steps, as in these systems: x 1 = f ; x2 =

Dx1

; x3 = dx2 ; x4 = Dx3

4 steps: iiii -> fiii -> ftii -> ftfi -> ftft jjjj -> fjjj -> ftjj -> ftfj -> ftft

x, = t ; x2 =

dx1

; x3 = Dx2 ; x4 = dx3 ; x5 = Dx4

5 steps: iiiii -> tiiii -> tfiii -> tftii -> tftfi -> tftft jjjj] -> tjjjj -> tfjjj -> tftjj -> tftfj -> tftft


Chapter 5. Fixedpoint Lattices. A. Relative Lattices

We can generalize these results to " seeds". Theorem. The minimum of two left seeds is a left seed.

Proof: Let c = a min b , where a and b are left seeds. Then c < a, and c < b, and c is the rightmost such vector. Therefore f(c) < f(a) < a ; f(c) < f(b) < b therefore f(c) < c , since c is the rightmost vector left of a and b. QED.

Similarly, the maximum of two right seeds is a right seed. Since all fixedpoints are seeds, their minima are left seeds, and their maxima are right seeds.

Theorem . The minimum of left seeds generates the minimum of the fixedpoints in the relative lattice: a min b generates fzn(a) minf fz"(b), if a and b are left seeds.

Proof: Let z = a min b, two left seeds. As noted above, 1) < z ; z is a left seed. Moreover, z is the rightmost vector left of a and of b.

Appendix. Relative Lattices 229

f2n(z) is a fixedpoint; it's left of f2"(a) and of f2n (b) because z is left of a and of b. If a fixedpoint c is to the left of f2n(a) and of f2(b), then c is to the left of a and of b:

c

Diamond: A Paradox Logic

Delta: A Paradox Logic

Delta: A Paradox Logic