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DIAMOND A Paradox Logic
SERIES ON KNOTS AND EVERYTHING Editor-in-charge: Louis H. Kauffman
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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
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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
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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
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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?
8 Diamond , A Paradox Logic
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.
12 Diamond, A Paradox Logic
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.
14 Diamond , A Paradox Logic
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.
16 Diamond , A Paradox Logic
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!
18 Diamond, A Paradox Logic
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.
20 Diamond, A Paradox Logic
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!
22 Diamond , A Paradox Logic
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,,^i1 = (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!)
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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.)
28 Diamond, A Paradox Logic
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":
I^I/ 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":
36 Diamond, A Paradox Logic
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.
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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.
48 Diamond , A Paradox Logic
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
Diamond , A Paradox Logic
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
56 Diamond, A Paradox Logic
B. Phase Order Now I define the concept of phase order: a
<
7
f
This structure is a lattice; it has a mutually distributive minimum and maximum.
Theorem :
min is the minimum operator for < ;
(X min Y):5 X; (X min Y) < Y; andZ <(XminY), if Z<XandZ
and(XmaxY)
Proof: X min (X min Y) = (X min X) min Y = X min Y ergo (X min Y) < X ; similarly, (X min Y):< Y. If Z < X and Z < Y then Z min X = Z ; also Z min Y = Z Therefore Z min (X min Y) = (Z min X) min Y = Z min Y = Z ThereforeZ<(XminY) ifZ<XandZ
4B. Phase Order Theorem : < is transitive and antisymmetric: a< b and b< c implies a< c a< b and b< a implies a= b
Proof: a < b and b < c implies amin b = a ; bminc=b ; so aminc = (aminb ) minc = amin (bminc)= = a min b = a ; therefore a < c. QED.
a < b and b < a implies a min b =a ; a min b = b; so a = b. QED.
Theorem: < is preserved by disjunction and conjunction: a< b implies a or c< b or c and a and c < b and c
Proof a < b implies a min b = a ; so (aorc)min(borc) = (aminb) orc = aorc; so (aorc)< (borc). Similarly (a and c) < (band c) . QED.
57
58 Diamond, A Paradox Logic Theorem : < is preserved by negation:
a < b implies not(a) < not(b) . Proof: a < b implies a min b = a ; so not(a) min not(b) = not ( a min b) = not a so not(a) < not(b) ; QED.
Theorem : < is preserved by any harmonic function: a < b implies F(a) < F(b) This follows by induction from the previous two results.
Theorem: For any harmonic f, f(x max y) f(x) max f(y) f(x min y) f(x) min f(y)
Proof: by lattice properties. xmaxy > x ; xmaxy > y ergo f(x max y) > f(x) and f(x max y) > f(y) ; so by definition of the max operator f(x max y) > f(y) max f(y) . We get the other half of the theorem the same way. f(x min y) < f(y) min f(y) . QED.
4B. Phase Order 59
These inequalities can be strict; for instance: =
f,
yet d(t min f) = i
Dt min Df =
t;
yet D(t min f) = i
dt min df
dt max df = f, yet d(t max f) = j Dt max Df = t; yet D(t max f) = j
Now we extend < to ordered form vectors: (x1,x2,x3,...,x,,) y if and only if ();:5 y1) for all i
Theorem : < has "limited chains", with limit 2N. That is, if x„ is an ordered chain of finite form vectors; x, <x2<x3... or x,>x2>x3...;
and if N is the dimension of these vectors, then for all n > 2N, N. = x 2N .
Proof: Any given component of the x's can move at most two steps before ending up at i or at j; at that point that component stops moving. For N components, this implies at most 2N steps in an ordered chain before it stops moving.
60 Diamond, A Paradox Logic Given any harmonic function fix), define a left seed for f is any vector a such that
f(a) < a;
a right seed for f is any vector a such that
a < f(a).
afixedpoint for f is any vector a such that
a = f(a).
A vector is a fixedpoint if and only if it is both a left seed and a right seed.
Left seeds generate fixedpoints, thus: If a is a left seed for f, then f( g) < A. Since f is harmonic , it preserves order; so f2(a) < f(a); and f3(a) (1):5 f'(a); and so on:
f(a) _>f2(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
=
^I
^I
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.
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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
Diamond , A Paradox Logic
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
70 Diamond , A Paradox Logic
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))...) .
72 Diamond , A Paradox Logic
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
Diamond , A Paradox Logic
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
^I 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.
Diamond , A Paradox Logic
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
a
= 1 min c = c.
It is a theorem of lattice theory that any non-distributive non-modular lattice contains N5 as a sublattice. Note that the FTF state is the ant's only boolean state; all others contain paradox. Assuming that gate C is boolean forces gates A and B to be in the FT state only. The "ant" thus resembles the "Santa" statements of Chapter 1; both attempt to use the threat of paradox to force values otherwise free.
80 Diamond, A Paradox Logic
Consider this form; "Brown's First Modulator":
I I I I I 1 1 1 1 1 1 I I I I I A I_I_I_I I A I-1-I-I I I I I _I I I I i I I I I I I I I
It is equivalent to the boundary-logic system:
A = input B
=
[KA]
C
=
[BD]
D
=
[BE]
E
_
[DF]
F
=
[HA]
G
=
[FE]
H
=
[KC]
K
=
[HG]
5C. Examples 81 If we symbolize the marked state by " I", curl by "i", and unmarked by "0", then this system has these fixedpoints: h
a
iiiiiiiii
b C
jj jj j j j j j
0iiiiiiii
/ g \
d e
0jjjjjjjj 10iii0iii
/ / \ \ f--d -- j a -- c--e
f
lojjjojjj
\ \ / /
9 h i j
101010001 000101001 100100110 010010010
\
/
/
\ j
Exercise for the reader : are these the only fixedpoints?
G.S.Brown, in his Laws of Form, claims that this circuit "counts to two"; i.e. when A oscillates twice between marked and unmarked, K oscillates once. Is this true? (Assume that the circuit cycles much faster than the input.)
Now consider this form ; "Brown's Second Modulator":
I I I I I II IIII IIIII I II II IIII IIIII Al IIIII II IIII IIIII I I I I-I-I_I-I_I_l_I-l I I I I I I I_I I_I_I I I I I-I I II I-I
82 Diamond , A Paradox Logic
It is equivalent to this system: A = input B = [ACE] C = [BD] D = [BCG] E _ [BCD] F = [BDG] G = [CDF] Exercise for the reader: find all fixedpoints for this system. Is this system a modulator?
Chapter 6
Limit Logic Limits Limit fixedpoints Diamond computation; Halting theorem
A. Limits
Diamond logic is continuous; it defines limit operators. These operators equal combinations of two more familiar limit operators; "infinity" and "cofinity":
Inf(x„) = (All N>O)(Exists n>N)(x„) (x, or x2 or x3 or x4 or... ) and (x2 or x3 or x4 or ... ) and (x3 or x4 or ... ) and (x4 or ... ) and ....
Cof(xo) = (Exists N>O)(All n>N)(x„)
( x, and x2 and x3 and x4 and ... ) or (x2 and x3 and x4 and ... ) or (x3 and x4 and ... ) or (x4 and...) or ....
83
84 Diamond , A Paradox Logic Inf, the "infinity" quantifier, says that x„ is true infinitely often. Cof, the "cofinity" operator, says that xp is false only finitely often. Obviously these are deeply implicated in the Paradox of Finitude.
Note that cofinity is a stricter condition; cofinite implies infinite, but not necessarily the reverse: Inf(xo) or Cof(x.) = Inf(x,J ; Inf(x,J and Cof(x.) = Cof(x,).
Now define a "directed limit" via majorities: lim'(x,J = M(Inf(x o), a, Cof(x ,J)
Note that:
limf(x,) = M(Inf(x,), f, Cof(x,)) =
Cof(x„)
lim`(x,,) = M(Inf(x,J, t, Cof(x,)) = Inf(x„)
The intermediate settings define "limit" operators; lim'(x,J M(Inf(x,), i, Cof(x,J) Inf(x,) min Cof(x.) Inf(x,J / Cof(x,J Max(N?0)Min(n>N) (x") x,minx2minx3minx4min... ) max (x2 min x3 min x4 min ... ) max (x3 min x4 min ... )
max (x4 min ... ) max ....
6A. Limits 85 lim+(xo) =
lim^(x„) M(Inf(xo), j, Cof(x,J) Inf(x„) max Cof(x,J Cof(x„) / Inf1x,J Min(N>O)Max(n>N) (x„ ) (x,maxx2maxx3maxx4max...) min (x2 max x3 max x4 max ... ) min (x3 max x4 max...) min (x4 max ... ) min ....
Note that:
M( lim-(Y,), t, lim+(x,) ) = lim-(x,) or lim+(x„) = Inl x,) = lim`(x„) M( lim-(x.), f, lim+(x,J) = lim-(x„) and lim+(x„) ) = CORY,) = limf(x„) and in general: M( lim-(xo), a, lim+(x„)) = lim'(x,)
Theorem : (lim- x„+,) _ (lim- x,) (lim+ x„+1) _ (lim+ x„) This is true because Inf and Cof have that property. Inf and Cof are about the long run, not about the beginning.
86 Diamond, A Paradox Logic Lim- is the rightmost value left of cofinitely many x„'s, and lim+ is the leftmost value right of cofinitely many x„'s: lim- xo < xN , for all but finitely many N, and lim- is the rightmost such value; lim+ xo > 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
88 Diamond, A Paradox Logic
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.
90 Diamond, A Paradox Logic
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<2*4" = FZ° ( max (000))) 4"<j<2*4"
This is because by 4" steps, the system has run through all possible different states; so between 4" and 2*4n it will traverse at least one cycle, and thus generate a seed. The minimum of stages 4" to 2*4", iterated 2n times more, yields a wavebracketing fixedpoint, in (2n + 2*4") steps.
In diamond logic, any computation with any input has an output; a wavebracketing fixedpoint. However, some computations take exponential time to find their wave, and thus are nonfeasible.
Most of the logic fixedpoints in the last few chapters exist thanks to the default value; paradox . In diamond logic, paradox doesn't refute reasoning; it grounds reasoning.
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Chapter 7
Paradox Resolved The Liar and the Anti Diagonal Russell's Paradox Santa Sentences Antistrephon Heap Paradoxes Game Paradoxes
A. The Liar and the Anti Diagonal "This sentence is false"; is that, the pseudomenon , true or false? Yes but no! Or, if you prefer, no but yes. Before we had no solutions at all; now we have more than one! Dear reader, I must confess to a sense of anticlimax in this resolution. So many logicians have treated paradox with respect bordering on terror ; surely the solution can't be that simple? Well, yes it can be ; for as you can see, yes it is!
Call an adjective "heterological" if and only if it does not apply to itself: "A" is heterological = "A" is not A. "Heterological" is heterological
Is "heterological " heterological?
= "Heterological" is not heterological.
Yes but no. (Or: no but yes.)
"'Is false when quined' is false when quined "; is it true? Yes but no.
93
94
Diamond , A Paradox Logic
B. Russell 's Paradox
Recall the definition of Russell's set R: R = { x I x not an element of x } So in general x in R = x not in x and therefore R in R = R not in R . Therefore R is paradoxical. Does R exist? In 2-valued logic, the answer must be "no"; yet there it is! In diamond logic, "R in R" equals i, or it equals j.
Recall also 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.
Russell's barber shaves all those - and only those - who do not shave themselves. Does the barber shave himself? Yes but no; which can be realized several ways. For instance, the barber might only partially shave himself. Or, if there are two barbers in town, then each can shave each other, but not themselves; then the two of them, as a team, shave all those who do not shave themselves.
7B. Russell 's Paradox 95 That village's watchmen watch all those, and only those, who do not watch themselves. But who watches the watchmen? Answer: they shall watch each other, but not themselves. Thus honesty in government is truly imaginary!
If you were to ask that village's veterans about the Great War (a war to end all wars, and only those wars, which do not end themselves), then they will laugh at your quaint name for a conflict now known as World War I. "Did the Great War end itself?" they will say, then scratch their heads. "Yes, it did; but no, it did not!"
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 himself? Answer: not this, not that . A mystery!
96 Diamond , A Paradox Logic
C. Santa Sentences
If a young child were to proclaim: "Santa Claus exists, if I'm not mistaken." and subsequent events were to refute his belief, then the poor child will be justified in exclaiming:
"I am mistaken!" Humbling moments like these are part of growing up. Note that this admission is formally identical to the Fool's paradox! Evidently Kris Kringle, in his departure, left behind some fool's gold. How generous!
Recall that we can create Santa sentences by Grelling's method, by Quine's method, and by Russell's method:
Grelling's Santa: Define the adjective "Santa-logical": "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.
7C. Santa Sentences 97
Quine's Santa is: "Implies that Santa exists when quined" implies that Santa exists when quined.
Russell's "Santa Set for sentence G" is: SG = (x I (x an element of x) implies G ) Therefore: x in SG
= (x in x) G.
and therefore: S. in SG = (SG in SG) G.
If there is no Santa Claus, then the above are all paradoxes.
Above I told Barber-like stories about Santa sets. For instance, in another Spanish village, the barber takes weekends off; 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. Does the barber shave himself? B shaves B = If B shaves B, then it's the weekend. When Monday rolls around, then (B shaves B) = paradox.
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. Who watches the watchmen? W watches W = if W watches W, then fortune smiles. If fortune ever frowns, then (W watches W) = paradox.
98 Diamond, A Paradox Logic 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? If not, then (CW ends CW) = paradox.
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. Does God bless God? G blesses G = If G blesses G, then there is peace.
Is there peace? If not, then (God blesses God) = paradox.
Recall Promenides the Cretan, who said; "If I am honest, then some Cretan is honest." How logical! But alas, this is equivalent to: "If all Cretans are liars, then so am I." Promenides sounds logical; but his statement still leaves open the possibility that every Cretan is a liar, including Promenides.
7D. Antistrephon 99
D. Antistrep/ion
In the next few paragraphs I take the role of judge, and address the shades of Protagoras and Euathius. Gentlemen, you have given me a dilemma. If Euathius is to win this case, then he must show that he has no obligation under the contract; but the contract says that he need not pay just if he loses the first case - which is this one. He wins if he loses and he loses if he wins; and the same goes for Protagoras. If I find for Protagoras, then the judgement should go for Euathius; and if I find for Euathius, then the judgement should go for Protagoras. You wish me to declare sentence, but any sentence I declare will be an incorrect sentence, a false sentence. Therefore I declare:
This sentence is false.
The Pseudomenon; a paradox, or half-truth. By the nature of this case, I can be only half-right; I can only half-satisfy you. In the interest of justice, I should take a position midway between yours, favoring neither side. Compromise is called for. I therefore reformulate this case. I say that it is actually two cases being decided simultaneously. The first case is about the second half of the fee, to be awarded only if the second case is lost; and the second case is about the first half of the fee, to be awarded only if the first case is lost.
100 Diamond, A Paradox Logic This is an artificial division of the original case; it would make no difference if the original case had an unequivocal solution . But here equivocation is necessary, and it works; for it is consistent for Protagoras to win the first case and Euathius to win the second . Upon recombining these results, we see that Protagoras can claim half the fee, having won but lost, and Euathius can keep the other half of the fee, having lost but won. One final legal note: in this case , as is usual, Protagoras won if and only if Euathius lost: i iff not j = t What is unusual about this case is that it's also true that Protagoras won if and only if Euathius won: i iffj = t
Stranger still: either Protagoras won and lost, or Euathius won and lost! (iandnoti ) or(jandnotj) = t
7E. Size Paradoxes 101
E. Size Paradoxes In Chapter 1, I heaped together the paradoxes of The Heap, The First Boring Number, Berry's Paradox, and Finitude. They all had in common the vagueness of the boundary between the interesting and the uninteresting. Surely both types of integers exist; but where do they meet? Assuming that we could find a number on the boundary (even though the search for such a number would be boringly long), then it would be interesting just as much as it is boring; which suggests an intermediate state. So is "the first boring number" boring or not? Yes but not And what is "the smallest number that cannot be defined in less than twenty syllables"? In standard decimal nomenclature, that would be 127,777. (However, other naming schemes might name 127,777 in fewer than twenty syllables. As ever, uncertainty reigns.) If you were to pile together 127,777 grains of sand, each 1 mm wide, then they will form a conical pile approximately 9.9 cm wide and half as tall; a small but respectable Heap. If you tried to move this Heap one grain at a time, laboring 5 seconds per grain, 8 hours per day, 5 days per week, then you will finish the job in approximately 4.5 weeks; a Heap of work. "One plus the largest number defineable in less than twenty syllables" might be one plus "Twelve googol googol googol googol googol googol googol googol googol," or I + 1.2 * 10901. (This is if you allow the use of the word "googol", for 10100. Other naming schemes yield even greater numbers.)
102 Diamond, A Paradox Logic
F. Game Paradoxes Recall the definition of Hypergame: its 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? Above I told the story of "the Mortal"; an unborn spirit who must now make this fatal choice; to choose some mortal form to incarnate as, and thus be be doomed to certain death. The Mortal has a choice of dooms. Is the Mortal doomed? The answer is that Hypergame is Finitude in disguise. Presumably the Mortal lives until the last interesting moment, then dies of boredom.
Recall my definition of the game Normalcy: The move N -J G is legal = the move G ' G is not legal. Is Normalcy normal? Let G = N: The move N -J N is legal = the move N -J N is not legal. This is a game-theory version of Russell's paradox. Normalcy is normal if and only if it is not. So is Normalcy normal? Yes but no. Above I told the story of the Rebel, who may become those, and only those, who do not remain themselves: R may become B = B may not become B . Can the Rebel remain a Rebel? Yes but no. Presumably Rebels play at Normalcy.
Chapter 8
The Continuum Cantor's Paradox Dedekind Splice Cantor's Dyadic The Line Within the Diamond Zeno's Theorem
A. Cantor's Paradox Cantor's proof of the "non-denumerability" of the continuum relies on an anti-diagonal . For suppose we had a countable list of the real numbers: R, = 0
.Dill
D 12, D 13, D 1 4 ...
R2 = 0. D21,D22,D23,D24
R3 = 0 . D31,
D32 , D33 , D34
where DNM is the Mth binary digit of the Nth number. Then we define Cantor' s anti-diagonal number: C = 0 . not Dl, , not D22, not D33, not D44 .. . If C = RN for any N, then DNx = not D,tx Therefore DNN = not DNN. The pivot bit buzzes. From this single buzzing bit Cantor deduces the existence of an infinity beyond infinity of real points! Was more ever made from less? In diamond logic, the continuum is "semi-countable"; countable listings are possible, but they all contain paradox bits. The continuum is intermediate!
103
104 Diamond , A Paradox Logic
B. Dedekind Splices Recall the "paradox of the boundary":
What day is midnight? Is noon A.M. or P.M.? Is dawn day or night? Is dusk? Which country owns the boundary? Is zero positive or negative? (± 0 ?) Is infinity odd or even? If a statement is true at point A and false at point B, then somewhere inbetween lies a boundary. At any point on the boundary, is the statement true, or is it false?
To solve the paradox ofthe boundary, put a paradox on the boundary:
X S Y = (X
This is the "dedekind splice" operator, equal to paradox at the boundary.
8B. Dedekind Splices 105 To make this a "continuous" function from R to diamond, we need to define a topology on diamond. Let the "open subsets" of diamond be the rightclosed subsets:
Open sets = { O : [(x in O) and (x < y)) 4, (y in O) } ( {}, {j), {t,j}, {fj}, {t,fj}, {i,t,f,j} )
In this topology, all harmonic functions are continuous, the Dedekind splice is continuous, all values are near i , and none are near j.
The Dedekind splice is anti-symmetric, transitive, and dense: For all x, y and z: (xSy) = not(ySx) if (x S z) and (z S y) then (x S y)
if ( x S y) , then there exists a z
such that (x S z) and (z S y)
If the sequence (x") approaches the limit x from "both sides ", as in an (x S y) = Jim- (x" S Y)
alternating series , then In general
((lim
x")
S y ) < lim- (x„ S Y)
The splice's anti-symmetry implies the paradox of the boundary:
(xSx) = not(xSx)
106 Diamond , A Paradox Logic
C. Cantor's Dyadic
Let us take a closer look at Cantor's anti-diagonal number; just what kind of quantity is it? This number is so fraught with mathematical significance that it forces us to postulate an infinity of infinities; so surely it must, within itself, contain an infinite amount of information about all those infinities. Otherwise the silly thing's just bluffing us! We know that C has a paradox bit at place N; so the question is, when does that happen? Graph rN vs. r:
r„ 4
I
I----I I----I I----I <-- true I I I I I I <-- paradox --->r --I I----I I----I I-- <-- false
As you can see, the most obvious place to put paradox values is at the boundary points; namely, the dyadics m/2N. Paradox denotes bit-flip; the blur when 0 becomes 1:
0.111111... <----> i.iiiiii ... <----> 1.000000...
Cantor's number has bit-flip at place N; this implies bit-flip at all higherprecision places. Therefore Cantor's Anti-Diagonal Number has this form: C = 0.0100101 ... 101011iiiiiiiiiiiiiiiiiii... N boolean region paradox region
8C. Cantor's Dyadic 107 Thus we see that Cantor's Number contains only a finite number of boolean bits. It's a measly dyadic! The silly thing was bluffing us! Far from being infinitary (this proof of infinities!) it is instead the most finite entity of all; a bounded bit string with round-off error! The dyadics can trick up Cantor's proof, even within boolean logic. For instance, the possibility exists that C, as an anti-diagonal, reads .0111111111..., while C, on the list, reads .100000000...! Cantor's Theorem is hereby exposed as not only superfluous, but actually ridiculous. The continuum is countable; Cantor's Paradox detects bit-flip at a dyadic. Therefore I propose a down-to-earth alternative to Cantor's tottering cardinal tower; a single countable infinity with paradox logic. A slightly subtler logic yields an infinitely simpler model. This is known as elegance; sign of a correct theory.
Even with paradox accounted for, Cantor's argument still has revolutionary implications. Consider C. for n > N : C. = i for n > N; but C. = not R. for all n: So R. = not C. = noti = i ,foralln>N.
In other words, every real number on the list after Cantor's Dyadic also has a paradox bit; and so is also a dyadic! By this account, at most a finite number of reals possess infinite precision!
108 Diamond , A Paradox Logic Rather than showing that most real numbers are, say, transcendental, Cantor's Dyadic instead demonstrates that most real numbers are dyadic! As in quantum mechanics, uncertainty quantizes the continuum. Indeed we have a classic quantum-style complementarity; finitely many infinite-precision reals, and infinitely many finite-precision reals. But what then oi; say, 1/3? 1/5? 1/7? 1/(2n+1), for all n? Do we only have room for finitely many full rationals - let alone finitely many transcendentals? Do all these need Cantor's tower? Perhaps C has a non-dyadic form: C = 0.0100101... 101011i10111000011... N
But what does that paradox bit at place N mean, given that higherprecision bits are boolean? Is C of the form c + 2'N ? What does such a dual number mean? Perhaps Cantor's Dyadic is telling us that the synchronized bit-flips of dyadic numeration produce masking noise. Or perhaps Cantor's Dyadic is there to remind us that approximation is inevitable. In practice, real numbers are dyadics. After all, dyadics are the numbers we really do, in fact, calculate with. Every single so-called real number in the socalled real world has finite precision. Even the Chudnovski brothers have computed only 2 billion digits of pi, and not infinity! Not one single infiniteprecision computer has ever come off the assembly line; nor ever shall, so long as human beings remain finite. Call this Math for Mortals.
8C. Cantor's Dyadic 109 The finite-precision reals are easy to count: Aiiii11iii... . Oiiiiiiiii... . liiiiiiiii... . 00iiiiiiii... . 0liiiiiiii... . 10iiiiiiii... . lliiiiiiii... . 000iiiiiii... . 001iiiiiii... . 010iiiiiii... . 011iiiiiii... . 100iiiiiii... . 101iiiiiii... . 110iiiiiii... . llliiiiiii... .0000iiiiii...
and so on, in binary! Note that by this counting C = iiiii... = 1/2 ; the first entry!
110 Diamond , A Paradox Logic
D. The Line Within The Diamond
The "approximate comparison" operator ^ is ideal for embedding the continuum in diamond logic. Consider the following mapping from R ( the continuum) to 0" ( the space of all infinite diamond-valued sequences) :
E(x) _ (x^_ g1,xZg2,xIq3,x^q4,...)
where q„ is an enumeration of the rationals. This function E sends R (the real number continuum) into 0', the space of all infinite diamond vectors. Its nth component, E", is comparison with the nth rational: x ^ q"
Theorem : This mapping E embeds R in 0': that is, R's topology is carried intact into 0°', the space of diamond vectors.
Proof: First, note that E is one-to -one; for if x
8D. The Line Within The Diamond 111
To complete proof of embedding , we need to prove this Lemma : the inverse of E is continuous. A function is continuous if the inverse image of an open set is an open set. The real line's topology is generated by the "half-lines":
(x,+-)
_ {y:x
(--,x) _ (y:y<x} so it suffices to prove that E sends each half-line to the intersection of an open set in 0" with the image E(R). E(x,+-)
Union[ n such that q„ > 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.
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Part Two
Advanced Diamond Logic
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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
120 Diamond, A Paradox Logic
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)
122 Diamond, A Paradox Logic
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.
126 Diamond , A Paradox Logic
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.
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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
130 Diamond , A Paradox Logic
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)
132 Diamond, A Paradox Logic
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
This has k2 terms.
Theorem. Quadrature in k terms, and n dimensions: If x,, x2, x3 ..., xk are all n-dimensional vectors;
and F(x) is from 0° to 0, then; F(Mn[1
1 OC. Quadrature
143
Theorem: Quadratic Continuity:
For all harmonic functions f(x) f(lim- xo) = lim- fix/x.) f(lim+ x„) = lim+ f(x/)1.) where the second limits are of the form: Max[all N] Min[all i,j > N] (x/.xx ) Min[all N] Max[all i,j > N] (x/xj ) i.e. "in the limit of large i and j".
Proof is via "cofinal-range" definition of the limit operators. Recall that the lim- of a sequence is the minimum of the "cofinal range", i.e. the set of all values that occur infinitely many times:
lim-{x") = Min cofinal{x"} lim+{x"} = Max cofinal{x"} where cofinal{x„} Y : x" = Y for infinitely many n } Therefore f(lim-{x„}) = f(Min cofinal{x„}) = Min f( (cofinal{x,})/(cofinal{xx}) ) = Min f( (cofinal{x/.xx}) ) = Min cofinal { f(x/xx) ) = lim- f(x/x)
Similarly, f(lim+{x„}) = lim+ f(x/xx)
144 Diamond, A Paradox Logic
Open question: when can we reduce Quadrature to two terms, yielding direct distribution: f(x min y) = f(x) min f(y) ; and similarly for max. Direct distribution is true for positive function (i.e. negation-free functions); yet it is also true for negation! However, it is false for d and D, the derivatives: d(t min f) = di = i < f = f min f = dt min df D(tminf) = Di = i < t = tmint = DtminDf
Conjecture: Directly Distributive Equals Semi-Negative. For all harmonic f(x), fix min y) = f(x) min f(X) for all x,y (& sim for max) iff f(x) min f(y) = f(x/y) min f(y/x) for all xy iff f = P(S(x)), where P is a positive function and each component of S is one of the forms { );, not x;, ),x;, px, }.
and each x; occurs in only one of these forms.
Conjecture: "Derivative Extraction". If f(a min b) < f(a) min f(b) , for some a, b then there exists a positive function g(x) such that f(g(x)) = dx or f(g(x)) = Dx
IOC. Quadrature 145
Quadrature of Majority:
f(M(a,b,c)) f((amin b)max (bminc)max (cmin a))
f(a min b) max f(b min c) max f(c min a) max f(a min b/c) max f(a min c/b) max f(b min a/c) max f(b min c/a) max f(c min a/b) max f(c min b/a)
( f(a) min f(b) min f(a/b) min f(b/a) ) max ( f(b) min f(c) min f(b/c) min f(c/b) ) max ( f(c) min f(a) min f(c/a) min f(a/c) ) max (f(a) min f(b/c) min f(a/c) min f(b/a) ) max (f(a) min f(c/b) min f(a/b) min f(c/a) ) max (f(b) min f(a/c) min f(b/c) min f(a/b) ) max (f(b) min f(c/a) min f(b/a) min f(c/b) ) max (f(c) min f(a/b) min f(c/b) min f(a/c) ) max (f(c) min f(b/a) min f(c/a) min f(b/c) )
Similarly with min and max swapped. This is the "permeable form" of majority (see below).
Corollary. If f is a positive function, f(M(a,b,c)) = M(f(a),f(b),f(c)) ( because f(x/y) = f(x)/f(y) for f positive.)
146 Diamond , A Paradox Logic
Corollary: Phased distribution over the positives
f( a and b) (f(a)
min f(alb) ) max
f(b) min f(b/a)
( f(a) max f(b/a) ) min f(b) max f(a/b)
f( a orb) ( f(a) min f(b/a) ) max (f(b) min f(a/b) ) ( f(a) max f(a/b) ) min (f(b) max f(b/a) )
These are "permeable forms of the positives".
If we juxtapose these formulas we get:
f( a and b)
(f(a) or f(a/b)/f(b/a) ) and ( f(b) or f(b/a)/f(a/b) ) ( f(a) and f(b/a)/f(a/b) ) or (f(b) and f(a/b)/f(b/a) )
f( a or b) _ ( f(a) or f(b/a)/f(a/b)) and (f(b) or f(a/b)/f(b/a) ) ( f(a) and f(a/b)/f(b/a)) or (f(b) and f(b/a)/f(a/b) )
1 OC. Quadrature
147
Let us define these diffraction functions: fL(a;b) = f(a/b) / f(b/a)
fR(a;b) = f(b/a) / f(a/b)
Then we get these rules:
f( a and b) (f(a) or fL(a;b)) and f(b) or fR(a;b) ) ( f(a) and fR(a;b)) or f(b) and fL(a;b)
f( a orb) (f(a) or fR(a;b)) and (f(b) or fL(a;b) ) ( f(a) and fL(a;b) ) or (f(b) and fR(a;b) )
These general Leibnitz rules imply the differential Leibnitz rules, the De Morgan laws, and the distribution laws.
By juxtaposing the quadrature laws, we get these phased quadrature laws: f(x) and f(y) and f(x/y) and f(y/x) =
f(x max y) / f(x min y) fL(xandy;xory)
f(x) or f(y) or f(x/y) or f(y/x) ftx mm y) / fx max y) fL(xory; xandy)
148 Diamond , A Paradox Logic
Any function g(x) can be put in disjunctive normal form: g(x) = OR[all s;] ( sl(x,) and ... and s„(x,J )
where each s(x) equals t, or x, or (not x), or dx. Use this fact, along with the Phased Distribution rules: f( a and b) _ (f(a) min f(a/b)) max (f(b) min f(b/a)) ( f(a) max f(b/a)) min (f(b) max f(a/b)) f( a or b) (f(a) min f(b/a)) max (f(b) min f(a/b)) ( f(a) max f(a/b)) min (f(b) max f(b/a)) Proceeding by iteration, we can distribute any f over g, until we get a lattice operation over the set
{ f(±x/±x) }, where ±x; is either (x;) or it is (not x;) Let N(x) be the function, (x,, not x,, x2, not x2, .... , x,,, not x" ) Let J(y)'s components be of the form (y/y,)
Then I(N(x)) has 4n2 components, all of the form (±x/±)j).
This gets us the Theorem: Permeable Form Theorem For any g(x), there exists a lattice function Lg (made from min, max) such that g(x) = Lg(J(N(x)) ; g's "permeable form" Furthermore, for any harmonic f,
f(g(x)) = Lg(f(J(N(x))) This is General Quadrature, or semi-distribution.
1 OC. Quadrature 149
Examples: the permeable forms for d, D, , xor, iff. Recall:
f( a and b) _ (f(a) min f(a/b)) max (f(b) min f(b/a)) _ (f(a) max f(b/a)) min (f(b) max f(a/b)) f( a or b) _ ( t(a) min f(b/a)) max ( f(b) min f(a/b)) _ (t(a) max f(a/b)) min (f(b) max f(b/a))
Therefore: f(dx) = f( xandnotx) _ ( f(x) min f(x/not(x))) max (f(not(x)) min f(not(x)/x) ) _ (f(x) max f(not(x)/x)) min (f(not(x)) max f(x/not(x)) ) _ (f(x) or fL(x; not x)) and ( f(not x) or fR(x; not x) ) _ ( f(x) and fL(not x; x)) or ( f(not x) and fR(not x; x) )
f(Dx) = f(xornotx) _ (f(x) max f(x/not(x))) min (f(not(x)) max f(not(x)/x) ) _ (f(x) min f(not(x)/x)) max (f(not(x)) min f(x/not(x)) ) _ (f(x) and fL(x; not x)) or ( f(not x) and fR(x; not x) ) _ (f(x) or fL(not x; x)) and ( f(not x) or fR(not x; x) )
f(x=y) = f(not(x)ory) _ (f(not(x)) max f(not(x)/y)) min (f(y) max f(y/not(x)) ) _ (f(not(x)) min f(y/not(x))) max (f(y) min f(not(x)/y) ) _ (f(not x) and fL(not x; y)) or ( f(y) and fR(not x; y) ) _ (f(not x) or fL(y; not x)) and (f(y) or fR(y; not x) )
150 Diamond, A Paradox Logic (For the next few pages I abbreviate "not" by "-S".)
f(xiffy) = f((-xory)and(-yorx)) ( f(-x or y) min f((-x or y)/(-y or x)) )
max (f(-y or x) min f((-y or x)/(-x or y)) )
( f(-x or y) min f( p(x) or Ja(y)) ) max ( f(-y or x) min f( I(x) or p(y)) )
{ [( f(-x) max f(-x/y)) min (f(y) max f(y/'x)) ] min[(f( px)maxf(px/ly))min ( f(Xy)maxf(Xy/px))] ) max { [( t(-y) max f(-y/x) ) min (f(x) max f(x/-y)) ] min [( f( py) max f(py/Ax) ) min ( f().x) max f(1lx/py) ) ] }
{ ( f(-x) max f(-x/y) ) min (f(y) max f(y/-x) ) ] min (f(-x/x) max f(-x/--y)) min (f(y/-y) max f(y/x) ) } max { (f(-y) max f(-y/x) ) min (f(x) max f(x/-y) )
min (f(-y/y) max f(-y/- x) ) min ( f(x/-x) max f(x/y) ) )
I OC. Quadrature 151
Here is a dual form:
f(xiffy) =
[ (f(-x) min f(y/-x) ) max ( f(y) min f('-x/y) ) max ( f(x/-x) min f(-y/- x) ) max ( f(-y/y) min f(x/y)) ] min [ (f(-y) min f(x/-'y)) max (f(x) min f(-y/x) ) max ( f(y/--y) min f(-x/-y) ) max (f(-x/x) min f(y/x)) ]
By dual calculations we derive:
f (x xor y) = f ((x and -y) or (y and -x) )
{ (f(-x) min f(-x/y) ) max ( f(y) min f(y/-x) ) ]
max ( f(-x/x) min f(--x/--y)) max ( f(y/-y) min f(y/x) ) } min { (f(-y) min f(-y/x) ) max ( f(x) min f(x/-y) )
max ( f(-y/y) min f( -y/-x) ) max ( f(x/-x) min f(x/y)) } [ (f(-x) max f(y//x)) min (f(y) max f(-x/y) ) min ( f(x/--x) max fr-y/-x)) mm (f(-y/y) max f(x/y)) ] max [ (f - y) max f( x/-y) ) min (f(x) max f(-y/x) min ( f(y/-y) max f(-x/- y)) min (f(Hx/x) max f(y/x) ) ]
152 Diamond , A Paradox Logic
D. Diffraction a.k.a. harmonic analysis
Recall these diffraction functions: fL(a;b) = f(a/b) / f(b/a) fR(a;b) = f(b/a) / f(a/b) = fL(b;a) These obey the rules: fL(a;b) / fR(a;b) = f(a/b) fR(a;b) / fL(a;b) = f(b/a) fL( a/b; b/a) = f(a) / f(b) fR( a/b; b/a) = f(b) / f(a)
Recall that I and r are the two analytic "side projection" operators; IN = x / *x ; r(x) x = lx/rx
*x/x
; *x = rx/lx
lx is always boolean ; so is rx.
Then: l(f(x)) = fL(lx; rx) r(f(x)) = fL(lx; rx) f(x) = fL(lx; rx) / fR(Ix; rx)
1 OD. Diffraction 153 fL and fR display diamond's typical phase-weaving: if f is a positive function, then: fL(x;Y) = gx) ; fR(x;Y) = f(Y) if f = (not g), then: fL(x;y) = not gR(x;y) ; fR(x;Y) = not gL(x;y) so positives preserve phase while negation reverses it.
We can use these rules iteratively to define fL and fR "syntactically". For instance: If f(x,y) = x xor y = (x and not y) or (y and not x), then fL(xL, YL; xR, yR) = (xL and (not yR)) or (yL and (not xR)) fR(xL, YL; XR, yR) = (xR and (not yL)) or (yR and (not xL))
The sides of x iffy are: iffL iffR
(XL or (not yR)) and (yL or (not xR))
(xR or (not yL) ) and (yR or (not xL))
Diffraction can be defined by the intermix function: J(a,b) = (a/b , b/a ) Note: J(J(a,b)) = (a,b) notJ(nota,notb) = (b,a) *J(*a,*b) = (b, a) and similarly with the rotation operators.
154 Diamond, A Paradox Logic In dual-rail wiring, J is a simple shuffle gate: x x/y
\ \ / / made purely of wires \_V / V ergo conserves information -/\
/ A \ also resembles Feynman diagram // \\ Y Y/x J is its own inverse; therefore we can dualize with J,thus: Jo(f,f)oJ (a,b) = (fL(a;b);fR(a;b)) Phase separation: a f f1(a;b) \/\/ ] ] /\/\ b f fR(a;b) Dual to this, by conjunction algebra, is: Jo(fL,fR)oJ (a,b) = ( f(a) , f(b) ) Phase recombination: a f1(a ; b) f(a) \/ \/ ] / \
] / \
b fR(a ; b) f(b) Phase separation resembles a 2-slit diffraction experiment, with J as the half-silvered mirror, and f as the filter. Similarly, phase recombination resembles a hologram, with phase data reshuffled to retrieve local data.
1 OD. Diffraction 155 If we make a phase-separation circuit with two functions f and g, we get (f/g)L and (g/fjR: a f (f/g ) 1(a;b) \ / \ / J 7 / \ / \
b g (9/f) R(a;b)
These recombine to get f and g back: a (f/g)L(a;b) f(a) \ / \ / 7 ] / \ / \ g (b) b (9 /f)R( a ; b)
J defines these "Diffraction Circuits": dL(x;y) = x and (not y); dL(x;y) = y and (not x) x d (x & not y) \/\/ 7 ] / \ / \ y d (y & not x)
( and their "or" = x xor y )
a=x/y ; c=da ; e=c/d = x-y b=y/x d=db f=d/c = y-x g = eorf = cord = xxory Dual to this is: x (a and not b) dx \/ \/ J 7 ( and their "or" = x/y xor y/x ) /\ /\ y (b and not a) dy
156 Diamond , A Paradox Logic a = x/y ; c = a and not b ; e = c/d = dx b = y/x d = b and not a f = d/c = dy g = e or f = c or d = x/y xor y/x Thus "lower differential is dual to difference".
DL(x;y) = y x ; DR(x;Y) = x Y x D y= x
7 ] and their "&"=xiffy) /\/\ y D x=y
a=x/y ; c=Da ; e= c/d = yx b=y/x d=Db f= d/c = x y
g = e&f = c&d = xiffy
Dual to this is: x (b=a) Dx \/ \/ ]
7 ( and their "&" = x/y if y/x )
y (a=4^b) Dy
a=x/y ; c= (bra) ; e=c/d = Dx b = y/x d = (ab) f = d/c = Dy g = e&f = c&d =(x/yiff y/x) Thus "upper differential is dual to implication".
1OD. Diffraction 157 DL(f;x) = DR(x;f) = not x dL(t;x) = dR(x;t) = not x f
D
not
3
x
D
x
t
7
t
d
not
x
3
x
d
f
not y = DL(*, y) DL(x;noty) = xory DR(x;noty) = not x or not y a=f/y; c=Da; a=c/d; f=x/e;h=Df;m=h/k b=y/f; d=Db; g=e/x; k=Dg; n=k/h
e = not y ; m =xory ; n = (not x) or (not y) x \ D xory J J
f D /\/\ \ / \ / D (not x ) or (not y) = x nand y 7 / \ / y D x \ d x and y \ / \ /
] 7 t d /\/\ \ / \ / d (not x) and (not y) = x nor y ] 7 / \ / y d
Thus diffraction and differential define all harmonic gates.
Diamond, A Paradox Logic
158
Diffraction of the rotations yield both negations: x L *y \ / \ / 3 3
\ / 3
x R not y \ / 3
y R not x y L * x L(x/y) / R(y/x) = * y R(x/y) / L(y/x) = not y
Harmonic Self-Analysis: For any harmonic f(x), there exists xL and xR, all of whose components are Boolean, such that = Jo(f,OoJ(XL,XR)
(XL,XR)
(fLfR)(XL , N R That is,
(XL,
XR)
y = (f(XL/X)/f`XR/X^ , fY lxR/xL) f(2L/xR) )
For any such xL and xR, we have: f(xL/xR) = (xL/xR) Finally: if f(x) = x, then 1(x) and r(x) fit the above equations.
Thus xL and xR are the sides of a diamond fixedpoint, themselves forming a two-fold fixedpoint. We can find this "diffracted fixedpoint" several ways:
* by iteration from (t,f) or (f,t). * by (and,or) and (or,and) on fixedpoints, then iteration * by (Inf,Cof) and (Cof,Inf) limits of iterations
I OD. Diffraction 159
Thus by doubling the size of any diamond system, you can harmonically define a boolean analysis of the old system . This is "splitting the circuit". To double is to analyse.
Buzzers-To-Toggle Theorem: If (AB) = J(a,b) i.e. A = a/b, B =h/g, a = AB , and b = B/A. then:
f(a) = a and f(b) = b if and only if
(A,B) = (f (A;B), WA&)) = JLf,fJ(AB) Also, the fixedpoint lattice of the intermixed pair is a junction of two
fixedpoint lattices:
Limn = J (414)
Thus two fixedpoints, when intermixed, become an interreferential pair.
The "buzzers-to-toggle " theorem gets its name from this example: (A,B) = J(a,b): a = not a
; A = not a / not b = not(b/a) = not B
b = not b ;
B = not b / not a = not(a/b) = not A
2 lattices : the buzzers I lattice : the toggle tf i
----
j
/
\
i
----
j
\
/ ft
160 Diamond, A Paradox Logic Note that the buzzers, though formally paradoxical, contain boolean information in their phases. In general, Lief, contains a copy of Lf; its diagonal. It also contains a boolean side-analysis of Lf; its equator. For instance, in the toggle lattice, the diagonal
( ii , jj ) is the lattice
for a = not a; and the equator { tf , ft } are the sides for that lattice.
If we "split the duck": i.e. if (A,B) = J(a,b), where a = da, b = db; then we get the "rabbit": A = (A&-B) ; B = (B&-A) The "Duck" has a linear lattice:
Therefore the "Rabbit" has a grid-like lattice: tf if ii
jf
ff
fi
jj
fj ft
The diagonal is {ii, ff, jj}, and the boolean equator is {tf,ff,ft}; duplicates and analysis of the Duck. The "duck" circuit, when split, yields the "rabbit" circuit; but the "rabbit", when split, just yields two "rabbits". In general any diffracted circuit duplicates when split.
IOD. Diffraction 161
Analytic Diffraction
Recall the Type Theorem:
Type Theorem If F does not preserve order, then F is analytic. Or, to be more specific: If F(a) I F(b) for some a < b then at least one of these two functions equals non(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)))
Recall the harmonic projection functions: ll(x)=x/notx ; p(x)=notx/x Then the functions are:
F(a max (b min )l(x))) / not F(a max (b min p(x))) not F(a max (b min A(x))) / F(a max (b min p (x)))
The functions F1 and F2 are diffraction circuits:
162
Diamond , A Paradox Logic * min b --> 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
164 Diamond , A Paradox Logic
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 <:
f <
i
<
t
(tf)[<]:
t
<
i
<
f
U[<]:
i <
t
<
f
(ti) [<] :
f <
t
<
i
D[<]:
t <
f
<
i
i <
f
<
t
(if)[<]:
We also define these permuted lattice operators:
- = (tf) , U[-] = (if) , D[-] = (ti) & = the "minimum " operator for f
& f i t
D[&] f i t
t i f
id f i t
f f f f
i f i i
t f i t
v f i t
f f i t
i i i t
t t t t
f f i t
i i i i
t t i t
U[V] f i t
f f f f
i f i t
t f t t
U[-] i f t
U i t f
f f f
i f i
t t t
D[V] f i
f f i
i i i
t f i
D[-]
f t
D t f
t
t
t
t
f
i
t
i
i
11 B. S3 Conjugation 167
Note that each "or" operator is the "and" operator of the reverse order; and therefore a permuted DeMorgan's Law applies:
P[-](x P[&] Y) = P[" ](x) P[V] PH(y) P[-](x P[V] Y) = P[-](x) P[&] P[-](y)
For any permutation P, P[-], P[&], and P[V ] is isomorphic to ternary logic. The "midpoint" of the lattice P[<] is a Liar paradox ; it has the equation L = P[-](L). Permuting three-logic generates three interlocking paradox logics.
168 Diamond , A Paradox Logic
C. Cyclic Distribution Note that { <, U[<], D[<] ) yields a political conundrum: 2/3 agree that f < i 2/3 agree that i < t 2/3 agree that t < f yet all agree that the order is linear! This is the "Condorcet" or "voter's" paradox.
This paradox is reflected in an extraordinary phenomenon which I call cyclic distributivity: & distributes over &, V, U[&], U[V], but not D[&] or D[V] V distributes over &, V, D[&], D[V],
but not U[&] or U[V] U[&] distributes over U[&], U[V], D[&], D[V], but not & or V U[V] distributes over U[&], U[V], &, V, but not D[&] or D[V] D[&] distributes over D[&], D[V], &, V,
but not U[&] or U[V] D[V] distributes over D[&], D[V], U[&], U[V], but not & or V
11 C. Cyclic Distribution
169
Define Vo = V ; V, = U[V] ; V2 = D[V] &o = & ; &, = U[&] &2 = D[&] U = ^. ; -1 = U[-] -2 = D[-]
Then for F equal to &, V, or -; U[F"] = F"+1 mod 3 D[F"] = F,1.1 mod 3
distributes over V", &,,, V,,., and but not V,,, or &"_,. That is, "&" distributes up the loop 0-01-02-00 V. distributes over V", &", V,-, and &"_,,
but not V"+, or &"+,. That is, "V" distributes down the loop 0-D1-D240
Proof of Cyclic Distributivily:
In this proof we ask, when does P[&] distribute over Q[&], if P and Q are permutations of {f,i,t}? Without loss of generality we will assume that P[&] _ &. Other cases will follow by group symmetry. So now our question is:
When is X & (Y Q[&] Z) = (X & Y) Q[&] (X & Z) ?
170 Diamond , A Paradox Logic Case 1 : {X,Y,Z} has only I element. Then the equation follows by the idempotence of every lattice operator; X R[&] X = X.
Case 2 : {X,Y,Z} has only 2 elements - say, {f,i}. Then the lattice operators & and Q[&] would be min or max operators on the 2-element lattice f
Case 3: {X,Y,Z} = {f,i,t}. This divides into subcases: Case 3A: X = f = &'s minimum : so Z & f = f ergo f & (i Q[&] t) = f and (f & i) Q[&] (f & t) = f Q[&] f = f CHECK. Case 3B : X = t = &'s maximum : so Z & t = Z ergot & (f Q[&] i) = f Q[&] i and (t & f) Q[&] (t & i) = f Q[&] i ; CHECK. Case 3C: X = i = &'s midpoint : so -(i) = i. ergo i & (f Q[&] t) = i & (f Q[&] t) ; itself and (i & f) Q[&] (i & t) = f Q[&] i
So now our question is: For what Q is i & (f Q[&] t) = f Q[&] i ?
11 C. Cyclic Distribution 171 Simply check all six lattices:
Q
Q[<]
i & (f Q[&] t)
f Q[&] i
id:
f
f
f
CHECK
U:
i
i
i
CHECK
D:
t
i
f
(tf):
t
i
i
CHECK
U(tf):
f
f
f
CHECK
D(tf):
i
f
i
NO!
NO!
Thus this part of the lattice cycle: & distributes over all but D[&] and D[V]; those " after" it in the cycle o=U4 D4o. Starting from &, you can generate the rest of the distribution cycle via group symmetry transformations . Thus & distributes up the loop 0c414240, and V distributes down the loop. I illustrate the system this way: x - ^ y means x distributes over y but not the reverse x -- y means x and y distribute over each other Then we get this diagram:
172 Diamond , A Paradox Logic Two counter-rotating distribution cycles. Also this:
The "eye in the pyramid" diagram. The edges are labelled with the appropriate conjugation function. These diagrams resemble magnetic fields:
X71&^ VIC! "Bar Magnet" "Electromagnet"
11 C. Cyclic Distribution 173 Here are two Stars of David - or octohedra:
And so we see that permuting three-valued logic generates three entangled paradox logics. Each 3-valued DeMorgan logic solves all paradoxes of self-reference and of continuity; however, they disagree as to which of {f,i,t) is minumum, maximum, or midpoint. The three logics interlock in a voter's paradox; S3 versions of the DeMorgan laws apply, and the lattice operators distribute over each other in two counter-rotating cycles of period 3. A Strange Loop indeed! Three values suffice to solve the liar's paradox, three ways; but this in turn generates a voter's paradox. Liar's paradox plus voter's paradox yields a logical knot; cyclic distributivity!
174 Diamond , A Paradox Logic
D. Voter's Paradox Recall that { <, U[<], D[<] } yields a voter's paradox: 2/3 agree that
f < i
2/3 agree that
i
<
t
2/3 agree that
t
<
f
yet all agree that the order is linear. The voter's paradox is the heart of Kenneth Arrow's Impossibility Theorem. It appears that such logic knots have a habit of bollixing political systems. These tiny tangles give politics its notorious perversity. To simplify presentation, I now introduce three fictional characters; none other than the Three Stooges. General Moe rules the Scissors Party with an iron hand. His politics are fascistic; he favors power over logic over fairness. He would rather be decisive than consistent, and he would rather be consistent than share power. Naturally he prefers monarchy, most preferably if the monarch is himself. Judge Larry is senior theoretician for the Paper Party. His politics are legalistic: he favors logic over fairness over power. Naturally he prefers to govern by consensus. Mayor Curly is lead singer for the Rock Party. His politics are populistic: he favors fairness over power over logic. Naturally he prefers to govern by
majority rule. Each single Stooge has a consistent linear ranking of fairness, power, and logic; but when you put them all together, something has got to go.
11 D. Voter's Paradox 175
Two-thirds of the Stooges (namely, Moe and Larry) put logic above fairness; Larry and Curly put fairness above power; and Curly and Moe put power above logic. Moe Larry Curly Power < Fairness?
no
yes
yes
Logic < Power? yes no yes Fairness < Logic? yes yes no
This gives us a Condorcet Election , or "Voter's Paradox":
logic < < 2/3 majority each fairness > power
- yet they all agree that the ranking is linear!
There are several partial resolutions to this.
If we appoint a single voter as tyrant (Moe, say) then we can decide this consistently; but this is not a fair system. If we attempt to decide by consensus (as Larry suggests) then that is fair and consistent; but we decide nothing, and that is a weak system. If we have faith in majority rule (as Curly professes) then we accept the non-linear order, and the linearity of the order. This is fair and decisive, but it is inconsistent.
176 Diamond, A Paradox Logic Finally, we can accept the non-linear ranking, and accept it as non-linear; this goes with every 2/3 majority, but reverses a consensus; and that is perverse.
This political knot is an instance of Arrow's Theorem, which says that no voting system has all four of these virtues:
it is fair: it gives all voters equal power it is decisive: it decides all questions posed to it it is logical: it does not believe contradictions it is responsive: it never defies a voter consensus.
In other words, any government is at least one of
cruel ; weak ; absurd ; perverse.
Moe prefers cruelty, Larry prefers weakness, and Curly prefers folly; none of them want perversity, but that of course is what they always get! The logic of Stooge elections is highly non-Aristotelian. Even though each Stooge is as logical as he can be, the system within which they operate adds error on its own.
11 D. Voter's Paradox 177
Systematic errors include : weak and, strong or, weak majority, strong majority, arithmetic glitch, equivalence glitch, implication glitch, modus ponens breakdown, barbarism, and Condorcet sets.
Weak and is this election: Moe Larry Curly Are you an ape? no yes yes Are you a bozo? yes no yes
Majorities agree to these propositions: * I am an ape.
* I am a bozo. * I am not both an ape and a bozo.
Strong or is this election: Moe Larry Curly Are you an ape? yes no no Are you a bozo? no yes no
Majorities agree to: I am not an ape.
* I am not a bozo. * I am either an ape or a bozo.
Diamond , A Paradox Logic
178
Here is a weak majority: Moe
Larry
Curly
Do you like ale?
yes
yes
no
Do you like beer?
no
yes
yes
Do you like cider?
no
yes
no
Majorities agree to: * I like ale.
* I like beer. * I don't like cider. * I don't like most of those three.
Here is a strong majority: Moe Larry Curly Do you like ale? no no yes Do you like beer? yes no no Do you like cider? yes no yes
Majorities agree to: * I don't like ale.
* I don't like beer. * I do like cider. * I do like most of those three.
II D. Voter's Paradox 179 Here is an arithmetic glitch: Moe Larry Curly x
=
1
0
0
y
=
0
1
0
Majorities believe: *x=0 *y=0 * x+y = 1
Here is an equivalence glitch: Moe Larry Curly Do you love Alice? no yes yes Do you love Bob? no no yes
Majorities believe: * I love Alice. * I love Alice and Bob equally. * I do not love Bob.
In the above election, this also passes: * If I love Alice, then I love Bob.
so this is also an implication glitch.
180 Diamond, A Paradox Logic Related to the implication glitch is modus ponens breakdown: Moe Larry Curly
Are all men fools?
yes yes no
Are all fools goons? yes no yes Are all men goons?
yes no no
Majorities believe: * All men are fools. * All fools are goons. * Not all men are goons.
I also call this Barbarism because it undermines the validity of that classic Aristotelian syllogism, BARBARA: All A are B, all B are C, therefore all A are C. The same sort of barbarism, only worse, shows up in this "Ouroboros", or Condorcet set election:
Moe Larry Curly Apes {Dr.O} {Dr.O, #1) { } Bozos
{Dr.O, #1) { } {Dr.O}
Crooks { } {Dr.O} {Dr.O, #1) Majorities agree that: * All apes are bozos. * All bozos are crooks. * All crooks are apes.
11 D. Voter's Paradox 181
Yet not the reverse ! That is: * Not all bozos are apes. * Not all crooks are bozos. * Not all apes are crooks.
And worst of all, every Stooge agrees: ** These three classes form a BARBARA syllogism: all X are Y; all Y are Z; therefore all X are Z.
So the Stooges, those bunglers, made a huge mess of BARBARA, in the very act of affirming it! These spinning Condorcet sets make mincemeat of classical logic . How barbaric!
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Chapter 12
Metamat hemics Godelian Quanta (via quining) Meta-Logic Dialectic Dialectical Dilemma
A. Gvdelian Quanta Diamond arises in Godelian meta-mathematics . In meta-math, sentences can refer to each other's provability, and to quining . This yields self-reference: T = "'is provable when quined' is provable when quined." D = "'is unprovable when quined ' is unprovable when quined." S = "'is refutable when quined' is refutable when quined." P = "'is irrefutable when quined' is irrefutable when quined."
These are self-referential "logic quanta": T = prv T = the "self-trust" statement D = not prv D = the "self-doubt" statement S = prv not S P = poss P
= the "self-shame" statement = the "self-proud " statement
where prv = provable and poss = possible.
183
184 Diamond , A Paradox Logic
T declares itself believable; it "trusts itself'. D declares itself unbelievable; it "doubts itself'. S declares itself refutable; it's "ashamed of itself'. P declares itself irrefutable; it's "proud of itself'.
According to Godel, D and S are complementary undecidables, corresponding to the paradox-values I and J: Logic is consistent if and only if D is true but unprovable, and S is false but irrefutable. Logic is inconsistent if and only if
D is false but provable, and S is true but refutable.
According Lob, T is provably true, and P is provably false; these correspond to the binary truth-values T and F. This is because of the existence of the Godelian paradoxes; they make the consistency of logic unprovable by any consistent logic ; and this collapses the P statement to falsehood.
We get these equations:
d=possT ; s = prvF ; p=F ; t = T .
d = not prv d = poss not d = poss s = poss T = not prv F s = not poss s = prv not s = prv d = prv F = not poss T p = poss p = not prv not p = not prv t = poss F = F t = prv t = not poss not t = not poss p = prv T = T
12B. Meta-Logic 185
B. Meta-Logic
Shame comes to us, proclaiming its own wrongness; it calls itself a liar. If logic is valid, it is wrong, but you cannot be sure. Shame accuses itself; and who are we to contradict it? But by that same token, who are we to believe it? Really it's best to doubt it! To accuse Shame would be to participate in Shame; why bother? But if we permit Shame, then all is well! To admit that self-shame is possible is not selfshame, but self-doubt. What a wonder! Shame is truly irrefutable in spite of itself. Therefore let there be Shame! Shamelessly allow Shame to exist! Admit its possibility; thus you dispel it!
Doubt comes to us, questioning itself; it calls itself a fool. Yet it is true, if all is well. It comes to us doubting its own good sense; yet if logic is valid, it is honest though dubious. It speaks of a marvel and a wonder, namely itself, and really it's best to wonder at such talk. Don't believe doubt! Distrust doubt! For the belief in self-doubt is not self-doubt but self-refutation. What a shame!
Thus Doubt is truly dubious. Doubt doubt! But perhaps it is true anyhow. Even Doubt may be possible.
186 Diamond , A Paradox Logic Pride comes to us, boasting of the great big mathematical model it possesses . But it is wrong! It claims that it will never be refuted; thus it is refuted. Pride is an illusion. All fantasies proclaim their existence; and they are indeed constructible on paper. In theory they're practical, but in practice they're theoretical. Pride is just such a fantasy. Pride cannot be wrong; that is its fatal error. Godel's Second Incompleteness Theorem describes a truly cosmic catastrophe; for it destabilizes Stability itself. Self-doubt exists, as do self-trust and self-shame, but self-construction does not. Absolute existence does not exist. Alas! Pride governs markets, churches, states, and empires! Entire civilizations have sold out for Pride's false promises of absolute existence! Pride boasts of its safety, its sanity, its security, its infallibility, its invulnerability, and indeed its immortality! But it is wrong!
Trust comes to us, proclaiming its implicit confidence in itself. Trust believes in Trust; and it is right! Why? Not because of what it says; but because of what it doesn't say: for in fact Trust says nothing at all. It is deductively empty. If you believe Trust, then you believe Trust; that's it. Trust is tautological, i.e. information-free; it proves nothing that was not already provable. Trust is a vanity. Trust is without content; that's why it's true. Trust can fly because it takes itself lightly. It operates according to the Law of Levity: Bubbles rise.
12B. Meta-Logic 187
What exists? In other words; what is constructible? What cannot be refuted? What is possible?
Pride declares itself possible, but that backfires; whereas the other three quanta are all equally possible: poss d = poss s = poss t = d.
poss s = d ; maybe logic is absurd. Maybe everything is impossible. Maybe nothing exists! I doubt it; but nonetheless the possibility that nothing exists does, itself, exist. I personally think it would be a terrible shame if nothing exists; yet on the other hand I think it wonderful that maybe nothing exists.
poss t = d ; maybe trust is possible. Maybe there is self-belief. Maybe necessity exists. In fact I consider that to be an understatement. Necessity doesn't just exist; necessity is necessary. Vanity is more than just possible; it's universal.
poss d = d ; maybe doubt is possible. Maybe the unexpected happens. Maybe there is chaos. Maybe wonders exist.
Thus existence, when not outright refutable, reduces to Doubt. What truly exists, then, exists in a state of wonder, a paradox; literally beyond belief. In short, a miracle.
188 Diamond , A Paradox Logic The Doubt statement is a meta-mathematical metaphor for a selforganizing, self-propagating system; its referential twist enables it to transcend stasis . The Doubt statement denotes an organic process whose inherent energy cannot be contained in any fixed stable form.. At every stage its paradoxes prevent closure. Doubt is unstable and unreliable; but for that very reason it is irrefutable. It represents Chaos; natural anarchy, which exists if anything does. The anarchy of nature undermines all power systems; arbitrary rules collapse, leaving only those rules which are truly necessary. Thus Natural Anarchy engenders Natural Law; order emerges from chaos. Because Doubt exists, Trust is valid; from the dynamics of paradox, universal laws derive structure.
Thus cosmos is created by chaos.
Now for an even stranger question: Do 1 exist? What a hazardous question that is! For I am trapped into mathematical error, whether I answer yes or no. If I deny that I exist, wouldn't that be a self-refutation? What a shame! But worse, if I affirm that I exist, then my affirmation would proclaim its own existence; and that would be quite a thing to be proud of. For consider; suppose that I were to prove that I exist; and suppose that people duly noted down my proof, and suppose that some scholar came along a century later to inspect my proof. Would it still be valid?
12B. Meta -Logic 189 And so I cannot prove that I exist. My existence is truly unprovable; and indeed, it is rather unlikely. When I contemplate the realities of my existence, I perceive that I am a most improbable person. The fact that I exist can only be called a marvel, a wonder, a mystery, and a miracle, utterly beyond belief. And the same goes for you, dear reader; for I doubt that your existence makes any more sense than mine. Do I exist? I doubt it! So unask the question. The real question is not if I exist (a dubious contingency); it is if I am necessary. And my answer to that question is'yes'. I gladly affirm that - to me at least - I am absolutely necessary! Does that sound vain? Of course it is! Such is the vanity of faith. But that very vanity makes it universal; for don't you, dear reader, consider yourself absolutely necessary - to yourself at least? Don't you have implicit trust in you? Blessed be the vanity of faith!
190 Diamond , A Paradox Logic
C. Dialectic
Now let us consider pairs of statements, referring to each other's provability. We get this table: Dialectic a , b
b = prv a a=_I I prv b l T, T
I
S, D
I
S, D
I
S, S
pose b l D, S
I
P , P
I
D, D
I
D, S
I
D, D
I
D, D
not I D , S
prv b l
I
not I S ,S poss bl
I I
I
I
I
I
I
I S,D
not poss a
not prv a
poss a
I
I I
I
P, T
I
T,P
I I
S,S
Thus GOdelian dialogs define a 4 by 4 game. How we score the 16 outcomes is a matter of taste. Generally we will say the "upper outcomes" T and D are worth more than the "lower outcomes" P and S. Below we will use the scoring systems: S < P < T < D ; "soft-edged" scoring; the 'soft' outcomes S and D are the extreme values, and P < S < D < T ; "hard-edged" scoring; the 'hard' outcomes P and T are the extreme values.
12C. Dialectic 191 Note the column b = not poss a. Player B has accused player A of being outright refutable . B said to A, "I am sure you are wrong !" If you were player A, what would be your best reply? Clearly it would be to respond a = poss b; that is, to say to player B, "You may be right !" For in that case A equals Doubt and B equals Shame. Thus "a soft answer turneth away wrath"!
192 Diamond, A Paradox Logic
D. Dialectical Dilemma
Consider these subgames of "Dialectic":
The Doubt-Shame Dilemma ( the GOdel game ) a,b not poss a
b = not prv a
a=- I
I
not I prv b I
I D, D
not I poss
bl I
I
P, T
I T, P
I
S, 5
I
"Hard-edged" scoring : P < S < D < T.
The Trust-Pride Dilemma (the Lob game) a,b b = prv a poss a a=-I I I prv b I T, T I S, D
I I I I poss bI D S I P P
I
I
" Soft-edged" scoring: S < P < T < D.
12D. Dilectical Dilemma 193 Both the "Godel game" and the "Lob game" are variants of the following game, "Prisoner's Dilemma": a
,
b
B
nice
-I
mean
I
I
I I I W = Win nice I T , T I L , W I T = Truce A I------------- I------------ I D = Draw mean I W , L I D , D I L = Lose I I I scoring: L < D < T < W ; also W+L = D+T For instance : (L,D,T,W) = (0,1,2,3)
In the "Godel game", nice = "not prv" and mean = "prv not". In the "Lob game", nice = "prv" and mean = "poss". If players A and B are nice to each other, then they both get T, which is more than D, which they would both get if they were mean to each other; but if only one is nice, that one loses big to the other. Both players are tempted to cheat; but if both do, neither wins! These games set individual gain against mutual gain. No matter what each player says, the other one's best retort is to be mean; but if both do that, then they do worse than if they both are nice. The best shared outcome requires cooperation, but that this cooperation is vulnerable to exploitation. These Godelian dialogs reveal meta-logical social dilemmas. Here metamathematics meets paradox logic and game theory.
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Chapter 13
Dilemma Prisoner's Dilemma Dilemma Diamond; star vs not Dilemma metric ofdiamond Banker's Dilemma The Unexpected Departure
A. Prisoner's Dilemma a,b
B nice
mean
I I I W = Win nice I T , T I L , W I T = Truce A I------------- I ------------ I D = Draw mean I W , L I D , D I L = Lose I I I scoring : L < D < T < W ; also W+L = D+T For instance : (L,D,T,W) = (0,1,2,3)
This non-zero-sum game presents a player's paradox. It exemplifies the central dilemma of any society; namely, how to get people to co-operate for mutual benefit, when competitive behavior yields a tactical advantage. Negotiation and reciprocation are possible in Dilemma, unlike in competitive games, where there is never anything to negotiate. Mutual profit gives incentive to mutual aid; but exploitation remains tempting.
195
196 Diamond , A Paradox Logic There are many different strategies for dilemma play. I call three of them the "Iron", "Gold", and "Silver" rules.
The Iron rule is the rule of rigid exploitation, justified in the name of expediency. Players ruled by the Iron rule see that no matter how the other player plays, exploitation always yields an advantage; they jump to the conclusion that no more thought is necessary, and play accordingly. This strategy is usually called "All D" (AD) for "Always Defect".
The Gold rule is the policy of absolute altruism. Gold rule players see that a society under Golden rule would be at peace, and thus prevail in the long run; they jump to the conclusion that the long run is already here, and play accordingly. This strategy is usually called "All C" (AC) for "Always Cooperate".
The Silver rule is the strategy of reciprocity. Silver players do unto others as those others have done unto them. They see that only exact imitation can ensure that the game's inner logic favors cooperation; they jump to the conclusion that the other player is aware of this, and play accordingly. This strategy is usually called "TFT", for "Tit For Tat", which starts by cooperating and continues by reciprocation.
Thus the Iron, Gold, and Silver rules are, respectively, vicious, vulnerable, and vain. Gold is for prey (or host) species, Iron for predator (or parasite) species, and Silver for social (or symbiotic) species. Gold says, "what's mine is yours"; Iron says, "greed is good"; and Silver says, "value for value".
13A. Prisoner's Dilemma 197
Negotiation, strategy, and tactics intermesh in the following two negotiation agendas; "Axial Play" and "The Generous Offer":
Axial Play: for players at balance. Tactic; a player limits play to truce-draw "axis". The board permits no advantage of one over another. Strategy; that player threatens draw unless truce. Appeal to principle. Firmness against exploitation. This is tactically soft-line cooperative and strategically hard-line competitive. This is the Justice agenda; soft actions, hard bargaining. It stands on shared principle. Its motto is: "Bribe, threaten, and emulate."
The Generous Offer: for player in position of strength. Tactic; the player limits play to truce-win "column". The board permits no adverse outcome for player. Strategy; the player offers to share his prosperity. Appeal to self-interest . Peace bought and paid for. This is tactically hard-line competitive and strategically soft-line cooperative. This is the Mercy agenda; hard actions, soft bargaining. It stands on shared privilege. Its motto is: "Make them an offer they can't refuse."
Each agenda requires tactical support (the facts on the board) and strategic negotiation (the offer on the table).
198 Diamond , A Paradox Logic Here is a dilemma version of a familiar game: Dilemma Tic-Tac-Toe
The grid # and the letters X and 0 are the same; but there are three new rules: * Player X starts first, but not in the center square; * X and 0 alternate, until they fill the grid; and * Truce = both XXX and 000 rows; win/lose = only one sort; draw = neither XXX nor 000 rows.
Here are some sample games. Numbers tell order of moves:
XS I 06 I X3 08 102 I X9 Xl I X7 1 04
X5 106 I X3 X7 102 I X9 X1 1 08 1 04
04 I X3 106 08 102 I X9 X1 I X7 I X5
Draw
Truce
X wins
X9 I X3 108 04 102 106 Xl I X7 I X5
X5 I X9 I X7 06 102 108 X1 1 04 1 X3
06 I X3 I X5 X7 102 I X9 04 1 X1 1 08
Truce
Truce
0 wins
06 I X1 104 X9 102 I X3 XS 1 08 1 X7
X9 I X1 104 08 102 106 X5 I X7 I X3
X5 I X7 I X3 08 102 I X9 X1 1 06 1 04
Draw
Truce
X
wins
13A. Prisoner's Dilemma 199
Even if we allow X to start in the center square, we might still get a truce such as this: 04 I X5 108 06 I X1 I X9 02 l X7 I X3
Compare the first two games: X5 1 06 1 X3 08 1 02 1 X9 X1 I X7 l 04 Draw
X5 1 06 I X3 X7 102 1 X9 X1 ( 08 I 04 Truce
On the second game's sixth move, 0 put down 06 in the top-center square (A2), and then told X, "If you block me at C2, I'll block you at B 1, and well draw. Better to grab the ABCI file now, and let me get ABC2." X agreed, and they truced. This is classic axial play.
200 Diamond, A Paradox Logic
B. Dilemma Diamond
The word "truce" rhymes with "true" for good reason . Dilemma games have a four-valued logic, thus: T = T/T = Truce "true but true" I = T/F = Win/Lose J = F/T = Lose/Win "true but false" "false but true" F = F/F = Draw "false but false"
This treats the payoff of a dilemma game as a diamond value. Each dilemma game's value is a pair of Boolean values, thus:
value of dilemma game G = (Left gets an upper outcome) / (Right gets an upper outcome) = "Left prospers but Right prospers"
- where truce and win are the upper outcomes, Left and Right are the two players, and to "prosper" means to truce or win.
13B. Dilemma Diamond 201
Given three dilemma games G, H, and K, we define the positive functions "and", "or", "min", "max" and "majority" these ways; (G and H) _ (Left prospers in G and H)/(Right prospers in G and H) (G or H) _ (Left prospers in G or H)/(Right prospers in G or H) (G min H) _ (Left prospers in G or H)/(Right prospers in G and H) (G max H) _ (Left prospers in G and H)/(Right prospers in G or H) M(G,H,K) _ (Left prospers in most of G, H, and K) / (Right prospers in most of G, H and K)
Recall that diamond has orthogonal reflections, along the horizontal and vertical axes. Horizontal reflection is "not"; vertical reflection is "star". not (A/B) _ (not B) / (not A) *(A/B) _ (*B)/(*A) Through these the two sides interact. Star is the exchange operator; it switches the payoffs for the two players. Not exchanges with a reversal; thus each side gets the opposite of the other's; "we won because they lost". Considered in dilemma game terms, "not" corresponds to "strategic" distinction. This form of thinking considers the good of the whole; it distinguishes truce and draw, and regards each victory with indifference. "Star" corresponds to "tactical" distinction; this form of thinking considers the good of the individual player; it distinguishes win and lose and regards zero-difference play with indifference. This matches the two incompatible ways of thinking about Dilemma with the two dual reflections of the logic diamond.
202 Diamond , A Paradox Logic
C. Dilemma As Diamond Metric Let Dilemma be given this scoring rule: Lose=O, Draw= l, Truce=2, Win=3. If we use the "right side" identification: Lose=i, Draw=f, Truce=t, Win j, then dilemma scoring defines a function N: N(i) = 0, N(f) = 1, N(t) = 2, N(j) = 3.
This function is a "norm"; that is, it obeys these rules: N(x) > 0; if x
From the norm N, we can define a "metric" D: D(x, y) = N(x max y) - N(x min y) It is a theorem of lattice theory that any such metric D defined from a norm N has these properties: D(x,y) > 0 for all x and y ; D(x,y) = 0 if and only if x = y ; D(x,y) = D(y,x) ; D(x,z) < D(x,y) + D(y,z) . The dilemma metric defines "distance" in diamond. Thus dilemma makes diamond not just a logic, but a space as well.
13D. Banker's Dilemma
203
D. Banker 's Dilemma
A Billiard-Marker, whose skill was immense might perhaps have won more than his share; But a Banker, engaged at enormous expense had the whole of their cash in his care. - Lewis Carroll , The Hunting Of The Shark
Consider a Dilemma game between players A and B; it is financed by a banker C, who gets to keep the remainder of the fund after the payoffs are distributed. Their payoffs are:
4 dollars invested
(A,B,C) payoff B nice
mean
------------- -------- -----nice I (2,2,0) I (0,3,1)
I I A I------------- I--------------------------I mean i (3,0,1) I (1,1,2)
I
I
204 Diamond, A Paradox Logic This makes Dilemma zero-sum again ; for the main player's cooperation is the banker's defeat. What is more , the banker has a vested interest in fostering distrust between the other two players. (Indeed, that is the the only thing the banker can actively do; for the other two players make all the moves.)
The three players rank win/lose, lose/win, truce, and draw in three different ways: A:
L/W <
D <
T <
W/L
B:
W/L <
D <
T <
L/W
C:
T <
W/L =
D
L/W <
These three preference rankings yield these majorities: 2/3 say:
W/L <
D
(Voters B and C)
2/3 say:
L/W <
D
(Voters A and C)
2/3 say:
D <
T
(Voters A and B)
2/3 say:
T <
W/L
(Voters A and C)
2/3 say:
T <
L/W
(Voters B and C)
L/W
W/L <'C
L/W
W/L
13D. Banker's Dilemma 205
This "Condorcet Crossing" diagram agrees with most - but not all - of each voter's preferences. It contains preference loops, yet every player agrees that the order relation is transitive! Thus we get a voter's paradox. The banker's financing makes Dilemma zero-sum, but non-Aristotelian. The glitch remains; to escalate order is to escalate chaos. Either non-modus-ponens or non-zero-sum; Dilemma's illogic is marked. It often displays paradoxical signs, for Dilemma is, so to speak, snarked.
206 Diamond , A Paradox Logic
E. The Unexpected Departure For truce to succeed requires certain conditions. One of them is that the expected number of plays be great enough; another is that the play not end at too definite at time . If it does, then a "backwards induction paradox" destroys truce, no matter how long the tournament. Consider the following scene: Curly is about to play with Moe in a dilemma tournament scheduled to last exactly 100 rounds. Curly, a Silver Rule player, is optimistic that he can convince Moe (an Iron Rule player) that it'll be in his own best interest to cooperate. But Moe said, "What about the 100th round? Won't that be the last one?" Curly said, "Yes." "There won't be any after the 100th?" "Yes," said Curly. Moe asked, "So in the very last play, what's to keep me from defecting?" Cause I'll defect the next..." Curly said, then slapped himself on the face. "Alright, nothing will stop you from defecting on the 100th play." "So you might as well defect too, right?" Moe said, smiling. "I guess so ," Curly said reluctantly. "On the 100th play." Moe continued, "And what about the 99th play? What's to keep me from defecting then?" Cause I'll defect the next..." Curly said, then slapped himself on the face. "But I'll defect on the 100th play anyhow."
13E. The Unexpected Departure 207
"That's right," Moe said, smiling. "So nothing's keeping you from defecting on the 99th play." "That's right," Moe said, smiling. "So I should defect on the 99th play also," said Curly. "That's right," Moe said. "Now, what about the 98th play?" And so they continued! Moe whittled down Curly's proposed truce, one play at a time, starting from the end. By the time the conversation was over, Moe had convinced Curly that the only logical course was for them to defect from each other 100 times, drawing the tournament. And so they did; yet when Curly played with Larry (a Gold Rule player) they cooperated 100 times, for a truce! Thus we deduce, by mathematical induction, that the prospect of abruptly terminated play, even if in the far future, poisons the relationship at its inception. That is the "backwards induction paradox". In dilemma play, cooperation requires continuity to the end. Departure should not be at an expected time lest that light the backwards-induction fuse; departure should be unannounced, at an unexpected time. We need an unexpected departure; but this yields a paradox. Consider this following story about an Unexpected Exam:
Once upon a time a professor told his students, " Sometime next week I will give you an exam ; and that exam will be at an unexpected time. Right up until the moment I give you the exam , you will have no way to deduce when it will happen, or even if it will happen . It will be an Unexpected Exam." One of the professor's student objected, "But then the exam couldn't happen on Friday; for by then it would be expected!"
208 Diamond , A Paradox Logic The professor said, " True."
The student continued, "So Friday's ruled out." Another student said, "But if Thursday's the last possible day for an Unexpected Exam, then it's ruled out too; for by Thursday the Thursday exam will be expected!"
The professor said, "True." And so on; by such steps the students concluded that the Unexpected Exam can't happen on Friday, Thursday, Wednesday, Tuesday, or Monday; so it can't happen at all! " So you don't expect it?" said the professor. His students said, "No!" The professor smiled... On the next Wednesday, he handed out an exam, to his students' surprise.
That's the Paradox of the Unexpected Exam . Here a backwards induction paradox also appears; but this time it yields a strangely false result rather than a strangely undesirable result. This match of methods suggest the following fable.
The same professor visited the Dean; he said, "I will depart this school sometime during the next month. To ensure cordial relations between us until that time, my departure will take place on an unexpected day. It will be an Unexpected Departure." The Dean retorted, "You couldn't leave on the 31st, for by then your Unexpected Departure would be expected." The professor agreed.
13E. The Unexpected Departure 209
The Dean added, "Having ruled out the 31st, the 30th is also ruled out; for it would be expected." The professor agreed to that too. And so the conversation continued; and in the end the Dean concluded, "Your Unexpected Departure can't happen on any day. Therefore I don't expect it." The professor agreed. On the seventeenth day of the month the professor departed, to the Dean's astonishment.
This Paradox of the Unexpected Departure is just what the doctor ordered; for here the failure of backwards induction (so puzzling to the reason) is precisely what is needed to defend the Axelrod equilibrium from its backwards induction proofl A dilemma tournament can use "open bounding"; replay only if a random device permits it. This ensures an Unexpected Departure; play will be finite, but there will be no definite last play during which the Iron player is safe from the danger of Silver retaliation. The paradox of the Unexpected Departure is related to the paradox of the First Boring Number; for presumably the tournament ends as soon as it stops being interesting.
The conclusion then is clear; let none of your social relationships end too definitely; let there be some possibility that you might encounter that person again, soon . (And conversely, when you must leave, slip away quietly!)
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Chapter 14
Speculations Diamond Types? Diamond Values for Dilemma? Null Quotients? General Lattices? General Waves?
A. Diamond Types? The very concept of a diamond type seems ironic; for the whole point of diamond is to create typeless fixedpoints. In complete self-reference, there is but one type, and it refers to itself, harmonically or paraharmonically. If we allow reference by analytic functions as well, then we need type theory; for analytic functions unearth diamond's boolean substrate. Stranger still, harmonic and paraharmonic functions are analytic relative to each other; that is, paraharmonic functions are analytic to harmonic systems, and vice versa. This is the "perpendicularity" of the two logics. Given two perpendicular systems, one must be of lower type than the other. Definition. A diamond type is a completely interreferential (harmonic or paraharmonic) system of statements. Definition . Type Order. Any statement in a given type may refer to any other statement in that or any lower type. Conjecture. Is type order a lattice? A fixedpoint lattice? Conjecture. Outer fixedpoints for typeless systems take linear time to evaluate. Can diamond types define polynomial-time fixedpoints?
211
212 Diamond , A Paradox Logic
B. Diamond Values for Dilemma?
Dear reader, I used to believe the following conjecture: Valuation Conjecture: Any dilemma game position's diamond value can be evaluated via iteration to a fixedpoint.
This fixedpoint is found by assigning a diamond value to each node of a "game tree"; the graph describing all game positions and moves. The diamond values can be computed for each node, in terms of the other nodes, following a formula given by the game tree. Thus we get an inter-referential system of diamond values; and any such system has at least one fixedpoint.
I call Valuation a "conjecture" rather than a "theorem" because of certain difficulties in the above "proof'. Specifically, it seems impossible to capture the entire Dilemma game-outcome valuation by either harmonic or paraharmonic functions. The trouble is that Dilemma evaluates the outcomes in a linear (numerical) lattice, but diamond is not such a lattice. Paraharmonic functions capture the zero-sum nature of competitive thought; likewise, harmonic functions capture the zero-difference nature of cooperative thought; but Dilemma transcends both. The value of the game depends on how two players think about the game; thus, by definition, no single system of reasoning can capture Dilemma's essence. Dilemma requires dialog.
14C. Null Quotients?
213
C. Null Quotients?
The null quotients are the result of division by zero. There are two of them:
1/0 = "infinity"; larger than any finite quantity 0/0 = "indefinity"; indistinguishable from any quantity
Consider these algebraic equations: x = 1/0 Ox = 1 0 = 1 Infinity leads us to an obvious absurdity. 1/0 is inherently inconsistent; "over-determined".
Consider these algebraic equations: x = 0/0 Ox = 0 0 = 0 Indefinity leads us to a vague tautology. 0/0 is inherently uninformative; "underdetermined". As noted in Chapter 2B, this connects us to diamond logic; for we can identify i with one, and j with the other.
214 Diamond , A Paradox Logic In terms of Size paradoxes, perhaps we can say: 0/0 = the Heap, and 1/0 = Finitude
Consider the "sign" function: sign(x) = IxI/x . Its graph is: * ----------------------- +1 - ----------------------0------------------------ + -1 ----------------------*
Note that sign(0) = 0/0 ; sign is undefined at zero. Note also the similarity of this graph to the Dedekind splice.
According to Godel's Theorem, any arithmetical deductive system is either inconsistent or incomplete. Inconsistent is overdetermined, like 1/0; incomplete is underdetermined , like 0/0; thus arithmetic, though it avoids using null quotients, itself resembles a null quotient!
14D. General Lattices? 215
D. General Lattices?
Is every lattice the fixedpoint lattice for some harmonic system? If so, then given a lattice, how do we find a system that emulates it?
Given a system, how can we improve it? (Reduce the number of variables, references, etc.)
How does a lattice change when you change its system? And vice versa?
Given a lattice and its system, can input leads into the system provide control over points in the lattice?
What practical computation tasks does lattice emulation permit? Could we (say) emulate a tree (for sorting, searching, etc.) as the bottom half of a lattice? Note that boolean logic usually corresponds to the equator of the lattice, or the nodes of the tree; the part with the fewest lattice constraints.
Can lattice emulation be useful in doing, say, "quantum logic"? And would this, plus the energy equation, suffice to do fast quantum-mechanical computation?
216 Diamond , A Paradox Logic
E. General Waves?
Diamond is merely the foot in the door! Diamond covers stable period-2 waves; but data streams can be period 3, period 4, ... or even no period at all. As above, we define "and", "or" and "not" on these binary streams as follows: (A and B )(n) =
A(n-2) and B(n-2)
(A or B)(n) =
A(n-2) or B(n-2)
(not A )(n) =
not ( A(n-1) )
In addition, define "star" as the delay operator: (* A)(n) = A(n-1) We therefore get these equations: not(not(x)) = **x = delay 2 *' = delay 2n
Define notN(x) = not(*N''(x)); delay-N negation. We can similarly define delay-n conjunction and disjunction.
Delay-n negation has its own paradox equation: P = notN(P ) Any such P is a period-2N wave. For instance, there is the "square wave": ...fFttt...ttfff..ftt... ; each block N units long. If P equals such a wave, then dP = P and not1 P equals a "blip" wave: ...f ff...ffff..fftf .. (one t every 2N spaces.)
14E. General Waves? 217 If P is a square wave, then it has 2N phases P;. Similarly, dP has 2N phases dP,,. Any period-2N wave is a disjunction of blip waves; thus we get these "wave integrations": W = OR[1
Consider the "Ant", or "toggled buzzer" circuit, as a general-wave circuit. First, write the "ant" circuit using period-n negation: A = notN (B ) B = notN (A) C = notN(BorC) When the ant's inner toggle is set one way, it freezes into its halt state; but when the toggle is reset , then the ant emits a square wave with period 2N. The square wave starts just when the reset signal came; thus the ant's signal retains the phase of the reset's arrival. Thus the Ant is a "phase -memory" circuit.
These, dear reader, are but the bare bones of any genuine theory of general-wave logics . Diamond suggests general-wave logic, just as 2-space suggests n-space. This text now points beyond Diamond , for it has reached the outer limit of Diamond. Therefore this text has reached an end.
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Appendix Notes and Proofs
Chapter 1. Paradox. D. Santa Sentences Suppose that some sarcastic Grinch were to proclaim: "Santa Claus exists, and I am a liar."
G = S and not G
If Boolean logic applies to this "Grinch Sentence", then it refutes both itself and Santa Claus! Why? Consider this line of argument:
G = (S and not G); assume that G is either true or false. If G is true, then G = (S and not true) = false. G = true implies G = false; therefore (by contradiction) G must be false. False = G = (S and not G) = (S and not false) = S Therefore S is false. Therefore Santa Claus does not exist!
219
220 Diamond, A Paradox Logic 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:
S is either true or false. If it's false, then so is G: G = false and not G = false. No problem. But if S is true, then G becomes a liar paradox: G = true and not G = not G If S is true, then G is non-boolean. Therefore; if G is boolean, then S is false. QED.
Call an adjective "Grinchian" if and only if it does not apply to itself, and Santa Claus exists: "A" is Grinchian = Santa exists, and "A" is not A. Is "Grinchian" Grinchian? "G" is G = Santa exists, and "G" is not G.
The "Grinch Set for sentence H" is:
GH
{ x I H is true, and x not an element of x }
G. in GH
H is true, and GH not in GH
In diamond logic, the threatened paradox need not affect any other truth value. If Santa Claus does exist after all, then the Grinch is exposed as a Liar!
Appendix. The Grinch 221
The Grinch sets suggest Grinch stories. Consider the Weekend Barber, who only shaves on the weekends, and only those who do not shave themselves: WB shave M = It's the weekend, and M does not shave M. Does the Weekend Barber shave himself? WB shave WB = It's the weekend, and WB does not shave WB.
Note that Epimenides's statement: "All Cretans are liars, including myself." - makes him the Grinch of Crete!
222 Diamond , A Paradox Logic
F. Size Paradoxes Standard (i.e. boolean) set theorists were so disturbed by Russell's paradox that they decided to acknowlege only "well-founded " sets; that is, sets without "infinite descending element chains". X is well-founded if and only if there is no infinite sequence of sets X,, X2, X3, ... such that X contains X,, X, contains X2, X2 contains X3, and so on. Well founded sets include { }; { { } }; { { { } }, { { }, { { } } } }; and even infinite sets such as { { }, { { } }, { { { } } }, { { { { } } } },... }; for well-founded sets can be infinitely "wide", so long as they are finitely "deep" along each "branch". On the other hand, well-foundedness excludes sets such as
A = { A) = {{{{{{...}}}}}} for it has the infinite descending element chain A, A, A, ...
Let WF be the set containing all well-founded sets. Is WF well-founded? If WF is well-founded, then WF is in WF; but this yields the infinite descending element chain WF, WF, WF, ... On the other hand, if WF is not well-founded, then any element of WF is well-founded, and element chains deriving from those will be finite. Thus all element chains from WF will be finite; and therefore WF would be well-founded. And so we see that the concept of "well-foundedness" leads us to the paradox of Finitude.
Appendix. Diamond Values 223
Chapter 2 . Diamond. B. Diamond Values By "underdetermined" I mean a statement which logic still hasn't decided; by "overdetermined" I mean a statement about which logic has derived opposite conclusions . Thus, an underdetermined statement is neither provable nor refutable, and an overdetermined statement is both provable and refutable. According to Godel's Theorem, any logic system is either incomplete or inconsistent; thus the equation; i or j = t that is; underdetermined or overdetermined = true is none other than Godel's Theorem, written as a diamond equation. Diamond harmonizes with meta-mathematics. Note that we have four interpretations for diamond logic: t j f or and
true ; true ; false ; false undet ; overdet ; undet ; overdet overdet ; undet ; overdet ; undet false ; false ; true ; true aorb ; aorb ; aandb ; aandb aandb; aandb ; aorb ; aorb
These are isomorphic to each other under conjugation by the four operations { identity, not, star, minus } - a Klein group. I and J are complementary paradoxes; the yin and yang of diamond logic. They oppose, yet reflect. Yang is not yang, yin is not yin, and the Tao is not the Tao!
224 Diamond, A Paradox Logic
Chapter 3. Diamond Algebra.
B. Normal Forms.
One can rewrite the Primary Normal Form in terms of differentials: F(x) = B,(x) or (B2 and dx ) = b,(x) and (b2 or Dx ) where each B. and each b; is a "derivative-free" expression . (That is, they have no dx or Dx terms.) This "derivative series" form can be extended to two variables: F(x,y) = B,(x,y) or (B2(y) and dx) or (B3(x) and dy) or (B4 and dx and dy) b,(x,y) and (b2(y) or Dx) and (b3(x) or Dy) and (b4 or Dx or Dy)
- and so on, for more variables, using higher order differentials.
Appendix. A Parenthetical Remark 225
Chapter 4. Self-Reference. A Parenthetical Remark About The Parenthetical Remark
Consider the Brownian form [[]] . It is equal to (i.e. confused with) the void; yet it is not itself void, being made of two nested marks. It therefore deserves names of its own; I suggest "doublecross", or the "remark". Doublecross denotes the void, but unlike the void, is visible. The remark is to forms as zero is to numbers; both name the nameless, and both are placeholders. In algebraic terms, the remark denotes parentheses:
(A)
=
A
I l
l
=
A
I use parentheses to distinguish these from the brackets of boundary logic. In fact (A) = [[A]]. The remark allows one to express the associative law: I I (AB)C = A B II C = ABC = A B C II = A(BC)
[[A]] remarks about A without marking A; it draws attention without changing values. It is literally a parenthetical remark.
226 Diamond, A Paradox Logic
Finally, consider this re-entrant remark:
It represents this system:
A = B I B = A I
A = (A) .
This is a toggle, or memory circuit. Thus memory remarks on itself.
Appendix. The Outer Fixedpoints 227
C. The Outer Fixedpoints
Consider the function f(x) = dx / t ; It takes two steps to go from i to the fixedpoint j; i ---> 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
228 Diamond, A Paradox Logic
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
therefore c < f 2n (z) < ... < f 2(z) < fl z) < Z. f 2n(z) is a fixedpoint left of fZ'(a) and of f 2(b), and it is the rightmost such fixedpoint . Therefore z generates the minimum in the relative lattice:
fen( a min b ) = f2"(a) minf f2"(b) QED.
Theorem . The maximum of two right seeds is the right seed of the relative maximum: f2n(A min b ) = f2"(a) Min f f2"(b) if a and b are right seeds. Proof is identical.
In summary: the minimum of left seeds is a left seed, one which generates the relative minimum of the generated fixedpoints. Since any fixedpoint is a seed, it follows that the minimum of fixedpoints is a seed generating the relative minimum of the fixedpoints: f2n(a min b ) = a min f if a and b are fixedpoints. f2n(a max b ) = a max f b , if a and b are fixedpoints.
230 Diamond , A Paradox Logic
C. Examples I gave these systems whimsical names based on the appearance of their circuits. The "Duck" has this circuit:
-I\___^IV \ I/ I/ This is equivalent to the fixedpoint: B = (Band not B) = dB A differential of itself! The "Truck" has this circuit:
UN L-\ 1\-I\-/ I/ I/ I/ The "Rabbit" has this circuit:
\- I \- I\ L 1 I \- I \-/
The "Triplet" has this circuit:
1 \ / / \ \ \.- I \,_ X_ I \1^ I \-/
Appendix. Examples 231
In general, the system a = not(b or c) ; b = not(c or a) ; c = not(a or b) ;
4 = (a & f(d)) or (b & g(d)) or (c & h(d)) or (a&b&c) will have this fixedpoint lattice: (Ll) / \ / \ iiii --(L3)--]]J1 \ / (L2)
- where L1 , L2, and L3 are the lattices for fI g, and h.
The "Ant" has this circuit:
\-I\-I\, L I\ J If these are "nand" gates: C = C A The Ant's a Santa! If these are "nor" gates: C = A - C The Ant's a Grinch!
To see the circuit diagrams for Brown 's two Modulators, see chapter 11 of his book, Laws of Form.
Diamond , A Paradox Logic
232
Consider this Brownian form; "two ducks in a box":
C
=
[ [a[a]]. [b[b]]b c ].
a = [a[a]]
I I I
a
=
b c
=
I
I
I
I I I IIII I-I_I
I I I IIII I-I_I
I
b= [b[b]]
I I
c= [abc]
da
"I am honest and a liar."
db
"I am honest and a liar."
a nor b nor c
"All of us are liars."
The fixedpoint lattice of this "Anti-Interference Grinch" is: ifi -------- ijf -------- fjj / \ / \ iii ffi ----------- ffj jjj \ / \ / fii -------- jif -------- jfj
Note the fixedpoints ijf and jif; these are the only ones where C has a boolean value; but this is due to Interference, an anti-boolean axiom! Were it not for those points, this lattice would be distributive; but due to them it contains N5.
Appendix. Cantor's Dyadic 233
Chapter 8. The Continuum. B. Cantor 's Dyadic
I noted that C, as an anti-diagonal , can read .0111111111..., while C, on the list, reads .100000000... The usual response to this is that we could separately list all dyadics in a countable list; but that the non-dyadic (i.e. nonterminating) reals are uncountable by Cantor's proof. This, however, is a non-sequitur; for the anti-diagonal of the nondyadic numbers would then be a dyadic. Another problem with this defense of Cantor's theorem is that it uses the word "dyadic"; for a real number is "dyadic" if and only if its binary expansion has afinite number of 1's, or of 0's. But this leads us straight back to the paradox of Finitude! Indeed, the smooth transition from small to large through paradox mimics the Dedekind splice's transition from false to true through paradox. The "first boring number " and the "last interesting number" are boundary paradoxes in the discrete domain. Note the complementarity between "finitely many infinite-precision numbers " and "infinitely many finite-precision numbers". Here we see finitude versus itself, second-order finitude. Cantor's Dyadic is the limit of real arithmetic.
234 Diamond , A Paradox Logic Diamond logic radically simplifies the theory of the infinite. In the diamond world, Cantor's Theorem does not apply; the power set of a set can be put in one-to-one correspondence with the set. (There will, of course, be paradoxes at the pivot bits.) The diamond world has no cardinal hierarchy: it has only one infinity, and that one tinged with paradox and finitude. I made it this way on purpose; for I seek a theory of infinity that is comprehensible to finite beings such as you or I. Who are we to speak of alephseventeen? This theory is to be computable by actual store-bought computers, not by Platonic ideals . I want this Diamond Logic to be like its namesake; crystalline, clear, adamantine, and (above all) down to earth. Set theorists speak of uncountable ordinals and measureable cardinals; but mathematics nowadays is more concerned with megahertz and gigabytes. Finitude is our style, indeed our birthright. Call this Math for Mortals.
Appendix. Diffraction
235
Chapter 10. Harmonic Analysis. D. Diffraction
The Buzzers- To-Toggle Theorem can be generalized to two functions: If (A,B) = J(a,b) then:
f(a) = a and g(b) = b if and only if
(A,B) = ((f/g)L(A;B), (g/-OR(A;B))
A f (f/g)L(A;B) ] 7 / \ / \
B g (9 /f) R( A;B)
a (f/g) L f(A) \/ \/ 3
11 (g/f)R 9(b)
236 Diamond , A Paradox Logic Kauffman , in his paper "Knot Automata", claimed the following as a reduction circuit;
a = [bjz] b = [aiz] c = [ad] d = [bc] = z/2 ; the half-period oscillator i = [bd] j = [ac]
Let V equal (a,b,c,d,i,j);
A=(110101),B=(101011),C= (111010),D= (010111) z=0 V=A,orV=C z=1 V=B,orV=D If z = 0,1,0,1,0,1,..., then V = A,B,C,D AB,CD,... d oscillates at half z's period: 1,0,0,1,1,0,0,...
If we diffract Kauffman's reductor: (A,B) = J(a,b) ; (C,D) = J(c,d) ; (I,J) = J(i,j) then we have this system: A = (AIz) B = (BIz) C = (BC) D = (AD) I = (AC) J = (BD) - almost, but not quite, two parallel circuits.
Appendix. 3-Logic 237
Chapter 11. 3-Logic. Consider these period-three permutations of diamond: (tit), (fit), (tjf), (ft), (ijf), (f i), (tij), (lit). They do not preserve adjacency, minus, or majority. D is a normal subgroup of S4; and modulo D, all elements of S4 are conjugate to one of { id, (tif), (fit) } ; the group Z3. At first I thought that these non-dihedral elements of S4 generate, as in 3-logic, three cyclically distributive lattices on the diamond. However, after checking the additional cases added by the fourth value, I found that this is not true. Consider the permutation U = (fit). (This is the same as the "up" rotation in 3-logic, extended to the diamond.) Thus U applied to the diamond yields this lattice : U[0]: f t
j i
and these equations: j&(tU[&]i) = j&i = f (j & t) U[&] (j & i) = j U[&] f = j j&(tU[V]i) = j&t = j (j & t) U[V] 0&0 = j U[V] f = f As in 3-logic, the period-3 rotations induce three isomorphic logics; however, "&" distributes over neither U[&] nor U[V]; so cyclic distributivity fails on the diamond. Naturally, cyclic distributivity still applies on the sublogic {fi,t).
238 Diamond , A Paradox Logic
Chapter 12. Metamathematics. A. Godelian Quanta.
x not x
prv x poss x not prv x not poss x
--- ------------------------------------------------------t
I
p
t
d
p
s
d
l
s
s
d
d
s
s
I
d
s
d
d
S
p
I
t
s
p
d
t
To create a diamond logic out of the four godelian quanta , we need to identify the "sides". Godel's analysis provides the key question; namely, is the logic system used consistent or not? If the system is consistent , then these comparisons apply: P < S < D < T P is proven false and so at the bottom; S is false but not refutable, and so second-from-bottom; D is true but not provably so, and so second-from-top; and T is provably true. If the system is inconsistent , then D is false but provable , and S is true but refutable; so the two change places: P
<
D
<
S
<
T
Appendix. Godelian Quanta 239
These two order relations agreed on the following: D < < P
<
T
S
If we define & and V as minimum and maximum relative to this lattice, then we get diamond's positive functions.
"Not" is boolean: it sends T to P, P to T, D to S, and S to D. Let "*" interchange D and S while leaving T and P fixed. This is the "consistency switch"; it mimics the effect of reversing the consistency of the system. If we define _(x) as "not * x", then we get diamond's negation. Thus &, V, and - define a diamond logic for Godel's quanta.
There are quanta beyond the Godelian quanta. For instance, there are the Tertullian quanta. I name these after Tertullian, who once declared that he believed a certain point of doctrine " because it is impossible ." Charmed by so defiant a folly, I decided to recast his little joke into meta-mathematical terms. Let a "Tertullian" quantum be of the form: "Believe this statement only if X is absurd". Trtl = prv(Trtl) implies (prv(X) and prv(not X) )
240 Diamond , A Paradox Logic This suggests the equation: Trtl = (Trtl dX )
- a differential Santa sentence!
Equivalent forms exist. The logical principle 'modus tollens' assures us that the statement "A B" equals "not B not A"; therefore the Tertullian quanta can be rephrased thus: "X is possible only if this sentence is unprovable." " X is possible only if I am incredible. "
These formulae seem more modernistically skeptical than Tertullian's aggressive faith, but they are logically equivalent. Therefore my reply to Tertullian is:
He is irrefutable because he is unbelievable. "Possibile est quia incredibile est." It is possible because it is unprovable. Tolerate me because I am dubious!
Appendix. Prisoner's Dilemma 241
Chapter 13. Dilemma. A. Prisoner's Dilemma There exist many dilemma strategies other than the Iron, Gold, and Silver rules. For instance, there is R, for Random play; TF2T, "Tit For Two Tats", which defects only after the other player defects twice in a row; 2TFT (two-titsfor-a-tat); "angry " TFT (TFT starting in an unfriendly state); TFT with occasional "testing" behavior; and TFT with "forgiveness factor", which occasionally (at random) forgives misbehavior on the other player's part; RTFT, "reverse tit-for-tat", which punishes cooperation and rewards punishment; and "Pavlov", which is nice on the next round if this round truced or drew, and is mean on the next round if this round won or lost. (That is, Pavlov repeats its present play if it came out truce or win, and switches if it came out draw or loss.) Which strategy is best? That depends on many factors; the other player's strategy, the expected length of the tournement, and the tactical position of the dilemma game itself. Thus dilemma games have a second level of play; strategic as well as tactical . How to play matters as much as what to play.
Many kinds of ordinary games can be "dilemmized". Prominent among these is chess. Dilemma chess is chess plus deterrence, with a dilemma payoff matrix. The board, pieces and moves are the same as in regular chess; but the game is allowed to end with mutual checkmate, called truce.
242 Diamond, A Paradox Logic This is the payoff matrix: B payoff for (A , B) checkmated I not checkmated
checkmated not checkmated
(truce, truce) (lose, win) (win, lose) (draw, draw)
Thus a dilemma. If competitive chess is the king of games, then dilemma chess is the queen; for truce opens up a new dimension of play; namely, between competition and cooperation. The basic innovation in dilemma chess is to allow the "reply" move. The reply move is a final move by the player whose king has been captured. If the other king can be captured in the reply move, then the first capture is "deterred". You may not capture if your check is deterred. You may move into deterred check, or respond to check with a deterrent . You may not cancel the other player's deterrent unless you also escape check (no "forced exchanges"). Mutual deterred check is "tryst"; both sides can capture and retaliate. Truce is mutual assured check (MAC), or inescapable tryst; one move after truce, both sides can still capture and retaliate . In tryst, capture is deterred; the other player could capture next move, but would suffer retaliation . Other forms of deterrence exist ; "pinned check", "delayed deterrent", even "temporary checkmate". If you have a deterrent, then your king is free to advance into enemy territory; the "brave king" phenomenon.
Appendix. Prisoner's Dilemma 243
For instance, consider this endgame:
I I al I
I I I I I
I I I I I BR
I I I I I I I I I I I I I I 71 BP I I I BQ I BP I BP I 61 I BP I I BP I I I BK
51 I I BP I WP I BB I WP
41
I
31
I I
WK
WP I
I
I
I
I
WP
21 WP I WP I I I WN
11 I I I I I WR I I WQ I I I I I I I I a
b
c
d
e
Black to move. Note that PxK is deterred. ... Kg6xf5 tryst Kf5-f4 tryst Qhl-f3 tryst Qd7-f5 tryst Kg4-h5 check KhS-g6 tryst Rg8-h8 truce
Two courageous kings!
f
g
h
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Bibliog r a phy Robert Axelrod The Evolution of Cooperation New York: Basic Books, 1984. G. Spencer Brown Laws of Form New York: the Julian Press, 1979. Nicholas Falletta The Paradoxicon Garden City, New York: Doubleday & Company, Inc., 1983. Nathaniel Hellerstein "Diamond , a four-valued approach to the problem of Paradox" U.C. Berkeley: Doctoral thesis, 1984. "Diamond, a Logic of Paradox" Cybernetic, Summer-Fall 1985, v.1. Laws of Flux, manuscript Wave Laws, manuscript Diamond, A Paradox Logic Singapore : World Scientific Publishing Co. Ltd., 1996 Douglas Hofstadter Godel, Escher, Bach New York: Basic Books, 1979.
245
246 Diamond, A Paradox Logic Patrick Hughes and George Brecht Vicious Circles and Infinity New York: Penguin Books, 1975. Louis Kauffman "De Morgan Algebras - completeness and recursion" Proceedings of the 8th International Symposium on MultipleValued Logic, 1978, pp.209-213 "Imaginary Values in Mathematical Logic" Proceedings of the 17th International Symposium on MultipleValued Logic, IEEE Publications, 1987. "Knot Automata" 24th Int. Symp. on Multiple-Valued Logic, IEEE Pub., 1994 "Self-Reference and recursive forms" Journal of Social and Biological Structure, 1987, v.10, pp.53-72 "Network Synthesis and Varela's Calculus" International Journal of General Systems, 1978, v.4, pp. 179-187 Ernest Nagel and James Newman Godel's Proof New York University Press, 1958. James Newman The World of Mathematics New York: Simon and Schuster, 1956. William Poundstone Labyrinths of Reason New York: Anchor Press, Doubleday, 1988.
Bibliography 247
Rudy Rucker Infinity and the Mind New York: Bantam Books, 1982 Richard Shoup "A Complex Logic for Computation" Interval Research Corporation. Raymond Smullyan This Book Needs No Title Englewoods Cliffs, NJ: Prentice-Hall, 1980. Forever Undecided New York: Knopf, 1987 To Mock A Mockingbird New York: Knopf, 1985 Stan Tenen;
Innumerable personal communications.
Francisco Varela and J. Goguen "The Arithmetic of Closure" Journal of Cybernetics; 1978, v.8 Francisco Varela and Louis Kauffman "Form Dynamics" Journal of Social and Biological Structure, 1980, v.3, pp.171-206. Francisco Varela "A calculus for self-reference" International Journal of General Systems, 1975, v.2, pp.5-24 "The extended calculus of indications interpreted as a three-valued logic" Notre Dame Journal of Formal Logic, 1976, v.17
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Index 2 by 2 matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 -125 Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41,43 analytic functions . . . . . . . . . . . . . . . . . . . . . . . . 119, 121 -122 Analytic Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . 161 ant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 , 217, 231 anti-boolean. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37, 42, 23 2 anti-symmetric . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57, 105 arithmetic glitch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Associativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41,43 asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41,43 Axial Play . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197, 199 backwards induction paradox . . . . . . . . . . . . . . . . . . . . 206 - 209 Banker's Dilemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 barbarism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 Barry's paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 boundary logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 - 39 brave king . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 Brown's First Modulator . . . . . . . . . . . . . . . . . . . . . . . . . . 80 Brown's Second Modulator . . . . . . . . . . . . . . . . . . . . . . . . . 81 Brownian forms . . . . . . . . . . .33 - 37, 53 - 55, 61 - 62, 68, 72 - 81, 232 but . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27-28
249
250 Diamond , A Paradox Logic buzzer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 - 26 Buzzers-To-Toggle Theorem . . . . . . . . . . . . . . . . . . . . . . . 159 Cantor's Dyadic ........................ 107 - 108, 233 Cantor's Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Coalition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 cofinal range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 cofinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Commutativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41,43 Complementarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Completeness . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 51 Condorcet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 Condorcet sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 180-181 Conjugation Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Conjunctive Normal Form . . . . . . . . . . . . . . . . . . . . . . . . .45 Cross-Transposition . . . . . . . . . . . . . . . . . . . . . . . . . 47 - 50 curl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 - 37, 62 - 63 Curly . . . . . . . . . . . . . . . . . . . . . . . . . . 174 - 180, 206 - 207 cyclic distributivity . . . . . . . . . . . . . . . . . . . . . . . . . 168 - 173 De Morgan laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41 Dedekind Splice . . . . . . . . . . . . . . . . . . . . . . . . . . 104 - 105 dense ...................................105 Derivative Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 derivative laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Dialectic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190, 192
diamond computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Index
251
diamond laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 diamond types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 differential . . . . . . . . . . . . . . . . 29, 44 - 45, 48, 51, 130 - 140, 157 Differential Normal Forms . . . . . . . . . . . . . . . . . . . . . . . 48,51 diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152- 162 Diffraction Circuits . . . . . . . . . . . . . . . . . . . . . . . 155 - 157 diffraction functions . . . . . . . . . . . . . . . . . . . . . . . . . 147, 152 dihedral conjugation . . . . . . . . . . . . . . . . . . . . . . . . 123 - 125 Dilemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 - 209, 212 dilemma chess . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 - 243 dilemma diamond . . . . . . . . . . . . . . . . . . . . . . . . . . 200 - 201 dilemma metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 dilemma Tic-Tac-Toe . . . . . . . . . . . . . . . . . . . . . . . . 198 - 199 direct distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 Disjunctive Normal Form . . . . . . . . . . . . . . . . . . . . . . . . . .45 Doubt ...................................185 Doubt-Shame Dilemma . . . . . . . . . . . . . . . . . . . . . . . . . . 192 dual rail circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 - 32 duck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73, 230 ducks ....................................74 embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110-114 entangled paradox logics . . . . . . . . . . . . . . . . . . . . . . . . . 173 equivalence glitch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 eye in the pyramid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 feasibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
252 Diamond , A Paradox Logic Finitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18 - 20, 222 fixedpoint lattices . . . . . . . . . . . . ...... . . . . . . . .. 67 - 82 fixedpoint
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 55, 60, 82
fool's gold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5, 96 general Leibnitz rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 general quadratic distribution . . . . . . . . . . . . . . . . . . . . . . . 141 General Quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Generous Offer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 glitches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177- 181 Gadel game . . . . . . . . . . . . . . . . . . . . . . . . 192- 193 GOdelian Quanta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Gold ...................................196 Grelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7, 96 Grinch . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 - 221 Grinchian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 Halting theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 harmonic functions . . . . . . . . . . . . . . . . . . . . . . . . . . 28 - 30 harmonic projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Harmonic Self-Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 158 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41,43 implication glitch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83, 90, 213 indefinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Interference . . . . . . . . . . . . . . . . . . . . . . . . 37, 39, 41 - 42, 232 intermix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135,138,153
Index
253
Iron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 Junction Diamond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Kenneth Arrow's Impossibility Theorem . . . . . . . . . . . . . . 174 - 176 Klein group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Larry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 - 180, 207 Late fixedpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 lattice laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 left and right side . . . . . . . . . . . . . . . . . . . . . . . . . . 119 -120 Leibnitz rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44, 141 limit fixedpoints
. . . . . . . . . . . . . . . . . . . . . . . . . . 88 - 89
limited chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 - 86 Lob game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192- 193 lower differential . . . . . . . . . . . . . . . . . . . . . . . . . . 29, 130 Lower Differential Santa . . . . . . . . . . . . . . . . . . . . . . . . . . 68 M3 .....................................78 majority . . . . . . . . . . . . . . . . . . . . . . . . . . 29, 35, 43, 175 majority laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 mark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 - 35 Math for Mortals . . . . . . . . . . . . . . . . . . . . . . . . . . 108, 234 Meta-Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 - 189 Modulation . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . . 43 modus ponens breakdown . . . . . . . . . . . . . . . . . . . . . . . . . 180 Moe . . . . . . . . . . . . . . . . . . . . . . . . . . 174 - 180, 206 - 207 Mutual Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
254 Diamond , A Paradox Logic NS ......................................79 negation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 non-Aristotelian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
non-distributive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 non-modular . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 non-zero-sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195, 205 nonassociativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 nondistributivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 norm ...................................202 Normal Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45 null quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Occultation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 7, 47 oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Outer Fixedpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 overdetermined . . . . . . . . . . . . . . . . . . . . . . . . . . 27, 36, 223 paraharmonic . . . . . . . . . . . . . . . . . . . . . . . . . . 127, 130, 211 Parallellism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Parenthetical Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Pavlov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 permeable forms . . . . . . . . . . . . . . . . . . . . . . . 145, 148 - 149 permuted lattices . . . . . . . . . . . . . . . . . . . . . . . 165 - 166, 237 Perpendicular Processing . . . . . . . . . . . . . . . . . . . . . . . . . 127 phase order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 phase recombination . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 phase separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Index
255
phased delay circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . .31 phased distribution over the positives . . . . . . . . . . . . . . . . . . . 146 Polarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Pride ................................... 186 primary algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Primary Normal Forms . . . . . . . . . . . . . . . . . . . . . . . . . 46,49 Prisoner's Dilemma . . . . . . . . . . . . . . . . . . . . . . . . . 194 - 195 productio ex absurdo . . . . . . . . . . . . . . . . . . . . . . . . . 62 - 63 provability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 pseudomenon . . . . . . . . . . . . . . . . . . . . . . . . . . .3-5,93,99 Quadratic Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Quadrature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Quadrature in k terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 Quadrature in k terms, and n dimensions . . . . . . . . . . . . . . . . . 142 Quadrature of Majority . . . . . . . . . . . . . . . . . . . . . . . . . . 145 Quine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7,97 rabbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 5 , 2 3 0 re-entrance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 - 55 Recall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 - 43 Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 relative lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Relocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 S3 Conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
256 Diamond , A Paradox Logic seeds . . . . . . . . . . . . . . . . . . . 60 - 61, 67, 88 - 89, 91, 228 - 229 Self-Reference Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 61 semi-countable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 semi-distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 Shame ..................................185 Shared Fixedpoints . .. . . . . . ... . . .. . . . . . . . . . . . . . . 70 Silver . . . . . . . ... . . . . . . . . . . . . . . . . . . . . . . . . . . 196 snarked . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 star logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 - 127 Stooges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 Strange Loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 strong majority . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 strong or . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Tertullian quanta . . . . . . . . . . . . . . . . . . . . . . . . . . 239 - 240 topology . . . . . . . . . . . . . . . . . . . . . . . . . . . 105, 110-111 transitive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57,105 Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Transparency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Transposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4, 3 7 triplet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78, 23 0 truce . . . . . . . . . . . . . . . . . . .193, 195, 198 - 203, 206, 242 - 243 truck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73,230 Trust ...................................186 Trust-Pride Dilemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
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two ducks in a box . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 type order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 type theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 uncurl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 5 - 3 7, 63 underdetermined . . . . . . . . . . . . . . . . . . . . . . . . . 27, 36, 223 Unexpected Departure . . . . . . . . . . . . . . . . . . . . . . . . . . 209 Unexpected Exam . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 upper differential . . . . . . . . . . . . . . . . . . . . . . . . . . . 29, 130 Upper Differential Grinch . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Valuation Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 void . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 - 35 Voter's Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 - 182 wave-bracketing fixedpoints . . . . . . . . . . . . . . . . . . . . . . . . 90 weak and .................................177 weak majority . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 well-founded . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 yang ...................................223 yin ....................................223 Zeno's Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
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