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If one assumes that Boolean logic applies to this sentence, then its mere existence would imply the existence of Santa Claus! 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: 1D. Santa and the Grinch 11 S is either true or false. No problem. If S is false, then R is non-boolean. If that statement is boolean, then Santa Claus exists. One could presumably tell Barber-like stories about Santa sets. One fine day someone asked: does the barber shave himself?

Has it been weekends there ever since? One fine day someone asked: who watches the watchmen? Does fortune smile on that village? Did the Cold War end itself? Does money talk? One fine day someone asked the priest: Does God bless God? Is there peace? Recall that Epimenides the Cretan accused all Cretans of being liars, including himself.

Epimenides: All Cretans are liars, and I am a liar. Epimenides: All Cretans are liars, including myself.

See a Problem?

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. Therefore S is false. Therefore Santa Claus does not exist! Here is another argument, one which confronts the paradox directly: S is either true or false. If S is true, then G is non-boolean.

Therefore; if G is boolean, then S is false. If Santa Claus does exist after all, then the Grinch is exposed as a Liar! The Grinch sets suggest Grinch stories. Does the Weekend Barber shave himself? Antistrephon 17 E. 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. How should the judge rule? Therefore Euathius wins the suit if and only if he loses it; ditto for Protagoras. Parity of Infinity What is the parity of infinity? Is infinity odd or even? Infinity has paradoxical parity. We encounter this paradox when we try to define the limit of an infinite oscillation. Lim xn equals what? If we cannot define this limit, in what sense does infinity have a parity at all?

And if no parity, why other arithmetical properties? We can illuminate the Parity of Infinity paradox with a fictional lamp; the Thompson Lamp, capable of infinitely making many powertoggles in a finite time. The Thompson Lamp clicks on for one minute, then off for a half-minute, then back on for a quarter-minute; then off for an eighth-minute; and so on, in geometrically decreasing intervals until the limit at two minutes, at which point the Lamp stops clicking.

After the second minute, is the Lamp on or off? The Heap 19 G.


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The Heap Surely one sand grain does not make 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 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? 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; the first uninteresting number. How interesting! Thus we find a contradiction; and this seems to imply that there are no uninteresting numbers! What then becomes of the above argument? Simply this; that the smallest boring number is inherently paradoxical. If this defines a number, then we have done so in 19 syllables!

So this defines a number if and only if it does not. These paradoxes connect to the paradox of the Heap by simple psychology. If, for some mad reason, you actually did try to count the number of grains in a sand heap, then you will eventually get bored with such an absurd task. Your attention would wander; you would lose track of all those sand grains; errors would accumulate, and the number would become indefinite.

The Heap arises at the onset of uncertainty. In practice, the Heap contains a boring number of sand grains; and the smallest Heap contains the smallest boring number of sand grains! Finitude H. 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. You may replace it with any finite expression.

Is Finity finite? 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 is just short of infinity! It is infinity seen from underneath. Then Finitude is the smallest large number; that is, the smallest number bigger than any interesting number. The Heap is the lower limit of boredom; Finitude is the upper limit of interest. We have these inequalities: small interesting numbers 1H.

Let WF be the set containing all well-founded sets. Is WF wellfounded? 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. A set is clear-founded if it has only interesting depth.

Is the class of clear-founded sets clear-founded? And just as infinity has paradoxical parity, so do Finitude and the Heap. Is the first boring number odd or even? Is the last interesting number? Is Hypergame short? If Hypergame is short, then the first move in Hypergame can be too — 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 definable in less than twenty syllables. Dear reader, allow me to dramatize this paradox by means of a fictional story about a mythical being. The Mortal has a choice of dooms. Is the Mortal doomed?

Is Normalcy normal? Normalcy is normal if and only if it is abnormal! Now consider this: The Rebel is a being who must become one who changes. Can the Rebel remain a Rebel? D11 D12 D13 D D21 D22 D23 D D31 D32 D33 D D41 D42 D43 D Thus a single buzzing bit implies infinities beyond infinities! Was more ever made from less?

Why not ask for Santa Claus? If the pivot bit is boolean, then Santa Claus exists! Paradox of the Boundary K. 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. Is dawn day or night? Is dusk? Which hemisphere is the equator on?

Which longitude are the poles at? Which country owns the border? Is zero plus or minus? 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! The Buzzer Chapter 1 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! Just wire together a spring-loaded relay, a switch, and a battery, using this childishly simple circuit: 29 30 Diamond 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 spring-loaded circuit will re-connect, and current will flow.

Therefore the relay is open if and only if it is closed. To find out, close the switch. What do you see? 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.

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. Diamond Values B. There are four such logic waves: t t t t t t …. It describes the logic waves of period 2. Thus paradox is possible in diamond logic. This, then, is Diamond; a logic containing the boolean values, plus paradoxes and lattice operators.

Gluts and Gaps D. Thus, an underdetermined statement is neither provable nor refutable, and an overdetermined statement is both provable and refutable. Diamond harmonizes with meta-mathematics. 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! The gates then are: 2E. Brownian Forms Make a mark. This act generates a form: A mark marks a space. Each mark is a call; to recall is to call.

A mark is a crossing, between marked and unmarked space. Each mark is a crossing; to recross is not to cross. The usual matching is: 42 Diamond There is a complementary interpretation: 2F. Brownian Forms 43 The standard interpretation is usually preferred because it has a simpler implication operator.

This exploits the symmetries of the diamond. These three axioms suffice to calculate all the truth tables, and all the algebraic identities, of diamond. Boundary Logic Boundary logic is Brownian form algebra, adapted for the typewriter. We can identify 6 with i, and 9 with j. Or vice versa. Inspection of tables shows that juxtaposition ab is isomorphic to diamond disjunction; and crossing [a] is isomorphic to diamond negation. These equations are implicit in the bracket notation. Brackets distinguish only inside from outside, not left from right. From the bracket axioms we can derive theorems: Reflexion.

Directly from Occultation. Bracket Algebra Fixity. The bracket axioms yield the bracket arithmetic. Modified Generation. Inverse Transposition. Modified Transposition. Bracket Algebra 53 Retransposition 3 terms. Proof is by induction on n. General Cross-Transposition. From right to left. Let M x, y, z denote [[xy][yz][zx]], or [[x][y]][[y][z]][[z][x]]. We can derive these theorems: Transmission. General Distribution. Bracket Algebra Coalition. General Associativity. These are the boolean laws, minus the Law of the Excluded Middle.

In Venn diagram terms, dx is the boundary of x, and Dx is everything else. We do this by distributing negations downwards, canceling double-negations, and distributing enough times. These normal forms are just like their counterparts in boolean logic, except that they allow differential terms.

Normal Forms 65 Theorem. 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 andf. Second proof. QED 3C. This separates the function into boolean and lattice components. It expresses the function in terms of its values. QED 3D. Completeness D. Any equational identity in diamond can be deduced from the diamond laws.

By induction on the number of variables. Initial step. Induction step. Let F x be F considered as an expression in its Nth variable x. This concludes the induction proof. Therefore any equational QED identity in diamond is provable from the diamond laws. 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?

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POLITICAL OPINIONS #61- The Diamond-Water Paradox. An Economic Puzzle.

Students of feasibility will recognize this as a variant of the Boolean Consistency Problem, and therefore NP-complete. That can be represented via re-entrance, thus: 71 72 Self-Reference 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: 4A. Re-Entrance and Fixedpoints 73 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: 74 Self-Reference B. Thus X min Y is the rightmost element left of both X and Y. Phase Order 75 Theorem. QED Theorem. This follows by induction from the previous two results. By lattice properties. We get the other half of the theorem the same way. QED 4B. For N components, this implies at most 2N steps in an ordered chain before it stops moving.

A vector is a fixedpoint if and only if it is both a left seed and a right seed. 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. This is the greatest fixedpoint left of a. Left seeds grow leftwards towards fixedpoints. All fixedpoints are both left and right seeds — of themselves. The Outer Fixedpoints 79 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.

Recall that all harmonic functions preserve order. Diamond logic begins where boolean logic ends. To see productio ex absurdo in action, consider this system: Iterate this system from curl: 4C. Thus the outer fixedpoints are the only ones. 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. Can the process take all of 2n steps? Relative Lattices Any harmonic function F x has the outer fixedpoints: Fn i , Fn j , the leftmost and rightmost fixedpoints. But often this is not all. In general, F has an entire lattice of fixedpoints. Therefore F2n a min b is a fixedpoint left of a and of b, and is moreover the rightmost such fixedpoint.

An Upper Differential Grinch! The minimum of two left seeds is a left seed. QED Since all fixedpoints are seeds, their minima are left seeds. The minimum of left seeds generates the minimum of the fixedpoints in the relative lattice: a min b generates f 2n a minf f 2n b , if a and b are left seeds. QED In summary: the minimum of left seeds is a left seed, one which generates the relative minimum of the generated fixedpoints. And dually: the maximum of right seeds is a right seed, which generates the relative maximum of the generated fixedpoints. The spiral coils leftwards until it reaches a limit cycle.

Spiral Theorem. The minimum of a left spiral coil is a left seed. Of course there are similar results for right spiral coils, etc. Shared Fixedpoints C. For instance: Theorem. F sends LG to itself. Therefore F is an order-preserving function from LG to itself. Its least element is F2n G2n i ; its greatest element is F2n G2n j ; and its relative minimum operator is F2n G2n a min b. These results can be extended to N functions: If F1 , F2 ,. Examples D. It is a theorem of lattice theory that any non-distributive modular lattice contains M3 as a sublattice.

It is a theorem of lattice theory that any non-distributive nonmodular lattice contains N5 as a sublattice. A Self and Buzzer Santa! The buzzer c seals itself off from the buzzer a. A Two-Buzzers Santa! Without those points, this lattice would be modular; but with them it contains N5. Limits Diamond logic is continuous; it defines limit operators.

Obviously these are deeply implicated in the Paradox of Finitude. This is true because Inf and Cof have that property. Inf and Cof are about the long run, not about the beginning. QED 6A. These inequalities can be strict. Limit Fixedpoints Fixedpoints can be found by transfinite induction on the limit operators.

If F has only finitely many components, then the descending sequence can only descend finitely many steps before coming to rest. If F has infinitely many components, then we must continue the iteration through more limits. And so on through the higher ordinals.

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. These are the left and right fixedpoints generated from s0 by iterating F twice-infinity times. We can regard each of these as the output of a computation process whose input is s0 and whose program is F. This is the rightmost fixedpoint left of cofinitely many Fi s0. This is the leftmost fixedpoint right of cofinitely many Fi s0.

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. Well, yes it can be; for as you can see, yes it is! Yes but no. Or: no but yes. 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. But who watches the watchmen?

Answer: they shall watch each other, but not themselves. Does God worship himself? Answer: not this, not that. A mystery! Humbling moments like these are part of growing up. How generous! If there is no Santa Claus, then the above are all paradoxes. Above I told Barber-like stories about Santa sets. Who watches the watchmen? Santa and the Grinch Did the Cold War end itself? Does God bless God? How logical! Promenides sounds logical; but his statement still leaves open the possibility that every Cretan is a liar, including Promenides.

Antistrephon 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.

Antistrephon 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. Stranger still: either Protagoras won and lost, or Euathius won and lost! Is it odd or even? Evidently it is in the nature of infinity to blur some details. This should come as no surprise; infinity is notoriously paradoxical. Even approaching infinity yields paradox. 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. In standard decimal nomenclature, that would be , However, other naming schemes might name , in fewer than 20 syllables. As ever, uncertainty reigns.

If you were to pile together , grains of sand, each 1 mm wide, then they will form a conical pile approximately 9. Infinity, Finitude and the Heap 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. Other naming schemes yield even greater numbers. The answer is that Hypergame is Finitude in disguise.

Presumably the Mortal lives until the last interesting moment, then dies of boredom. Normalcy is normal if and only if it is not. So is Normalcy normal? Presumably Rebels play at Normalcy. From this single buzzing bit Cantor deduces the existence of an infinity beyond infinity of real numbers! The continuum is intermediate! Which country owns the boundary?

Is zero positive or negative? Null Quotients The null quotients are the result of division by zero. Infinity leads us to an obvious absurdity. Indefinity leads us to a vague tautology. As noted in Chapter 2, this connects us to diamond logic; for we can identify i with one, and j with the other. Note also the similarity of this graph to the Dedekind splice. This number is so fraught with mathematical significance that it forces us to postulate a transfinite infinity of infinities; so surely it must, within itself, contain a transfinite amount of information about all those infinities.

We know that C has a buzz-bit. What does that bit mean? And how many such bits are there? Only one? A finite number? Infinitely many? Cofinitely many? This is paradox as intermediate; a convergent compromise. This is paradox as uncertainty; a divergent blur. This implies higher-order carries:. C does not have I bits in cofinitely many digits; for an infinity of reals are all boolean; so the anti-diagonal has infinitely many boolean digits. C is uncleared!

This endless motion between 0 and 1 is represented by I. The number 0. Three I digits indicate eight numbers; N represent 2N numbers; and a dispersed infinity of I digits represent a Cantor dust of real numbers. So in this interpretation, C is a strange attractor; and its I bits tell where the other strange attractors are. The silly thing was bluffing us! A slightly subtler logic yields an infinitely simpler model. This is known as elegance; sign of a correct theory.

Therefore the diagonal of the list is also dyadic; it ends in an infinite or I therefore propose this counter-Cantorian axiom: Diagonal Limit. The diagonal of any complete list of boolean reals has a boolean limit. The Line within the Diamond E. Next note that E is continuous; for each of its components is continuous. The inverse of E is continuous. A function is continuous if the inverse image of an open set is an open set. The Line within the Diamond Proof. Let F x be a continuous function from R to diamond. Call the first set A and the second set B. These are the characteristic functions for A and B; made strictly from the Dedekind splice and diamond logic.

But F is a continuous function; so it equals t in the interior of A, f in the interior of B, and i at the boundary. The Line within the Diamond Theorem. QED Thus the real continuum embeds and extends into the space of diamond vectors. The continuum reduces to harmonic form. But every harmonic function on diamond space has a fixedpoint. 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. Fuzzy Chaos G. As the previous section demonstrated, continuous real functions like these can be embedded into diamond space via the Dedekind splice. This has an attracting fixedpoint at 0, and a repelling fixedpoint at 1. This has an attracting fixedpoint at 1, and a repelling fixedpoint at 0.

This has an attracting fixedpoint at 0. This is none other than the logistic map, most studied of all chaotic dynamical systems. A complex version of this map yields the Mandelbrot set. This system has a limit attractor in the form of a loop:. It has a fractal attractor. My thanks to Lou Kauffman for the following printout. In boolean logic, that means that R does not exist; which implies unspecified restrictions on set-building. By accepting paradox, we defuse the Russell set. In diamond logic, any property of sets is itself a set; even properties defined in terms of themselves.

Diamond-valued sets are different enough from boolean sets that I give them another name; cliques. All cliques are paradox elements for S. The empty clique is the falsehood predicate. The Liar clique contains all cliques not in the Liar clique. Cliques In general, any property of cliques defined as a harmonic function of the epsilon relation is also a clique. That is, if H a, b, c,. If the first statement is boolean, then Santa Claus exists. If the second statement is boolean, then Santa Claus does not exist.

Either way, one of these sentences is a Liar. The Groucho clique is the negation of its description. It contains its containers. Therefore SD is like an ultrafilter on the clique world. Therefore the descriptions of the cliques have the same epsilon relations as the cliques. Des is an epsilon—morphism. Clique Equality The preceding was about the Weak Clique world, defined by harmonic self-reference on epsilon. But what of the equality property? Also, shall we define equality extensionally — that is, in terms of clique elements—or intensionally—that is, in terms of clique containers?

In boolean set theory, they are identical; sets with the same elements are the same; but not so in clique theory. Therefore extensional equality fails for cliques; cliques can have the same elements yet still differ. In clique theory, equality is intensional; cliques are the same if they have the same containers. Yes, we can! To construct a clique world, start with approximations for epsilon and equality.

Therefore once cliques separate from each other, they remain separated. Since terms can only become less equal, and never more equal, this iteration must stop sometime before every term becomes different from every other term. Cliques are defined simultaneously, in terms of each other, with default unity of equality and default paradox containment.

You merely define it as a property of cliques, and its elements and properties grow organically with the rest of the clique world. The Zermelo—Frankel boolean set universe is like a tower; the clique world is like a web. In general, cliques given isomorphic definitions never differentiate, and so remain equal. Note that the differentiation term has no effect after differentiation; it is as if it is no longer there. Nonetheless the division it caused persists. A and B are unequal because they are unequal; a self-perpetuating separation that I call arbitrary difference.

Of course we can make quadruplets and up. It posits epsilon as based on paradoxical self-reference. Clique theory fulfills some, but not all, of the axioms of Zermelo—Frankel set theory. It satisfies the limited comprehension, null set, pairs, unions, replacement, and infinity axioms; but not choice, foundation, or extensionality. And unlike the ZF sets, cliques include the world clique, negations, and iffs. The Choice axiom also fails.

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In ZF set theory, the set universe is not a set itself. In general any harmonic function of cliques is a clique. Clique Axioms Define a logomorphism as a function from the clique terms to the clique terms that preserve definitions. But since their definitions are equivalent, A and B never differentiate, and they remain equal. That is a special case of another clique theory axiom: Logomorphic Equality The only logomorphism is the identity. Graph Cliques Because you can create a clique merely by invoking it, any finite graph can be made into a clique.

The differentiation term is there to ensure separation. You can of course define longer cycles. Graph Cliques Infinite graphs also define graph cliques, given clear definitions. For instance we can define the order type of Z in cliques:. There are even cliques with the order type of R; continuous cliques. Clique Circuits Starting from a well-founded set, no infinite descending element chains exist. This suggests a game; Clique Nim. In Clique Nim, the first player chooses an element of C; call that C1. The second player chooses an element of C1 ; call that C2.

And so on. If the chain stops, then the player who cannot choose further loses. Let U be the clique of all cliques for which the first player has a winning strategy; let V be the clique of all cliques for which the second player has a winning strategy. If the cliques a, b, c, d,. These are the clique circuits. These inclusions are somewhat symmetrical; the definition would require differentiation terms. Analytic Functions If a function is harmonic, then it preserves order; but not all functions preserve lattice order, so not all function are harmonic.

Function Types B. Minus does not preserve lattice order; so no analytic function does either. The converse is also true: If F does not preserve order, then F is analytic. QED If F is harmonic, then it preserves order. The converse is also true: If F preserves order, then F is harmonic.

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. Dihedral Conjugation C. These are F and R conjugated by P. Conjugation Theorems. 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. D permutes functions and relations as well as elements, by conjugation. Star Logic Most of the best properties of diamond are shared by 3-valued logic, a simpler sublogic; T, I and F.

Or equivalently; T, J and F. So what is the second paradox value doing? Star reverses order. It exchanges i and j, leaving t and f fixed. In dual-rail circuits star equals swap wires. 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. Clique theory. Diamond A The Buzzer. Diamond Algebra A Laws. SelfReference A Reentrance and Fixedpoints. Fixedpoint Lattices A Relative Lattices.

A The Liar and the AntiDiagonal. E Zenos Theorem. Harmonic Analysis A Harmonic Projection. Threelogic A Ternary Logic Embeds. Metamathematics A Godelian Quanta. Dilemma A Prisoners Dilemma. Speculations A Diamond Types?