Appendix: Diamond Logic
Diamond logic has a new operator: /, called “juxtaposition”, pronounced “but”. This defines four truth values:
T/T = T
F/F = F ; these are the “Boolean” values.
T/F = I
F/T = J ; these are the “paradox” values.
In diamond logic, the positive operators Ù (“and”) and Ú (“or”) operate side-wise:
(a/b) Ù (c/d) = (a Ù c) / (b Ù d)
(a/b) Ú (c/d) = (a Ú c) / (b Ú d)
Negation flips sides:
~ (a/b) = (~ b) / (~ a)
This creates fixedpoints for negation:
~(t/f) = (~ f) / (~ t) = t/f
~(f/t) = (~ t) / (~ f) = f/t
Thus paradox is possible in diamond logic.
Call a function “harmonic” if it can be defined from Ù, Ú, ~, and the four values t, i, j, f. They include:
a nor b = ~ (a Ú b) = ~a Ù ~ b
a nand b = ~ (a Ù b) = ~a Ú ~ b
Here are diamond’s truth tables for ‘not’, ‘and’, ‘or’, and ‘but’:
x: ~ x: Ù y: Ú y: but y:
t f i j t f i j t f i j
t f t f i j t t t t t i i t
f t f f f f t f i j j f f j
i i i f i f t i i t t i i t
j j j f f j t j t j j f f j
Diamond logic has a Brownian-style notation: “bracket-form algebra”, which has four forms ( [], [[]], 6, and 9 ), and two operations (ab and [a]), with these rules:
ab = ba
[] a = []
[[]] a = a
a a = a
69 = []
[[[]]] = []
[6] = 6
[9] = 9
Bracket forms match diamond’s logic tables these ways:
[] :: T ; T ; F ; F
[[]] :: F ; F ; T ; T
6 :: I ; J ; I ; J
9 :: J ; I ; J ; I
ab :: aÚb ; aÚb ; aÙb ; aÙb
[[a][b]] :: aÙb ; aÙb ; aÚb ; aÚb
[a] :: ~ a ; ~a ; ~a ; ~a
Logic circuits can operate in diamond logic, via the “dual-rail” system. Replace each wire by a pair of wires, and each gate by a pair of gates. AND and OR gate-pairs operate separately on the sides; NOT, NOR and NAND gate-pairs swap sides after operating.
Juxtaposition defines the intermix function:
J(a,b) = ( a/b , b/a )
Note: J(J(a,b)) = (a,b)
~ J ( a , b ) = J ( ~ b , ~ a )
J(a,b) Ú J(A,B) = J(a Ú A, b Ú B)
J(a,b) Ù J(A,B) = J(a Ù A, b Ù B)
In terms of bracket forms:
[ (a/b) ] = [b] / [a]
(a/b)(c/d) = (ac)/(bd)
Diamond logic obeys the DeMorgan Laws: commutativity, associativity, identities, attractors and distribution for ‘and’ and ‘or’; double negation and the De Morgan identities for ‘not’.
But not the Law of the Excluded Middle:
(a Ù ~ a) = F ; (a Ú ~ a) = T
Instead diamond logic obeys Complementarity:
(I Ù ~ I) Ú (J Ù ~ J) = T ; (I Ú ~ I) Ù (J Ú ~ J) = F
Complementarity defies the Law of the Excluded Middle, but not directly enough to collapse logic to a single value. Instead it shares contradiction between two paradox values.
Theorem: Complete Deduction
The De Morgan laws, plus Complementarity, prove all identity equations in diamond logic.
Theorem: Complete Self-Reference
If F(x) is a n-component harmonic function on n variables:
F(x) = ( F1(x1,…,xn) , … , Fn(x1,…,xn) )
then F has a fixedpoint; that is, a vector s = (s1,…,sn) such that
F(s) = s
So any re-entrant form can be given a value. For instance, here is the “First Brownian Modulator”:
_________________________________________
___________________________________ |
_____________________________ | |
_______________________ | | |
_________________ | | | |
_____________ | | | | |
_________ | | | | | |
_____ | | | | | | |
| | | | | | | | | | | | | |
| Z |_|_|_| | | | Z |__|__|___| | | |
| | |_____|_|________| | | |
| | | |___________|________| |
| |_______|_____________| |
|_____________|_________________________|
It equals this bracket-form system:
a = [ zd ]
b = [ ag ]
c = [ db ] = z/2
d = [ cf ]
e = [ cz ]
f = [ eh ]
g = [ ah ]
h = [ eg ]
Let (A,E) = J(a,e) ; so that A = a/e, and E = e/a.
Also let (B,F) = J(b,f) ; (C,D) = J(c,d) ; (G,H) = J(g,h)
This is “diffracting the modulator against itself”.
Then: A = [ C z ]
E = [ D z ]
B = [ E H ]
F = [ A G ]
C = [ C F ]
D = [ D B ]
G = [ G E ]
H = [ H A ]
z/2 = C/D
Write this in a cycle to get a “Brownian Rotor”:
A = [ C z ]
H = [ A H ]
B = [ H E ]
D = [ B D ]
E = [ D z ]
G = [ E G ]
F = [ G A ]
C = [ F C ]
z/2 = C/D
See Illustrations 1, 2, 3.
Now consider Kauffman’s Modulator:
a = [ bdz ]
b = [ ae ]
c = [ df ]
d = [ acz ]
e = [ af ] = z/2
f = [ de ]
Let (A,D) = J(a,d) ; so that A = a/d, and D = d/a.
Also let (B,C) = J(b,c) ; (E,F) = J(e,f)
This too is diffracting the modulator against itself.
Then: A = [ A C z ]
D = [ D B z ]
B = [ D F ]
C = [ A E ]
E = [ D E ]
F = [ A F ]
z/2 = E/F
Write this in a cycle to get a “Kauffman Rotor”:
A = [ C A z ]
F = [ A F ]
B = [ F D ]
D = [ B D z ]
E = [ D E ]
C = [ E A ]
z/2 = E/F
See Illustrations 4, 5, 6.
Illustrations
1. Brownian Modulator
In this circuit diagram, the triangles denote NOR gates:
2. Brownian Rotor
This circuit diagram is dual-rail, for diamond logic. The triangles denote diamond NOR gates. J denotes an intermix gate. The table shows four stable states of the system, which are successive in a period-4 cycle, when the input z toggles between T and F four times.
z A H B D E G F C C/D
---------------------------------------------------------------------------
T F i i i F J J J F
F J J F i i i F J F
T F J J J F i i i T
F i i F J J J F i T
3. Brownian Rotor Rotating
3. Kauffman Modulator
4. Kauffman Rotor
z A F B D E C E/F
----------------------------------------------------------------
F i i F J J F F
T F i i F J J F
F J J F i i F T
T F J J F i i T
6. Kauffman Rotor Rotating
Bibliography
Nathaniel Hellerstein
“Diamond, a four-valued approach to the problem of Paradox”
U.C. Berkeley: Doctoral thesis, 1984.
Diamond, A Paradox Logic
Singapore: World Scientific Publishing Co. Ltd., 1997
Delta, A Paradox Logic
Singapore: World Scientific Publishing Co. Ltd., 1997
Diamond, A Paradox Logic, 2nd Edition
Singapore: World Scientific Publishing Co. Ltd., 2010
Louis Kauffman
“Knot Automata”
Boston: 24th International Symp. on Multiple Valued Logic, 1994
Louis Kauffman and Francisco Varela
“Form Dynamics”
Journal of Social and Biological Structure, 1980, v.3, pp.171-206.
George Spencer-Brown
Laws of Form
New York: the Julian Press, 1979.
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