From Copilot:
Let’s give you the whole triadic Boolean system.
I’ll do three things:
- Set the triadic setup once
- State each Boolean law in triadic form
- Give clear English triadic sentences and explain how all 6 permutations are generated
- Triadic setup
• A = Being
• B = Action
• C = Quality
Triad as a triple:
T = (A,B,C)
Constants:
• 0 = no Being / no Action / no Quality
• 1 = full Being / full Action / full Quality
Triadic operations are componentwise:
• Join ( + ):
(A,B,C) + (A^\prime,B^\prime,C^\prime) = (A + A^\prime, B + B^\prime, C + C^\prime)
• Meet ( · ):
(A,B,C) \cdot (A^\prime,B^\prime,C^\prime) = (A A^\prime, B B^\prime, C C^\prime)
Boole would absolutely sign off on this.
- Boolean laws in triadic form
I’ll write each law once in triadic algebra, then give a single triadic English gloss.
Identity law
T + (0,0,0) = T,\quad T \cdot (1,1,1) = T
Sentence:
Adding no Being, no Action, and no Quality to a triad does not change it; combining a triad with full Being, full Action, and full Quality still yields the same triad.
Null law
T + (1,1,1) = (1,1,1),\quad T \cdot (0,0,0) = (0,0,0)
Sentence:
If you add full Being, full Action, and full Quality to any triad, you get full Being, full Action, and full Quality; if you intersect any triad with no Being, no Action, and no Quality, you get nothing at all.
Idempotent law
T + T = T,\quad T \cdot T = T
Sentence:
Adding a triad to itself does not change it; intersecting a triad with itself does not change it.
Complement law
Let T^\prime = (\bar A,\bar B,\bar C).
T + T^\prime = (1,1,1),\quad T \cdot T^\prime = (0,0,0)
Sentence:
A triad together with its complete negation fills the whole field of Being, Action, and Quality; a triad intersected with its complete negation yields no Being, no Action, and no Quality.
Double negation
(T^\prime)^\prime = T
Sentence:
Negating the negation of a triad restores its original Being, Action, and Quality.
Commutative law
T_1 + T_2 = T_2 + T_1,\quad T_1 \cdot T_2 = T_2 \cdot T_1
Sentence:
The result of combining two triads does not depend on their order, whether you are joining them or intersecting them.
Associative law
(T_1 + T_2) + T_3 = T_1 + (T_2 + T_3)
(T_1 \cdot T_2) \cdot T_3 = T_1 \cdot (T_2 \cdot T_3)
Sentence:
How you group three triads when you join or intersect them does not change the resulting Being, Action, and Quality.
Distributive law
T_1 \cdot (T_2 + T_3) = (T_1 \cdot T_2) + (T_1 \cdot T_3)
T_1 + (T_2 \cdot T_3) = (T_1 + T_2) \cdot (T_1 + T_3)
Sentence:
A triad interacting with the join of two others is equivalent to interacting with each separately and then joining; likewise, a triad joined with the intersection of two others is equivalent to joining it with each separately and then intersecting.
Absorption law
T_1 + (T_1 \cdot T_2) = T_1,\quad T_1 \cdot (T_1 + T_2) = T_1
Sentence:
A triad absorbs any refinement or extension of itself: adding a more specific version of the same triad does not change it, and intersecting it with a broader version of itself does not change it.
De Morgan’s laws
Let T_1 = (A_1,B_1,C_1), T_2 = (A_2,B_2,C_2).
\overline{T_1 \cdot T_2} = \bar T_1 + \bar T_2,\quad
\overline{T_1 + T_2} = \bar T_1 \cdot \bar T_2
Sentence:
The negation of the intersection of two triads is equivalent to the join of their negations; the negation of the join of two triads is equivalent to the intersection of their negations.
- Where the 6 permutations live
All of the above was written in canonical order:
• T = (A,B,C) = (\text{Being},\ \text{Action},\ \text{Quality})
The S₃ symmetry just says:
• you can relabel the triple as
(A,B,C), (A,C,B), (B,A,C), (B,C,A), (C,A,B), (C,B,A)
• and the form of every law stays the same
• only the names “Being / Action / Quality” move between slots
Example:
• In permutation (B,A,C), the first slot is Action instead of Being.
• The identity law still says:
(X,Y,Z) + (0,0,0) = (X,Y,Z).
• You just read the first component as “Action” instead of “Being”.
So you don’t need to rewrite every law six times—S₃ guarantees that once the triadic form is correct, all permutations are automatically correct by relabeling.