User:Jon Awbrey/TABLE
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Table 1. Propositional Forms On Two Variables o---------o---------o---------o----------o------------------o----------o | L_1 | L_2 | L_3 | L_4 | L_5 | L_6 | | | | | | | | | Decimal | Binary | Vector | Cactus | English | Ordinary | o---------o---------o---------o----------o------------------o----------o | | x : 1 1 0 0 | | | | | | y : 1 0 1 0 | | | | o---------o---------o---------o----------o------------------o----------o | | | | | | | | f_0 | f_0000 | 0 0 0 0 | () | false | 0 | | | | | | | | | f_1 | f_0001 | 0 0 0 1 | (x)(y) | neither x nor y | ~x & ~y | | | | | | | | | f_2 | f_0010 | 0 0 1 0 | (x) y | y and not x | ~x & y | | | | | | | | | f_3 | f_0011 | 0 0 1 1 | (x) | not x | ~x | | | | | | | | | f_4 | f_0100 | 0 1 0 0 | x (y) | x and not y | x & ~y | | | | | | | | | f_5 | f_0101 | 0 1 0 1 | (y) | not y | ~y | | | | | | | | | f_6 | f_0110 | 0 1 1 0 | (x, y) | x not equal to y | x + y | | | | | | | | | f_7 | f_0111 | 0 1 1 1 | (x y) | not both x and y | ~x v ~y | | | | | | | | | f_8 | f_1000 | 1 0 0 0 | x y | x and y | x & y | | | | | | | | | f_9 | f_1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y | | | | | | | | | f_10 | f_1010 | 1 0 1 0 | y | y | y | | | | | | | | | f_11 | f_1011 | 1 0 1 1 | (x (y)) | not x without y | x => y | | | | | | | | | f_12 | f_1100 | 1 1 0 0 | x | x | x | | | | | | | | | f_13 | f_1101 | 1 1 0 1 | ((x) y) | not y without x | x <= y | | | | | | | | | f_14 | f_1110 | 1 1 1 0 | ((x)(y)) | x or y | x v y | | | | | | | | | f_15 | f_1111 | 1 1 1 1 | (()) | true | 1 | | | | | | | | o---------o---------o---------o----------o------------------o----------o
Table 2. Ef Expanded Over Ordinary Features {x, y}
o------o------------o------------o------------o------------o------------o
| | | | | | |
| | f | Ef | xy | Ef | x(y) | Ef | (x)y | Ef | (x)(y)|
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_0 | () | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_1 | (x)(y) | dx dy | dx (dy) | (dx) dy | (dx)(dy) |
| | | | | | |
| f_2 | (x) y | dx (dy) | dx dy | (dx)(dy) | (dx) dy |
| | | | | | |
| f_4 | x (y) | (dx) dy | (dx)(dy) | dx dy | dx (dy) |
| | | | | | |
| f_8 | x y | (dx)(dy) | (dx) dy | dx (dy) | dx dy |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_3 | (x) | dx | dx | (dx) | (dx) |
| | | | | | |
| f_12 | x | (dx) | (dx) | dx | dx |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_6 | (x, y) | (dx, dy) | ((dx, dy)) | ((dx, dy)) | (dx, dy) |
| | | | | | |
| f_9 | ((x, y)) | ((dx, dy)) | (dx, dy) | (dx, dy) | ((dx, dy)) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_5 | (y) | dy | (dy) | dy | (dy) |
| | | | | | |
| f_10 | y | (dy) | dy | (dy) | dy |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_7 | (x y) | ((dx)(dy)) | ((dx) dy) | (dx (dy)) | (dx dy) |
| | | | | | |
| f_11 | (x (y)) | ((dx) dy) | ((dx)(dy)) | (dx dy) | (dx (dy)) |
| | | | | | |
| f_13 | ((x) y) | (dx (dy)) | (dx dy) | ((dx)(dy)) | ((dx) dy) |
| | | | | | |
| f_14 | ((x)(y)) | (dx dy) | (dx (dy)) | ((dx) dy) | ((dx)(dy)) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_15 | (()) | (()) | (()) | (()) | (()) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
Table 3. Df Expanded Over Ordinary Features {x, y}
o------o------------o------------o------------o------------o------------o
| | | | | | |
| | f | Df | xy | Df | x(y) | Df | (x)y | Df | (x)(y)|
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_0 | () | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_1 | (x)(y) | dx dy | dx (dy) | (dx) dy | ((dx)(dy)) |
| | | | | | |
| f_2 | (x) y | dx (dy) | dx dy | ((dx)(dy)) | (dx) dy |
| | | | | | |
| f_4 | x (y) | (dx) dy | ((dx)(dy)) | dx dy | dx (dy) |
| | | | | | |
| f_8 | x y | ((dx)(dy)) | (dx) dy | dx (dy) | dx dy |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_3 | (x) | dx | dx | dx | dx |
| | | | | | |
| f_12 | x | dx | dx | dx | dx |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_6 | (x, y) | (dx, dy) | (dx, dy) | (dx, dy) | (dx, dy) |
| | | | | | |
| f_9 | ((x, y)) | (dx, dy) | (dx, dy) | (dx, dy) | (dx, dy) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_5 | (y) | dy | dy | dy | dy |
| | | | | | |
| f_10 | y | dy | dy | dy | dy |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_7 | (x y) | ((dx)(dy)) | (dx) dy | dx (dy) | dx dy |
| | | | | | |
| f_11 | (x (y)) | (dx) dy | ((dx)(dy)) | dx dy | dx (dy) |
| | | | | | |
| f_13 | ((x) y) | dx (dy) | dx dy | ((dx)(dy)) | (dx) dy |
| | | | | | |
| f_14 | ((x)(y)) | dx dy | dx (dy) | (dx) dy | ((dx)(dy)) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_15 | (()) | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
Table 4. Ef Expanded Over Differential Features {dx, dy}
o------o------------o------------o------------o------------o------------o
| | | | | | |
| | f | T_11 f | T_10 f | T_01 f | T_00 f |
| | | | | | |
| | | Ef| dx dy | Ef| dx(dy) | Ef| (dx)dy | Ef|(dx)(dy)|
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_0 | () | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_1 | (x)(y) | x y | x (y) | (x) y | (x)(y) |
| | | | | | |
| f_2 | (x) y | x (y) | x y | (x)(y) | (x) y |
| | | | | | |
| f_4 | x (y) | (x) y | (x)(y) | x y | x (y) |
| | | | | | |
| f_8 | x y | (x)(y) | (x) y | x (y) | x y |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_3 | (x) | x | x | (x) | (x) |
| | | | | | |
| f_12 | x | (x) | (x) | x | x |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_6 | (x, y) | (x, y) | ((x, y)) | ((x, y)) | (x, y) |
| | | | | | |
| f_9 | ((x, y)) | ((x, y)) | (x, y) | (x, y) | ((x, y)) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_5 | (y) | y | (y) | y | (y) |
| | | | | | |
| f_10 | y | (y) | y | (y) | y |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_7 | (x y) | ((x)(y)) | ((x) y) | (x (y)) | (x y) |
| | | | | | |
| f_11 | (x (y)) | ((x) y) | ((x)(y)) | (x y) | (x (y)) |
| | | | | | |
| f_13 | ((x) y) | (x (y)) | (x y) | ((x)(y)) | ((x) y) |
| | | | | | |
| f_14 | ((x)(y)) | (x y) | (x (y)) | ((x) y) | ((x)(y)) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_15 | (()) | (()) | (()) | (()) | (()) |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | |
| Fixed Point Total | 4 | 4 | 4 | 16 |
| | | | | |
o-------------------o------------o------------o------------o------------o
Table 5. Df Expanded Over Differential Features {dx, dy}
o------o------------o------------o------------o------------o------------o
| | | | | | |
| | f | Df| dx dy | Df| dx(dy) | Df| (dx)dy | Df|(dx)(dy)|
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_0 | () | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_1 | (x)(y) | ((x, y)) | (y) | (x) | () |
| | | | | | |
| f_2 | (x) y | (x, y) | y | (x) | () |
| | | | | | |
| f_4 | x (y) | (x, y) | (y) | x | () |
| | | | | | |
| f_8 | x y | ((x, y)) | y | x | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_3 | (x) | (()) | (()) | () | () |
| | | | | | |
| f_12 | x | (()) | (()) | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_6 | (x, y) | () | (()) | (()) | () |
| | | | | | |
| f_9 | ((x, y)) | () | (()) | (()) | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_5 | (y) | (()) | () | (()) | () |
| | | | | | |
| f_10 | y | (()) | () | (()) | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_7 | (x y) | ((x, y)) | y | x | () |
| | | | | | |
| f_11 | (x (y)) | (x, y) | (y) | x | () |
| | | | | | |
| f_13 | ((x) y) | (x, y) | y | (x) | () |
| | | | | | |
| f_14 | ((x)(y)) | ((x, y)) | (y) | (x) | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
| | | | | | |
| f_15 | (()) | () | () | () | () |
| | | | | | |
o------o------------o------------o------------o------------o------------o
Wiki Tables
| L1 | L2 | L3 | L4 | L5 | L6 |
|---|---|---|---|---|---|
| x : | 1 1 0 0 | ||||
| y : | 1 0 1 0 | ||||
| f0 | f0000 | 0 0 0 0 | ( ) | false | 0 |
| f1 | f0001 | 0 0 0 1 | (x)(y) | neither x nor y | ¬x ∧ ¬y |
| f2 | f0010 | 0 0 1 0 | (x) y | y and not x | ¬x ∧ y |
| f3 | f0011 | 0 0 1 1 | (x) | not x | ¬x |
| f4 | f0100 | 0 1 0 0 | x (y) | x and not y | x ∧ ¬y |
| f5 | f0101 | 0 1 0 1 | (y) | not y | ¬y |
| f6 | f0110 | 0 1 1 0 | (x, y) | x not equal to y | x ≠ y |
| f7 | f0111 | 0 1 1 1 | (x y) | not both x and y | ¬x ∨ ¬y |
| f8 | f1000 | 1 0 0 0 | x y | x and y | x ∧ y |
| f9 | f1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y |
| f10 | f1010 | 1 0 1 0 | y | y | y |
| f11 | f1011 | 1 0 1 1 | (x (y)) | not x without y | x → y |
| f12 | f1100 | 1 1 0 0 | x | x | x |
| f13 | f1101 | 1 1 0 1 | ((x) y) | not y without x | x ← y |
| f14 | f1110 | 1 1 1 0 | ((x)(y)) | x or y | x ∨ y |
| f15 | f1111 | 1 1 1 1 | (( )) | true | 1 |
Differential Logic and Dynamic Systems
Table 1. Syntax & Semantics of a Calculus for Propositional Logic
Table 1. Syntax & Semantics of a Calculus for Propositional Logic o-------------------o-------------------o-------------------o | Expression | Interpretation | Other Notations | o-------------------o-------------------o-------------------o | " " | True. | 1 | o-------------------o-------------------o-------------------o | () | False. | 0 | o-------------------o-------------------o-------------------o | A | A. | A | o-------------------o-------------------o-------------------o | (A) | Not A. | A' | | | | ~A | o-------------------o-------------------o-------------------o | A B C | A and B and C. | A & B & C | o-------------------o-------------------o-------------------o | ((A)(B)(C)) | A or B or C. | A v B v C | o-------------------o-------------------o-------------------o | (A (B)) | A implies B. | A => B | | | If A then B. | | o-------------------o-------------------o-------------------o | (A, B) | A not equal to B. | A =/= B | | | A exclusive-or B. | A + B | o-------------------o-------------------o-------------------o | ((A, B)) | A is equal to B. | A = B | | | A if & only if B. | A <=> B | o-------------------o-------------------o-------------------o | (A, B, C) | Just one of | A'B C v | | | A, B, C | A B'C v | | | is false. | A B C' | o-------------------o-------------------o-------------------o | ((A),(B),(C)) | Just one of | A B'C' v | | | A, B, C | A'B C' v | | | is true. | A'B'C | | | | | | | Partition all | | | | into A, B, C. | | o-------------------o-------------------o-------------------o | ((A, B), C) | Oddly many of | A + B + C | | (A, (B, C)) | A, B, C | | | | are true. | A B C v | | | | A B'C' v | | | | A'B C' v | | | | A'B'C | o-------------------o-------------------o-------------------o | (Q, (A),(B),(C)) | Partition Q | Q'A'B'C' v | | | into A, B, C. | Q A B'C' v | | | | Q A'B C' v | | | Genus Q comprises | Q A'B'C | | | species A, B, C. | | o-------------------o-------------------o-------------------o
| Expression | Interpretation | Other Notations |
|---|---|---|
| " " | True. | 1 |
| ( ) | False. | 0 |
| A | A. | A |
| (A) | Not A. | A’ ~A ¬A |
| A B C | A and B and C. | A ∧ B ∧ C |
| ((A)(B)(C)) | A or B or C. | A ∨ B ∨ C |
| (A (B)) | A implies B. If A then B. |
A ⇒ B |
| (A, B) | A not equal to B. A exclusive-or B. |
A ≠ B A + B |
| ((A, B)) | A is equal to B. A if & only if B. |
A = B A ⇔ B |
| (A, B, C) | Just one of A, B, C is false. |
A’B C ∨ |
| ((A),(B),(C)) | Just one of A, B, C is true. Partition all |
A B’C’ ∨ |
| ((A, B), C) (A, (B, C)) |
Oddly many of A, B, C are true. |
A + B + C |
| (Q, (A),(B),(C)) | Partition Q into A, B, C. Genus Q comprises |
Q’A’B’C’ ∨ |
Table 2. Fundamental Notations for Propositional Calculus
Table 2. Fundamental Notations for Propositional Calculus
o---------o-------------------o-------------------o-------------------o
| Symbol | Notation | Description | Type |
o---------o-------------------o-------------------o-------------------o
| !A! | {a_1, ..., a_n} | Alphabet | [n] = #n# |
o---------o-------------------o-------------------o-------------------o
| A_i | {(a_i), a_i} | Dimension i | B |
o---------o-------------------o-------------------o-------------------o
| A | <|!A!|> | Set of cells, | B^n |
| | <|a_i, ..., a_n|> | coordinate tuples,| |
| | {<a_i, ..., a_n>} | interpretations, | |
| | A_1 x ... x A_n | points, or vectors| |
| | Prod_i A_i | in the universe | |
o---------o-------------------o-------------------o-------------------o
| A* | (hom : A -> B) | Linear functions | (B^n)* = B^n |
o---------o-------------------o-------------------o-------------------o
| A^ | (A -> B) | Boolean functions | B^n -> B |
o---------o-------------------o-------------------o-------------------o
| A% | [!A!] | Universe of Disc. | (B^n, (B^n -> B)) |
| | (A, A^) | based on features | (B^n +-> B) |
| | (A +-> B) | {a_1, ..., a_n} | [B^n] |
| | (A, (A -> B)) | | |
| | [a_1, ..., a_n] | | |
o---------o-------------------o-------------------o-------------------o
| Symbol | Notation | Description | Type |
|---|---|---|---|
| A | {a1, …, an} | Alphabet | [n] = n |
| Ai | {(ai), ai} | Dimension i | B |
| A |
〈A〉 |
Set of cells, |
Bn |
| A* | (hom : A → B) | Linear functions | (Bn)* = Bn |
| A^ | (A → B) | Boolean functions | Bn → B |
| A• |
[A] |
Universe of discourse |
(Bn, (Bn → B)) |
Table 3. Analogy of Real and Boolean Types
Table 3. Analogy of Real and Boolean Types o-------------------------o-------------------------o-------------------------o | Real Domain R | <-> | Boolean Domain B | o-------------------------o-------------------------o-------------------------o | R^n | Basic Space | B^n | o-------------------------o-------------------------o-------------------------o | R^n -> R | Function Space | B^n -> B | o-------------------------o-------------------------o-------------------------o | (R^n -> R) -> R | Tangent Vector | (B^n -> B) -> B | o-------------------------o-------------------------o-------------------------o | R^n -> ((R^n -> R) -> R)| Vector Field | B^n -> ((B^n -> B) -> B)| o-------------------------o-------------------------o-------------------------o | (R^n x (R^n -> R)) -> R | ditto | (B^n x (B^n -> B)) -> B | o-------------------------o-------------------------o-------------------------o | ((R^n -> R) x R^n) -> R | ditto | ((B^n -> B) x B^n) -> B | o-------------------------o-------------------------o-------------------------o | (R^n -> R) -> (R^n -> R)| Derivation | (B^n -> B) -> (B^n -> B)| o-------------------------o-------------------------o-------------------------o | R^n -> R^m | Basic Transformation | B^n -> B^m | o-------------------------o-------------------------o-------------------------o | (R^n -> R) -> (R^m -> R)| Function Transformation | (B^n -> B) -> (B^m -> B)| o-------------------------o-------------------------o-------------------------o | ... | ... | ... | o-------------------------o-------------------------o-------------------------o
| Real Domain R | ←→ | Boolean Domain B |
|---|---|---|
| Rn | Basic Space | Bn |
| Rn → R | Function Space | Bn → B |
| (Rn→R) → R | Tangent Vector | (Bn→B) → B |
| Rn → ((Rn→R)→R) | Vector Field | Bn → ((Bn→B)→B) |
| (Rn × (Rn→ R)) → R | ditto | (Bn × (Bn→ B)) → B |
| ((Rn→R) × Rn) → R | ditto | ((Bn→B) × Bn) → B |
| (Rn→R) → (Rn→R) | Derivation | (Bn→B) → (Bn→B) |
| Rn → Rm | Basic Transformation | Bn → Bm |
| (Rn→R) → (Rm→R) | Function Transformation | (Bn→B) → (Bm→B) |
| ... | ... | ... |
Table 4. An Equivalence Based on the Propositions as Types Analogy
Table 4. An Equivalence Based on the Propositions as Types Analogy o-------------------------o------------------------o--------------------------o | Pattern | Construction | Instance | o-------------------------o------------------------o--------------------------o | X -> (Y -> Z) | Vector Field | K^n -> ((K^n -> K) -> K) | o-------------------------o------------------------o--------------------------o | (X x Y) -> Z | | (K^n x (K^n -> K)) -> K | o-------------------------o------------------------o--------------------------o | (Y x X) -> Z | | ((K^n -> K) x K^n) -> K | o-------------------------o------------------------o--------------------------o | Y -> (X -> Z) | Derivation | (K^n -> K) -> (K^n -> K) | o-------------------------o------------------------o--------------------------o
| Pattern | Construction | Instance |
|---|---|---|
| X → (Y → Z) | Vector Field | Kn → ((Kn → K) → K) |
| (X × Y) → Z | (Kn × (Kn → K)) → K | |
| (Y × X) → Z | ((Kn → K) × Kn) → K | |
| Y → (X → Z) | Derivation | (Kn → K) → (Kn → K) |
Table 5. A Bridge Over Troubled Waters
Table 5. A Bridge Over Troubled Waters
o-------------------------o-------------------------o-------------------------o
| Linear Space | Liminal Space | Logical Space |
o-------------------------o-------------------------o-------------------------o
| | | |
| !X! | !`X`! | !A! |
| | | |
| {x_1, ..., x_n} | {`x`_1, ..., `x`_n} | {a_1, ..., a_n} |
| | | |
| cardinality n | cardinality n | cardinality n |
o-------------------------o-------------------------o-------------------------o
| | | |
| X_i | `X`_i | A_i |
| | | |
| <|x_i|> | {(`x`_i), `x`_i} | {(a_i), a_i} |
| | | |
| isomorphic to K | isomorphic to B | isomorphic to B |
o-------------------------o-------------------------o-------------------------o
| | | |
| X | `X` | A |
| | | |
| <|!X!|> | <|!`X`!|> | <|!A!|> |
| | | |
| <|x_1, ..., x_n|> | <|`x`_1, ..., `x`_n|> | <|a_1, ..., a_n|> |
| | | |
| {<x_1, ..., x_n>} | {<`x`_1, ..., `x`_n>} | {<a_1, ..., a_n>} |
| | | |
| X_1 x ... x X_n | `X`_1 x ... x `X`_n | A_1 x ... x A_n |
| | | |
| Prod_i X_i | Prod_i `X`_i | Prod_i A_i |
| | | |
| isomorphic to K^n | isomorphic to B^n | isomorphic to B^n |
o-------------------------o-------------------------o-------------------------o
| | | |
| X* | `X`* | A* |
| | | |
| (hom : X -> K) | (hom : `X` -> B) | (hom : A -> B) |
| | | |
| isomorphic to K^n | isomorphic to B^n | isomorphic to B^n |
o-------------------------o-------------------------o-------------------------o
| | | |
| X^ | `X`^ | A^ |
| | | |
| (X -> K) | (`X` -> B) | (A -> B) |
| | | |
| isomorphic to (K^n -> K)| isomorphic to (B^n -> B)| isomorphic to (B^n -> B)|
o-------------------------o-------------------------o-------------------------o
| | | |
| X% | `X`% | A% |
| | | |
| [!X!] | [!`X`!] | [!A!] |
| | | |
| [x_1, ..., x_n] | [`x`_1, ..., `x`_n] | [a_1, ..., a_n] |
| | | |
| (X, X^) | (`X`, `X`^) | (A, A^) |
| | | |
| (X +-> K) | (`X` +-> B) | (A +-> B) |
| | | |
| (X, (X -> K)) | (`X`, (`X` -> B)) | (A, (A -> B)) |
| | | |
| isomorphic to: | isomorphic to: | isomorphic to: |
| | | |
| (K^n, (K^n -> K)) | (B^n, (B^n -> B)) | (B^n, (B^n -> K)) |
| | | |
| (K^n +-> K) | (B^n +-> B) | (B^n +-> B) |
| | | |
| [K^n] | [B^n] | [B^n] |
o-------------------------o-------------------------o-------------------------o
| Linear Space | Liminal Space | Logical Space |
|---|---|---|
|
X |
X |
A |
|
Xi |
Xi |
Ai |
|
X |
X |
A |
|
X* |
X* |
A* |
|
X^ |
X^ |
A^ |
|
X• |
X• |
A• |
Table 6. Propositional Forms on One Variable
Table 6. Propositional Forms on One Variable o---------o---------o---------o----------o------------------o----------o | L_1 | L_2 | L_3 | L_4 | L_5 | L_6 | | | | | | | | | Decimal | Binary | Vector | Cactus | English | Ordinary | o---------o---------o---------o----------o------------------o----------o | | x : 1 0 | | | | o---------o---------o---------o----------o------------------o----------o | | | | | | | | f_0 | f_00 | 0 0 | ( ) | false | 0 | | | | | | | | | f_1 | f_01 | 0 1 | (x) | not x | ~x | | | | | | | | | f_2 | f_10 | 1 0 | x | x | x | | | | | | | | | f_3 | f_11 | 1 1 | (( )) | true | 1 | | | | | | | | o---------o---------o---------o----------o------------------o----------o
| L1 Decimal |
L2 Binary |
L3 Vector |
L4 Cactus |
L5 English |
L6 Ordinary |
|---|---|---|---|---|---|
| x : | 1 0 | ||||
| f0 | f00 | 0 0 | ( ) | false | 0 |
| f1 | f01 | 0 1 | (x) | not x | ~x |
| f2 | f10 | 1 0 | x | x | x |
| f3 | f11 | 1 1 | (( )) | true | 1 |
Table 7. Propositional Forms on Two Variables
Table 7. Propositional Forms on Two Variables o---------o---------o---------o----------o------------------o----------o | L_1 | L_2 | L_3 | L_4 | L_5 | L_6 | | | | | | | | | Decimal | Binary | Vector | Cactus | English | Ordinary | o---------o---------o---------o----------o------------------o----------o | | x : 1 1 0 0 | | | | | | y : 1 0 1 0 | | | | o---------o---------o---------o----------o------------------o----------o | | | | | | | | f_0 | f_0000 | 0 0 0 0 | () | false | 0 | | | | | | | | | f_1 | f_0001 | 0 0 0 1 | (x)(y) | neither x nor y | ~x & ~y | | | | | | | | | f_2 | f_0010 | 0 0 1 0 | (x) y | y and not x | ~x & y | | | | | | | | | f_3 | f_0011 | 0 0 1 1 | (x) | not x | ~x | | | | | | | | | f_4 | f_0100 | 0 1 0 0 | x (y) | x and not y | x & ~y | | | | | | | | | f_5 | f_0101 | 0 1 0 1 | (y) | not y | ~y | | | | | | | | | f_6 | f_0110 | 0 1 1 0 | (x, y) | x not equal to y | x + y | | | | | | | | | f_7 | f_0111 | 0 1 1 1 | (x y) | not both x and y | ~x v ~y | | | | | | | | | f_8 | f_1000 | 1 0 0 0 | x y | x and y | x & y | | | | | | | | | f_9 | f_1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y | | | | | | | | | f_10 | f_1010 | 1 0 1 0 | y | y | y | | | | | | | | | f_11 | f_1011 | 1 0 1 1 | (x (y)) | not x without y | x => y | | | | | | | | | f_12 | f_1100 | 1 1 0 0 | x | x | x | | | | | | | | | f_13 | f_1101 | 1 1 0 1 | ((x) y) | not y without x | x <= y | | | | | | | | | f_14 | f_1110 | 1 1 1 0 | ((x)(y)) | x or y | x v y | | | | | | | | | f_15 | f_1111 | 1 1 1 1 | (()) | true | 1 | | | | | | | | o---------o---------o---------o----------o------------------o----------o
| L1 Decimal |
L2 Binary |
L3 Vector |
L4 Cactus |
L5 English |
L6 Ordinary |
|---|---|---|---|---|---|
| x : | 1 1 0 0 | ||||
| y : | 1 0 1 0 | ||||
| f0 | f0000 | 0 0 0 0 | ( ) | false | 0 |
| f1 | f0001 | 0 0 0 1 | (x)(y) | neither x nor y | ¬x ∧ ¬y |
| f2 | f0010 | 0 0 1 0 | (x) y | y and not x | ¬x ∧ y |
| f3 | f0011 | 0 0 1 1 | (x) | not x | ¬x |
| f4 | f0100 | 0 1 0 0 | x (y) | x and not y | x ∧ ¬y |
| f5 | f0101 | 0 1 0 1 | (y) | not y | ¬y |
| f6 | f0110 | 0 1 1 0 | (x, y) | x not equal to y | x ≠ y |
| f7 | f0111 | 0 1 1 1 | (x y) | not both x and y | ¬x ∨ ¬y |
| f8 | f1000 | 1 0 0 0 | x y | x and y | x ∧ y |
| f9 | f1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y |
| f10 | f1010 | 1 0 1 0 | y | y | y |
| f11 | f1011 | 1 0 1 1 | (x (y)) | not x without y | x → y |
| f12 | f1100 | 1 1 0 0 | x | x | x |
| f13 | f1101 | 1 1 0 1 | ((x) y) | not y without x | x ← y |
| f14 | f1110 | 1 1 1 0 | ((x)(y)) | x or y | x ∨ y |
| f15 | f1111 | 1 1 1 1 | (( )) | true | 1 |
Table 8. Notation for the Differential Extension of Propositional Calculus
Table 8. Notation for the Differential Extension of Propositional Calculus
o---------o-------------------o-------------------o-------------------o
| Symbol | Notation | Description | Type |
o---------o-------------------o-------------------o-------------------o
| d!A! | {da_1, ..., da_n} | Alphabet of | [n] = #n# |
| | | differential | |
| | | features | |
o---------o-------------------o-------------------o-------------------o
| dA_i | {(da_i), da_i} | Differential | D |
| | | dimension i | |
o---------o-------------------o-------------------o-------------------o
| dA | <|d!A!|> | Tangent space | D^n |
| | <|da_i,...,da_n|> | at a point: | |
| | {<da_i,...,da_n>} | Set of changes, | |
| | dA_1 x ... x dA_n | motions, steps, | |
| | Prod_i dA_i | tangent vectors | |
| | | at a point | |
o---------o-------------------o-------------------o-------------------o
| dA* | (hom : dA -> B) | Linear functions | (D^n)* ~=~ D^n |
| | | on dA | |
o---------o-------------------o-------------------o-------------------o
| dA^ | (dA -> B) | Boolean functions | D^n -> B |
| | | on dA | |
o---------o-------------------o-------------------o-------------------o
| dA% | [d!A!] | Tangent universe | (D^n, (D^n -> B)) |
| | (dA, dA^) | at a point of A%, | (D^n +-> B) |
| | (dA +-> B) | based on the | [D^n] |
| | (dA, (dA -> B)) | tangent features | |
| | [da_1, ..., da_n] | {da_1, ..., da_n} | |
o---------o-------------------o-------------------o-------------------o
| Symbol | Notation | Description | Type |
|---|---|---|---|
| dA | {da1, …, dan} |
Alphabet of |
[n] = n |
| dAi | {(dai), dai} |
Differential |
D |
| dA |
〈dA〉 |
Tangent space |
Dn |
| dA* | (hom : dA → B) |
Linear functions |
(Dn)* = Dn |
| dA^ | (dA → B) |
Boolean functions |
Dn → B |
| dA• |
[dA] |
Tangent universe |
(Dn, (Dn → B)) |
Table 9. Higher Order Differential Features
Table 9. Higher Order Differential Features
o----------------------------------------o----------------------------------------o
| | |
| !A! = d^0.!A! = {a_1, ..., a_n} | E^0.!A! = d^0.!A! |
| | |
| d!A! = d^1.!A! = {da_1, ..., da_n} | E^1.!A! = d^0.!A! |_| d^1.!A! |
| | |
| d^k.!A! = {d^k.a_1,...,d^k.a_n}| E^k.!A! = d^0.!A! |_| ... |_| d^k.!A! |
| | |
| d*!A! = {d^0.!A!, ..., d^k.!A!, ...} | E^oo.!A! = |_| d*!A! |
| | |
o----------------------------------------o----------------------------------------o
|
A = d0A = {a1, …, an} |
E0A = d0A |
|
|
Table 10. A Realm of Intentional Features
Table 10. A Realm of Intentional Features
o---------------------------------------o----------------------------------------o
| | |
| p^0.!A! = !A! = {a_1, ..., a_n} | Q^0.!A! = !A! |
| | |
| p^1.!A! = !A!' = {a_1', ..., a_n'} | Q^1.!A! = !A! |_| !A!' |
| | |
| p^2.!A! = !A!" = {a_1", ..., a_n"} | Q^2.!A! = !A! |_| !A!' |_| !A!" |
| | |
| ... ... ... | ... ... |
| | |
| p^k.!A! = {p^k.a_1, ..., p^k.a_n} | Q^k.!A! = !A! |_| ... |_| p^k.!A! |
| | |
o---------------------------------------o----------------------------------------o
|
|
Formula Display 1
o-------------------------------------------------o | | | From (A) & (dA) infer (A) next. | | | | From (A) & dA infer A next. | | | | From A & (dA) infer A next. | | | | From A & dA infer (A) next. | | | o-------------------------------------------------o
|
Table 11. A Pair of Commodious Trajectories
Table 11. A Pair of Commodious Trajectories o---------o-------------------o-------------------o | Time | Trajectory 1 | Trajectory 2 | o---------o-------------------o-------------------o | | | | | 0 | A dA (d^2.A) | (A) (dA) d^2.A | | | | | | 1 | (A) dA d^2.A | (A) dA d^2.A | | | | | | 2 | A (dA) (d^2.A) | A (dA) (d^2.A) | | | | | | 3 | A (dA) (d^2.A) | A (dA) (d^2.A) | | | | | | 4 | " " " | " " " | | | | | o---------o-------------------o-------------------o
| Time | Trajectory 1 | Trajectory 2 | |||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
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|
Figure 12. The Anchor
o-------------------------------------------------o | E^2.X | | | | o-------------o | | / \ | | / A \ | | / \ | | / ->- \ | | o / \ o | | | \ / | | | | -o- | | | | ^ | | | o---o---------o | o---------o---o | | / \ \|/ / \ | | / \ o | / \ | | / \ | /|\ / \ | | / \ | / | \ / \ | | o o-|-o--|--o---o o | | | | | | | | | | | ---->o<----o | | | | | | | | | o dA o o d^2.A o | | \ \ / / | | \ \ / / | | \ o / | | \ / \ / | | o-------------o o-------------o | | | | | o-------------------------------------------------o Figure 12. The Anchor
Figure 13. The Tiller
o-------------------------------------------------o | | | ->- | | / \ | | \ / | | o-------------o -o- | | / \ ^ | | / dA \/ A | | / /\ | | / / \ | | o o / o | | | \ / | | | | \ / | | o------------|-------\-------/-------|------------o | | \ / | | | | \ / | | | o v / o | | \ o / | | \ ^ / | | \ | / d^2.A | | \ | / | | o------|------o | | | | | | | | o | | | o-------------------------------------------------o Figure 13. The Tiller
Table 14. Differential Propositions
Table 14. Differential Propositions o-------o--------o---------o-----------o-------------------o----------o | | A : 1 1 0 0 | | | | | | dA : 1 0 1 0 | | | | o-------o--------o---------o-----------o-------------------o----------o | | | | | | | | f_0 | g_0 | 0 0 0 0 | () | False | 0 | | | | | | | | o-------o--------o---------o-----------o-------------------o----------o | | | | | | | | | g_1 | 0 0 0 1 | (A)(dA) | Neither A nor dA | ~A & ~dA | | | | | | | | | | g_2 | 0 0 1 0 | (A) dA | Not A but dA | ~A & dA | | | | | | | | | | g_4 | 0 1 0 0 | A (dA) | A but not dA | A & ~dA | | | | | | | | | | g_8 | 1 0 0 0 | A dA | A and dA | A & dA | | | | | | | | o-------o--------o---------o-----------o-------------------o----------o | | | | | | | | f_1 | g_3 | 0 0 1 1 | (A) | Not A | ~A | | | | | | | | | f_2 | g_12 | 1 1 0 0 | A | A | A | | | | | | | | o-------o--------o---------o-----------o-------------------o----------o | | | | | | | | | g_6 | 0 1 1 0 | (A, dA) | A not equal to dA | A + dA | | | | | | | | | | g_9 | 1 0 0 1 | ((A, dA)) | A equal to dA | A = dA | | | | | | | | o-------o--------o---------o-----------o-------------------o----------o | | | | | | | | | g_5 | 0 1 0 1 | (dA) | Not dA | ~dA | | | | | | | | | | g_10 | 1 0 1 0 | dA | dA | dA | | | | | | | | o-------o--------o---------o-----------o-------------------o----------o | | | | | | | | | g_7 | 0 1 1 1 | (A dA) | Not both A and dA | ~A v ~dA | | | | | | | | | | g_11 | 1 0 1 1 | (A (dA)) | Not A without dA | A => dA | | | | | | | | | | g_13 | 1 1 0 1 | ((A) dA) | Not dA without A | A <= dA | | | | | | | | | | g_14 | 1 1 1 0 | ((A)(dA)) | A or dA | A v dA | | | | | | | | o-------o--------o---------o-----------o-------------------o----------o | | | | | | | | f_3 | g_15 | 1 1 1 1 | (()) | True | 1 | | | | | | | | o-------o--------o---------o-----------o-------------------o----------o
| A : | 1 1 0 0 | ||||
| dA : | 1 0 1 0 | ||||
| f0 | g0 | 0 0 0 0 | ( ) | False | 0 |
| g1 | 0 0 0 1 | (A)(dA) | Neither A nor dA | ¬A ∧ ¬dA | |
| g2 | 0 0 1 0 | (A) dA | Not A but dA | ¬A ∧ dA | |
| g4 | 0 1 0 0 | A (dA) | A but not dA | A ∧ ¬dA | |
| g8 | 1 0 0 0 | A dA | A and dA | A ∧ dA | |
| f1 | g3 | 0 0 1 1 | (A) | Not A | ¬A |
| f2 | g12 | 1 1 0 0 | A | A | A |
| g6 | 0 1 1 0 | (A, dA) | A not equal to dA | A ≠ dA | |
| g9 | 1 0 0 1 | ((A, dA)) | A equal to dA | A = dA | |
| g5 | 0 1 0 1 | (dA) | Not dA | ¬dA | |
| g10 | 1 0 1 0 | dA | dA | dA | |
| g7 | 0 1 1 1 | (A dA) | Not both A and dA | ¬A ∨ ¬dA | |
| g11 | 1 0 1 1 | (A (dA)) | Not A without dA | A → dA | |
| g13 | 1 1 0 1 | ((A) dA) | Not dA without A | A ← dA | |
| g14 | 1 1 1 0 | ((A)(dA)) | A or dA | A ∨ dA | |
| f3 | g15 | 1 1 1 1 | (( )) | True | 1 |
| A : | 1 1 0 0 | ||||||||||
| dA : | 1 0 1 0 | ||||||||||
| f0 | g0 | 0 0 0 0 | ( ) | False | 0 | ||||||
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| f3 | g15 | 1 1 1 1 | (( )) | True | 1 |
Table 15. Tacit Extension of [A] to [A, dA]
Table 15. Tacit Extension of [A] to [A, dA] o---------------------------------------------------------------------o | | | 0 = 0 . ((dA), dA) = 0 | | | | (A) = (A) . ((dA), dA) = (A)(dA) + (A) dA | | | | A = A . ((dA), dA) = A (dA) + A dA | | | | 1 = 1 . ((dA), dA) = 1 | | | o---------------------------------------------------------------------o
|
Figure 16-a. A Couple of Fourth Gear Orbits: 1
o-------------------------------------------------o | | | o | | / \ | | / \ | | / \ | | / \ | | o o | | / \ / \ | | / \ / \ | | / \ / \ | | / \ / \ | | o o o | | / \ / \ / \ | | / \ / \ / \ | | / \ / \ / \ | | / \ / \ / \ | | o 5 o 7 o o | | / \ ^| / \ ^| / \ / \ | | / \/ | / \/ | / \ / \ | | / /\ | / /\ | / \ / \ | | / / \|/ / \|/ \ / \ | | o 4<---|----/----|----3 o o | | |\ /|\ / /|\ ^ / \ /| | | | \ / | \/ / | \/ / \ / | | | | \ / | /\ / | /\ / \ / | | | | \ / v/ \ / |/ \ / \ / | | | | o 6 o | o o | | | | |\ / \ /| / \ /| | | | | | \ / \/ | / \ / | | | | | | \ / /\ | / \ / | | | | | d^0.A \ / / \|/ \ / d^1.A | | | o----+----o 2<---|----1 o----+----o | | | \ /|\ ^ / | | | | \ / | \/ / | | | | \ / | /\ / | | | | d^2.A \ / v/ \ / d^3.A | | | o---------o 0 o---------o | | \ / | | \ / | | \ / | | \ / | | o | | | o-------------------------------------------------o Figure 16-a. A Couple of Fourth Gear Orbits: 1
Figure 16-b. A Couple of Fourth Gear Orbits: 2
o-------------------------------------------------o | | | o | | / \ | | / \ | | / \ | | / \ | | o 0 o | | / \ / \ | | / \ / \ | | / \ / \ | | / \ / \ | | o 5 o 2 o | | / \ / \ / \ | | / \ / \ / \ | | / \ / \ / \ | | / \ / \ / \ | | o o o 6 o | | / \ / \ / \ / \ | | / \ / \ / \ / \ | | / \ / \ / \ / \ | | / \ / \ / \ / \ | | o o 7 o o 4 o | | |\ / \ / \ / \ /| | | | \ / \ / \ / \ / | | | | \ / \ / \ / \ / | | | | \ / \ / \ / \ / | | | | o o 3 o 1 o | | | | |\ / \ / \ /| | | | | | \ / \ / \ / | | | | | | \ / \ / \ / | | | | | d^0.A \ / \ / \ / d^1.A | | | o----+----o o o----+----o | | | \ / \ / | | | | \ / \ / | | | | \ / \ / | | | | d^2.A \ / \ / d^3.A | | | o---------o o---------o | | \ / | | \ / | | \ / | | \ / | | o | | | o-------------------------------------------------o Figure 16-b. A Couple of Fourth Gear Orbits: 2
Formula Display 2
o-------------------------------------------------------------------------------o | | | r(q) = Sum_k d_k . 2^(-k) = Sum_k d^k.A(q) . 2^(-k) | | | | = | | | | s(q)/t = (Sum_k d_k . 2^(m-k)) / 2^m = (Sum_k d^k.A(q) . 2^(m-k)) / 2^m | | | o-------------------------------------------------------------------------------o
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Table 17-a. A Couple of Orbits in Fourth Gear: Orbit 1
Table 17-a. A Couple of Orbits in Fourth Gear: Orbit 1 o---------o---------o---------o---------o---------o---------o---------o | Time | State | A | dA | | | | | p_i | q_j | d^0.A | d^1.A | d^2.A | d^3.A | d^4.A | o---------o---------o---------o---------o---------o---------o---------o | | | | | p_0 | q_01 | 0. 0 0 0 1 | | | | | | p_1 | q_03 | 0. 0 0 1 1 | | | | | | p_2 | q_05 | 0. 0 1 0 1 | | | | | | p_3 | q_15 | 0. 1 1 1 1 | | | | | | p_4 | q_17 | 1. 0 0 0 1 | | | | | | p_5 | q_19 | 1. 0 0 1 1 | | | | | | p_6 | q_21 | 1. 0 1 0 1 | | | | | | p_7 | q_31 | 1. 1 1 1 1 | | | | | o---------o---------o---------o---------o---------o---------o---------o
| Time | State | A | dA | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| pi | qj | d0A | d1A | d2A | d3A | d4A | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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Table 17-b. A Couple of Orbits in Fourth Gear: Orbit 2
Table 17-b. A Couple of Orbits in Fourth Gear: Orbit 2 o---------o---------o---------o---------o---------o---------o---------o | Time | State | A | dA | | | | | p_i | q_j | d^0.A | d^1.A | d^2.A | d^3.A | d^4.A | o---------o---------o---------o---------o---------o---------o---------o | | | | | p_0 | q_25 | 1. 1 0 0 1 | | | | | | p_1 | q_11 | 0. 1 0 1 1 | | | | | | p_2 | q_29 | 1. 1 1 0 1 | | | | | | p_3 | q_07 | 0. 0 1 1 1 | | | | | | p_4 | q_09 | 0. 1 0 0 1 | | | | | | p_5 | q_27 | 1. 1 0 1 1 | | | | | | p_6 | q_13 | 0. 1 1 0 1 | | | | | | p_7 | q_23 | 1. 0 1 1 1 | | | | | o---------o---------o---------o---------o---------o---------o---------o
| Time | State | A | dA | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| pi | qj | d0A | d1A | d2A | d3A | d4A | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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Figure 18-a. Extension from 1 to 2 Dimensions: Areal
o-----------------------------------------------------------o | | | o o | | / \ / \ | | / \ / \ | | / \ / \ | | / \ / \ | | / o o 1 1 o | | / / \ / \ / \ | | / / \ / \ / \ | | / 1 / \ / \ / \ | | / / \ !e! / \ / \ | | o / o ----> o 1 0 o 0 1 o | | |\ / / |\ / \ /| | | | \ / 0 / | \ / \ / | | | | \ / / | \ / \ / | | | |x_1\ / / |x_1\ / \ /x_2| | | o----o / o----o 0 0 o----o | | \ / \ / | | \ / \ / | | \ / \ / | | \ / \ / | | o o | | | o-----------------------------------------------------------o Figure 18-a. Extension from 1 to 2 Dimensions: Areal
Figure 18-b. Extension from 1 to 2 Dimensions: Bundle
o-----------------------------o o-------------------o | | | | | | | o-------o | | o---------o | | / \ | | / \ | | o o | | / o------------------------| | dx | | | / \ | | o o | | / \ | | \ / | | o o | | o-------o | | | | | | | | | | | o-------------------o | | x | | | | | | o-------------------o | | | | | | | o o | | o-------o | | \ / | | / \ | | \ / | | o o | | \ / o------------| | dx | | | \ / | | o o | | o---------o | | \ / | | | | o-------o | | | | | o-----------------------------o o-------------------o Figure 18-b. Extension from 1 to 2 Dimensions: Bundle
Figure 18-c. Extension from 1 to 2 Dimensions: Compact
o-----------------------------------------------------------o | | | | | o-----------------o | | / o \ | | / (dx) / \ \ dx | | / v o--------------------->o | | / \ / \ | | / o \ | | o o | | | | | | | | | | | x | (x) | | | | | | | | | | o o | | \ / o | | \ / / \ | | \ o<---------------------o v | | \ / dx \ / (dx) | | \ / o | | o-----------------o | | | | | o-----------------------------------------------------------o Figure 18-c. Extension from 1 to 2 Dimensions: Compact
Figure 18-d. Extension from 1 to 2 Dimensions: Digraph
o-----------------------------------------------------------o | | | | | dx | | .--. .---------->----------. .--. | | | \ / \ / | | | (dx) ^ @ x (x) @ v (dx) | | | / \ / \ | | | *--* *----------<----------* *--* | | dx | | | | | o-----------------------------------------------------------o Figure 18-d. Extension from 1 to 2 Dimensions: Digraph
Figure 19-a. Extension from 2 to 4 Dimensions: Areal
o-------------------------------------------------------------------------------o | | | o o | | / \ / \ | | / \ / \ | | / \ / \ | | / \ o 1100 o | | / \ / \ / \ | | / \ / \ / \ | | / \ !e! / \ / \ | | o 1 1 o ----> o 1101 o 1110 o | | / \ / \ / \ / \ / \ | | / \ / \ / \ / \ / \ | | / \ / \ / \ / \ / \ | | / \ / \ o 1001 o 1111 o 0110 o | | / \ / \ / \ / \ / \ / \ | | / \ / \ / \ / \ / \ / \ | | / \ / \ / \ / \ / \ / \ | | o 1 0 o 0 1 o o 1000 o 1011 o 0111 o 0100 o | | |\ / \ /| |\ / \ / \ / \ /| | | | \ / \ / | | \ / \ / \ / \ / | | | | \ / \ / | | \ / \ / \ / \ / | | | | \ / \ / | | o 1010 o 0011 o 0101 o | | | | \ / \ / | | |\ / \ / \ /| | | | | \ / \ / | | | \ / \ / \ / | | | | | x_1 \ / \ / x_2 | |x_1| \ / \ / \ / |x_2| | | o-------o 0 0 o-------o o---+---o 0010 o 0001 o---+---o | | \ / | \ / \ / | | | \ / | \ / \ / | | | \ / | x_3 \ / \ / x_4 | | | \ / o-------o 0000 o-------o | | \ / \ / | | \ / \ / | | \ / \ / | | o o | | | o-------------------------------------------------------------------------------o Figure 19-a. Extension from 2 to 4 Dimensions: Areal
Figure 19-b. Extension from 2 to 4 Dimensions: Bundle
o-----------------------------o
| o-----o o-----o |
| / \ / \ |
| / o \ |
| / / \ \ |
| o o o o |
@ | du | | dv | |
/| o o o o |
/ | \ \ / / |
/ | \ o / |
/ | \ / \ / |
/ | o-----o o-----o |
/ o-----------------------------o
/
o-----------------------------------------/---o o-----------------------------o
| / | | o-----o o-----o |
| @ | | / \ / \ |
| o---------o o---------o | | / o \ |
| / \ / \ | | / / \ \ |
| / o \ | | o o o o |
| / / \ @-------\-----------@ | du | | dv | |
| / / @ \ \ | | o o o o |
| / / \ \ \ | | \ \ / / |
| / / \ \ \ | | \ o / |
| o o \ o o | | \ / \ / |
| | | \| | | | o-----o o-----o |
| | | | | | o-----------------------------o
| | u | |\ v | |
| | | | \ | | o-----------------------------o
| | | | \ | | | o-----o o-----o |
| o o o \ o | | / \ / \ |
| \ \ / \ / | | / o \ |
| \ \ / \ / | | / / \ \ |
| \ \ / \ / | | o o o o |
| \ @-----\-/-----------\-------------@ | du | | dv | |
| \ o / | | o o o o |
| \ / \ / \ | | \ \ / / |
| o---------o o---------o \ | | \ o / |
| \ | | \ / \ / |
| \ | | o-----o o-----o |
o-----------------------------------------\---o o-----------------------------o
\
\ o-----------------------------o
\ | o-----o o-----o |
\ | / \ / \ |
\ | / o \ |
\ | / / \ \ |
\| o o o o |
@ | du | | dv | |
| o o o o |
| \ \ / / |
| \ o / |
| \ / \ / |
| o-----o o-----o |
o-----------------------------o
Figure 19-b. Extension from 2 to 4 Dimensions: Bundle
Figure 19-c. Extension from 2 to 4 Dimensions: Compact
o---------------------------------------------------------------------o | | | | | o-------------------o o-------------------o | | / \ / \ | | / o \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | o o (du).(dv) o o | | | | -->-- | | | | | | \ / | | | | | dv .(du) | \ / | du .(dv) | | | | u <---------------@---------------> v | | | | | | | | | | | | | | | | | | | | | | | | o o | o o | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \|/ / | | \ | / | | \ /|\ / | | o-------------------o | o-------------------o | | | | | du . dv | | | | | V | | | o---------------------------------------------------------------------o Figure 19-c. Extension from 2 to 4 Dimensions: Compact
Figure 19-d. Extension from 2 to 4 Dimensions: Digraph
o-----------------------------------------------------------o | | | .->-. | | | | | | * * | | \ / | | .-->--@--<--. | | / / \ \ | | / / \ \ | | / . . \ | | / | | \ | | / | | \ | | / | | \ | | . | | . | | | | | | | | v | | v | | .--. | .---------->----------. | .--. | | | \|/ | | \|/ | | | ^ @ ^ v @ v | | | /|\ | | /|\ | | | *--* | *----------<----------* | *--* | | ^ | | ^ | | | | | | | | * | | * | | \ | | / | | \ | | / | | \ | | / | | \ . . / | | \ \ / / | | \ \ / / | | *-->--@--<--* | | / \ | | . . | | | | | | *-<-* | | | o-----------------------------------------------------------o Figure 19-d. Extension from 2 to 4 Dimensions: Digraph
Figure 20-i. Thematization of Conjunction (Stage 1)
o-------------------------------o o-------------------------------o
| | | |
| o-----o o-----o | | o-----o o-----o |
| / \ / \ | | / \ / \ |
| / o \ | | / o \ |
| / /`\ \ | | / /`\ \ |
| o o```o o | | o o```o o |
| | u |```| v | | | | u |```| v | |
| o o```o o | | o o```o o |
| \ \`/ / | | \ \`/ / |
| \ o / | | \ o / |
| \ / \ / | | \ / \ / |
| o-----o o-----o | | o-----o o-----o |
| | | |
o-------------------------------o o-------------------------------o
\ /
\ /
\ /
u v \ J /
\ /
\ /
\ /
\ /
o
Figure 20-i. Thematization of Conjunction (Stage 1)
Figure 20-ii. Thematization of Conjunction (Stage 2)
o-------------------------------o o-------------------------------o
| | | |
| o-----o o-----o | | o-----o o-----o |
| / \ / \ | | / \ / \ |
| / o \ | | / o \ |
| / /`\ \ | | / /`\ \ |
| o o```o o | | o o```o o |
| | u |```| v | | | | u |```| v | |
| o o```o o | | o o```o o |
| \ \`/ / | | \ \`/ / |
| \ o / | | \ o / |
| \ / \ / | | \ / \ / |
| o-----o o-----o | | o-----o o-----o |
| | | |
o-------------------------------o o-------------------------------o
\ / \ /
\ / \ /
\ / \ J /
\ / \ /
\ / \ /
o----------\---------/----------o o----------\---------/----------o
| \ / | | \ / |
| \ / | | \ / |
| o-----@-----o | | o-----@-----o |
| /`````````````\ | | /`````````````\ |
| /```````````````\ | | /```````````````\ |
| /`````````````````\ | | /`````````````````\ |
| o```````````````````o | | o```````````````````o |
| |```````````````````| | | |```````````````````| |
| |```````` J ````````| | | |```````` x ````````| |
| |```````````````````| | | |```````````````````| |
| o```````````````````o | | o```````````````````o |
| \`````````````````/ | | \`````````````````/ |
| \```````````````/ | | \```````````````/ |
| \`````````````/ | | \`````````````/ |
| o-----------o | | o-----------o |
| | | |
| | | |
o-------------------------------o o-------------------------------o
J = u v x = J<u, v>
Figure 20-ii. Thematization of Conjunction (Stage 2)
Figure 20-iii. Thematization of Conjunction (Stage 3)
o-------------------------------o o-------------------------------o
| | |```````````````````````````````|
| | |````````````o-----o````````````|
| | |```````````/ \```````````|
| | |``````````/ \``````````|
| | |`````````/ \`````````|
| | |````````/ \````````|
| J | |```````o x o```````|
| | |```````| |```````|
| | |```````| |```````|
| | |```````| |```````|
| o-----o o-----o | |```````o-----o o-----o```````|
| / \ / \ | |``````/`\ \ / /`\``````|
| / o \ | |`````/```\ o /```\`````|
| / /`\ \ | |````/`````\ /`\ /`````\````|
| / /```\ \ | |```/```````\ /```\ /```````\```|
| o o`````o o | |``o`````````o-----o`````````o``|
| | u |`````| v | | |``|`````````| |`````````|``|
o--o---------o-----o---------o--o |``|``` u ```| |``` v ```|``|
|``|`````````| |`````````|``| |``|`````````| |`````````|``|
|``o`````````o o`````````o``| |``o`````````o o`````````o``|
|```\`````````\ /`````````/```| |```\`````````\ /`````````/```|
|````\`````````\ /`````````/````| |````\`````````\ /`````````/````|
|`````\`````````o`````````/`````| |`````\`````````o`````````/`````|
|``````\```````/`\```````/``````| |``````\```````/`\```````/``````|
|```````o-----o```o-----o```````| |```````o-----o```o-----o```````|
|```````````````````````````````| |```````````````````````````````|
o-------------------------------o o-------------------------------o
\ /
\ /
J = u v \ /
\ !j! /
\ /
!j! = (( x , u v )) \ /
\ /
\ /
@
Figure 20-iii. Thematization of Conjunction (Stage 3)
Figure 21. Thematization of Disjunction and Equality
f g
o-------------------------------o o-------------------------------o
| | |```````````````````````````````|
| o-----o o-----o | |```````o-----o```o-----o```````|
| /```````\ /```````\ | |``````/ \`/ \``````|
| /`````````o`````````\ | |`````/ o \`````|
| /`````````/`\`````````\ | |````/ /`\ \````|
| /`````````/```\`````````\ | |```/ /```\ \```|
| o`````````o`````o```````` o | |``o o`````o o``|
| |`````````|`````|`````````| | |``| |`````| |``|
| |``` u ```|`````|``` v ```| | |``| u |`````| v |``|
| |`````````|`````|`````````| | |``| |`````| |``|
| o`````````o`````o`````````o | |``o o`````o o``|
| \`````````\```/`````````/ | |```\ \```/ /```|
| \`````````\`/`````````/ | |````\ \`/ /````|
| \`````````o`````````/ | |`````\ o /`````|
| \```````/ \```````/ | |``````\ /`\ /``````|
| o-----o o-----o | |```````o-----o```o-----o```````|
| | |```````````````````````````````|
o-------------------------------o o-------------------------------o
((u)(v)) ((u , v))
| |
| |
theta theta
| |
| |
v v
!f! !g!
o-------------------------------o o-------------------------------o
|```````````````````````````````| | |
|````````````o-----o````````````| | o-----o |
|```````````/ \```````````| | /```````\ |
|``````````/ \``````````| | /`````````\ |
|`````````/ \`````````| | /```````````\ |
|````````/ \````````| | /`````````````\ |
|```````o f o```````| | o`````` g ``````o |
|```````| |```````| | |```````````````| |
|```````| |```````| | |```````````````| |
|```````| |```````| | |```````````````| |
|```````o-----o o-----o```````| | o-----o```o-----o |
|``````/ \`````\ /`````/ \``````| | /`\ \`/ /`\ |
|`````/ \`````o`````/ \`````| | /```\ o /```\ |
|````/ \```/`\```/ \````| | /`````\ /`\ /`````\ |
|```/ \`/```\`/ \```| | /```````\ /```\ /```````\ |
|``o o-----o o``| | o`````````o-----o`````````o |
|``| | | |``| | |`````````| |`````````| |
|``| u | | v |``| | |``` u ```| |``` v ```| |
|``| | | |``| | |`````````| |`````````| |
|``o o o o``| | o`````````o o`````````o |
|```\ \ / /```| | \`````````\ /`````````/ |
|````\ \ / /````| | \`````````\ /`````````/ |
|`````\ o /`````| | \`````````o`````````/ |
|``````\ /`\ /``````| | \```````/ \```````/ |
|```````o-----o```o-----o```````| | o-----o o-----o |
|```````````````````````````````| | |
o-------------------------------o o-------------------------------o
((f , ((u)(v)) )) ((g , ((u , v)) ))
Figure 21. Thematization of Disjunction and Equality
Table 22. Disjunction f and Equality g
Table 22. Disjunction f and Equality g o-------------------o-------------------o | u v | f g | o-------------------o-------------------o | | | | 0 0 | 0 1 | | | | | 0 1 | 1 0 | | | | | 1 0 | 1 0 | | | | | 1 1 | 1 1 | | | | o-------------------o-------------------o
|
| ||||||||||||||||
|
|
Tables 23-i and 23-ii. Thematics of Disjunction and Equality (1)
Tables 23-i and 23-ii. Thematics of Disjunction and Equality (1) o-----------------o-----------o o-----------------o-----------o | u v f | x !f! | | u v g | y !g! | o-----------------o-----------o o-----------------o-----------o | | | | | | | 0 0 --> | 0 1 | | 0 0 --> | 1 1 | | | | | | | | 0 1 --> | 1 1 | | 0 1 --> | 0 1 | | | | | | | | 1 0 --> | 1 1 | | 1 0 --> | 0 1 | | | | | | | | 1 1 --> | 1 1 | | 1 1 --> | 1 1 | | | | | | | o-----------------o-----------o o-----------------o-----------o | | | | | | | 0 0 | 1 0 | | 0 0 | 0 0 | | | | | | | | 0 1 | 0 0 | | 0 1 | 1 0 | | | | | | | | 1 0 | 0 0 | | 1 0 | 1 0 | | | | | | | | 1 1 | 0 0 | | 1 1 | 0 0 | | | | | | | o-----------------o-----------o o-----------------o-----------o
|
|
Tables 24-i and 24-ii. Thematics of Disjunction and Equality (2)
Tables 24-i and 24-ii. Thematics of Disjunction and Equality (2) o-----------------------o-----o o-----------------------o-----o | u v f x | !f! | | u v g y | !g! | o-----------------------o-----o o-----------------------o-----o | | | | | | | 0 0 --> 0 | 1 | | 0 0 0 | 0 | | | | | | | | 0 0 1 | 0 | | 0 0 --> 1 | 1 | | | | | | | | 0 1 0 | 0 | | 0 1 --> 0 | 1 | | | | | | | | 0 1 --> 1 | 1 | | 0 1 1 | 0 | | | | | | | o-----------------------o-----o o-----------------------o-----o | | | | | | | 1 0 0 | 0 | | 1 0 --> 0 | 1 | | | | | | | | 1 0 --> 1 | 1 | | 1 0 1 | 0 | | | | | | | | 1 1 0 | 0 | | 1 1 0 | 0 | | | | | | | | 1 1 --> 1 | 1 | | 1 1 --> 1 | 1 | | | | | | | o-----------------------o-----o o-----------------------o-----o
|
|
Tables 25-i and 25-ii. Thematics of Disjunction and Equality (3)
Tables 25-i and 25-ii. Thematics of Disjunction and Equality (3) o-----------------------o-----o o-----------------------o-----o | u v f x | !f! | | u v g y | !g! | o-----------------------o-----o o-----------------------o-----o | | | | | | | 0 0 --> 0 | 1 | | 0 0 0 | 0 | | | | | | | | 0 1 0 | 0 | | 0 1 --> 0 | 1 | | | | | | | | 1 0 0 | 0 | | 1 0 --> 0 | 1 | | | | | | | | 1 1 0 | 0 | | 1 1 0 | 0 | | | | | | | o-----------------------o-----o o-----------------------o-----o | | | | | | | 0 0 1 | 0 | | 0 0 --> 1 | 1 | | | | | | | | 0 1 --> 1 | 1 | | 0 1 1 | 0 | | | | | | | | 1 0 --> 1 | 1 | | 1 0 1 | 0 | | | | | | | | 1 1 --> 1 | 1 | | 1 1 --> 1 | 1 | | | | | | | o-----------------------o-----o o-----------------------o-----o
|
|
Tables 26-i and 26-ii. Tacit Extension and Thematization
Tables 26-i and 26-ii. Tacit Extension and Thematization o-----------------o-----------o o-----------------o-----------o | u v x | !e!f !f! | | u v y | !e!g !g! | o-----------------o-----------o o-----------------o-----------o | | | | | | | 0 0 0 | 0 1 | | 0 0 0 | 1 0 | | | | | | | | 0 0 1 | 0 0 | | 0 0 1 | 1 1 | | | | | | | | 0 1 0 | 1 0 | | 0 1 0 | 0 1 | | | | | | | | 0 1 1 | 1 1 | | 0 1 1 | 0 0 | | | | | | | o-----------------o-----------o o-----------------o-----------o | | | | | | | 1 0 0 | 1 0 | | 1 0 0 | 0 1 | | | | | | | | 1 0 1 | 1 1 | | 1 0 1 | 0 0 | | | | | | | | 1 1 0 | 1 0 | | 1 1 0 | 1 0 | | | | | | | | 1 1 1 | 1 1 | | 1 1 1 | 1 1 | | | | | | | o-----------------o-----------o o-----------------o-----------o
|
|
Table 27. Thematization of Bivariate Propositions
Table 27. Thematization of Bivariate Propositions o---------o---------o----------o--------------------o--------------------o | u : 1 1 0 0 | f | theta (f) | theta (f) | | v : 1 0 1 0 | | | | o---------o---------o----------o--------------------o--------------------o | | | | | | | f_0 | 0 0 0 0 | () | (( f , () )) | f + 1 | | | | | | | | f_1 | 0 0 0 1 | (u)(v) | (( f , (u)(v) )) | f + u + v + uv | | | | | | | | f_2 | 0 0 1 0 | (u) v | (( f , (u) v )) | f + v + uv + 1 | | | | | | | | f_3 | 0 0 1 1 | (u) | (( f , (u) )) | f + u | | | | | | | | f_4 | 0 1 0 0 | u (v) | (( f , u (v) )) | f + u + uv + 1 | | | | | | | | f_5 | 0 1 0 1 | (v) | (( f , (v) )) | f + v | | | | | | | | f_6 | 0 1 1 0 | (u, v) | (( f , (u, v) )) | f + u + v + 1 | | | | | | | | f_7 | 0 1 1 1 | (u v) | (( f , (u v) )) | f + uv | | | | | | | o---------o---------o----------o--------------------o--------------------o | | | | | | | f_8 | 1 0 0 0 | u v | (( f , u v )) | f + uv + 1 | | | | | | | | f_9 | 1 0 0 1 | ((u, v)) | (( f , ((u, v)) )) | f + u + v | | | | | | | | f_10 | 1 0 1 0 | v | (( f , v )) | f + v + 1 | | | | | | | | f_11 | 1 0 1 1 | (u (v)) | (( f , (u (v)) )) | f + u + uv | | | | | | | | f_12 | 1 1 0 0 | u | (( f , u )) | f + u + 1 | | | | | | | | f_13 | 1 1 0 1 | ((u) v) | (( f , ((u) v) )) | f + v + uv | | | | | | | | f_14 | 1 1 1 0 | ((u)(v)) | (( f , ((u)(v)) )) | f + u + v + uv + 1 | | | | | | | | f_15 | 1 1 1 1 | (()) | (( f , (()) )) | f | | | | | | | o---------o---------o----------o--------------------o--------------------o
Table 28. Propositions on Two Variables
Table 28. Propositions on Two Variables o-------o-----o----------------------------------------------------------------o | u v | | f f f f f f f f f f f f f f f f | | | | 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 | o-------o-----o----------------------------------------------------------------o | | | | | 0 0 | | 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 | | | | | | 0 1 | | 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 | | | | | | 1 0 | | 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 | | | | | | 1 1 | | 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 | | | | | o-------o-----o----------------------------------------------------------------o
Table 29. Thematic Extensions of Bivariate Propositions
Table 29. Thematic Extensions of Bivariate Propositions o-------o-----o----------------------------------------------------------------o | u v | f^¢ |!f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! | | | | 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 | o-------o-----o----------------------------------------------------------------o | | | | | 0 0 | 0 | 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 | | | | | | 0 0 | 1 | 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 | | | | | | 0 1 | 0 | 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 | | | | | | 0 1 | 1 | 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 | | | | | | 1 0 | 0 | 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 | | | | | | 1 0 | 1 | 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 | | | | | | 1 1 | 0 | 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 | | | | | | 1 1 | 1 | 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 | | | | | o-------o-----o----------------------------------------------------------------o
Figure 30. Generic Frame of a Logical Transformation
o-------------------------------------------------------o
| U |
| |
| o-----------o o-----------o |
| / \ / \ |
| / o \ |
| / / \ \ |
| / / \ \ |
| o o o o |
| | | | | |
| | u | | v | |
| | | | | |
| o o o o |
| \ \ / / |
| \ \ / / |
| \ o / |
| \ / \ / |
| o-----------o o-----------o |
| |
| |
o---------------------------o---------------------------o
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
o-------------------------o o-------------------------o o-------------------------o
| U | | U | | U |
| o---o o---o | | o---o o---o | | o---o o---o |
| / \ / \ | | / \ / \ | | / \ / \ |
| / o \ | | / o \ | | / o \ |
| / / \ \ | | / / \ \ | | / / \ \ |
| o o o o | | o o o o | | o o o o |
| | u | | v | | | | u | | v | | | | u | | v | |
| o o o o | | o o o o | | o o o o |
| \ \ / / | | \ \ / / | | \ \ / / |
| \ o / | | \ o / | | \ o / |
| \ / \ / | | \ / \ / | | \ / \ / |
| o---o o---o | | o---o o---o | | o---o o---o |
| | | | | |
o-------------------------o o-------------------------o o-------------------------o
\ | \ / | /
\ | \ / | /
\ | \ / | /
\ | \ / | /
\ g | \ f / | h /
\ | \ / | /
\ | \ / | /
\ | \ / | /
\ | \ / | /
\ o----------|-----------\-----/-----------|----------o /
\ | X | \ / | | /
\ | | \ / | | /
\ | | o-----o-----o | | /
\| | / \ | |/
\ | / \ | /
|\ | / \ | /|
| \ | / \ | / |
| \ | / \ | / |
| \ | o x o | / |
| \ | | | | / |
| \ | | | | / |
| \ | | | | / |
| \ | | | | / |
| \ | | | | / |
| \| | | |/ |
| o--o--------o o--------o--o |
| / \ \ / / \ |
| / \ \ / / \ |
| / \ o / \ |
| / \ / \ / \ |
| / \ / \ / \ |
| o o--o-----o--o o |
| | | | | |
| | | | | |
| | | | | |
| | y | | z | |
| | | | | |
| | | | | |
| o o o o |
| \ \ / / |
| \ \ / / |
| \ o / |
| \ / \ / |
| \ / \ / |
| o-----------o o-----------o |
| |
| |
o---------------------------------------------------o
\ /
\ /
\ /
\ /
\ /
\ p , q /
\ /
\ /
\ /
\ /
\ /
\ /
\ /
o
Figure 30. Generic Frame of a Logical Transformation
Formula Display 3
o-------------------------------------------------o | | | x = f<u, v> | | | | y = g<u, v> | | | | z = h<u, v> | | | o-------------------------------------------------o
|
Figure 31. Operator Diagram (1)
o---------------------------------------o | | | | | U% F X% | | o------------------>o | | | | | | | | | | | | | | | | | | !W! | | !W! | | | | | | | | | | | | | | v v | | o------------------>o | | !W!U% !W!F !W!X% | | | | | o---------------------------------------o Figure 31. Operator Diagram (1)
Figure 32. Operator Diagram (2)
o---------------------------------------o | | | | | U% !W! !W!U% | | o------------------>o | | | | | | | | | | | | | | | | | | F | | !W!F | | | | | | | | | | | | | | v v | | o------------------>o | | X% !W! !W!X% | | | | | o---------------------------------------o Figure 32. Operator Diagram (2)
Figure 33-i. Analytic Diagram (1)
U% $E$ $E$U% $E$U% $E$U% o------------------>o============o============o | | | | | | | | | | | | | | | | F | | $E$F = | $d$^0.F + | $r$^0.F | | | | | | | | | | | | v v v v o------------------>o============o============o X% $E$ $E$X% $E$X% $E$X% Figure 33-i. Analytic Diagram (1)
Figure 33-ii. Analytic Diagram (2)
U% $E$ $E$U% $E$U% $E$U% $E$U% o------------------>o============o============o============o | | | | | | | | | | | | | | | | | | | | F | | $E$F = | $d$^0.F + | $d$^1.F + | $r$^1.F | | | | | | | | | | | | | | | v v v v v o------------------>o============o============o============o X% $E$ $E$X% $E$X% $E$X% $E$X% Figure 33-ii. Analytic Diagram (2)
Formula Display 4
o--------------------------------------------------------------------------------------o | | | x_1 = !e!F_1 <u_1, ..., u_n, du_1, ..., du_n> = F_1 <u_1, ..., u_n> | | | | ... | | | | x_k = !e!F_k <u_1, ..., u_n, du_1, ..., du_n> = F_k <u_1, ..., u_n> | | | | | | dx_1 = EF_1 <u_1, ..., u_n, du_1, ..., du_n> = F_1 <u_1 + du_1, ..., u_n + du_n> | | | | ... | | | | dx_k = EF_k <u_1, ..., u_n, du_1, ..., du_n> = F_k <u_1 + du_1, ..., u_n + du_n> | | | o--------------------------------------------------------------------------------------o
| |||||||||||
|
Formula Display 5
o--------------------------------------------------------------------------------o | | | x_1 = !e!F_1 <u_1, ..., u_n, du_1, ..., du_n> = F_1 <u_1, ..., u_n> | | | | ... | | | | x_k = !e!F_k <u_1, ..., u_n, du_1, ..., du_n> = F_k <u_1, ..., u_n> | | | | | | dx_1 = !e!F_1 <u_1, ..., u_n, du_1, ..., du_n> = F_1 <u_1, ..., u_n> | | | | ... | | | | dx_k = !e!F_k <u_1, ..., u_n, du_1, ..., du_n> = F_k <u_1, ..., u_n> | | | o--------------------------------------------------------------------------------o
| |||||||||||
|
Formula Display 6
o--------------------------------------------------------------------------------o | | | dx_1 = !e!F_1 <u_1, ..., u_n, du_1, ..., du_n> = F_1 <u_1, ..., u_n> | | | | ... | | | | dx_k = !e!F_k <u_1, ..., u_n, du_1, ..., du_n> = F_k <u_1, ..., u_n> | | | o--------------------------------------------------------------------------------o
|
Formula Display 7
o-------------------------------------------------o | | | $D$ = $E$ - $e$ | | | | = $r$^0 | | | | = $d$^1 + $r$^1 | | | | = $d$^1 + ... + $d$^m + $r$^m | | | | = Sum_(i = 1 ... m) $d$^i + $r$^m | | | o-------------------------------------------------o
|
Figure 34. Tangent Functor Diagram
U% $T$ $T$U% $T$U% o------------------>o============o | | | | | | | | | | | | F | | $T$F = | <!e!, d> F | | | | | | | | | v v v o------------------>o============o X% $T$ $T$X% $T$X% Figure 34. Tangent Functor Diagram
Figure 35. Conjunction as Transformation
o---------------------------------------o
| |
| |
| o---------o o---------o |
| / \ / \ |
| / o \ |
| / /`\ \ |
| / /```\ \ |
| o o`````o o |
| | |`````| | |
| | u |`````| v | |
| | |`````| | |
| o o`````o o |
| \ \```/ / |
| \ \`/ / |
| \ o / |
| \ / \ / |
| o---------o o---------o |
| |
| |
o---------------------------------------o
\ /
\ /
\ /
\ J /
\ /
\ /
\ /
o--------------\---------/--------------o
| \ / |
| \ / |
| o------@------o |
| /```````````````\ |
| /`````````````````\ |
| /```````````````````\ |
| /`````````````````````\ |
| o```````````````````````o |
| |```````````````````````| |
| |`````````` x ``````````| |
| |```````````````````````| |
| o```````````````````````o |
| \`````````````````````/ |
| \```````````````````/ |
| \`````````````````/ |
| \```````````````/ |
| o-------------o |
| |
| |
o---------------------------------------o
Figure 35. Conjunction as Transformation
Table 36. Computation of !e!J
Table 36. Computation of !e!J o---------------------------------------------------------------------o | | | !e!J = J<u, v> | | | | = u v | | | | = u v (du)(dv) + u v (du) dv + u v du (dv) + u v du dv | | | o---------------------------------------------------------------------o | | | !e!J = u v (du)(dv) + | | u v (du) dv + | | u v du (dv) + | | u v du dv | | | o---------------------------------------------------------------------o
Figure 37-a. Tacit Extension of J (Areal)
o---------------------------------------o | | | o | | /%\ | | /%%%\ | | /%%%%%\ | | o%%%%%%%o | | /%\%%%%%/%\ | | /%%%\%%%/%%%\ | | /%%%%%\%/%%%%%\ | | o%%%%%%%o%%%%%%%o | | / \%%%%%/%\%%%%%/ \ | | / \%%%/%%%\%%%/ \ | | / \%/%%%%%\%/ \ | | o o%%%%%%%o o | | / \ / \%%%%%/ \ / \ | | / \ / \%%%/ \ / \ | | / \ / \%/ \ / \ | | o o o o o | | |\ / \ / \ / \ /| | | | \ / \ / \ / \ / | | | | \ / \ / \ / \ / | | | | o o o o | | | | |\ / \ / \ /| | | | | | \ / \ / \ / | | | | | u | \ / \ / \ / | v | | | o---+---o o o---+---o | | | \ / \ / | | | | \ / \ / | | | | du \ / \ / dv | | | o-------o o-------o | | \ / | | \ / | | \ / | | o | | | o---------------------------------------o Figure 37-a. Tacit Extension of J (Areal)
Figure 37-b. Tacit Extension of J (Bundle)
o-----------------------------o
| |
| o-----o o-----o |
| / \ / \ |
| / o \ |
| / / \ \ |
| o o o o |
@ | du | | dv | |
/| o o o o |
/ | \ \ / / |
/ | \ o / |
/ | \ / \ / |
/ | o-----o o-----o |
/ | |
/ o-----------------------------o
/
o----------------------------------------/----o o-----------------------------o
| / | | |
| @ | | o-----o o-----o |
| | | / \ / \ |
| o---------o o---------o | | / o \ |
| / \ / \ | | / / \ \ |
| / o \ | | o o o o |
| / /`\ @------\-----------@ | du | | dv | |
| / /```\ \ | | o o o o |
| / /`````\ \ | | \ \ / / |
| / /```````\ \ | | \ o / |
| o o`````````o o | | \ / \ / |
| | |````@````| | | | o-----o o-----o |
| | |`````\```| | | | |
| | |``````\``| | | o-----------------------------o
| | u |```````\`| v | |
| | |````````\| | | o-----------------------------o
| | |`````````| | | | |
| | |`````````|\ | | | o-----o o-----o |
| o o`````````o \ o | | / \ / \ |
| \ \```````/ \ / | | / o \ |
| \ \`````/ \ / | | / / \ \ |
| \ \```/ \ / | | o o o o |
| \ @------\-/---------\---------------@ | du | | dv | |
| \ o \ / | | o o o o |
| \ / \ / | | \ \ / / |
| o---------o o---------o \ | | \ o / |
| \ | | \ / \ / |
| \ | | o-----o o-----o |
| \ | | |
o----------------------------------------\----o o-----------------------------o
\
\ o-----------------------------o
\ |`````````````````````````````|
\ |````` o-----o```o-----o``````|
\ |`````/```````\`/```````\`````|
\ |````/`````````o`````````\````|
\ |```/`````````/`\`````````\```|
\|``o`````````o```o`````````o``|
@``|```du````|```|````dv```|``|
|``o`````````o```o`````````o``|
|```\`````````\`/`````````/```|
|````\`````````o`````````/````|
|`````\```````/`\```````/`````|
|``````o-----o```o-----o``````|
|`````````````````````````````|
o-----------------------------o
Figure 37-b. Tacit Extension of J (Bundle)
Figure 37-c. Tacit Extension of J (Compact)
o---------------------------------------------------------------------o | | | | | o-------------------o o-------------------o | | / \ / \ | | / o \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | o o (du).(dv) o o | | | | -->-- | | | | | | \ / | | | | | dv .(du) | \ / | du .(dv) | | | | u <---------------@---------------> v | | | | | | | | | | | | | | | | | | | | | | | | o o | o o | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \|/ / | | \ | / | | \ /|\ / | | o-------------------o | o-------------------o | | | | | du . dv | | | | | V | | | o---------------------------------------------------------------------o Figure 37-c. Tacit Extension of J (Compact)
Figure 37-d. Tacit Extension of J (Digraph)
o-----------------------------------------------------------o | | | (du).(dv) | | --->--- | | \ / | | \ / | | \ / | | u @ v | | /|\ | | / | \ | | / | \ | | / | \ | | / | \ | | (du) dv / | \ du (dv) | | / | \ | | / | \ | | / | \ | | / | \ | | v | v | | @ | @ | | u (v) | (u) v | | | | | | | | | | | | | | du . dv | | | | | | | | | | | | | | v | | @ | | | | (u).(v) | | | o-----------------------------------------------------------o Figure 37-d. Tacit Extension of J (Digraph)
Table 38. Computation of EJ (Method 1)
Table 38. Computation of EJ (Method 1) o-------------------------------------------------------------------------------o | | | EJ = J<u + du, v + dv> | | | | = (u, du)(v, dv) | | | | = u v J<1 + du, 1 + dv> + | | | | u (v) J<1 + du, 0 + dv> + | | | | (u) v J<0 + du, 1 + dv> + | | | | (u)(v) J<0 + du, 0 + dv> | | | | = u v J<(du), (dv)> + | | | | u (v) J<(du), dv > + | | | | (u) v J< du , (dv)> + | | | | (u)(v) J< du , dv > | | | o-------------------------------------------------------------------------------o | | | EJ = u v (du)(dv) | | + u (v)(du) dv | | + (u) v du (dv) | | + (u)(v) du dv | | | o-------------------------------------------------------------------------------o
Table 39. Computation of EJ (Method 2)
Table 39. Computation of EJ (Method 2) o-------------------------------------------------------------------------------o | | | EJ = <u + du> <v + dv> | | | | = u v + u dv + v du + du dv | | | | EJ = u v (du)(dv) + u (v)(du) dv + (u) v du (dv) + (u)(v) du dv | | | o-------------------------------------------------------------------------------o
Figure 40-a. Enlargement of J (Areal)
o---------------------------------------o | | | o | | /%\ | | /%%%\ | | /%%%%%\ | | o%%%%%%%o | | / \%%%%%/ \ | | / \%%%/ \ | | / \%/ \ | | o o o | | /%\ / \ /%\ | | /%%%\ / \ /%%%\ | | /%%%%%\ / \ /%%%%%\ | | o%%%%%%%o o%%%%%%%o | | / \%%%%%/ \ / \%%%%%/ \ | | / \%%%/ \ / \%%%/ \ | | / \%/ \ / \%/ \ | | o o o o o | | |\ / \ /%\ / \ /| | | | \ / \ /%%%\ / \ / | | | | \ / \ /%%%%%\ / \ / | | | | o o%%%%%%%o o | | | | |\ / \%%%%%/ \ /| | | | | | \ / \%%%/ \ / | | | | | u | \ / \%/ \ / | v | | | o---+---o o o---+---o | | | \ / \ / | | | | \ / \ / | | | | du \ / \ / dv | | | o-------o o-------o | | \ / | | \ / | | \ / | | o | | | o---------------------------------------o Figure 40-a. Enlargement of J (Areal)
Figure 40-b. Enlargement of J (Bundle)
o-----------------------------o
| |
| o-----o o-----o |
| / \ / \ |
| / o \ |
| / /%\ \ |
| o o%%%o o |
@ | du |%%%| dv | |
/| o o%%%o o |
/ | \ \%/ / |
/ | \ o / |
/ | \ / \ / |
/ | o-----o o-----o |
/ | |
/ o-----------------------------o
/
o----------------------------------------/----o o-----------------------------o
| / | | |
| @ | | o-----o o-----o |
| | | /%%%%%%%\ / \ |
| o---------o o---------o | | /%%%%%%%%%o \ |
| / \ / \ | | /%%%%%%%%%/ \ \ |
| / o \ | | o%%%%%%%%%o o o |
| / /`\ @------\-----------@ |%% du %%%| | dv | |
| / /```\ \ | | o%%%%%%%%%o o o |
| / /`````\ \ | | \%%%%%%%%%\ / / |
| / /```````\ \ | | \%%%%%%%%%o / |
| o o`````````o o | | \%%%%%%%/ \ / |
| | |````@````| | | | o-----o o-----o |
| | |`````\```| | | | |
| | |``````\``| | | o-----------------------------o
| | u |```````\`| v | |
| | |````````\| | | o-----------------------------o
| | |`````````| | | | |
| | |`````````|\ | | | o-----o o-----o |
| o o`````````o \ o | | / \ /%%%%%%%\ |
| \ \```````/ \ / | | / o%%%%%%%%%\ |
| \ \`````/ \ / | | / / \%%%%%%%%%\ |
| \ \```/ \ / | | o o o%%%%%%%%%o |
| \ @------\-/---------\---------------@ | du | |%%% dv %%| |
| \ o \ / | | o o o%%%%%%%%%o |
| \ / \ / | | \ \ /%%%%%%%%%/ |
| o---------o o---------o \ | | \ o%%%%%%%%%/ |
| \ | | \ / \%%%%%%%/ |
| \ | | o-----o o-----o |
| \ | | |
o----------------------------------------\----o o-----------------------------o
\
\ o-----------------------------o
\ |%%%%%%%%%%%%%%%%%%%%%%%%%%%%%|
\ |%%%%%%o-----o%%%o-----o%%%%%%|
\ |%%%%%/ \%/ \%%%%%|
\ |%%%%/ o \%%%%|
\ |%%%/ / \ \%%%|
\|%%o o o o%%|
@%%| du | | dv |%%|
|%%o o o o%%|
|%%%\ \ / /%%%|
|%%%%\ o /%%%%|
|%%%%%\ /%\ /%%%%%|
|%%%%%%o-----o%%%o-----o%%%%%%|
|%%%%%%%%%%%%%%%%%%%%%%%%%%%%%|
o-----------------------------o
Figure 40-b. Enlargement of J (Bundle)
Figure 40-c. Enlargement of J (Compact)
o---------------------------------------------------------------------o | | | | | o-------------------o o-------------------o | | / \ / \ | | / o \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | o o (du).(dv) o o | | | | -->-- | | | | | | \ / | | | | | dv .(du) | \ / | du .(dv) | | | | u o---------------->@<----------------o v | | | | | ^ | | | | | | | | | | | | | | | | | | o o | o o | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \|/ / | | \ | / | | \ /|\ / | | o-------------------o | o-------------------o | | | | | du . dv | | | | | o | | | o---------------------------------------------------------------------o Figure 40-c. Enlargement of J (Compact)
Figure 40-d. Enlargement of J (Digraph)
o-----------------------------------------------------------o | | | (du).(dv) | | --->--- | | \ / | | \ / | | \ / | | u @ v | | ^^^ | | / | \ | | / | \ | | / | \ | | / | \ | | (du) dv / | \ du (dv) | | / | \ | | / | \ | | / | \ | | / | \ | | / | \ | | @ | @ | | u (v) | (u) v | | | | | | | | | | | | | | du . dv | | | | | | | | | | | | | | | | | @ | | | | (u).(v) | | | o-----------------------------------------------------------o Figure 40-d. Enlargement of J (Digraph)
Table 41. Computation of DJ (Method 1)
Table 41. Computation of DJ (Method 1) o-------------------------------------------------------------------------------o | | | DJ = EJ + !e!J | | | | = J<u + du, v + dv> + J<u, v> | | | | = (u, du)(v, dv) + u v | | | o-------------------------------------------------------------------------------o | | | DJ = 0 | | | | + u v (du) dv + u (v)(du) dv | | | | + u v du (dv) + (u) v du (dv) | | | | + u v du dv + (u)(v) du dv | | | o-------------------------------------------------------------------------------o | | | DJ = u v ((du)(dv)) + u (v)(du) dv + (u) v du (dv) + (u)(v) du dv | | | o-------------------------------------------------------------------------------o
Table 42. Computation of DJ (Method 2)
Table 42. Computation of DJ (Method 2) o-------------------------------------------------------------------------------o | | | DJ = !e!J + EJ | | | | = J<u, v> + J<u + du, v + dv> | | | | = u v + (u, du)(v, dv) | | | | = 0 + u dv + v du + du dv | | | | = 0 + u (du) dv + v du (dv) + ((u, v)) du dv | | | o-------------------------------------------------------------------------------o
Table 43. Computation of DJ (Method 3)
Table 43. Computation of DJ (Method 3) o-------------------------------------------------------------------------------o | | | DJ = !e!J + EJ | | | o-------------------------------------------------------------------------------o | | | !e!J = u v (du)(dv) + u v (du) dv + u v du (dv) + u v du dv | | | | EJ = u v (du)(dv) + u (v)(du) dv + (u) v du (dv) + (u)(v) du dv | | | o-------------------------------------------------------------------------------o | | | DJ = 0 . (du)(dv) + u . (du) dv + v . du (dv) + ((u, v)) du dv | | | o-------------------------------------------------------------------------------o
Formula Display 8
o-------------------------------------------------------------------------------o
| |
| !e!J = {Dispositions from J to J } + {Dispositions from J to (J)} |
| |
| EJ = {Dispositions from J to J } + {Dispositions from (J) to J } |
| |
| DJ = (!e!J, EJ) |
| |
| DJ = {Dispositions from J to (J)} + {Dispositions from (J) to J } |
| |
o-------------------------------------------------------------------------------o
Figure 44-a. Difference Map of J (Areal)
o---------------------------------------o | | | o | | / \ | | / \ | | / \ | | o o | | /%\ /%\ | | /%%%\ /%%%\ | | /%%%%%\ /%%%%%\ | | o%%%%%%%o%%%%%%%o | | /%\%%%%%/%\%%%%%/%\ | | /%%%\%%%/%%%\%%%/%%%\ | | /%%%%%\%/%%%%%\%/%%%%%\ | | o%%%%%%%o%%%%%%%o%%%%%%%o | | / \%%%%%/ \%%%%%/ \%%%%%/ \ | | / \%%%/ \%%%/ \%%%/ \ | | / \%/ \%/ \%/ \ | | o o o o o | | |\ / \ /%\ / \ /| | | | \ / \ /%%%\ / \ / | | | | \ / \ /%%%%%\ / \ / | | | | o o%%%%%%%o o | | | | |\ / \%%%%%/ \ /| | | | | | \ / \%%%/ \ / | | | | | u | \ / \%/ \ / | v | | | o---+---o o o---+---o | | | \ / \ / | | | | \ / \ / | | | | du \ / \ / dv | | | o-------o o-------o | | \ / | | \ / | | \ / | | o | | | o---------------------------------------o Figure 44-a. Difference Map of J (Areal)
Figure 44-b. Difference Map of J (Bundle)
o-----------------------------o
| |
| o-----o o-----o |
| / \ / \ |
| / o \ |
| / /%\ \ |
| o o%%%o o |
@ | du |%%%| dv | |
/| o o%%%o o |
/ | \ \%/ / |
/ | \ o / |
/ | \ / \ / |
/ | o-----o o-----o |
/ | |
/ o-----------------------------o
/
o----------------------------------------/----o o-----------------------------o
| / | | |
| @ | | o-----o o-----o |
| | | /%%%%%%%\ / \ |
| o---------o o---------o | | /%%%%%%%%%o \ |
| / \ / \ | | /%%%%%%%%%/ \ \ |
| / o \ | | o%%%%%%%%%o o o |
| / /`\ @------\-----------@ |%% du %%%| | dv | |
| / /```\ \ | | o%%%%%%%%%o o o |
| / /`````\ \ | | \%%%%%%%%%\ / / |
| / /```````\ \ | | \%%%%%%%%%o / |
| o o`````````o o | | \%%%%%%%/ \ / |
| | |````@````| | | | o-----o o-----o |
| | |`````\```| | | | |
| | |``````\``| | | o-----------------------------o
| | u |```````\`| v | |
| | |````````\| | | o-----------------------------o
| | |`````````| | | | |
| | |`````````|\ | | | o-----o o-----o |
| o o`````````o \ o | | / \ /%%%%%%%\ |
| \ \```````/ \ / | | / o%%%%%%%%%\ |
| \ \`````/ \ / | | / / \%%%%%%%%%\ |
| \ \```/ \ / | | o o o%%%%%%%%%o |
| \ @------\-/---------\---------------@ | du | |%%% dv %%| |
| \ o \ / | | o o o%%%%%%%%%o |
| \ / \ / | | \ \ /%%%%%%%%%/ |
| o---------o o---------o \ | | \ o%%%%%%%%%/ |
| \ | | \ / \%%%%%%%/ |
| \ | | o-----o o-----o |
| \ | | |
o----------------------------------------\----o o-----------------------------o
\
\ o-----------------------------o
\ | |
\ | o-----o o-----o |
\ | /%%%%%%%\ /%%%%%%%\ |
\ | /%%%%%%%%%o%%%%%%%%%\ |
\ | /%%%%%%%%%/%\%%%%%%%%%\ |
\| o%%%%%%%%%o%%%o%%%%%%%%%o |
@ |%% du %%%|%%%|%%% dv %%| |
| o%%%%%%%%%o%%%o%%%%%%%%%o |
| \%%%%%%%%%\%/%%%%%%%%%/ |
| \%%%%%%%%%o%%%%%%%%%/ |
| \%%%%%%%/ \%%%%%%%/ |
| o-----o o-----o |
| |
o-----------------------------o
Figure 44-b. Difference Map of J (Bundle)
Figure 44-c. Difference Map of J (Compact)
o---------------------------------------------------------------------o | | | | | o-------------------o o-------------------o | | / \ / \ | | / o \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | o o o o | | | | | | | | | | | | | | | dv .(du) | | du .(dv) | | | | u @<--------------->@<--------------->@ v | | | | | ^ | | | | | | | | | | | | | | | | | | o o | o o | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \|/ / | | \ | / | | \ /|\ / | | o-------------------o | o-------------------o | | | | | du . dv | | | | | v | | @ | | | o---------------------------------------------------------------------o Figure 44-c. Difference Map of J (Compact)
Figure 44-d. Difference Map of J (Digraph)
o-----------------------------------------------------------o | | | u v | | | | @ | | ^^^ | | / | \ | | / | \ | | / | \ | | / | \ | | (du) dv / | \ du (dv) | | / | \ | | / | \ | | / | \ | | / | \ | | v | v | | @ | @ | | u (v) | (u) v | | | | | | | | | | | | | | du | dv | | | | | | | | | | | | | | v | | @ | | | | (u) (v) | | | o-----------------------------------------------------------o Figure 44-d. Difference Map of J (Digraph)
Table 45. Computation of dJ
Table 45. Computation of dJ o-------------------------------------------------------------------------------o | | | DJ = u v ((du)(dv)) + u (v)(du) dv + (u) v du (dv) + (u)(v) du dv | | | | => | | | | dj = u v (du, dv) + u (v) dv + (u) v du + (u)(v) . 0 | | | o-------------------------------------------------------------------------------o
Figure 46-a. Differential of J (Areal)
o---------------------------------------o | | | o | | / \ | | / \ | | / \ | | o o | | /%\ /%\ | | /%%%\ /%%%\ | | /%%%%%\ /%%%%%\ | | o%%%%%%%o%%%%%%%o | | /%\%%%%%/ \%%%%%/%\ | | /%%%\%%%/ \%%%/%%%\ | | /%%%%%\%/ \%/%%%%%\ | | o%%%%%%%o o%%%%%%%o | | / \%%%%%/%\ /%\%%%%%/ \ | | / \%%%/%%%\ /%%%\%%%/ \ | | / \%/%%%%%\ /%%%%%\%/ \ | | o o%%%%%%%o%%%%%%%o o | | |\ / \%%%%%/ \%%%%%/ \ /| | | | \ / \%%%/ \%%%/ \ / | | | | \ / \%/ \%/ \ / | | | | o o o o | | | | |\ / \ / \ /| | | | | | \ / \ / \ / | | | | | u | \ / \ / \ / | v | | | o---+---o o o---+---o | | | \ / \ / | | | | \ / \ / | | | | du \ / \ / dv | | | o-------o o-------o | | \ / | | \ / | | \ / | | o | | | o---------------------------------------o Figure 46-a. Differential of J (Areal)
Figure 46-b. Differential of J (Bundle)
o-----------------------------o
| |
| o-----o o-----o |
| / \ / \ |
| / o \ |
| / / \ \ |
| o o o o |
@ | du | | dv | |
/| o o o o |
/ | \ \ / / |
/ | \ o / |
/ | \ / \ / |
/ | o-----o o-----o |
/ | |
/ o-----------------------------o
/
o----------------------------------------/----o o-----------------------------o
| / | | |
| @ | | o-----o o-----o |
| | | /%%%%%%%\ / \ |
| o---------o o---------o | | /%%%%%%%%%o \ |
| / \ / \ | | /%%%%%%%%%/%\ \ |
| / o \ | | o%%%%%%%%%o%%%o o |
| / /`\ @------\-----------@ |%% du %%%|%%%| dv | |
| / /```\ \ | | o%%%%%%%%%o%%%o o |
| / /`````\ \ | | \%%%%%%%%%\%/ / |
| / /```````\ \ | | \%%%%%%%%%o / |
| o o`````````o o | | \%%%%%%%/ \ / |
| | |````@````| | | | o-----o o-----o |
| | |`````\```| | | | |
| | |``````\``| | | o-----------------------------o
| | u |```````\`| v | |
| | |````````\| | | o-----------------------------o
| | |`````````| | | | |
| | |`````````|\ | | | o-----o o-----o |
| o o`````````o \ o | | / \ /%%%%%%%\ |
| \ \```````/ \ / | | / o%%%%%%%%%\ |
| \ \`````/ \ / | | / /%\%%%%%%%%%\ |
| \ \```/ \ / | | o o%%%o%%%%%%%%%o |
| \ @------\-/---------\---------------@ | du |%%%|%%% dv %%| |
| \ o \ / | | o o%%%o%%%%%%%%%o |
| \ / \ / | | \ \%/%%%%%%%%%/ |
| o---------o o---------o \ | | \ o%%%%%%%%%/ |
| \ | | \ / \%%%%%%%/ |
| \ | | o-----o o-----o |
| \ | | |
o----------------------------------------\----o o-----------------------------o
\
\ o-----------------------------o
\ | |
\ | o-----o o-----o |
\ | /%%%%%%%\ /%%%%%%%\ |
\ | /%%%%%%%%%o%%%%%%%%%\ |
\ | /%%%%%%%%%/ \%%%%%%%%%\ |
\| o%%%%%%%%%o o%%%%%%%%%o |
@ |%% du %%%| |%%% dv %%| |
| o%%%%%%%%%o o%%%%%%%%%o |
| \%%%%%%%%%\ /%%%%%%%%%/ |
| \%%%%%%%%%o%%%%%%%%%/ |
| \%%%%%%%/ \%%%%%%%/ |
| o-----o o-----o |
| |
o-----------------------------o
Figure 46-b. Differential of J (Bundle)
Figure 46-c. Differential of J (Compact)
o---------------------------------------------------------------------o | | | | | o-------------------o o-------------------o | | / \ / \ | | / o \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / @ \ \ | | / / ^ ^ \ \ | | o o / \ o o | | | | / \ | | | | | | / \ | | | | | |/ \| | | | | u (du)/ dv du \(dv) v | | | | /| |\ | | | | / | | \ | | | | / | | \ | | | o / o o \ o | | \ / \ / \ / | | \ v \ du dv / v / | | \ @<----------------------->@ / | | \ \ / / | | \ \ / / | | \ o / | | \ / \ / | | o-------------------o o-------------------o | | | | | o---------------------------------------------------------------------o Figure 46-c. Differential of J (Compact)
Figure 46-d. Differential of J (Digraph)
o-----------------------------------------------------------o | | | u v | | @ | | ^ ^ | | / \ | | / \ | | / \ | | / \ | | (du) dv / \ du (dv) | | / \ | | / \ | | / \ | | / \ | | v v | | u (v) @<--------------------->@ (u) v | | du dv | | | | | | | | | | | | | | | | | | | | | | @ | | (u) (v) | | | o-----------------------------------------------------------o Figure 46-d. Differential of J (Digraph)
Table 47. Computation of rJ
Table 47. Computation of rJ o-------------------------------------------------------------------------------o | | | rJ = DJ + dJ | | | o-------------------------------------------------------------------------------o | | | DJ = u v ((du)(dv)) + u (v)(du) dv + (u) v du (dv) + (u)(v) du dv | | | | dJ = u v (du, dv) + u (v) dv + (u) v du + (u)(v) . 0 | | | o-------------------------------------------------------------------------------o | | | rJ = u v du dv + u (v) du dv + (u) v du dv + (u)(v) du dv | | | o-------------------------------------------------------------------------------o
Figure 48-a. Remainder of J (Areal)
o---------------------------------------o | | | o | | / \ | | / \ | | / \ | | o o | | / \ / \ | | / \ / \ | | / \ / \ | | o o o | | / \ /%\ / \ | | / \ /%%%\ / \ | | / \ /%%%%%\ / \ | | o o%%%%%%%o o | | / \ /%\%%%%%/%\ / \ | | / \ /%%%\%%%/%%%\ / \ | | / \ /%%%%%\%/%%%%%\ / \ | | o o%%%%%%%o%%%%%%%o o | | |\ / \%%%%%/%\%%%%%/ \ /| | | | \ / \%%%/%%%\%%%/ \ / | | | | \ / \%/%%%%%\%/ \ / | | | | o o%%%%%%%o o | | | | |\ / \%%%%%/ \ /| | | | | | \ / \%%%/ \ / | | | | | u | \ / \%/ \ / | v | | | o---+---o o o---+---o | | | \ / \ / | | | | \ / \ / | | | | du \ / \ / dv | | | o-------o o-------o | | \ / | | \ / | | \ / | | o | | | o---------------------------------------o Figure 48-a. Remainder of J (Areal)
Figure 48-b. Remainder of J (Bundle)
o-----------------------------o
| |
| o-----o o-----o |
| / \ / \ |
| / o \ |
| / /%\ \ |
| o o%%%o o |
@ | du |%%%| dv | |
/| o o%%%o o |
/ | \ \%/ / |
/ | \ o / |
/ | \ / \ / |
/ | o-----o o-----o |
/ | |
/ o-----------------------------o
/
o----------------------------------------/----o o-----------------------------o
| / | | |
| @ | | o-----o o-----o |
| | | / \ / \ |
| o---------o o---------o | | / o \ |
| / \ / \ | | / /%\ \ |
| / o \ | | o o%%%o o |
| / /`\ @------\-----------@ | du |%%%| dv | |
| / /```\ \ | | o o%%%o o |
| / /`````\ \ | | \ \%/ / |
| / /```````\ \ | | \ o / |
| o o`````````o o | | \ / \ / |
| | |````@````| | | | o-----o o-----o |
| | |`````\```| | | | |
| | |``````\``| | | o-----------------------------o
| | u |```````\`| v | |
| | |````````\| | | o-----------------------------o
| | |`````````| | | | |
| | |`````````|\ | | | o-----o o-----o |
| o o`````````o \ o | | / \ / \ |
| \ \```````/ \ / | | / o \ |
| \ \`````/ \ / | | / /%\ \ |
| \ \```/ \ / | | o o%%%o o |
| \ @------\-/---------\---------------@ | du |%%%| dv | |
| \ o \ / | | o o%%%o o |
| \ / \ / | | \ \%/ / |
| o---------o o---------o \ | | \ o / |
| \ | | \ / \ / |
| \ | | o-----o o-----o |
| \ | | |
o----------------------------------------\----o o-----------------------------o
\
\ o-----------------------------o
\ | |
\ | o-----o o-----o |
\ | / \ / \ |
\ | / o \ |
\ | / /%\ \ |
\| o o%%%o o |
@ | du |%%%| dv | |
| o o%%%o o |
| \ \%/ / |
| \ o / |
| \ / \ / |
| o-----o o-----o |
| |
o-----------------------------o
Figure 48-b. Remainder of J (Bundle)
Figure 48-c. Remainder of J (Compact)
o---------------------------------------------------------------------o | | | | | o-------------------o o-------------------o | | / \ / \ | | / o \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | / / \ \ | | o o o o | | | | | | | | | | | | | | | | du dv | | | | | u @<------------------------->@ v | | | | | | | | | | | | | | | | | | | | | o o @ o o | | \ \ ^ / / | | \ \ | / / | | \ \ | / / | | \ \ | / / | | \ \|/ / | | \ du | dv / | | \ /|\ / | | o-------------------o | o-------------------o | | | | | | | | v | | @ | | | o---------------------------------------------------------------------o Figure 48-c. Remainder of J (Compact)
Figure 48-d. Remainder of J (Digraph)
o-----------------------------------------------------------o | | | u v | | @ | | ^ | | | | | | | | | | | | | | | | | | | | | | | | | | | | | du | dv | | u (v) @<----------|---------->@ (u) v | | du | dv | | | | | | | | | | | | | | | | | | | | | | | | | | | | | v | | @ | | (u) (v) | | | o-----------------------------------------------------------o Figure 48-d. Remainder of J (Digraph)
Table 49. Computation Summary for J
Table 49. Computation Summary for J o-------------------------------------------------------------------------------o | | | !e!J = uv . 1 + u(v) . 0 + (u)v . 0 + (u)(v) . 0 | | | | EJ = uv . (du)(dv) + u(v) . (du)dv + (u)v . du(dv) + (u)(v) . du dv | | | | DJ = uv . ((du)(dv)) + u(v) . (du)dv + (u)v . du(dv) + (u)(v) . du dv | | | | dJ = uv . (du, dv) + u(v) . dv + (u)v . du + (u)(v) . 0 | | | | rJ = uv . du dv + u(v) . du dv + (u)v . du dv + (u)(v) . du dv | | | o-------------------------------------------------------------------------------o
Table 50. Computation of an Analytic Series in Terms of Coordinates
Table 50. Computation of an Analytic Series in Terms of Coordinates o-----------o-------------o-------------oo-------------o---------o-------------o | u v | du dv | u' v' || !e!J EJ | DJ | dJ d^2.J | o-----------o-------------o-------------oo-------------o---------o-------------o | | | || | | | | 0 0 | 0 0 | 0 0 || 0 0 | 0 | 0 0 | | | | || | | | | | 0 1 | 0 1 || 0 | 0 | 0 0 | | | | || | | | | | 1 0 | 1 0 || 0 | 0 | 0 0 | | | | || | | | | | 1 1 | 1 1 || 1 | 1 | 0 1 | | | | || | | | o-----------o-------------o-------------oo-------------o---------o-------------o | | | || | | | | 0 1 | 0 0 | 0 1 || 0 0 | 0 | 0 0 | | | | || | | | | | 0 1 | 0 0 || 0 | 0 | 0 0 | | | | || | | | | | 1 0 | 1 1 || 1 | 1 | 1 0 | | | | || | | | | | 1 1 | 1 0 || 0 | 0 | 1 1 | | | | || | | | o-----------o-------------o-------------oo-------------o---------o-------------o | | | || | | | | 1 0 | 0 0 | 1 0 || 0 0 | 0 | 0 0 | | | | || | | | | | 0 1 | 1 1 || 1 | 1 | 1 0 | | | | || | | | | | 1 0 | 0 0 || 0 | 0 | 0 0 | | | | || | | | | | 1 1 | 0 1 || 0 | 0 | 1 1 | | | | || | | | o-----------o-------------o-------------oo-------------o---------o-------------o | | | || | | | | 1 1 | 0 0 | 1 1 || 1 1 | 0 | 0 0 | | | | || | | | | | 0 1 | 1 0 || 0 | 1 | 1 0 | | | | || | | | | | 1 0 | 0 1 || 0 | 1 | 1 0 | | | | || | | | | | 1 1 | 0 0 || 0 | 1 | 0 1 | | | | || | | | o-----------o-------------o-------------oo-------------o---------o-------------o
Formula Display 9
o-------------------------------------------------o | | | u' = u + du = (u, du) | | | | v' = v + du = (v, dv) | | | o-------------------------------------------------o
Formula Display 10
o--------------------------------------------------------------o | | | EJ<u, v, du, dv> = J<u + du, v + dv> = J<u', v'> | | | o--------------------------------------------------------------o
Table 51. Computation of an Analytic Series in Symbolic Terms
Table 51. Computation of an Analytic Series in Symbolic Terms o-----------o---------o------------o------------o------------o-----------o | u v | J | EJ | DJ | dJ | d^2.J | o-----------o---------o------------o------------o------------o-----------o | | | | | | | | 0 0 | 0 | du dv | du dv | () | du dv | | | | | | | | | 0 1 | 0 | du (dv) | du (dv) | du | du dv | | | | | | | | | 1 0 | 0 | (du) dv | (du) dv | dv | du dv | | | | | | | | | 1 1 | 1 | (du)(dv) | ((du)(dv)) | (du, dv) | du dv | | | | | | | | o-----------o---------o------------o------------o------------o-----------o
Figure 52. Decomposition of the Enlarged Conjunction EJ = (J, DJ)
o o o
/%\ /%\ / \
/%%%\ /%%%\ / \
o%%%%%o o%%%%%o o o
/ \%%%/ \ /%\%%%/%\ /%\ /%\
/ \%/ \ /%%%\%/%%%\ /%%%\ /%%%\
o o o o%%%%%o%%%%%o o%%%%%o%%%%%o
/%\ / \ /%\ / \%%%/%\%%%/ \ /%\%%%/%\%%%/%\
/%%%\ / \ /%%%\ / \%/%%%\%/ \ /%%%\%/%%%\%/%%%\
o%%%%%o o%%%%%o o o%%%%%o o o%%%%%o%%%%%o%%%%%o
/ \%%%/ \ / \%%%/ \ / \ / \%%%/ \ / \ / \%%%/ \%%%/ \%%%/ \
/ \%/ \ / \%/ \ / \ / \%/ \ / \ / \%/ \%/ \%/ \
o o o o o o o o o o o o o o o
|\ / \ /%\ / \ /| |\ / \ / \ / \ /| |\ / \ /%\ / \ /|
| \ / \ /%%%\ / \ / | | \ / \ / \ / \ / | | \ / \ /%%%\ / \ / |
| o o%%%%%o o | | o o o o | | o o%%%%%o o |
| |\ / \%%%/ \ /| | | |\ / \ / \ /| | | |\ / \%%%/ \ /| |
|u | \ / \%/ \ / | v| |u | \ / \ / \ / | v| |u | \ / \%/ \ / | v|
o--+--o o o--+--o o--+--o o o--+--o o--+--o o o--+--o
| \ / \ / | | \ / \ / | | \ / \ / |
| du \ / \ / dv | | du \ / \ / dv | | du \ / \ / dv |
o-----o o-----o o-----o o-----o o-----o o-----o
\ / \ / \ /
\ / \ / \ /
o o o
EJ = J + DJ
o-----------------------o o-----------------------o o-----------------------o
| | | | | |
| o--o o--o | | o--o o--o | | o--o o--o |
| / \ / \ | | / \ / \ | | / \ / \ |
| / o \ | | / o \ | | / o \ |
| / u / \ v \ | | / u / \ v \ | | / u / \ v \ |
| o /->-\ o | | o /->-\ o | | o / \ o |
| | o \ / o | | | | o \ / o | | | | o o | |
| | @--|->@<-|--@ | | | | @<-|--@--|->@ | | | | @<-|->@<-|->@ | |
| | o ^ o | | | | o | o | | | | o ^ o | |
| o \ | / o | | o \ | / o | | o \ | / o |
| \ \|/ / | | \ \|/ / | | \ \|/ / |
| \ | / | | \ | / | | \ | / |
| \ /|\ / | | \ /|\ / | | \ /|\ / |
| o--o | o--o | | o--o v o--o | | o--o v o--o |
| @ | | @ | | @ |
o-----------------------o o-----------------------o o-----------------------o
Figure 52. Decomposition of the Enlarged Conjunction EJ = (J, DJ)
Figure 53. Decomposition of the Differed Conjunction DJ = (dJ, ddJ)
o o o
/ \ / \ / \
/ \ / \ / \
o o o o o o
/%\ /%\ /%\ /%\ / \ / \
/%%%\ /%%%\ /%%%\%/%%%\ / \ / \
o%%%%%o%%%%%o o%%%%%o%%%%%o o o o
/%\%%%/%\%%%/%\ /%\%%%/ \%%%/%\ / \ /%\ / \
/%%%\%/%%%\%/%%%\ /%%%\%/ \%/%%%\ / \ /%%%\ / \
o%%%%%o%%%%%o%%%%%o o%%%%%o o%%%%%o o o%%%%%o o
/ \%%%/ \%%%/ \%%%/ \ / \%%%/%\ /%\%%%/ \ / \ /%\%%%/%\ / \
/ \%/ \%/ \%/ \ / \%/%%%\ /%%%\%/ \ / \ /%%%\%/%%%\ / \
o o o o o o o%%%%%o%%%%%o o o o%%%%%o%%%%%o o
|\ / \ /%\ / \ /| |\ / \%%%/ \%%%/ \ /| |\ / \%%%/%\%%%/ \ /|
| \ / \ /%%%\ / \ / | | \ / \%/ \%/ \ / | | \ / \%/%%%\%/ \ / |
| o o%%%%%o o | | o o o o | | o o%%%%%o o |
| |\ / \%%%/ \ /| | | |\ / \ / \ /| | | |\ / \%%%/ \ /| |
|u | \ / \%/ \ / | v| |u | \ / \ / \ / | v| |u | \ / \%/ \ / | v|
o--+--o o o--+--o o--+--o o o--+--o o--+--o o o--+--o
| \ / \ / | | \ / \ / | | \ / \ / |
| du \ / \ / dv | | du \ / \ / dv | | du \ / \ / dv |
o-----o o-----o o-----o o-----o o-----o o-----o
\ / \ / \ /
\ / \ / \ /
o o o
DJ = dJ + ddJ
o-----------------------o o-----------------------o o-----------------------o
| | | | | |
| o--o o--o | | o--o o--o | | o--o o--o |
| / \ / \ | | / \ / \ | | / \ / \ |
| / o \ | | / o \ | | / o \ |
| / u / \ v \ | | / u / \ v \ | | / u / \ v \ |
| o / \ o | | o / \ o | | o / \ o |
| | o o | | | | o o | | | | o o | |
| | @<-|->@<-|->@ | | | | @<-|->@<-|->@ | | | | @<-|-----|->@ | |
| | o ^ o | | | | ^ o o ^ | | | | o @ o | |
| o \ | / o | | o \ \ / / o | | o \ ^ / o |
| \ \|/ / | | \ --\-/-- / | | \ \|/ / |
| \ | / | | \ o / | | \ | / |
| \ /|\ / | | \ / \ / | | \ /|\ / |
| o--o v o--o | | o--o o--o | | o--o v o--o |
| @ | | @ | | @ |
o-----------------------o o-----------------------o o-----------------------o
Figure 53. Decomposition of the Differed Conjunction DJ = (dJ, ddJ)
Table 54. Cast of Characters: Expansive Subtypes of Objects and Operators
Table 54. Cast of Characters: Expansive Subtypes of Objects and Operators o------o-------------------------o------------------o----------------------------o | Item | Notation | Description | Type | o------o-------------------------o------------------o----------------------------o | | | | | | U% | = [u, v] | Source Universe | [B^2] | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | X% | = [x] | Target Universe | [B^1] | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | EU% | = [u, v, du, dv] | Extended | [B^2 x D^2] | | | | Source Universe | | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | EX% | = [x, dx] | Extended | [B^1 x D^1] | | | | Target Universe | | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | J | J : U -> B | Proposition | (B^2 -> B) c [B^2] | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | J | J : U% -> X% | Transformation, | [B^2] -> [B^1] | | | | or Mapping | | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | W | W : | Operator | | | | U% -> EU%, | | [B^2] -> [B^2 x D^2], | | | X% -> EX%, | | [B^1] -> [B^1 x D^1], | | | (U%->X%)->(EU%->EX%), | | ([B^2] -> [B^1]) | | | for each W among: | | -> | | | e!, !h!, E, D, d | | ([B^2 x D^2]->[B^1 x D^1]) | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | !e! | | Tacit Extension Operator !e! | | !h! | | Trope Extension Operator !h! | | E | | Enlargement Operator E | | D | | Difference Operator D | | d | | Differential Operator d | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | $W$ | $W$ : | Operator | | | | U% -> $T$U% = EU%, | | [B^2] -> [B^2 x D^2], | | | X% -> $T$X% = EX%, | | [B^1] -> [B^1 x D^1], | | | (U%->X%)->($T$U%->$T$X%)| | ([B^2] -> [B^1]) | | | for each $W$ among: | | -> | | | $e$, $E$, $D$, $T$ | | ([B^2 x D^2]->[B^1 x D^1]) | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | $e$ | | Radius Operator $e$ = <!e!, !h!> | | $E$ | | Secant Operator $E$ = <!e!, E > | | $D$ | | Chord Operator $D$ = <!e!, D > | | $T$ | | Tangent Functor $T$ = <!e!, d > | | | | | o------o-------------------------o-----------------------------------------------o
Table 55. Synopsis of Terminology: Restrictive and Alternative Subtypes
Table 55. Synopsis of Terminology: Restrictive and Alternative Subtypes o--------------o----------------------o--------------------o----------------------o | | Operator | Proposition | Map | o--------------o----------------------o--------------------o----------------------o | | | | | | Tacit | !e! : | !e!J : | !e!J : | | Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> B | [u,v,du,dv]->[x] | | | (U%->X%)->(EU%->X%) | B^2 x D^2 -> B | [B^2 x D^2]->[B^1] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Trope | !h! : | !h!J : | !h!J : | | Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] | | | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Enlargement | E : | EJ : | EJ : | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] | | | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Difference | D : | DJ : | DJ : | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] | | | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Differential | d : | dJ : | dJ : | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] | | | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Remainder | r : | rJ : | rJ : | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] | | | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Radius | $e$ = <!e!, !h!> : | | $e$J : | | Operator | U%->EU%, X%->EX%, | | [u,v,du,dv]->[x, dx] | | | (U%->X%)->(EU%->EX%) | | [B^2 x D^2]->[B x D] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Secant | $E$ = <!e!, E> : | | $E$J : | | Operator | U%->EU%, X%->EX%, | | [u,v,du,dv]->[x, dx] | | | (U%->X%)->(EU%->EX%) | | [B^2 x D^2]->[B x D] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Chord | $D$ = <!e!, D> : | | $D$J : | | Operator | U%->EU%, X%->EX%, | | [u,v,du,dv]->[x, dx] | | | (U%->X%)->(EU%->EX%) | | [B^2 x D^2]->[B x D] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Tangent | $T$ = <!e!, d> : | dJ : | $T$J : | | Functor | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[x, dx] | | | (U%->X%)->(EU%->EX%) | B^2 x D^2 -> D | [B^2 x D^2]->[B x D] | | | | | | o--------------o----------------------o--------------------o----------------------o
Figure 56-a1. Radius Map of the Conjunction J = uv
o
/X\
/XXX\
oXXXXXo
/X\XXX/X\
/XXX\X/XXX\
oXXXXXoXXXXXo
/ \XXX/X\XXX/ \
/ \X/XXX\X/ \
o oXXXXXo o
/ \ / \XXX/ \ / \
/ \ / \X/ \ / \
o o o o o
=|\ / \ / \ / \ /|=
= | \ / \ / \ / \ / | =
= | o o o o | =
= | |\ / \ / \ /| | =
= |u | \ / \ / \ / | v| =
o o--+--o o o--+--o o
//\ | \ / \ / | /\\
////\ | du \ / \ / dv | /\\\\
o/////o o-----o o-----o o\\\\\o
//\/////\ \ / /\\\\\/\\
////\/////\ \ / /\\\\\/\\\\
o/////o/////o o o\\\\\o\\\\\o
/ \/////\//// \ = = / \\\\/\\\\\/ \
/ \/////\// \ = = / \\/\\\\\/ \
o o/////o o = = o o\\\\\o o
/ \ / \//// \ / \ = = / \ / \\\\/ \ / \
/ \ / \// \ / \ = = / \ / \\/ \ / \
o o o o o o o o o o
|\ / \ / \ / \ /| |\ / \ / \ / \ /|
| \ / \ / \ / \ / | | \ / \ / \ / \ / |
| o o o o | | o o o o |
| |\ / \ / \ /| | | |\ / \ / \ /| |
|u | \ / \ / \ / | v| |u | \ / \ / \ / | v|
o--+--o o o--+--o o o--+--o o o--+--o
. | \ / \ / | /X\ | \ / \ / | .
.| du \ / \ / dv | /XXX\ | du \ / \ / dv |.
o-----o o-----o /XXXXX\ o-----o o-----o
. \ / /XXXXXXX\ \ / .
. \ / /XXXXXXXXX\ \ / .
. o oXXXXXXXXXXXo o .
. //\XXXXXXXXX/\\ .
. ////\XXXXXXX/\\\\ .
!e!J //////\XXXXX/\\\\\\ !h!J
. ////////\XXX/\\\\\\\\ .
. //////////\X/\\\\\\\\\\ .
. o///////////o\\\\\\\\\\\o .
. |\////////// \\\\\\\\\\/| .
. | \//////// \\\\\\\\/ | .
. | \////// \\\\\\/ | .
. | \//// \\\\/ | .
.| x \// \\/ dx |.
o-----o o-----o
\ /
\ /
x = uv \ / dx = uv
\ /
\ /
o
Figure 56-a1. Radius Map of the Conjunction J = uv
Figure 56-a2. Secant Map of the Conjunction J = uv
o
/X\
/XXX\
oXXXXXo
//\XXX//\
////\X////\
o/////o/////o
/\\/////\////\\
/\\\\/////\//\\\\
o\\\\\o/////o\\\\\o
/ \\\\/ \//// \\\\/ \
/ \\/ \// \\/ \
o o o o o
=|\ / \ /\\ / \ /|=
= | \ / \ /\\\\ / \ / | =
= | o o\\\\\o o | =
= | |\ / \\\\/ \ /| | =
= |u | \ / \\/ \ / | v| =
o o--+--o o o--+--o o
//\ | \ / \ / | /\\
////\ | du \ / \ / dv | /\\\\
o/////o o-----o o-----o o\\\\\o
//\/////\ \ / / \\\\/ \
////\/////\ \ / / \\/ \
o/////o/////o o o o o
/ \/////\//// \ = = /\\ / \ /\\
/ \/////\// \ = = /\\\\ / \ /\\\\
o o/////o o = = o\\\\\o o\\\\\o
/ \ / \//// \ / \ = = / \\\\/ \ / \\\\/ \
/ \ / \// \ / \ = = / \\/ \ / \\/ \
o o o o o o o o o o
|\ / \ / \ / \ /| |\ / \ /\\ / \ /|
| \ / \ / \ / \ / | | \ / \ /\\\\ / \ / |
| o o o o | | o o\\\\\o o |
| |\ / \ / \ /| | | |\ / \\\\/ \ /| |
|u | \ / \ / \ / | v| |u | \ / \\/ \ / | v|
o--+--o o o--+--o o o--+--o o o--+--o
. | \ / \ / | /X\ | \ / \ / | .
.| du \ / \ / dv | /XXX\ | du \ / \ / dv |.
o-----o o-----o /XXXXX\ o-----o o-----o
. \ / /XXXXXXX\ \ / .
. \ / /XXXXXXXXX\ \ / .
. o oXXXXXXXXXXXo o .
. //\XXXXXXXXX/\\ .
. ////\XXXXXXX/\\\\ .
!e!J //////\XXXXX/\\\\\\ EJ
. ////////\XXX/\\\\\\\\ .
. //////////\X/\\\\\\\\\\ .
. o///////////o\\\\\\\\\\\o .
. |\////////// \\\\\\\\\\/| .
. | \//////// \\\\\\\\/ | .
. | \////// \\\\\\/ | .
. | \//// \\\\/ | .
.| x \// \\/ dx |.
o-----o o-----o
\ /
\ / dx = (u, du)(v, dv)
x = uv \ /
\ / dx = uv + u dv + v du + du dv
\ /
o
Figure 56-a2. Secant Map of the Conjunction J = uv
Figure 56-a3. Chord Map of the Conjunction J = uv
o
//\
////\
o/////o
/X\////X\
/XXX\//XXX\
oXXXXXoXXXXXo
/\\XXX/X\XXX/\\
/\\\\X/XXX\X/\\\\
o\\\\\oXXXXXo\\\\\o
/ \\\\/ \XXX/ \\\\/ \
/ \\/ \X/ \\/ \
o o o o o
=|\ / \ /\\ / \ /|=
= | \ / \ /\\\\ / \ / | =
= | o o\\\\\o o | =
= | |\ / \\\\/ \ /| | =
= |u | \ / \\/ \ / | v| =
o o--+--o o o--+--o o
//\ | \ / \ / | / \
////\ | du \ / \ / dv | / \
o/////o o-----o o-----o o o
//\/////\ \ / /\\ /\\
////\/////\ \ / /\\\\ /\\\\
o/////o/////o o o\\\\\o\\\\\o
/ \/////\//// \ = = /\\\\\/\\\\\/\\
/ \/////\// \ = = /\\\\\/\\\\\/\\\\
o o/////o o = = o\\\\\o\\\\\o\\\\\o
/ \ / \//// \ / \ = = / \\\\/ \\\\/ \\\\/ \
/ \ / \// \ / \ = = / \\/ \\/ \\/ \
o o o o o o o o o o
|\ / \ / \ / \ /| |\ / \ /\\ / \ /|
| \ / \ / \ / \ / | | \ / \ /\\\\ / \ / |
| o o o o | | o o\\\\\o o |
| |\ / \ / \ /| | | |\ / \\\\/ \ /| |
|u | \ / \ / \ / | v| |u | \ / \\/ \ / | v|
o--+--o o o--+--o o o--+--o o o--+--o
. | \ / \ / | /X\ | \ / \ / | .
.| du \ / \ / dv | /XXX\ | du \ / \ / dv |.
o-----o o-----o /XXXXX\ o-----o o-----o
. \ / /XXXXXXX\ \ / .
. \ / /XXXXXXXXX\ \ / .
. o oXXXXXXXXXXXo o .
. //\XXXXXXXXX/\\ .
. ////\XXXXXXX/\\\\ .
!e!J //////\XXXXX/\\\\\\ DJ
. ////////\XXX/\\\\\\\\ .
. //////////\X/\\\\\\\\\\ .
. o///////////o\\\\\\\\\\\o .
. |\////////// \\\\\\\\\\/| .
. | \//////// \\\\\\\\/ | .
. | \////// \\\\\\/ | .
. | \//// \\\\/ | .
.| x \// \\/ dx |.
o-----o o-----o
\ /
\ / dx = (u, du)(v, dv) - uv
x = uv \ /
\ / dx = u dv + v du + du dv
\ /
o
Figure 56-a3. Chord Map of the Conjunction J = uv
Figure 56-a4. Tangent Map of the Conjunction J = uv
o
//\
////\
o/////o
/X\////X\
/XXX\//XXX\
oXXXXXoXXXXXo
/\\XXX//\XXX/\\
/\\\\X////\X/\\\\
o\\\\\o/////o\\\\\o
/ \\\\/\\////\\\\\/ \
/ \\/\\\\//\\\\\/ \
o o\\\\\o\\\\\o o
=|\ / \\\\/ \\\\/ \ /|=
= | \ / \\/ \\/ \ / | =
= | o o o o | =
= | |\ / \ / \ /| | =
= |u | \ / \ / \ / | v| =
o o--+--o o o--+--o o
//\ | \ / \ / | / \
////\ | du \ / \ / dv | / \
o/////o o-----o o-----o o o
//\/////\ \ / /\\ /\\
////\/////\ \ / /\\\\ /\\\\
o/////o/////o o o\\\\\o\\\\\o
/ \/////\//// \ = = /\\\\\/ \\\\/\\
/ \/////\// \ = = /\\\\\/ \\/\\\\
o o/////o o = = o\\\\\o o\\\\\o
/ \ / \//// \ / \ = = / \\\\/\\ /\\\\\/ \
/ \ / \// \ / \ = = / \\/\\\\ /\\\\\/ \
o o o o o o o\\\\\o\\\\\o o
|\ / \ / \ / \ /| |\ / \\\\/ \\\\/ \ /|
| \ / \ / \ / \ / | | \ / \\/ \\/ \ / |
| o o o o | | o o o o |
| |\ / \ / \ /| | | |\ / \ / \ /| |
|u | \ / \ / \ / | v| |u | \ / \ / \ / | v|
o--+--o o o--+--o o o--+--o o o--+--o
. | \ / \ / | /X\ | \ / \ / | .
.| du \ / \ / dv | /XXX\ | du \ / \ / dv |.
o-----o o-----o /XXXXX\ o-----o o-----o
. \ / /XXXXXXX\ \ / .
. \ / /XXXXXXXXX\ \ / .
. o oXXXXXXXXXXXo o .
. //\XXXXXXXXX/\\ .
. ////\XXXXXXX/\\\\ .
!e!J //////\XXXXX/\\\\\\ dJ
. ////////\XXX/\\\\\\\\ .
. //////////\X/\\\\\\\\\\ .
. o///////////o\\\\\\\\\\\o .
. |\////////// \\\\\\\\\\/| .
. | \//////// \\\\\\\\/ | .
. | \////// \\\\\\/ | .
. | \//// \\\\/ | .
.| x \// \\/ dx |.
o-----o o-----o
\ /
\ /
x = uv \ / dx = u dv + v du
\ /
\ /
o
Figure 56-a4. Tangent Map of the Conjunction J = uv
Figure 56-b1. Radius Map of the Conjunction J = uv
o-----------------------o
| |
| |
| |
| o--o o--o |
| / \ / \ |
| / o \ |
| / du / \ dv \ |
| o / \ o |
| | o o | |
| | | | | |
| | o o | |
| o \ / o |
| \ \ / / |
| \ o / |
| \ / \ / |
| o--o o--o |
| |
| |
| |
o-----------------------@
\
o-----------------------o \
| | \
| | \
| | \
| o--o o--o | \
| / \ / \ | \
| / o \ | \
| / du / \ dv \ | \
| o / \ o | \
| | o o | @ \
| | | | | |\ \
| | o o | | \ \
| o \ / o | \ \
| \ \ / / | \ \
| \ o / | \ \
| \ / \ / | \ \
| o--o o--o | \ \
| | \ \
| | \ \
| | \ \
o-----------------------o \ \
\ \
o-----------------------@ o--------\----------\---o o-----------------------o
| |\ | \ \ | |```````````````````````|
| | \ | \ @ | |```````````````````````|
| | \| \ | |```````````````````````|
| o--o o--o | \ o--o \o--o | |``````o--o```o--o``````|
| / \ / \ | |\ / \ /\ \ | |`````/````\`/````\`````|
| / o \ | | \ / o @ \ | |````/``````o``````\````|
| / du / \ dv \ | | \/ du /`\ dv \ | |```/``du``/`\``dv``\```|
| o / \ o | | o\ /```\ o | |``o``````/```\``````o``|
| | o o | | | | \ o`````o | | |``|`````o`````o`````|``|
| | | | | | | | @ |``@--|-----|------@``|`````|`````|`````|``|
| | o o | | | | o`````o | | |``|`````o`````o`````|``|
| o \ / o | | o \```/ o | |``o``````\```/``````o``|
| \ \ / / | | \ \`/ / | |```\``````\`/``````/```|
| \ o / | | \ o / | |````\``````o``````/````|
| \ / \ / | | \ / \ / | |`````\````/`\````/`````|
| o--o o--o | | o--o o--o | |``````o--o```o--o``````|
| | | | |```````````````````````|
| | | | |```````````````````````|
| | | | |```````````````````````|
o-----------------------o o-----------------------o o-----------------------o
\ / \ / \ /
\ !h!J / \ J / \ !h!J /
\ / \ / \ /
\ / o----------\---------/----------o \ /
\ / | \ / | \ /
\ / | \ / | \ /
\ / | o-----o-----o | \ /
\ / | /`````````````\ | \ /
\ / | /```````````````\ | \ /
o------\---/------o | /`````````````````\ | o------\---/------o
| \ / | | /```````````````````\ | | \ / |
| o--o--o | | /`````````````````````\ | | o--o--o |
| /```````\ | | o```````````````````````o | | /```````\ |
| /`````````\ | | |```````````````````````| | | /`````````\ |
| o```````````o | | |```````````````````````| | | o```````````o |
| |````dx`````| @----@ |```````````x`````@-----|------@ |``` dx ````| |
| o```````````o | | |```````````````````````| | | o```````````o |
| \`````````/ | | |```````````````````````| | | \`````````/ |
| \```````/ | | o```````````````````````o | | \```````/ |
| o-----o | | \`````````````````````/ | | o-----o |
| | | \```````````````````/ | | |
o-----------------o | \`````````````````/ | o-----------------o
| \```````````````/ |
| \`````````````/ |
| o-----------o |
| |
| |
o-------------------------------o
Figure 56-b1. Radius Map of the Conjunction J = uv
Figure 56-b2. Secant Map of the Conjunction J = uv
o-----------------------o
| |
| |
| |
| o--o o--o |
| / \ / \ |
| / o \ |
| / du /`\ dv \ |
| o /```\ o |
| | o`````o | |
| | |`````| | |
| | o`````o | |
| o \```/ o |
| \ \`/ / |
| \ o / |
| \ / \ / |
| o--o o--o |
| |
| |
| |
o-----------------------@
\
o-----------------------o \
| | \
| | \
| | \
| o--o o--o | \
| /````\ / \ | \
| /``````o \ | \
| /``du``/ \ dv \ | \
| o``````/ \ o | \
| |`````o o | @ \
| |`````| | | |\ \
| |`````o o | | \ \
| o``````\ / o | \ \
| \``````\ / / | \ \
| \``````o / | \ \
| \````/ \ / | \ \
| o--o o--o | \ \
| | \ \
| | \ \
| | \ \
o-----------------------o \ \
\ \
o-----------------------@ o--------\----------\---o o-----------------------o
| |\ | \ \ | |```````````````````````|
| | \ | \ @ | |```````````````````````|
| | \| \ | |```````````````````````|
| o--o o--o | \ o--o \o--o | |``````o--o```o--o``````|
| / \ /````\ | |\ / \ /\ \ | |`````/ \`/ \`````|
| / o``````\ | | \ / o @ \ | |````/ o \````|
| / du / \``dv``\ | | \/ du /`\ dv \ | |```/ du / \ dv \```|
| o / \``````o | | o\ /```\ o | |``o / \ o``|
| | o o`````| | | | \ o`````o | | |``| o o |``|
| | | |`````| | | | @ |``@--|-----|------@``| | | |``|
| | o o`````| | | | o`````o | | |``| o o |``|
| o \ /``````o | | o \```/ o | |``o \ / o``|
| \ \ /``````/ | | \ \`/ / | |```\ \ / /```|
| \ o``````/ | | \ o / | |````\ o /````|
| \ / \````/ | | \ / \ / | |`````\ /`\ /`````|
| o--o o--o | | o--o o--o | |``````o--o```o--o``````|
| | | | |```````````````````````|
| | | | |```````````````````````|
| | | | |```````````````````````|
o-----------------------o o-----------------------o o-----------------------o
\ / \ / \ /
\ EJ / \ J / \ EJ /
\ / \ / \ /
\ / o----------\---------/----------o \ /
\ / | \ / | \ /
\ / | \ / | \ /
\ / | o-----o-----o | \ /
\ / | /`````````````\ | \ /
\ / | /```````````````\ | \ /
o------\---/------o | /`````````````````\ | o------\---/------o
| \ / | | /```````````````````\ | | \ / |
| o--o--o | | /`````````````````````\ | | o--o--o |
| /```````\ | | o```````````````````````o | | /```````\ |
| /`````````\ | | |```````````````````````| | | /`````````\ |
| o```````````o | | |```````````````````````| | | o```````````o |
| |````dx`````| @----@ |```````````x`````@-----|------@ |``` dx ````| |
| o```````````o | | |```````````````````````| | | o```````````o |
| \`````````/ | | |```````````````````````| | | \`````````/ |
| \```````/ | | o```````````````````````o | | \```````/ |
| o-----o | | \`````````````````````/ | | o-----o |
| | | \```````````````````/ | | |
o-----------------o | \`````````````````/ | o-----------------o
| \```````````````/ |
| \`````````````/ |
| o-----------o |
| |
| |
o-------------------------------o
Figure 56-b2. Secant Map of the Conjunction J = uv
Figure 56-b3. Chord Map of the Conjunction J = uv
o-----------------------o
| |
| |
| |
| o--o o--o |
| / \ / \ |
| / o \ |
| / du /`\ dv \ |
| o /```\ o |
| | o`````o | |
| | |`````| | |
| | o`````o | |
| o \```/ o |
| \ \`/ / |
| \ o / |
| \ / \ / |
| o--o o--o |
| |
| |
| |
o-----------------------@
\
o-----------------------o \
| | \
| | \
| | \
| o--o o--o | \
| /````\ / \ | \
| /``````o \ | \
| /``du``/ \ dv \ | \
| o``````/ \ o | \
| |`````o o | @ \
| |`````| | | |\ \
| |`````o o | | \ \
| o``````\ / o | \ \
| \``````\ / / | \ \
| \``````o / | \ \
| \````/ \ / | \ \
| o--o o--o | \ \
| | \ \
| | \ \
| | \ \
o-----------------------o \ \
\ \
o-----------------------@ o--------\----------\---o o-----------------------o
| |\ | \ \ | | |
| | \ | \ @ | | |
| | \| \ | | |
| o--o o--o | \ o--o \o--o | | o--o o--o |
| / \ /````\ | |\ / \ /\ \ | | /````\ /````\ |
| / o``````\ | | \ / o @ \ | | /``````o``````\ |
| / du / \``dv``\ | | \/ du /`\ dv \ | | /``du``/`\``dv``\ |
| o / \``````o | | o\ /```\ o | | o``````/```\``````o |
| | o o`````| | | | \ o`````o | | | |`````o`````o`````| |
| | | |`````| | | | @ |``@--|-----|------@ |`````|`````|`````| |
| | o o`````| | | | o`````o | | | |`````o`````o`````| |
| o \ /``````o | | o \```/ o | | o``````\```/``````o |
| \ \ /``````/ | | \ \`/ / | | \``````\`/``````/ |
| \ o``````/ | | \ o / | | \``````o``````/ |
| \ / \````/ | | \ / \ / | | \````/ \````/ |
| o--o o--o | | o--o o--o | | o--o o--o |
| | | | | |
| | | | | |
| | | | | |
o-----------------------o o-----------------------o o-----------------------o
\ / \ / \ /
\ DJ / \ J / \ DJ /
\ / \ / \ /
\ / o----------\---------/----------o \ /
\ / | \ / | \ /
\ / | \ / | \ /
\ / | o-----o-----o | \ /
\ / | /`````````````\ | \ /
\ / | /```````````````\ | \ /
o------\---/------o | /`````````````````\ | o------\---/------o
| \ / | | /```````````````````\ | | \ / |
| o--o--o | | /`````````````````````\ | | o--o--o |
| /```````\ | | o```````````````````````o | | /```````\ |
| /`````````\ | | |```````````````````````| | | /`````````\ |
| o```````````o | | |```````````````````````| | | o```````````o |
| |````dx`````| @----@ |```````````x`````@-----|------@ |``` dx ````| |
| o```````````o | | |```````````````````````| | | o```````````o |
| \`````````/ | | |```````````````````````| | | \`````````/ |
| \```````/ | | o```````````````````````o | | \```````/ |
| o-----o | | \`````````````````````/ | | o-----o |
| | | \```````````````````/ | | |
o-----------------o | \`````````````````/ | o-----------------o
| \```````````````/ |
| \`````````````/ |
| o-----------o |
| |
| |
o-------------------------------o
Figure 56-b3. Chord Map of the Conjunction J = uv
Figure 56-b4. Tangent Map of the Conjunction J = uv
o-----------------------o
| |
| |
| |
| o--o o--o |
| / \ / \ |
| / o \ |
| / du / \ dv \ |
| o / \ o |
| | o o | |
| | | | | |
| | o o | |
| o \ / o |
| \ \ / / |
| \ o / |
| \ / \ / |
| o--o o--o |
| |
| |
| |
o-----------------------@
\
o-----------------------o \
| | \
| | \
| | \
| o--o o--o | \
| /````\ / \ | \
| /``````o \ | \
| /``du``/`\ dv \ | \
| o``````/```\ o | \
| |`````o`````o | @ \
| |`````|`````| | |\ \
| |`````o`````o | | \ \
| o``````\```/ o | \ \
| \``````\`/ / | \ \
| \``````o / | \ \
| \````/ \ / | \ \
| o--o o--o | \ \
| | \ \
| | \ \
| | \ \
o-----------------------o \ \
\ \
o-----------------------@ o--------\----------\---o o-----------------------o
| |\ | \ \ | | |
| | \ | \ @ | | |
| | \| \ | | |
| o--o o--o | \ o--o \o--o | | o--o o--o |
| / \ /````\ | |\ / \ /\ \ | | /````\ /````\ |
| / o``````\ | | \ / o @ \ | | /``````o``````\ |
| / du /`\``dv``\ | | \/ du /`\ dv \ | | /``du``/ \``dv``\ |
| o /```\``````o | | o\ /```\ o | | o``````/ \``````o |
| | o`````o`````| | | | \ o`````o | | | |`````o o`````| |
| | |`````|`````| | | | @ |``@--|-----|------@ |`````| |`````| |
| | o`````o`````| | | | o`````o | | | |`````o o`````| |
| o \```/``````o | | o \```/ o | | o``````\ /``````o |
| \ \`/``````/ | | \ \`/ / | | \``````\ /``````/ |
| \ o``````/ | | \ o / | | \``````o``````/ |
| \ / \````/ | | \ / \ / | | \````/ \````/ |
| o--o o--o | | o--o o--o | | o--o o--o |
| | | | | |
| | | | | |
| | | | | |
o-----------------------o o-----------------------o o-----------------------o
\ / \ / \ /
\ dJ / \ J / \ dJ /
\ / \ / \ /
\ / o----------\---------/----------o \ /
\ / | \ / | \ /
\ / | \ / | \ /
\ / | o-----o-----o | \ /
\ / | /`````````````\ | \ /
\ / | /```````````````\ | \ /
o------\---/------o | /`````````````````\ | o------\---/------o
| \ / | | /```````````````````\ | | \ / |
| o--o--o | | /`````````````````````\ | | o--o--o |
| /```````\ | | o```````````````````````o | | /```````\ |
| /`````````\ | | |```````````````````````| | | /`````````\ |
| o```````````o | | |```````````````````````| | | o```````````o |
| |````dx`````| @----@ |```````````x`````@-----|------@ |``` dx ````| |
| o```````````o | | |```````````````````````| | | o```````````o |
| \`````````/ | | |```````````````````````| | | \`````````/ |
| \```````/ | | o```````````````````````o | | \```````/ |
| o-----o | | \`````````````````````/ | | o-----o |
| | | \```````````````````/ | | |
o-----------------o | \`````````````````/ | o-----------------o
| \```````````````/ |
| \`````````````/ |
| o-----------o |
| |
| |
o-------------------------------o
Figure 56-b4. Tangent Map of the Conjunction J = uv
Figure 57-1. Radius Operator Diagram for the Conjunction J = uv
o o
//\ /X\
////\ /XXX\
//////\ oXXXXXo
////////\ /X\XXX/X\
//////////\ /XXX\X/XXX\
o///////////o oXXXXXoXXXXXo
/ \////////// \ / \XXX/X\XXX/ \
/ \//////// \ / \X/XXX\X/ \
/ \////// \ o oXXXXXo o
/ \//// \ / \ / \XXX/ \ / \
/ \// \ / \ / \X/ \ / \
o o o o o o o o
|\ / \ /| |\ / \ / \ / \ /|
| \ / \ / | | \ / \ / \ / \ / |
| \ / \ / | | o o o o |
| \ / \ / | | |\ / \ / \ /| |
| u \ / \ / v | |u | \ / \ / \ / | v|
o-----o o-----o o--+--o o o--+--o
\ / | \ / \ / |
\ / | du \ / \ / dv |
\ / o-----o o-----o
\ / \ /
\ / \ /
o o
U% $e$ $E$U%
o------------------>o
| |
| |
| |
| |
J | | $e$J
| |
| |
| |
v v
o------------------>o
X% $e$ $E$X%
o o
//\ /X\
////\ /XXX\
//////\ /XXXXX\
////////\ /XXXXXXX\
//////////\ /XXXXXXXXX\
////////////o oXXXXXXXXXXXo
///////////// \ //\XXXXXXXXX/\\
///////////// \ ////\XXXXXXX/\\\\
///////////// \ //////\XXXXX/\\\\\\
///////////// \ ////////\XXX/\\\\\\\\
///////////// \ //////////\X/\\\\\\\\\\
o//////////// o o///////////o\\\\\\\\\\\o
|\////////// / |\////////// \\\\\\\\\\/|
| \//////// / | \//////// \\\\\\\\/ |
| \////// / | \////// \\\\\\/ |
| \//// / | \//// \\\\/ |
| x \// / | x \// \\/ dx |
o-----o / o-----o o-----o
\ / \ /
\ / \ /
\ / \ /
\ / \ /
\ / \ /
o o
Figure 57-1. Radius Operator Diagram for the Conjunction J = uv
Figure 57-2. Secant Operator Diagram for the Conjunction J = uv
o o
//\ /X\
////\ /XXX\
//////\ oXXXXXo
////////\ //\XXX//\
//////////\ ////\X////\
o///////////o o/////o/////o
/ \////////// \ /\\/////\////\\
/ \//////// \ /\\\\/////\//\\\\
/ \////// \ o\\\\\o/////o\\\\\o
/ \//// \ / \\\\/ \//// \\\\/ \
/ \// \ / \\/ \// \\/ \
o o o o o o o o
|\ / \ /| |\ / \ /\\ / \ /|
| \ / \ / | | \ / \ /\\\\ / \ / |
| \ / \ / | | o o\\\\\o o |
| \ / \ / | | |\ / \\\\/ \ /| |
| u \ / \ / v | |u | \ / \\/ \ / | v|
o-----o o-----o o--+--o o o--+--o
\ / | \ / \ / |
\ / | du \ / \ / dv |
\ / o-----o o-----o
\ / \ /
\ / \ /
o o
U% $E$ $E$U%
o------------------>o
| |
| |
| |
| |
J | | $E$J
| |
| |
| |
v v
o------------------>o
X% $E$ $E$X%
o o
//\ /X\
////\ /XXX\
//////\ /XXXXX\
////////\ /XXXXXXX\
//////////\ /XXXXXXXXX\
////////////o oXXXXXXXXXXXo
///////////// \ //\XXXXXXXXX/\\
///////////// \ ////\XXXXXXX/\\\\
///////////// \ //////\XXXXX/\\\\\\
///////////// \ ////////\XXX/\\\\\\\\
///////////// \ //////////\X/\\\\\\\\\\
o//////////// o o///////////o\\\\\\\\\\\o
|\////////// / |\////////// \\\\\\\\\\/|
| \//////// / | \//////// \\\\\\\\/ |
| \////// / | \////// \\\\\\/ |
| \//// / | \//// \\\\/ |
| x \// / | x \// \\/ dx |
o-----o / o-----o o-----o
\ / \ /
\ / \ /
\ / \ /
\ / \ /
\ / \ /
o o
Figure 57-2. Secant Operator Diagram for the Conjunction J = uv
Figure 57-3. Chord Operator Diagram for the Conjunction J = uv
o o
//\ //\
////\ ////\
//////\ o/////o
////////\ /X\////X\
//////////\ /XXX\//XXX\
o///////////o oXXXXXoXXXXXo
/ \////////// \ /\\XXX/X\XXX/\\
/ \//////// \ /\\\\X/XXX\X/\\\\
/ \////// \ o\\\\\oXXXXXo\\\\\o
/ \//// \ / \\\\/ \XXX/ \\\\/ \
/ \// \ / \\/ \X/ \\/ \
o o o o o o o o
|\ / \ /| |\ / \ /\\ / \ /|
| \ / \ / | | \ / \ /\\\\ / \ / |
| \ / \ / | | o o\\\\\o o |
| \ / \ / | | |\ / \\\\/ \ /| |
| u \ / \ / v | |u | \ / \\/ \ / | v|
o-----o o-----o o--+--o o o--+--o
\ / | \ / \ / |
\ / | du \ / \ / dv |
\ / o-----o o-----o
\ / \ /
\ / \ /
o o
U% $D$ $E$U%
o------------------>o
| |
| |
| |
| |
J | | $D$J
| |
| |
| |
v v
o------------------>o
X% $D$ $E$X%
o o
//\ /X\
////\ /XXX\
//////\ /XXXXX\
////////\ /XXXXXXX\
//////////\ /XXXXXXXXX\
////////////o oXXXXXXXXXXXo
///////////// \ //\XXXXXXXXX/\\
///////////// \ ////\XXXXXXX/\\\\
///////////// \ //////\XXXXX/\\\\\\
///////////// \ ////////\XXX/\\\\\\\\
///////////// \ //////////\X/\\\\\\\\\\
o//////////// o o///////////o\\\\\\\\\\\o
|\////////// / |\////////// \\\\\\\\\\/|
| \//////// / | \//////// \\\\\\\\/ |
| \////// / | \////// \\\\\\/ |
| \//// / | \//// \\\\/ |
| x \// / | x \// \\/ dx |
o-----o / o-----o o-----o
\ / \ /
\ / \ /
\ / \ /
\ / \ /
\ / \ /
o o
Figure 57-3. Chord Operator Diagram for the Conjunction J = uv
Figure 57-4. Tangent Functor Diagram for the Conjunction J = uv
o o
//\ //\
////\ ////\
//////\ o/////o
////////\ /X\////X\
//////////\ /XXX\//XXX\
o///////////o oXXXXXoXXXXXo
/ \////////// \ /\\XXX//\XXX/\\
/ \//////// \ /\\\\X////\X/\\\\
/ \////// \ o\\\\\o/////o\\\\\o
/ \//// \ / \\\\/\\////\\\\\/ \
/ \// \ / \\/\\\\//\\\\\/ \
o o o o o\\\\\o\\\\\o o
|\ / \ /| |\ / \\\\/ \\\\/ \ /|
| \ / \ / | | \ / \\/ \\/ \ / |
| \ / \ / | | o o o o |
| \ / \ / | | |\ / \ / \ /| |
| u \ / \ / v | |u | \ / \ / \ / | v|
o-----o o-----o o--+--o o o--+--o
\ / | \ / \ / |
\ / | du \ / \ / dv |
\ / o-----o o-----o
\ / \ /
\ / \ /
o o
U% $T$ $E$U%
o------------------>o
| |
| |
| |
| |
J | | $T$J
| |
| |
| |
v v
o------------------>o
X% $T$ $E$X%
o o
//\ /X\
////\ /XXX\
//////\ /XXXXX\
////////\ /XXXXXXX\
//////////\ /XXXXXXXXX\
////////////o oXXXXXXXXXXXo
///////////// \ //\XXXXXXXXX/\\
///////////// \ ////\XXXXXXX/\\\\
///////////// \ //////\XXXXX/\\\\\\
///////////// \ ////////\XXX/\\\\\\\\
///////////// \ //////////\X/\\\\\\\\\\
o//////////// o o///////////o\\\\\\\\\\\o
|\////////// / |\////////// \\\\\\\\\\/|
| \//////// / | \//////// \\\\\\\\/ |
| \////// / | \////// \\\\\\/ |
| \//// / | \//// \\\\/ |
| x \// / | x \// \\/ dx |
o-----o / o-----o o-----o
\ / \ /
\ / \ /
\ / \ /
\ / \ /
\ / \ /
o o
Figure 57-4. Tangent Functor Diagram for the Conjunction J = uv
Formula Display 11
o-----------------------------------------------------------o | | | F = <f, g> = <F_1, F_2> : [u, v] -> [x, y] | | | | where f = F_1 : [u, v] -> [x] | | | | and g = F_2 : [u, v] -> [y] | | | o-----------------------------------------------------------o
Table 58. Cast of Characters: Expansive Subtypes of Objects and Operators
Table 58. Cast of Characters: Expansive Subtypes of Objects and Operators o------o-------------------------o------------------o----------------------------o | Item | Notation | Description | Type | o------o-------------------------o------------------o----------------------------o | | | | | | U% | = [u, v] | Source Universe | [B^n] | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | X% | = [x, y] | Target Universe | [B^k] | | | = [f, g] | | | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | EU% | = [u, v, du, dv] | Extended | [B^n x D^n] | | | | Source Universe | | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | EX% | = [x, y, dx, dy] | Extended | [B^k x D^k] | | | = [f, g, df, dg] | Target Universe | | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | F | F = <f, g> : U% -> X% | Transformation, | [B^n] -> [B^k] | | | | or Mapping | | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | | f, g : U -> B | Proposition, | B^n -> B | | | | special case | | | f | f : U -> [x] c X% | of a mapping, | c (B^n, B^n -> B) | | | | or component | | | g | g : U -> [y] c X% | of a mapping. | = (B^n +-> B) = [B^n] | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | W | W : | Operator | | | | U% -> EU%, | | [B^n] -> [B^n x D^n], | | | X% -> EX%, | | [B^k] -> [B^k x D^k], | | | (U%->X%)->(EU%->EX%), | | ([B^n] -> [B^k]) | | | for each W among: | | -> | | | !e!, !h!, E, D, d | | ([B^n x D^n]->[B^k x D^k]) | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | !e! | | Tacit Extension Operator !e! | | !h! | | Trope Extension Operator !h! | | E | | Enlargement Operator E | | D | | Difference Operator D | | d | | Differential Operator d | | | | | o------o-------------------------o------------------o----------------------------o | | | | | | $W$ | $W$ : | Operator | | | | U% -> $T$U% = EU%, | | [B^n] -> [B^n x D^n], | | | X% -> $T$X% = EX%, | | [B^k] -> [B^k x D^k], | | | (U%->X%)->($T$U%->$T$X%)| | ([B^n] -> [B^k]) | | | for each $W$ among: | | -> | | | $e$, $E$, $D$, $T$ | | ([B^n x D^n]->[B^k x D^k]) | | | | | | o------o-------------------------o------------------o----------------------------o | | | | | $e$ | | Radius Operator $e$ = <!e!, !h!> | | $E$ | | Secant Operator $E$ = <!e!, E > | | $D$ | | Chord Operator $D$ = <!e!, D > | | $T$ | | Tangent Functor $T$ = <!e!, d > | | | | | o------o-------------------------o-----------------------------------------------o
Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes
Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes o--------------o----------------------o--------------------o----------------------o | | Operator | Proposition | Transformation | | | or | or | or | | | Operand | Component | Mapping | o--------------o----------------------o--------------------o----------------------o | | | | | | Operand | F = <F_1, F_2> | F_i : <|u,v|> -> B | F : [u, v] -> [x, y] | | | | | | | | F = <f, g> : U -> X | F_i : B^n -> B | F : B^n -> B^k | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Tacit | !e! : | !e!F_i : | !e!F : | | Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> B | [u,v,du,dv]->[x, y] | | | (U%->X%)->(EU%->X%) | B^n x D^n -> B | [B^n x D^n]->[B^k] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Trope | !h! : | !h!F_i : | !h!F : | | Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] | | | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Enlargement | E : | EF_i : | EF : | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] | | | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Difference | D : | DF_i : | DF : | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] | | | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Differential | d : | dF_i : | dF : | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] | | | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Remainder | r : | rF_i : | rF : | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] | | | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Radius | $e$ = <!e!, !h!> : | | $e$F : | | Operator | | | | | | U%->EU%, X%->EX%, | | [u, v, du, dv] -> | | | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], | | | | | | | | | | [B^n x D^n] -> | | | | | [B^k x D^k] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Secant | $E$ = <!e!, E> : | | $E$F : | | Operator | | | | | | U%->EU%, X%->EX%, | | [u, v, du, dv] -> | | | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], | | | | | | | | | | [B^n x D^n] -> | | | | | [B^k x D^k] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Chord | $D$ = <!e!, D> : | | $D$F : | | Operator | | | | | | U%->EU%, X%->EX%, | | [u, v, du, dv] -> | | | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], | | | | | | | | | | [B^n x D^n] -> | | | | | [B^k x D^k] | | | | | | o--------------o----------------------o--------------------o----------------------o | | | | | | Tangent | $T$ = <!e!, d> : | dF_i : | $T$F : | | Functor | | | | | | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u, v, du, dv] -> | | | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], | | | | | | | | | B^n x D^n -> D | [B^n x D^n] -> | | | | | [B^k x D^k] | | | | | | o--------------o----------------------o--------------------o----------------------o
Formula Display 12
o-----------------------------------------------------------o | | | x = f(u, v) = ((u)(v)) | | | | y = g(u, v) = ((u, v)) | | | o-----------------------------------------------------------o
Formula Display 13
o-----------------------------------------------------------o | | | <x, y> = F<u, v> = <((u)(v)), ((u, v))> | | | o-----------------------------------------------------------o
Table 60. Propositional Transformation
Table 60. Propositional Transformation o-------------o-------------o-------------o-------------o | u | v | f | g | o-------------o-------------o-------------o-------------o | | | | | | 0 | 0 | 0 | 1 | | | | | | | 0 | 1 | 1 | 0 | | | | | | | 1 | 0 | 1 | 0 | | | | | | | 1 | 1 | 1 | 1 | | | | | | o-------------o-------------o-------------o-------------o | | | ((u)(v)) | ((u, v)) | o-------------o-------------o-------------o-------------o
Figure 61. Propositional Transformation
o-----------------------------------------------------o
| U |
| |
| o-----------o o-----------o |
| / \ / \ |
| / o \ |
| / / \ \ |
| / / \ \ |
| o o o o |
| | | | | |
| | u | | v | |
| | | | | |
| o o o o |
| \ \ / / |
| \ \ / / |
| \ o / |
| \ / \ / |
| o-----------o o-----------o |
| |
| |
o-----------------------------------------------------o
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
/ \ / \
o-------------------------o o-------------------------o
| U | |\U \\\\\\\\\\\\\\\\\\\\\\|
| o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|
| //////\ //////\ | |\\\\\/ \\/ \\\\\\|
| ////////o///////\ | |\\\\/ o \\\\\|
| //////////\///////\ | |\\\/ /\\ \\\\|
| o///////o///o///////o | |\\o o\\\o o\\|
| |// u //|///|// v //| | |\\| u |\\\| v |\\|
| o///////o///o///////o | |\\o o\\\o o\\|
| \///////\////////// | |\\\\ \\/ /\\\|
| \///////o//////// | |\\\\\ o /\\\\|
| \////// \////// | |\\\\\\ /\\ /\\\\\|
| o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|
| | |\\\\\\\\\\\\\\\\\\\\\\\\\|
o-------------------------o o-------------------------o
\ | | /
\ | | /
\ | | /
\ f | | g /
\ | | /
\ | | /
\ | | /
\ | | /
\ | | /
\ | | /
o-------\----|---------------------------|----/-------o
| X \ | | / |
| \| |/ |
| o-----------o o-----------o |
| //////////////\ /\\\\\\\\\\\\\\ |
| ////////////////o\\\\\\\\\\\\\\\\ |
| /////////////////X\\\\\\\\\\\\\\\\\ |
| /////////////////XXX\\\\\\\\\\\\\\\\\ |
| o///////////////oXXXXXo\\\\\\\\\\\\\\\o |
| |///////////////|XXXXX|\\\\\\\\\\\\\\\| |
| |////// x //////|XXXXX|\\\\\\ y \\\\\\| |
| |///////////////|XXXXX|\\\\\\\\\\\\\\\| |
| o///////////////oXXXXXo\\\\\\\\\\\\\\\o |
| \///////////////\XXX/\\\\\\\\\\\\\\\/ |
| \///////////////\X/\\\\\\\\\\\\\\\/ |
| \///////////////o\\\\\\\\\\\\\\\/ |
| \////////////// \\\\\\\\\\\\\\/ |
| o-----------o o-----------o |
| |
| |
o-----------------------------------------------------o
Figure 61. Propositional Transformation
Figure 62. Propositional Transformation (Short Form)
o-------------------------o o-------------------------o
| U | |\U \\\\\\\\\\\\\\\\\\\\\\|
| o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|
| //////\ //////\ | |\\\\\/ \\/ \\\\\\|
| ////////o///////\ | |\\\\/ o \\\\\|
| //////////\///////\ | |\\\/ /\\ \\\\|
| o///////o///o///////o | |\\o o\\\o o\\|
| |// u //|///|// v //| | |\\| u |\\\| v |\\|
| o///////o///o///////o | |\\o o\\\o o\\|
| \///////\////////// | |\\\\ \\/ /\\\|
| \///////o//////// | |\\\\\ o /\\\\|
| \////// \////// | |\\\\\\ /\\ /\\\\\|
| o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|
| | |\\\\\\\\\\\\\\\\\\\\\\\\\|
o-------------------------o o-------------------------o
\ / \ /
\ / \ /
\ / \ /
\ f / \ g /
\ / \ /
\ / \ /
\ / \ /
\ / \ /
\ / \ /
o---------\-----/---------------------\-----/---------o
| X \ / \ / |
| \ / \ / |
| o-----------o o-----------o |
| //////////////\ /\\\\\\\\\\\\\\ |
| ////////////////o\\\\\\\\\\\\\\\\ |
| /////////////////X\\\\\\\\\\\\\\\\\ |
| /////////////////XXX\\\\\\\\\\\\\\\\\ |
| o///////////////oXXXXXo\\\\\\\\\\\\\\\o |
| |///////////////|XXXXX|\\\\\\\\\\\\\\\| |
| |////// x //////|XXXXX|\\\\\\ y \\\\\\| |
| |///////////////|XXXXX|\\\\\\\\\\\\\\\| |
| o///////////////oXXXXXo\\\\\\\\\\\\\\\o |
| \///////////////\XXX/\\\\\\\\\\\\\\\/ |
| \///////////////\X/\\\\\\\\\\\\\\\/ |
| \///////////////o\\\\\\\\\\\\\\\/ |
| \////////////// \\\\\\\\\\\\\\/ |
| o-----------o o-----------o |
| |
| |
o-----------------------------------------------------o
Figure 62. Propositional Transformation (Short Form)
Figure 63. Transformation of Positions
o-----------------------------------------------------o
|`U` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
|` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
|` ` ` ` ` ` o-----------o ` o-----------o ` ` ` ` ` `|
|` ` ` ` ` `/' ' ' ' ' ' '\`/' ' ' ' ' ' '\` ` ` ` ` `|
|` ` ` ` ` / ' ' ' ' ' ' ' o ' ' ' ' ' ' ' \ ` ` ` ` `|
|` ` ` ` `/' ' ' ' ' ' ' '/^\' ' ' ' ' ' ' '\` ` ` ` `|
|` ` ` ` / ' ' ' ' ' ' ' /^^^\ ' ' ' ' ' ' ' \ ` ` ` `|
|` ` ` `o' ' ' ' ' ' ' 'o^^^^^o' ' ' ' ' ' ' 'o` ` ` `|
|` ` ` `|' ' ' ' ' ' ' '|^^^^^|' ' ' ' ' ' ' '|` ` ` `|
|` ` ` `|' ' ' ' u ' ' '|^^^^^|' ' ' v ' ' ' '|` ` ` `|
|` ` ` `|' ' ' ' ' ' ' '|^^^^^|' ' ' ' ' ' ' '|` ` ` `|
|` `@` `o' ' ' ' @ ' ' 'o^^@^^o' ' ' @ ' ' ' 'o` ` ` `|
|` ` \ ` \ ' ' ' | ' ' ' \^|^/ ' ' ' | ' ' ' / ` ` ` `|
|` ` `\` `\' ' ' | ' ' ' '\|/' ' ' ' | ' ' '/` ` ` ` `|
|` ` ` \ ` \ ' ' | ' ' ' ' | ' ' ' ' | ' ' / ` ` ` ` `|
|` ` ` `\` `\' ' | ' ' ' '/|\' ' ' ' | ' '/` ` ` ` ` `|
|` ` ` ` \ ` o---|-------o | o-------|---o ` ` ` ` ` `|
|` ` ` ` `\` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` `|
|` ` ` ` ` \ ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` `|
o-----------\----|---------|---------|----------------o
" " \ | | | " "
" " \ | | | " "
" " \ | | | " "
" " \| | | " "
o-------------------------o \ | | o-------------------------o
| U | |\ | | |`U```````````````````````|
| o---o o---o | | \ | | |``````o---o```o---o``````|
| /'''''\ /'''''\ | | \ | | |`````/ \`/ \`````|
| /'''''''o'''''''\ | | \ | | |````/ o \````|
| /'''''''/'\'''''''\ | | \ | | |```/ /`\ \```|
| o'''''''o'''o'''''''o | | \ | | |``o o```o o``|
| |'''u'''|'''|'''v'''| | | \ | | |``| u |```| v |``|
| o'''''''o'''o'''''''o | | \ | | |``o o```o o``|
| \'''''''\'/'''''''/ | | \| | |```\ \`/ /```|
| \'''''''o'''''''/ | | \ | |````\ o /````|
| \'''''/ \'''''/ | | |\ | |`````\ /`\ /`````|
| o---o o---o | | | \ | |``````o---o```o---o``````|
| | | | \ * |`````````````````````````|
o-------------------------o | | \ / o-------------------------o
\ | | | \ / | /
\ ((u)(v)) | | | \/ | ((u, v)) /
\ | | | /\ | /
\ | | | / \ | /
\ | | | / \ | /
\ | | | / * | /
\ | | | / | | /
\ | | |/ | | /
\ | | / | | /
\ | | /| | | /
o-------\----|---|-------/-|---------|---|----/-------o
| X \ | | / | | | / |
| \| | / | | |/ |
| o---|----/--o | o-------|---o |
| /' ' | ' / ' '\|/` ` ` ` | ` `\ |
| / ' ' | '/' ' ' | ` ` ` ` | ` ` \ |
| /' ' ' | / ' ' '/|\` ` ` ` | ` ` `\ |
| / ' ' ' |/' ' ' /^|^\ ` ` ` | ` ` ` \ |
| @ o' ' ' ' @ ' ' 'o^^@^^o` ` ` @ ` ` ` `o |
| |' ' ' ' ' ' ' '|^^^^^|` ` ` ` ` ` ` `| |
| |' ' ' ' f ' ' '|^^^^^|` ` ` g ` ` ` `| |
| |' ' ' ' ' ' ' '|^^^^^|` ` ` ` ` ` ` `| |
| o' ' ' ' ' ' ' 'o^^^^^o` ` ` ` ` ` ` `o |
| \ ' ' ' ' ' ' ' \^^^/ ` ` ` ` ` ` ` / |
| \' ' ' ' ' ' ' '\^/` ` ` ` ` ` ` `/ |
| \ ' ' ' ' ' ' ' o ` ` ` ` ` ` ` / |
| \' ' ' ' ' ' '/ \` ` ` ` ` ` `/ |
| o-----------o o-----------o |
| |
| |
o-----------------------------------------------------o
Figure 63. Transformation of Positions
Table 64. Transformation of Positions
Table 64. Transformation of Positions o-----o----------o----------o-------o-------o--------o--------o-------------o | u v | x | y | x y | x(y) | (x)y | (x)(y) | X% = [x, y] | o-----o----------o----------o-------o-------o--------o--------o-------------o | | | | | | | | ^ | | 0 0 | 0 | 1 | 0 | 0 | 1 | 0 | | | | | | | | | | | | | 0 1 | 1 | 0 | 0 | 1 | 0 | 0 | F | | | | | | | | | = | | 1 0 | 1 | 0 | 0 | 1 | 0 | 0 | <f , g> | | | | | | | | | | | 1 1 | 1 | 1 | 1 | 0 | 0 | 0 | ^ | | | | | | | | | | | o-----o----------o----------o-------o-------o--------o--------o-------------o | | ((u)(v)) | ((u, v)) | u v | (u,v) | (u)(v) | 0 | U% = [u, v] | o-----o----------o----------o-------o-------o--------o--------o-------------o
Table 65. Induced Transformation on Propositions
Table 65. Induced Transformation on Propositions o------------o---------------------------------o------------o | X% | <--- F = <f , g> <--- | U% | o------------o----------o-----------o----------o------------o | | u = | 1 1 0 0 | = u | | | | v = | 1 0 1 0 | = v | | | f_i <x, y> o----------o-----------o----------o f_j <u, v> | | | x = | 1 1 1 0 | = f<u,v> | | | | y = | 1 0 0 1 | = g<u,v> | | o------------o----------o-----------o----------o------------o | | | | | | | f_0 | () | 0 0 0 0 | () | f_0 | | | | | | | | f_1 | (x)(y) | 0 0 0 1 | () | f_0 | | | | | | | | f_2 | (x) y | 0 0 1 0 | (u)(v) | f_1 | | | | | | | | f_3 | (x) | 0 0 1 1 | (u)(v) | f_1 | | | | | | | | f_4 | x (y) | 0 1 0 0 | (u, v) | f_6 | | | | | | | | f_5 | (y) | 0 1 0 1 | (u, v) | f_6 | | | | | | | | f_6 | (x, y) | 0 1 1 0 | (u v) | f_7 | | | | | | | | f_7 | (x y) | 0 1 1 1 | (u v) | f_7 | | | | | | | o------------o----------o-----------o----------o------------o | | | | | | | f_8 | x y | 1 0 0 0 | u v | f_8 | | | | | | | | f_9 | ((x, y)) | 1 0 0 1 | u v | f_8 | | | | | | | | f_10 | y | 1 0 1 0 | ((u, v)) | f_9 | | | | | | | | f_11 | (x (y)) | 1 0 1 1 | ((u, v)) | f_9 | | | | | | | | f_12 | x | 1 1 0 0 | ((u)(v)) | f_14 | | | | | | | | f_13 | ((x) y) | 1 1 0 1 | ((u)(v)) | f_14 | | | | | | | | f_14 | ((x)(y)) | 1 1 1 0 | (()) | f_15 | | | | | | | | f_15 | (()) | 1 1 1 1 | (()) | f_15 | | | | | | | o------------o----------o-----------o----------o------------o
Formula Display 14
o-------------------------------------------------o | | | EG_i = G_i <u + du, v + dv> | | | o-------------------------------------------------o
Formula Display 15
o-------------------------------------------------o | | | DG_i = G_i <u, v> + EG_i <u, v, du, dv> | | | | = G_i <u, v> + G_i <u + du, v + dv> | | | o-------------------------------------------------o
Formula Display 16
o-------------------------------------------------o | | | Ef = ((u + du)(v + dv)) | | | | Eg = ((u + du, v + dv)) | | | o-------------------------------------------------o
Formula Display 17
o-------------------------------------------------o | | | Df = ((u)(v)) + ((u + du)(v + dv)) | | | | Dg = ((u, v)) + ((u + du, v + dv)) | | | o-------------------------------------------------o
Table 66-i. Computation Summary for f‹u, v› = ((u)(v))
Table 66-i. Computation Summary for f<u, v> = ((u)(v)) o--------------------------------------------------------------------------------o | | | !e!f = uv. 1 + u(v). 1 + (u)v. 1 + (u)(v). 0 | | | | Ef = uv. (du dv) + u(v). (du (dv)) + (u)v.((du) dv) + (u)(v).((du)(dv)) | | | | Df = uv. du dv + u(v). du (dv) + (u)v. (du) dv + (u)(v).((du)(dv)) | | | | df = uv. 0 + u(v). du + (u)v. dv + (u)(v). (du, dv) | | | | rf = uv. du dv + u(v). du dv + (u)v. du dv + (u)(v). du dv | | | o--------------------------------------------------------------------------------o
Table 66-ii. Computation Summary for g‹u, v› = ((u, v))
Table 66-ii. Computation Summary for g<u, v> = ((u, v)) o--------------------------------------------------------------------------------o | | | !e!g = uv. 1 + u(v). 0 + (u)v. 0 + (u)(v). 1 | | | | Eg = uv.((du, dv)) + u(v). (du, dv) + (u)v. (du, dv) + (u)(v).((du, dv)) | | | | Dg = uv. (du, dv) + u(v). (du, dv) + (u)v. (du, dv) + (u)(v). (du, dv) | | | | dg = uv. (du, dv) + u(v). (du, dv) + (u)v. (du, dv) + (u)(v). (du, dv) | | | | rg = uv. 0 + u(v). 0 + (u)v. 0 + (u)(v). 0 | | | o--------------------------------------------------------------------------------o
Table 67. Computation of an Analytic Series in Terms of Coordinates
Table 67. Computation of an Analytic Series in Terms of Coordinates o--------o-------o-------o--------o-------o-------o-------o-------o | u v | du dv | u' v' | f g | Ef Eg | Df Dg | df dg | rf rg | o--------o-------o-------o--------o-------o-------o-------o-------o | | | | | | | | | | 0 0 | 0 0 | 0 0 | 0 1 | 0 1 | 0 0 | 0 0 | 0 0 | | | | | | | | | | | | 0 1 | 0 1 | | 1 0 | 1 1 | 1 1 | 0 0 | | | | | | | | | | | | 1 0 | 1 0 | | 1 0 | 1 1 | 1 1 | 0 0 | | | | | | | | | | | | 1 1 | 1 1 | | 1 1 | 1 0 | 0 0 | 1 0 | | | | | | | | | | o--------o-------o-------o--------o-------o-------o-------o-------o | | | | | | | | | | 0 1 | 0 0 | 0 1 | 1 0 | 1 0 | 0 0 | 0 0 | 0 0 | | | | | | | | | | | | 0 1 | 0 0 | | 0 1 | 1 1 | 1 1 | 0 0 | | | | | | | | | | | | 1 0 | 1 1 | | 1 1 | 0 1 | 0 1 | 0 0 | | | | | | | | | | | | 1 1 | 1 0 | | 1 0 | 0 0 | 1 0 | 1 0 | | | | | | | | | | o--------o-------o-------o--------o-------o-------o-------o-------o | | | | | | | | | | 1 0 | 0 0 | 1 0 | 1 0 | 1 0 | 0 0 | 0 0 | 0 0 | | | | | | | | | | | | 0 1 | 1 1 | | 1 1 | 0 1 | 0 1 | 0 0 | | | | | | | | | | | | 1 0 | 0 0 | | 0 1 | 1 1 | 1 1 | 0 0 | | | | | | | | | | | | 1 1 | 0 1 | | 1 0 | 0 0 | 1 0 | 1 0 | | | | | | | | | | o--------o-------o-------o--------o-------o-------o-------o-------o | | | | | | | | | | 1 1 | 0 0 | 1 1 | 1 1 | 1 1 | 0 0 | 0 0 | 0 0 | | | | | | | | | | | | 0 1 | 1 0 | | 1 0 | 0 1 | 0 1 | 0 0 | | | | | | | | | | | | 1 0 | 0 1 | | 1 0 | 0 1 | 0 1 | 0 0 | | | | | | | | | | | | 1 1 | 0 0 | | 0 1 | 1 0 | 0 0 | 1 0 | | | | | | | | | | o--------o-------o-------o--------o-------o-------o-------o-------o
Table 68. Computation of an Analytic Series in Symbolic Terms
Table 68. Computation of an Analytic Series in Symbolic Terms o-----o-----o------------o----------o----------o----------o----------o----------o | u v | f g | Df | Dg | df | dg | rf | rf | o-----o-----o------------o----------o----------o----------o----------o----------o | | | | | | | | | | 0 0 | 0 1 | ((du)(dv)) | (du, dv) | (du, dv) | (du, dv) | du dv | () | | | | | | | | | | | 0 1 | 1 0 | (du) dv | (du, dv) | dv | (du, dv) | du dv | () | | | | | | | | | | | 1 0 | 1 0 | du (dv) | (du, dv) | du | (du, dv) | du dv | () | | | | | | | | | | | 1 1 | 1 1 | du dv | (du, dv) | () | (du, dv) | du dv | () | | | | | | | | | | o-----o-----o------------o----------o----------o----------o----------o----------o
Formula Display 18
o-------------------------------------------------------------------------o
| |
| Df = uv. du dv + u(v). du (dv) + (u)v.(du) dv + (u)(v).((du)(dv)) |
| |
| Dg = uv.(du, dv) + u(v).(du, dv) + (u)v.(du, dv) + (u)(v). (du, dv) |
| |
o-------------------------------------------------------------------------o
===Figure 69. Difference Map of F = <f, g> = <((u)(v)), ((u, v))>===
o-----------------------------------o o-----------------------------------o
| U | |`U`````````````````````````````````|
| | |```````````````````````````````````|
| ^ | |```````````````````````````````````|
| | | |```````````````````````````````````|
| o-------o | o-------o | |```````o-------o```o-------o```````|
| ^ /`````````\|/`````````\ ^ | | ^ ```/ ^ \`/ ^ \``` ^ |
| \ /```````````|```````````\ / | |``\``/ \ o / \``/``|
| \/`````u`````/|\`````v`````\/ | |```\/ u \/`\/ v \/```|
| /\``````````/`|`\``````````/\ | |```/\ /\`/\ /\```|
| o``\````````o``@``o````````/``o | |``o \ o``@``o / o``|
| |```\```````|`````|```````/```| | |``| \ |`````| / |``|
| |````@``````|`````|``````@````| | |``| @-------->`<--------@ |``|
| |```````````|`````|```````````| | |``| |`````| |``|
| o```````````o` ^ `o```````````o | |``o o`````o o``|
| \```````````\`|`/```````````/ | |```\ \```/ /```|
| \```` ^ ````\|/```` ^ ````/ | |````\ ^ \`/ ^ /````|
| \`````\`````|`````/`````/ | |`````\ \ o / /`````|
| \`````\```/|\```/`````/ | |``````\ \ /`\ / /``````|
| o-----\-o | o-/-----o | |```````o-----\-o```o-/-----o```````|
| \ | / | |``````````````\`````/``````````````|
| \ | / | |```````````````\```/```````````````|
| \|/ | |````````````````\`/````````````````|
| @ | |`````````````````@`````````````````|
o-----------------------------------o o-----------------------------------o
\ / \ /
\ / \ /
\ ((u)(v)) / \ ((u, v)) /
\ / \ /
\ / \ /
o----------\-------------/-----------------------\-------------/----------o
| X \ / \ / |
| \ / \ / |
| \ / \ / |
| o----------------o o----------------o |
| / \ / \ |
| / o \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| / / \ \ |
| o o o o |
| | | | | |
| | | | | |
| | f | | g | |
| | | | | |
| | | | | |
| o o o o |
| \ \ / / |
| \ \ / / |
| \ \ / / |
| \ \ / / |
| \ \ / / |
| \ o / |
| \ / \ / |
| o----------------o o----------------o |
| |
| |
| |
o-------------------------------------------------------------------------o
Figure 69. Difference Map of F = <f, g> = <((u)(v)), ((u, v))>
Figure 70-a. Tangent Functor Diagram for F‹u, v› = ‹((u)(v)), ((u, v))›
o-------------------------------------------------------------------------------o
| |
| df = uv. 0 + u(v). du + (u)v. dv + (u)(v).(du, dv) |
| |
| dg = uv.(du, dv) + u(v).(du, dv) + (u)v.(du, dv) + (u)(v).(du, dv) |
| |
o-------------------------------------------------------------------------------o
o o
/ \ / \
/ \ / \
/ \ / O \
/ \ o /@\ o
/ \ / \ / \
/ \ / \ / \
/ O \ / O \ / O \
o /@\ o o /@\ o /@\ o
/ \ / \ / \ \ / \ \ / \
/ \ / \ / \ / \ / \
/ \ / \ / O \ / O \ / O \
/ \ / \ o /@ o /@\ o /@ o
/ \ / \ / \ \ / \ / \ \ / \
/ \ / \ / \ / \ / \ / \
/ O \ / O \ / O \ / O \ / O \ / O \
o /@ o /@ o o /@ o /@ o /@ o /@ o
|\ / \ /| |\ / \ / / \ / / \ /|
| \ / \ / | | \ / \ / \ / \ / |
| \ / \ / | | \ / O \ / O \ / O \ / |
| \ / \ / | | o /@ o @\ o /@ o |
| \ / \ / | | |\ / \ / \ / \ / \ /| |
| \ / \ / | | | \ / \ / \ / | |
| u \ / O \ / v | | u | \ / O \ / O \ / | v |
o-------o @\ o-------o o---+---o @\ o @\ o---+---o
\ / | \ / \ / \ / \ / |
\ / | \ / \ / |
\ / | du \ / O \ / dv |
\ / o-------o @\ o-------o
\ / \ /
\ / \ /
\ / \ /
o o
U% $T$ $E$U%
o------------------>o
| |
| |
| |
| |
F | | $T$F
| |
| |
| |
v v
o------------------>o
X% $T$ $E$X%
o o
/ \ / \
/ \ / \
/ \ / O \
/ \ o /@\ o
/ \ / \ / \
/ \ / \ / \
/ O \ / O \ / O \
o /@\ o o /@\ o /@\ o
/ \ / \ / \ \ / \ / / \
/ \ / \ / \ / \ / \
/ \ / \ / O \ / O \ / O \
/ \ / \ o /@ o /@\ o @\ o
/ \ / \ / \ \ / \ / \ / \ / / \
/ \ / \ / \ / \ / \ / \
/ O \ / O \ / O \ / O \ / O \ / O \
o /@ o @\ o o /@ o /@ o @\ o @\ o
|\ / \ /| |\ / \ / \ / \ / \ / \ /|
| \ / \ / | | \ / \ / \ / \ / |
| \ / \ / | | \ / O \ / O \ / O \ / |
| \ / \ / | | o /@ o @ o @\ o |
| \ / \ / | | |\ / / \ / \ / \ \ /| |
| \ / \ / | | | \ / \ / \ / | |
| x \ / O \ / y | | x | \ / O \ / O \ / | y |
o-------o @ o-------o o---+---o @ o @ o---+---o
\ / | \ / / \ \ / |
\ / | \ / \ / |
\ / | dx \ / O \ / dy |
\ / o-------o @ o-------o
\ / \ /
\ / \ /
\ / \ /
o o
Figure 70-a. Tangent Functor Diagram for F‹u, v› = <((u)(v)), ((u, v))>
Figure 70-b. Tangent Functor Ferris Wheel for F‹u, v› = ‹((u)(v)), ((u, v))›
o-----------------------o o-----------------------o o-----------------------o
| dU | | dU | | dU |
| o--o o--o | | o--o o--o | | o--o o--o |
| /////\ /////\ | | /XXXX\ /XXXX\ | | /\\\\\ /\\\\\ |
| ///////o//////\ | | /XXXXXXoXXXXXX\ | | /\\\\\\o\\\\\\\ |
| //////// \//////\ | | /XXXXXX/ \XXXXXX\ | | /\\\\\\/ \\\\\\\\ |
| o/////// \//////o | | oXXXXXX/ \XXXXXXo | | o\\\\\\/ \\\\\\\o |
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |
| |/du//| |//dv/| | | |XXXXX| |XXXXX| | | |\du\\| |\\dv\| |
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |
| o//////\ ///////o | | oXXXXXX\ /XXXXXXo | | o\\\\\\\ /\\\\\\o |
| \//////\ //////// | | \XXXXXX\ /XXXXXX/ | | \\\\\\\\ /\\\\\\/ |
| \//////o/////// | | \XXXXXXoXXXXXX/ | | \\\\\\\o\\\\\\/ |
| \///// \///// | | \XXXX/ \XXXX/ | | \\\\\/ \\\\\/ |
| o--o o--o | | o--o o--o | | o--o o--o |
| | | | | |
o-----------------------o o-----------------------o o-----------------------o
= du' @ (u)(v) o-----------------------o dv' @ (u)(v) =
= | dU' | =
= | o--o o--o | =
= | /////\ /\\\\\ | =
= | ///////o\\\\\\\ | =
= | ////////X\\\\\\\\ | =
= | o///////XXX\\\\\\\o | =
= | |/////oXXXXXo\\\\\| | =
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
| |/////oXXXXXo\\\\\| |
| o//////\XXX/\\\\\\o |
| \//////\X/\\\\\\/ |
| \//////o\\\\\\/ |
| \///// \\\\\/ |
| o--o o--o |
| |
o-----------------------o
o-----------------------o o-----------------------o o-----------------------o
| dU | | dU | | dU |
| o--o o--o | | o--o o--o | | o--o o--o |
| / \ /////\ | | /\\\\\ /XXXX\ | | /\\\\\ /\\\\\ |
| / o//////\ | | /\\\\\\oXXXXXX\ | | /\\\\\\o\\\\\\\ |
| / //\//////\ | | /\\\\\\//\XXXXXX\ | | /\\\\\\/ \\\\\\\\ |
| o ////\//////o | | o\\\\\\////\XXXXXXo | | o\\\\\\/ \\\\\\\o |
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |
| | du |/////|//dv/| | | |\\\\\|/////|XXXXX| | | |\du\\| |\\dv\| |
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |
| o \//////////o | | o\\\\\\\////XXXXXXo | | o\\\\\\\ /\\\\\\o |
| \ \///////// | | \\\\\\\\//XXXXXX/ | | \\\\\\\\ /\\\\\\/ |
| \ o/////// | | \\\\\\\oXXXXXX/ | | \\\\\\\o\\\\\\/ |
| \ / \///// | | \\\\\/ \XXXX/ | | \\\\\/ \\\\\/ |
| o--o o--o | | o--o o--o | | o--o o--o |
| | | | | |
o-----------------------o o-----------------------o o-----------------------o
= du' @ (u) v o-----------------------o dv' @ (u) v =
= | dU' | =
= | o--o o--o | =
= | /////\ /\\\\\ | =
= | ///////o\\\\\\\ | =
= | ////////X\\\\\\\\ | =
= | o///////XXX\\\\\\\o | =
= | |/////oXXXXXo\\\\\| | =
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
| |/////oXXXXXo\\\\\| |
| o//////\XXX/\\\\\\o |
| \//////\X/\\\\\\/ |
| \//////o\\\\\\/ |
| \///// \\\\\/ |
| o--o o--o |
| |
o-----------------------o
o-----------------------o o-----------------------o o-----------------------o
| dU | | dU | | dU |
| o--o o--o | | o--o o--o | | o--o o--o |
| /////\ / \ | | /XXXX\ /\\\\\ | | /\\\\\ /\\\\\ |
| ///////o \ | | /XXXXXXo\\\\\\\ | | /\\\\\\o\\\\\\\ |
| /////////\ \ | | /XXXXXX//\\\\\\\\ | | /\\\\\\/ \\\\\\\\ |
| o//////////\ o | | oXXXXXX////\\\\\\\o | | o\\\\\\/ \\\\\\\o |
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |
| |/du//|/////| dv | | | |XXXXX|/////|\\\\\| | | |\du\\| |\\dv\| |
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |
| o//////\//// o | | oXXXXXX\////\\\\\\o | | o\\\\\\\ /\\\\\\o |
| \//////\// / | | \XXXXXX\//\\\\\\/ | | \\\\\\\\ /\\\\\\/ |
| \//////o / | | \XXXXXXo\\\\\\/ | | \\\\\\\o\\\\\\/ |
| \///// \ / | | \XXXX/ \\\\\/ | | \\\\\/ \\\\\/ |
| o--o o--o | | o--o o--o | | o--o o--o |
| | | | | |
o-----------------------o o-----------------------o o-----------------------o
= du' @ u (v) o-----------------------o dv' @ u (v) =
= | dU' | =
= | o--o o--o | =
= | /////\ /\\\\\ | =
= | ///////o\\\\\\\ | =
= | ////////X\\\\\\\\ | =
= | o///////XXX\\\\\\\o | =
= | |/////oXXXXXo\\\\\| | =
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
| |/////oXXXXXo\\\\\| |
| o//////\XXX/\\\\\\o |
| \//////\X/\\\\\\/ |
| \//////o\\\\\\/ |
| \///// \\\\\/ |
| o--o o--o |
| |
o-----------------------o
o-----------------------o o-----------------------o o-----------------------o
| dU | | dU | | dU |
| o--o o--o | | o--o o--o | | o--o o--o |
| / \ / \ | | /\\\\\ /\\\\\ | | /\\\\\ /\\\\\ |
| / o \ | | /\\\\\\o\\\\\\\ | | /\\\\\\o\\\\\\\ |
| / / \ \ | | /\\\\\\/ \\\\\\\\ | | /\\\\\\/ \\\\\\\\ |
| o / \ o | | o\\\\\\/ \\\\\\\o | | o\\\\\\/ \\\\\\\o |
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |
| | du | | dv | | | |\\\\\| |\\\\\| | | |\du\\| |\\dv\| |
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |
| o \ / o | | o\\\\\\\ /\\\\\\o | | o\\\\\\\ /\\\\\\o |
| \ \ / / | | \\\\\\\\ /\\\\\\/ | | \\\\\\\\ /\\\\\\/ |
| \ o / | | \\\\\\\o\\\\\\/ | | \\\\\\\o\\\\\\/ |
| \ / \ / | | \\\\\/ \\\\\/ | | \\\\\/ \\\\\/ |
| o--o o--o | | o--o o--o | | o--o o--o |
| | | | | |
o-----------------------o o-----------------------o o-----------------------o
= du' @ u v o-----------------------o dv' @ u v =
= | dU' | =
= | o--o o--o | =
= | /////\ /\\\\\ | =
= | ///////o\\\\\\\ | =
= | ////////X\\\\\\\\ | =
= | o///////XXX\\\\\\\o | =
= | |/////oXXXXXo\\\\\| | =
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
| |/////oXXXXXo\\\\\| |
| o//////\XXX/\\\\\\o |
| \//////\X/\\\\\\/ |
| \//////o\\\\\\/ |
| \///// \\\\\/ |
| o--o o--o |
| |
o-----------------------o
o-----------------------o o-----------------------o o-----------------------o
| U | |\U\\\\\\\\\\\\\\\\\\\\\| |\U\\\\\\\\\\\\\\\\\\\\\|
| o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\|
| /////\ /////\ | |\\\\\/////\\/////\\\\\\| |\\\\\/ \\/ \\\\\\|
| ///////o//////\ | |\\\\///////o//////\\\\\| |\\\\/ o \\\\\|
| /////////\//////\ | |\\\////////X\//////\\\\| |\\\/ /\\ \\\\|
| o//////////\//////o | |\\o///////XXX\//////o\\| |\\o /\\\\ o\\|
| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|
| |//u//|/////|//v//| | |\\|//u//|XXXXX|//v//|\\| |\\| u |\\\\\| v |\\|
| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|
| o//////\//////////o | |\\o//////\XXX///////o\\| |\\o \\\\/ o\\|
| \//////\///////// | |\\\\//////\X////////\\\| |\\\\ \\/ /\\\|
| \//////o/////// | |\\\\\//////o///////\\\\| |\\\\\ o /\\\\|
| \///// \///// | |\\\\\\/////\\/////\\\\\| |\\\\\\ /\\ /\\\\\|
| o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\|
| | |\\\\\\\\\\\\\\\\\\\\\\\| |\\\\\\\\\\\\\\\\\\\\\\\|
o-----------------------o o-----------------------o o-----------------------o
= u' o-----------------------o v' =
= | U' | =
= | o--o o--o | =
= | /////\ /\\\\\ | =
= | ///////o\\\\\\\ | =
= | ////////X\\\\\\\\ | =
= | o///////XXX\\\\\\\o | =
= | |/////oXXXXXo\\\\\| | =
= = = = = = = = = = =|/u'//|XXXXX|\\v'\|= = = = = = = = = = =
| |/////oXXXXXo\\\\\| |
| o//////\XXX/\\\\\\o |
| \//////\X/\\\\\\/ |
| \//////o\\\\\\/ |
| \///// \\\\\/ |
| o--o o--o |
| |
o-----------------------o
Figure 70-b. Tangent Functor Ferris Wheel for F<u, v> = <((u)(v)), ((u, v))>
Logical Tables
Higher Order Propositions
| \ x | 1 0 | F | m | m | m | m | m | m | m | m | m | m | m | m | m | m | m | m |
| F \ | 00 | 01 | 02 | 03 | 04 | 05 | 06 | 07 | 08 | 09 | 10 | 11 | 12 | 13 | 14 | 15 | ||
| F0 | 0 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
| F1 | 0 1 | (x) | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| F2 | 1 0 | x | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| F3 | 1 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| Measure | Happening | Exactness | Existence | Linearity | Uniformity | Information |
| m0 | nothing happens | |||||
| m1 | just false | nothing exists | ||||
| m2 | just not x | |||||
| m3 | nothing is x | |||||
| m4 | just x | |||||
| m5 | everything is x | F is linear | ||||
| m6 | F is not uniform | F is informed | ||||
| m7 | not just true | |||||
| m8 | just true | |||||
| m9 | F is uniform | F is not informed | ||||
| m10 | something is not x | F is not linear | ||||
| m11 | not just x | |||||
| m12 | something is x | |||||
| m13 | not just not x | |||||
| m14 | not just false | something exists | ||||
| m15 | anything happens |
| x : | 1100 | f | m | m | m | m | m | m | m | m | m | m | m | m | m | m | m | m | m | m | m | m | m | m | m | m |
| y : | 1010 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
| f0 | 0000 | ( ) | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
| f1 | 0001 | (x)(y) | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | ||
| f2 | 0010 | (x) y | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | ||||
| f3 | 0011 | (x) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||||||||
| f4 | 0100 | x (y) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||||||||||
| f5 | 0101 | (y) | ||||||||||||||||||||||||
| f6 | 0110 | (x, y) | ||||||||||||||||||||||||
| f7 | 0111 | (x y) | ||||||||||||||||||||||||
| f8 | 1000 | x y | ||||||||||||||||||||||||
| f9 | 1001 | ((x, y)) | ||||||||||||||||||||||||
| f10 | 1010 | y | ||||||||||||||||||||||||
| f11 | 1011 | (x (y)) | ||||||||||||||||||||||||
| f12 | 1100 | x | ||||||||||||||||||||||||
| f13 | 1101 | ((x) y) | ||||||||||||||||||||||||
| f14 | 1110 | ((x)(y)) | ||||||||||||||||||||||||
| f15 | 1111 | (( )) |
| x : | 1100 | f | α | α | α | α | α | α | α | α | α | α | α | α | α | α | α | α |
| y : | 1010 | 15 | 14 | 13 | 12 | 11 | 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 | |
| f0 | 0000 | ( ) | 1 | |||||||||||||||
| f1 | 0001 | (x)(y) | 1 | 1 | ||||||||||||||
| f2 | 0010 | (x) y | 1 | 1 | ||||||||||||||
| f3 | 0011 | (x) | 1 | 1 | 1 | 1 | ||||||||||||
| f4 | 0100 | x (y) | 1 | 1 | ||||||||||||||
| f5 | 0101 | (y) | 1 | 1 | 1 | 1 | ||||||||||||
| f6 | 0110 | (x, y) | 1 | 1 | 1 | 1 | ||||||||||||
| f7 | 0111 | (x y) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||
| f8 | 1000 | x y | 1 | 1 | ||||||||||||||
| f9 | 1001 | ((x, y)) | 1 | 1 | 1 | 1 | ||||||||||||
| f10 | 1010 | y | 1 | 1 | 1 | 1 | ||||||||||||
| f11 | 1011 | (x (y)) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||
| f12 | 1100 | x | 1 | 1 | 1 | 1 | ||||||||||||
| f13 | 1101 | ((x) y) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||
| f14 | 1110 | ((x)(y)) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||
| f15 | 1111 | (( )) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| x : | 1100 | f | β | β | β | β | β | β | β | β | β | β | β | β | β | β | β | β |
| y : | 1010 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | |
| f0 | 0000 | ( ) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| f1 | 0001 | (x)(y) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||
| f2 | 0010 | (x) y | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||
| f3 | 0011 | (x) | 1 | 1 | 1 | 1 | ||||||||||||
| f4 | 0100 | x (y) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||
| f5 | 0101 | (y) | 1 | 1 | 1 | 1 | ||||||||||||
| f6 | 0110 | (x, y) | 1 | 1 | 1 | 1 | ||||||||||||
| f7 | 0111 | (x y) | 1 | 1 | ||||||||||||||
| f8 | 1000 | x y | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ||||||||
| f9 | 1001 | ((x, y)) | 1 | 1 | 1 | 1 | ||||||||||||
| f10 | 1010 | y | 1 | 1 | 1 | 1 | ||||||||||||
| f11 | 1011 | (x (y)) | 1 | 1 | ||||||||||||||
| f12 | 1100 | x | 1 | 1 | 1 | 1 | ||||||||||||
| f13 | 1101 | ((x) y) | 1 | 1 | ||||||||||||||
| f14 | 1110 | ((x)(y)) | 1 | 1 | ||||||||||||||
| f15 | 1111 | (( )) | 1 |
| A | Universal Affirmative | All | x | is | y | Indicator of " x (y)" = 0 |
| E | Universal Negative | All | x | is | (y) | Indicator of " x y " = 0 |
| I | Particular Affirmative | Some | x | is | y | Indicator of " x y " = 1 |
| O | Particular Negative | Some | x | is | (y) | Indicator of " x (y)" = 1 |
| Mnemonic | Category | Classical Form | Alternate Form | Symmetric Form | Operator |
| E Exclusive |
Universal Negative |
All x is (y) | No x is y | (L11) | |
| A Absolute |
Universal Affirmative |
All x is y | No x is (y) | (L10) | |
| All y is x | No y is (x) | No (x) is y | (L01) | ||
| All (y) is x | No (y) is (x) | No (x) is (y) | (L00) | ||
| Some (x) is (y) | Some (x) is (y) | L00 | |||
| Some (x) is y | Some (x) is y | L01 | |||
| O Obtrusive |
Particular Negative |
Some x is (y) | Some x is (y) | L10 | |
| I Indefinite |
Particular Affirmative |
Some x is y | Some x is y | L11 |
| x : | 1100 | f | (L11) | (L10) | (L01) | (L00) | L00 | L01 | L10 | L11 |
| y : | 1010 | no x is y |
no x is (y) |
no (x) is y |
no (x) is (y) |
some (x) is (y) |
some (x) is y |
some x is (y) |
some x is y | |
| f0 | 0000 | ( ) | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 |
| f1 | 0001 | (x)(y) | 1 | 1 | 1 | 0 | 1 | 0 | 0 | 0 |
| f2 | 0010 | (x) y | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 0 |
| f3 | 0011 | (x) | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 |
| f4 | 0100 | x (y) | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 0 |
| f5 | 0101 | (y) | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |
| f6 | 0110 | (x, y) | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 |
| f7 | 0111 | (x y) | 1 | 0 | 0 | 0 | 1 | 1 | 1 | 0 |
| f8 | 1000 | x y | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 1 |
| f9 | 1001 | ((x, y)) | 0 | 1 | 1 | 0 | 1 | 0 | 0 | 1 |
| f10 | 1010 | y | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
| f11 | 1011 | (x (y)) | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 1 |
| f12 | 1100 | x | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| f13 | 1101 | ((x) y) | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 1 |
| f14 | 1110 | ((x)(y)) | 0 | 0 | 0 | 1 | 0 | 1 | 1 | 1 |
| f15 | 1111 | (( )) | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
Table 7. Higher Order Propositions (n = 1) o------o-----o-----o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o---o | \ x | 1 0 | F |m |m |m |m |m |m |m |m |m |m |m |m |m |m |m |m | | F \ | | |00|01|02|03|04|05|06|07|08|09|10|11|12|13|14|15 | o------o-----o-----o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o---o | | | | | | F_0 | 0 0 | 0 | 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 | | | | | | | F_1 | 0 1 | (x) | 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 | | | | | | | F_2 | 1 0 | x | 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 | | | | | | | F_3 | 1 1 | 1 | 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 | | | | | | o------o-----o-----o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o---o
Table 8. Interpretive Categories for Higher Order Propositions (n = 1) o-------o----------o------------o------------o----------o----------o-----------o |Measure| Happening| Exactness | Existence | Linearity|Uniformity|Information| o-------o----------o------------o------------o----------o----------o-----------o | m_0 | nothing | | | | | | | | happens | | | | | | o-------o----------o------------o------------o----------o----------o-----------o | m_1 | | | nothing | | | | | | | just false | exists | | | | o-------o----------o------------o------------o----------o----------o-----------o | m_2 | | | | | | | | | | just not x | | | | | o-------o----------o------------o------------o----------o----------o-----------o | m_3 | | | nothing | | | | | | | | is x | | | | o-------o----------o------------o------------o----------o----------o-----------o | m_4 | | | | | | | | | | just x | | | | | o-------o----------o------------o------------o----------o----------o-----------o | m_5 | | | everything | F is | | | | | | | is x | linear | | | o-------o----------o------------o------------o----------o----------o-----------o | m_6 | | | | | F is not | F is | | | | | | | uniform | informed | o-------o----------o------------o------------o----------o----------o-----------o | m_7 | | not | | | | | | | | just true | | | | | o-------o----------o------------o------------o----------o----------o-----------o | m_8 | | | | | | | | | | just true | | | | | o-------o----------o------------o------------o----------o----------o-----------o | m_9 | | | | | F is | F is not | | | | | | | uniform | informed | o-------o----------o------------o------------o----------o----------o-----------o | m_10 | | | something | F is not | | | | | | | is not x | linear | | | o-------o----------o------------o------------o----------o----------o-----------o | m_11 | | not | | | | | | | | just x | | | | | o-------o----------o------------o------------o----------o----------o-----------o | m_12 | | | something | | | | | | | | is x | | | | o-------o----------o------------o------------o----------o----------o-----------o | m_13 | | not | | | | | | | | just not x | | | | | o-------o----------o------------o------------o----------o----------o-----------o | m_14 | | not | something | | | | | | | just false | exists | | | | o-------o----------o------------o------------o----------o----------o-----------o | m_15 | anything | | | | | | | | happens | | | | | | o-------o----------o------------o------------o----------o----------o-----------o
Table 9. Higher Order Propositions (n = 2) o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o | | x | 1100 | f |m|m|m|m|m|m|m|m|m|m|m|m|m|m|m|m|.| | | y | 1010 | |0|0|0|0|0|0|0|0|0|0|1|1|1|1|1|1|.| | f \ | | |0|1|2|3|4|5|6|7|8|9|0|1|2|3|4|5|.| o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o | | | | | | f_0 | 0000 | () |0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 | | | | | | | f_1 | 0001 | (x)(y) | 1 1 0 0 1 1 0 0 1 1 0 0 1 1 | | | | | | | f_2 | 0010 | (x) y | 1 1 1 1 0 0 0 0 1 1 1 1 | | | | | | | f_3 | 0011 | (x) | 1 1 1 1 1 1 1 1 | | | | | | | f_4 | 0100 | x (y) | | | | | | | | f_5 | 0101 | (y) | | | | | | | | f_6 | 0110 | (x, y) | | | | | | | | f_7 | 0111 | (x y) | | | | | | | o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o | | | | | | f_8 | 1000 | x y | | | | | | | | f_9 | 1001 | ((x, y)) | | | | | | | | f_10 | 1010 | y | | | | | | | | f_11 | 1011 | (x (y)) | | | | | | | | f_12 | 1100 | x | | | | | | | | f_13 | 1101 | ((x) y) | | | | | | | | f_14 | 1110 | ((x)(y)) | | | | | | | | f_15 | 1111 | (()) | | | | | | | o------o------o----------o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o-o
Table 10. Qualifiers of Implication Ordering: !a!_i f = !Y!(f_i => f) o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o | | x | 1100 | f |a |a |a |a |a |a |a |a |a |a |a |a |a |a |a |a | | | y | 1010 | |1 |1 |1 |1 |1 |1 |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 | | f \ | | |5 |4 |3 |2 |1 |0 |9 |8 |7 |6 |5 |4 |3 |2 |1 |0 | o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o | | | | | | f_0 | 0000 | () | 1 | | | | | | | f_1 | 0001 | (x)(y) | 1 1 | | | | | | | f_2 | 0010 | (x) y | 1 1 | | | | | | | f_3 | 0011 | (x) | 1 1 1 1 | | | | | | | f_4 | 0100 | x (y) | 1 1 | | | | | | | f_5 | 0101 | (y) | 1 1 1 1 | | | | | | | f_6 | 0110 | (x, y) | 1 1 1 1 | | | | | | | f_7 | 0111 | (x y) | 1 1 1 1 1 1 1 1 | | | | | | | f_8 | 1000 | x y | 1 1 | | | | | | | f_9 | 1001 | ((x, y)) | 1 1 1 1 | | | | | | | f_10 | 1010 | y | 1 1 1 1 | | | | | | | f_11 | 1011 | (x (y)) | 1 1 1 1 1 1 1 1 | | | | | | | f_12 | 1100 | x | 1 1 1 1 | | | | | | | f_13 | 1101 | ((x) y) | 1 1 1 1 1 1 1 1 | | | | | | | f_14 | 1110 | ((x)(y)) | 1 1 1 1 1 1 1 1 | | | | | | | f_15 | 1111 | (()) |1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 | | | | | | o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
Table 11. Qualifiers of Implication Ordering: !b!_i f = !Y!(f => f_i) o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o | | x | 1100 | f |b |b |b |b |b |b |b |b |b |b |b |b |b |b |b |b | | | y | 1010 | |0 |0 |0 |0 |0 |0 |0 |0 |0 |0 |1 |1 |1 |1 |1 |1 | | f \ | | |0 |1 |2 |3 |4 |5 |6 |7 |8 |9 |0 |1 |2 |3 |4 |5 | o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o | | | | | | f_0 | 0000 | () |1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 | | | | | | | f_1 | 0001 | (x)(y) | 1 1 1 1 1 1 1 1 | | | | | | | f_2 | 0010 | (x) y | 1 1 1 1 1 1 1 1 | | | | | | | f_3 | 0011 | (x) | 1 1 1 1 | | | | | | | f_4 | 0100 | x (y) | 1 1 1 1 1 1 1 1 | | | | | | | f_5 | 0101 | (y) | 1 1 1 1 | | | | | | | f_6 | 0110 | (x, y) | 1 1 1 1 | | | | | | | f_7 | 0111 | (x y) | 1 1 | | | | | | | f_8 | 1000 | x y | 1 1 1 1 1 1 1 1 | | | | | | | f_9 | 1001 | ((x, y)) | 1 1 1 1 | | | | | | | f_10 | 1010 | y | 1 1 1 1 | | | | | | | f_11 | 1011 | (x (y)) | 1 1 | | | | | | | f_12 | 1100 | x | 1 1 1 1 | | | | | | | f_13 | 1101 | ((x) y) | 1 1 | | | | | | | f_14 | 1110 | ((x)(y)) | 1 1 | | | | | | | f_15 | 1111 | (()) | 1 | | | | | | o------o------o----------o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o--o
Table 13. Syllogistic Premisses as Higher Order Indicator Functions o---o------------------------o-----------------o---------------------------o | | | | | | A | Universal Affirmative | All x is y | Indicator of " x (y)" = 0 | | | | | | | E | Universal Negative | All x is (y) | Indicator of " x y " = 0 | | | | | | | I | Particular Affirmative | Some x is y | Indicator of " x y " = 1 | | | | | | | O | Particular Negative | Some x is (y) | Indicator of " x (y)" = 1 | | | | | | o---o------------------------o-----------------o---------------------------o
Table 14. Relation of Quantifiers to Higher Order Propositions o------------o------------o-----------o-----------o-----------o-----------o | Mnemonic | Category | Classical | Alternate | Symmetric | Operator | | | | Form | Form | Form | | o============o============o===========o===========o===========o===========o | E | Universal | All x | | No x | (L_11) | | Exclusive | Negative | is (y) | | is y | | o------------o------------o-----------o-----------o-----------o-----------o | A | Universal | All x | | No x | (L_10) | | Absolute | Affrmtve | is y | | is (y) | | o------------o------------o-----------o-----------o-----------o-----------o | | | All y | No y | No (x) | (L_01) | | | | is x | is (x) | is y | | o------------o------------o-----------o-----------o-----------o-----------o | | | All (y) | No (y) | No (x) | (L_00) | | | | is x | is (x) | is (y) | | o------------o------------o-----------o-----------o-----------o-----------o | | | Some (x) | | Some (x) | L_00 | | | | is (y) | | is (y) | | o------------o------------o-----------o-----------o-----------o-----------o | | | Some (x) | | Some (x) | L_01 | | | | is y | | is y | | o------------o------------o-----------o-----------o-----------o-----------o | O | Particular | Some x | | Some x | L_10 | | Obtrusive | Negative | is (y) | | is (y) | | o------------o------------o-----------o-----------o-----------o-----------o | I | Particular | Some x | | Some x | L_11 | | Indefinite | Affrmtve | is y | | is y | | o------------o------------o-----------o-----------o-----------o-----------o
Table 15. Simple Qualifiers of Propositions (n = 2) o------o------o----------o-----o-----o-----o-----o-----o-----o-----o-----o | | x | 1100 | f |(L11)|(L10)|(L01)|(L00)| L00 | L01 | L10 | L11 | | | y | 1010 | |no x|no x|no ~x|no ~x|sm ~x|sm ~x|sm x|sm x| | f \ | | |is y|is ~y|is y|is ~y|is ~y|is y|is ~y|is y| o------o------o----------o-----o-----o-----o-----o-----o-----o-----o-----o | | | | | | f_0 | 0000 | () | 1 1 1 1 0 0 0 0 | | | | | | | f_1 | 0001 | (x)(y) | 1 1 1 0 1 0 0 0 | | | | | | | f_2 | 0010 | (x) y | 1 1 0 1 0 1 0 0 | | | | | | | f_3 | 0011 | (x) | 1 1 0 0 1 1 0 0 | | | | | | | f_4 | 0100 | x (y) | 1 0 1 1 0 0 1 0 | | | | | | | f_5 | 0101 | (y) | 1 0 1 0 1 0 1 0 | | | | | | | f_6 | 0110 | (x, y) | 1 0 0 1 0 1 1 0 | | | | | | | f_7 | 0111 | (x y) | 1 0 0 0 1 1 1 0 | | | | | | | f_8 | 1000 | x y | 0 1 1 1 0 0 0 1 | | | | | | | f_9 | 1001 | ((x, y)) | 0 1 1 0 1 0 0 1 | | | | | | | f_10 | 1010 | y | 0 1 0 1 0 1 0 1 | | | | | | | f_11 | 1011 | (x (y)) | 0 1 0 0 1 1 0 1 | | | | | | | f_12 | 1100 | x | 0 0 1 1 0 0 1 1 | | | | | | | f_13 | 1101 | ((x) y) | 0 0 1 0 1 0 1 1 | | | | | | | f_14 | 1110 | ((x)(y)) | 0 0 0 1 0 1 1 1 | | | | | | | f_15 | 1111 | (()) | 0 0 0 0 1 1 1 1 | | | | | | o------o------o----------o-----o-----o-----o-----o-----o-----o-----o-----o
Zeroth Order Logic
| L1 | L2 | L3 | L4 | L5 | L6 |
|---|---|---|---|---|---|
| x : | 1 1 0 0 | ||||
| y : | 1 0 1 0 | ||||
| f0 | f0000 | 0 0 0 0 | ( ) | false | 0 |
| f1 | f0001 | 0 0 0 1 | (x)(y) | neither x nor y | ¬x ∧ ¬y |
| f2 | f0010 | 0 0 1 0 | (x) y | y and not x | ¬x ∧ y |
| f3 | f0011 | 0 0 1 1 | (x) | not x | ¬x |
| f4 | f0100 | 0 1 0 0 | x (y) | x and not y | x ∧ ¬y |
| f5 | f0101 | 0 1 0 1 | (y) | not y | ¬y |
| f6 | f0110 | 0 1 1 0 | (x, y) | x not equal to y | x ≠ y |
| f7 | f0111 | 0 1 1 1 | (x y) | not both x and y | ¬x ∨ ¬y |
| f8 | f1000 | 1 0 0 0 | x y | x and y | x ∧ y |
| f9 | f1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y |
| f10 | f1010 | 1 0 1 0 | y | y | y |
| f11 | f1011 | 1 0 1 1 | (x (y)) | not x without y | x → y |
| f12 | f1100 | 1 1 0 0 | x | x | x |
| f13 | f1101 | 1 1 0 1 | ((x) y) | not y without x | x ← y |
| f14 | f1110 | 1 1 1 0 | ((x)(y)) | x or y | x ∨ y |
| f15 | f1111 | 1 1 1 1 | (( )) | true | 1 |
| L1 | L2 | L3 | L4 | L5 | L6 |
|---|---|---|---|---|---|
| x : | 1 1 0 0 | ||||
| y : | 1 0 1 0 | ||||
| f0 | f0000 | 0 0 0 0 | ( ) | false | 0 |
| f1 | f0001 | 0 0 0 1 | (x)(y) | neither x nor y | ¬x ∧ ¬y |
| f2 | f0010 | 0 0 1 0 | (x) y | y and not x | ¬x ∧ y |
| f3 | f0011 | 0 0 1 1 | (x) | not x | ¬x |
| f4 | f0100 | 0 1 0 0 | x (y) | x and not y | x ∧ ¬y |
| f5 | f0101 | 0 1 0 1 | (y) | not y | ¬y |
| f6 | f0110 | 0 1 1 0 | (x, y) | x not equal to y | x ≠ y |
| f7 | f0111 | 0 1 1 1 | (x y) | not both x and y | ¬x ∨ ¬y |
| f8 | f1000 | 1 0 0 0 | x y | x and y | x ∧ y |
| f9 | f1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y |
| f10 | f1010 | 1 0 1 0 | y | y | y |
| f11 | f1011 | 1 0 1 1 | (x (y)) | not x without y | x → y |
| f12 | f1100 | 1 1 0 0 | x | x | x |
| f13 | f1101 | 1 1 0 1 | ((x) y) | not y without x | x ← y |
| f14 | f1110 | 1 1 1 0 | ((x)(y)) | x or y | x ∨ y |
| f15 | f1111 | 1 1 1 1 | (( )) | true | 1 |
Template Draft
| L1 | L2 | L3 | L4 | L5 | L6 | Name |
|---|---|---|---|---|---|---|
| x : | 1 1 0 0 | |||||
| y : | 1 0 1 0 | |||||
| f0 | f0000 | 0 0 0 0 | ( ) | false | 0 | Falsity |
| f1 | f0001 | 0 0 0 1 | (x)(y) | neither x nor y | ¬x ∧ ¬y | NNOR |
| f2 | f0010 | 0 0 1 0 | (x) y | y and not x | ¬x ∧ y | Insuccede |
| f3 | f0011 | 0 0 1 1 | (x) | not x | ¬x | Not One |
| f4 | f0100 | 0 1 0 0 | x (y) | x and not y | x ∧ ¬y | Imprecede |
| f5 | f0101 | 0 1 0 1 | (y) | not y | ¬y | Not Two |
| f6 | f0110 | 0 1 1 0 | (x, y) | x not equal to y | x ≠ y | Inequality |
| f7 | f0111 | 0 1 1 1 | (x y) | not both x and y | ¬x ∨ ¬y | NAND |
| f8 | f1000 | 1 0 0 0 | x y | x and y | x ∧ y | Conjunction |
| f9 | f1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y | Equality |
| f10 | f1010 | 1 0 1 0 | y | y | y | Two |
| f11 | f1011 | 1 0 1 1 | (x (y)) | not x without y | x → y | Implication |
| f12 | f1100 | 1 1 0 0 | x | x | x | One |
| f13 | f1101 | 1 1 0 1 | ((x) y) | not y without x | x ← y | Involution |
| f14 | f1110 | 1 1 1 0 | ((x)(y)) | x or y | x ∨ y | Disjunction |
| f15 | f1111 | 1 1 1 1 | (( )) | true | 1 | Tautology |
Truth Tables
Logical negation
Logical negation is an operation on one logical value, typically the value of a proposition, that produces a value of true when its operand is false and a value of false when its operand is true.
The truth table of NOT p (also written as ~p or ¬p) is as follows:
| p | ¬p |
|---|---|
| F | T |
| T | F |
The logical negation of a proposition p is notated in different ways in various contexts of discussion and fields of application. Among these variants are the following:
| Notation | Vocalization |
|---|---|
| \(\bar{p}\) | bar p |
| \(p'\!\) | p prime, p complement |
| \(!p\!\) | bang p |
No matter how it is notated or symbolized, the logical negation ¬p is read as "it is not the case that p", or usually more simply as "not p".
- Within a system of classical logic, double negation, that is, the negation of the negation of a proposition p, is logically equivalent to the initial proposition p. Expressed in symbolic terms, ¬(¬p) ⇔ p.
- Within a system of intuitionistic logic, however, ¬¬p is a weaker statement than p. On the other hand, the logical equivalence ¬¬¬p ⇔ ¬p remains valid.
Logical negation can be defined in terms of other logical operations. For example, ~p can be defined as p → F, where → is material implication and F is absolute falsehood. Conversely, one can define F as p & ~p for any proposition p, where & is logical conjunction. The idea here is that any contradiction is false. While these ideas work in both classical and intuitionistic logic, they don't work in Brazilian logic, where contradictions are not necessarily false. But in classical logic, we get a further identity: p → q can be defined as ~p ∨ q, where ∨ is logical disjunction.
Algebraically, logical negation corresponds to the complement in a Boolean algebra (for classical logic) or a Heyting algebra (for intuitionistic logic).
Logical conjunction
Logical conjunction is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both of its operands are true.
The truth table of p AND q (also written as p ∧ q, p & q, or p\(\cdot\)q) is as follows:
| p | q | p ∧ q |
|---|---|---|
| F | F | F |
| F | T | F |
| T | F | F |
| T | T | T |
Logical disjunction
Logical disjunction is an operation on two logical values, typically the values of two propositions, that produces a value of false if and only if both of its operands are false.
The truth table of p OR q (also written as p ∨ q) is as follows:
| p | q | p ∨ q |
|---|---|---|
| F | F | F |
| F | T | T |
| T | F | T |
| T | T | T |
Logical equality
Logical equality is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both operands are false or both operands are true.
The truth table of p EQ q (also written as p = q, p ↔ q, or p ≡ q) is as follows:
| p | q | p = q |
|---|---|---|
| F | F | T |
| F | T | F |
| T | F | F |
| T | T | T |
Exclusive disjunction
Exclusive disjunction is an operation on two logical values, typically the values of two propositions, that produces a value of true just in case exactly one of its operands is true.
The truth table of p XOR q (also written as p + q, p ⊕ q, or p ≠ q) is as follows:
| p | q | p XOR q |
|---|---|---|
| F | F | F |
| F | T | T |
| T | F | T |
| T | T | F |
The following equivalents can then be deduced:
\[\begin{matrix} p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q) \\ \\ & = & (p \lor q) & \land & (\lnot p \lor \lnot q) \\ \\ & = & (p \lor q) & \land & \lnot (p \land q) \end{matrix}\]
Generalized or n-ary XOR is true when the number of 1-bits is odd.
Logical implication
The material conditional and logical implication are both associated with an operation on two logical values, typically the values of two propositions, that produces a value of false if and only if the first operand is true and the second operand is false.
The truth table associated with the material conditional if p then q (symbolized as p → q) and the logical implication p implies q (symbolized as p ⇒ q) is as follows:
| p | q | p ⇒ q |
|---|---|---|
| F | F | T |
| F | T | T |
| T | F | F |
| T | T | T |
Logical NAND
The NAND operation is a logical operation on two logical values, typically the values of two propositions, that produces a value of false if and only if both of its operands are true. In other words, it produces a value of true if and only if at least one of its operands is false.
The truth table of p NAND q (also written as p | q or p ↑ q) is as follows:
| p | q | p ↑ q |
|---|---|---|
| F | F | T |
| F | T | T |
| T | F | T |
| T | T | F |
Logical NNOR
The NNOR operation is a logical operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both of its operands are false. In other words, it produces a value of false if and only if at least one of its operands is true.
The truth table of p NNOR q (also written as p ⊥ q or p ↓ q) is as follows:
| p | q | p ↓ q |
|---|---|---|
| F | F | T |
| F | T | F |
| T | F | F |
| T | T | F |
Exclusive Disjunction
A + B = (A ∧ !B) ∨ (!A ∧ B)
= {(A ∧ !B) ∨ !A} ∧ {(A ∧ !B) ∨ B}
= {(A ∨ !A) ∧ (!B ∨ !A)} ∧ {(A ∨ B) ∧ (!B ∨ B)}
= (!A ∨ !B) ∧ (A ∨ B)
= !(A ∧ B) ∧ (A ∨ B)
p + q = (p ∧ !q) ∨ (!p ∧ B)
= {(p ∧ !q) ∨ !p} ∧ {(p ∧ !q) ∨ q}
= {(p ∨ !q) ∧ (!q ∨ !p)} ∧ {(p ∨ q) ∧ (!q ∨ q)}
= (!p ∨ !q) ∧ (p ∨ q)
= !(p ∧ q) ∧ (p ∨ q)
p + q = (p ∧ ~q) ∨ (~p ∧ q)
= ((p ∧ ~q) ∨ ~p) ∧ ((p ∧ ~q) ∨ q)
= ((p ∨ ~q) ∧ (~q ∨ ~p)) ∧ ((p ∨ q) ∧ (~q ∨ q))
= (~p ∨ ~q) ∧ (p ∨ q)
= ~(p ∧ q) ∧ (p ∨ q)
\[\begin{matrix} p + q & = & (p \land \lnot q) & \lor & (\lnot p \land q) \\ & = & ((p \land \lnot q) \lor \lnot p) & \and & ((p \land \lnot q) \lor q) \\ & = & ((p \lor \lnot q) \land (\lnot q \lor \lnot p)) & \land & ((p \lor q) \land (\lnot q \lor q)) \\ & = & (\lnot p \lor \lnot q) & \land & (p \lor q) \\ & = & \lnot (p \land q) & \land & (p \lor q) \end{matrix}\]
Relational Tables
Sign Relations
| O | = | Object Domain | |
| S | = | Sign Domain | |
| I | = | Interpretant Domain |
| O | = | {Ann, Bob} | = | {A, B} | |
| S | = | {"Ann", "Bob", "I", "You"} | = | {"A", "B", "i", "u"} | |
| I | = | {"Ann", "Bob", "I", "You"} | = | {"A", "B", "i", "u"} |
| Object | Sign | Interpretant |
|---|---|---|
| A | "A" | "A" |
| A | "A" | "i" |
| A | "i" | "A" |
| A | "i" | "i" |
| B | "B" | "B" |
| B | "B" | "u" |
| B | "u" | "B" |
| B | "u" | "u" |
| Object | Sign | Interpretant |
|---|---|---|
| A | "A" | "A" |
| A | "A" | "u" |
| A | "u" | "A" |
| A | "u" | "u" |
| B | "B" | "B" |
| B | "B" | "i" |
| B | "i" | "B" |
| B | "i" | "i" |
Triadic Relations
Algebraic Examples
| X | Y | Z |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
| X | Y | Z |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
Semiotic Examples
| Object | Sign | Interpretant |
|---|---|---|
| A | "A" | "A" |
| A | "A" | "i" |
| A | "i" | "A" |
| A | "i" | "i" |
| B | "B" | "B" |
| B | "B" | "u" |
| B | "u" | "B" |
| B | "u" | "u" |
| Object | Sign | Interpretant |
|---|---|---|
| A | "A" | "A" |
| A | "A" | "u" |
| A | "u" | "A" |
| A | "u" | "u" |
| B | "B" | "B" |
| B | "B" | "i" |
| B | "i" | "B" |
| B | "i" | "i" |
Dyadic Projections
| LOS | = | projOS(L) | = | { (o, s) ∈ O × S : (o, s, i) ∈ L for some i ∈ I } | |
| LSO | = | projSO(L) | = | { (s, o) ∈ S × O : (o, s, i) ∈ L for some i ∈ I } | |
| LIS | = | projIS(L) | = | { (i, s) ∈ I × S : (o, s, i) ∈ L for some o ∈ O } | |
| LSI | = | projSI(L) | = | { (s, i) ∈ S × I : (o, s, i) ∈ L for some o ∈ O } | |
| LOI | = | projOI(L) | = | { (o, i) ∈ O × I : (o, s, i) ∈ L for some s ∈ S } | |
| LIO | = | projIO(L) | = | { (i, o) ∈ I × O : (o, s, i) ∈ L for some s ∈ S } |
Method 1 : Subtitles as Captions
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Method 2 : Subtitles as Top Rows
projOS(LA)
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projOS(LB)
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projSI(LA)
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projSI(LB)
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projOI(LA)
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projOI(LB)
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Relation Reduction
Method 1 : Subtitles as Captions
| X | Y | Z |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
| X | Y | Z |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
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| projXY(L0) = projXY(L1) | projXZ(L0) = projXZ(L1) | projYZ(L0) = projYZ(L1) |
| Object | Sign | Interpretant |
|---|---|---|
| A | "A" | "A" |
| A | "A" | "i" |
| A | "i" | "A" |
| A | "i" | "i" |
| B | "B" | "B" |
| B | "B" | "u" |
| B | "u" | "B" |
| B | "u" | "u" |
| Object | Sign | Interpretant |
|---|---|---|
| A | "A" | "A" |
| A | "A" | "u" |
| A | "u" | "A" |
| A | "u" | "u" |
| B | "B" | "B" |
| B | "B" | "i" |
| B | "i" | "B" |
| B | "i" | "i" |
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| projXY(LA) ≠ projXY(LB) | projXZ(LA) ≠ projXZ(LB) | projYZ(LA) ≠ projYZ(LB) |
Method 2 : Subtitles as Top Rows
| X | Y | Z |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
| X | Y | Z |
|---|---|---|
| 0 | 0 | 1 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 1 | 1 | 1 |
projXY(L0)
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projXZ(L0)
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projYZ(L0)
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projXY(L1)
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projXZ(L1)
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projYZ(L1)
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| projXY(L0) = projXY(L1) | projXZ(L0) = projXZ(L1) | projYZ(L0) = projYZ(L1) |
| Object | Sign | Interpretant |
|---|---|---|
| A | "A" | "A" |
| A | "A" | "i" |
| A | "i" | "A" |
| A | "i" | "i" |
| B | "B" | "B" |
| B | "B" | "u" |
| B | "u" | "B" |
| B | "u" | "u" |
| Object | Sign | Interpretant |
|---|---|---|
| A | "A" | "A" |
| A | "A" | "u" |
| A | "u" | "A" |
| A | "u" | "u" |
| B | "B" | "B" |
| B | "B" | "i" |
| B | "i" | "B" |
| B | "i" | "i" |
projXY(LA)
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projXZ(LA)
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projYZ(LA)
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projXY(LB)
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projXZ(LB)
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projYZ(LB)
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| projXY(LA) ≠ projXY(LB) | projXZ(LA) ≠ projXZ(LB) | projYZ(LA) ≠ projYZ(LB) |
Formatted Text Display
- So in a triadic fact, say, the example
| A gives B to C |
- we make no distinction in the ordinary logic of relations between the subject nominative, the direct object, and the indirect object. We say that the proposition has three logical subjects. We regard it as a mere affair of English grammar that there are six ways of expressing this:
| A gives B to C | A benefits C with B |
| B enriches C at expense of A | C receives B from A |
| C thanks A for B | B leaves A for C |
- These six sentences express one and the same indivisible phenomenon. (C.S. Peirce, "The Categories Defended", MS 308 (1903), EP 2, 170-171).
Work Area
| x0 | x1 | 2f0 | 2f1 | 2f2 | 2f3 | 2f4 | 2f5 | 2f6 | 2f7 | 2f8 | 2f9 | 2f10 | 2f11 | 2f12 | 2f13 | 2f14 | 2f15 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
| 1 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
Draft 1
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Draft 2
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