ONT Re: Differential Logic
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
DLOG. Note D31
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Thematization: Truth Tables
| That which distorts honest shapes or which creates unearthly
| beings or places or contingencies is a nuisance and a revolt.
|
| Walt Whitman, 'Leaves of Grass', [Whi, 19]
Tables 22 through 25 outline a method for computing
the thematic extensions of propositions in terms of
their coordinate values.
A preliminary step, as illustrated in Table 22, is to write out
the truth table representations of the propositional forms whose
thematic extensions one wants to compute, in the present instance,
the functions f<u, v> = ((u)(v)) and g<u, v> = ((u, v)).
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
Next, each propositional form is individually represented in the fashion
shown in Tables 23-i and 23-ii, using "f" and "g" as function names and
creating new variables x and y to hold the associated functional values.
This pair of Tables outlines the first stage in the transition from the
2-dimensional universes of f and g to the 3-dimensional universes of
theta(f) and theta(g). The top halves of the Tables replicate the
truth table patterns for f and g in the form f : [u, v] -> [x]
and g : [u, v] -> [y]. The bottom halves of the tables print
the negatives of these pictures, as it were, and paste the
truth tables for (f) and (g) under the copies for f and g.
At this stage, the columns for theta(f) and theta(g) are
appended almost as afterthoughts, amounting to indicator
functions for the sets of ordered triples that make up
the functions f and g.
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
All the data is now in place to give the truth tables for theta(f) and theta(g).
In the remaining steps all we do is to permute the rows and change the roles of
x and y from dependent to independent variables. In Tables 24-i and 24-ii the
rows are arranged in such a way as to put the 3-tuples <u, v, x> and <u, v, y>
in binary numerical order, suitable for viewing as the arguments of the maps
theta(f) = !f! : [u, v, x] -> B and theta(g) = !g! : [u, v, y]->B. Moreover,
the structure of the tables is altered slightly, allowing the now vestigial
functions f and g to be passed over without further attention and shifting
the heavy vertical bars a notch to the right. In effect, this clinches
the fact that the thematic variables x := f^¢ and y := g^¢ are now to
be regarded as independent variables.
Tables 24-i and 24-ii. Thematics of Disjunction & 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
An optional reshuffling of the rows brings additional features of the thematic
extensions to light. Leaving the columns in place for the sake of comparison,
Tables 25-i and 25-ii sorts the rows in a different order, in effect treating
x and y as the primary variables in their respective 3-tuples. Regarding the
thematic extensions in the form !f! : [x, u, v] -> B and !g! : [y, u, v] -> B
makes it easier to see in this tabular setting a property that was graphically
obvious in the venn diagrams above. Specifically, when the thematic variable
F^¢ is true then theta(F) exhibits the pattern of the original F, and when
F^¢ is false then theta(F) exhibits the pattern of its negation (F).
Tables 25-i and 25-ii. Thematics of Disjunction & 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
Finally, Tables 26-i and 26-ii compare the tacit extensions
!e! : [u, v] -> [u, v, x] and !e! : [u, v]->[u, v, y] with
the thematic extensions of the same types, as applied to
the propositions f and g, respectively.
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
Jon Awbrey
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o