[Inquiry] Re: Differential Logic
Jon Awbrey
jawbrey at oakland.edu
Tue May 20 13:00:05 CDT 2003
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
DLOG. Note D29
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Thematization: Venn Diagrams
| The known universe has one complete lover and that is the greatest poet.
| He consumes an eternal passion and is indifferent which chance happens
| and which possible contingency of fortune or misfortune and persuades
| daily and hourly his delicious pay.
|
| Walt Whitman, 'Leaves of Grass', [Whi, 11-12]
Figure 20-i traces the first couple of steps in this order of "thematic" progression,
that will gradually run the gamut through a complete series of degrees of functional
explicitness in the expression of logical propositions. The first venn diagram
represents a situation where the function is indicated by a shaded figure and
a logical expression. At this stage one may be thinking of the proposition
only as expressed by a formula in a particular language and its content
only as a subset of the universe of discourse, as when one considers
the proposition u.v in [u, v].
The second venn diagram depicts a situation in which two significant
steps have been taken. First, one has taken the trouble to give the
proposition u.v a distinctive functional name "J". Second, one has
come to think explicitly about the target domain that contains the
functional values of J, as when one writes J : <|u, v|> -> B.
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)
In Figure 20-ii the proposition J is viewed explicitly as
a transformation from one universe of discourse to another.
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----- at -----o | | o----- at -----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)
In the first venn diagram the name that is assigned to a composite proposition,
function, or region in the source universe is delegated to a simple feature in
the target universe. This can result in a single character or term exceeding
the responsibilities it can carry off well. Allowing the name of a function
J : <|u, v|> -> B to serve as the name of its dependent variable J : B does
not mean that one has to confuse a function with any of its values, but it
does put one at risk for a number of obvious problems, and we should not
be surprised, on numerous and limiting occasions, when quibbling arises
from the attempts of a too original syntax to serve these two masters.
The second venn diagram circumvents these difficulties by introducing a new
variable name for each basic feature of the target universe, as when one
writes J : <|u, v|> -> <|x|> and thereby assigns a concrete type <|x|>
to the abstract codomain B. To make this induction of variables more
formal one can append subscripts, as in x_J, to indicate the origin
or the derivation of these parvenu characters. However, it is not
always convenient to keep inventing new variable names in this way.
For use at these times, I introduce a lexical operator "¢", read
"cents" or "obelus", that converts a function name into a variable
name. For example, one may think of x = x_J = ¢(J) = J¢ = J^¢ as
"the cache variable of J", "J circumscript", "J made circumstantial",
or "J considered as a contingent variable".
In Figure 20-iii we arrive at a stage where the functional equations,
J = u.v and x = u.v, are regarded as propositions in their own right,
reigning in and ruling over 3-feature universes of discourse, [u, v, J]
and [u, v, x], respectively. Subject to the cautions already noted, the
function name "J" can be reinterpreted as the name of a feature J^¢, and
the equation J = u.v can be read as the logical equivalence ((J, u v)).
To give it a generic name let us call this newly expressed, collateral
proposition the "thematization" or the "thematic extension" of the
original proposition J.
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)
The first venn diagram represents the thematization of the conjunction J
with shading in the appropriate regions of the universe [u, v, J]. Also,
it illustrates a quick way of constructing a thematic extension. First,
draw a line, in practice or the imagination, that bisects every cell of
the original universe, placing half of each cell under the aegis of the
thematized proposition and the other half under its antithesis. Next,
on the scene where the theme applies leave the shade wherever it lies,
and off the stage, where it plays otherwise, stagger the pattern in
a harlequin guise.
In the final venn diagram of this sequence the thematic progression comes
full circle and completes one round of its development. The ambiguities
that were occasioned by the changing role of the name "J" are resolved
by introducing a new variable name "x" to take the place of J^¢, and
the region that represents this fresh featured x is circumscribed
in a more conventional symmetry of form and placement. Just as
we once gave the name "J" to the proposition u.v, we now give
the name "!j!" to its thematization ((x, u v)). Already,
again, at this culminating stage of reflection, we begin
to think of the newly named proposition as a distinctive
individual, a particular function !j! : <|u, v, x|> -> B.
More information about the Inquiry
mailing list