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, 20:45, 11 August 2009→Variations on a theme of transitivity: center figures & tables
To study the differences between these two versions of transitivity within what is locally a familiar context, let's view the propositional forms involved as if they were elementary cellular automaton rules, resulting in the following Table.
To study the differences between these two versions of transitivity within what is locally a familiar context, let's view the propositional forms involved as if they were elementary cellular automaton rules, resulting in the following Table.
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<pre>
<pre>
Table 21. Composite and Compiled Order Relations
Table 21. Composite and Compiled Order Relations
o---------o------------o-----------------o----------------o-------------o
o---------o------------o-----------------o----------------o-------------o
</pre>
</pre>
|}
Taking up another angle of incidence by way of extra perspective, let us now reflect on the venn diagrams of our four propositions.
Taking up another angle of incidence by way of extra perspective, let us now reflect on the venn diagrams of our four propositions.
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<pre>
<pre>
o-----------------------------------------------------------o
o-----------------------------------------------------------o
q_207. (p (q))
q_207. (p (q))
</pre>
</pre>
|-
|
<pre>
<pre>
o-----------------------------------------------------------o
o-----------------------------------------------------------o
q_187. (q (r))
q_187. (q (r))
</pre>
</pre>
|-
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<pre>
<pre>
o-----------------------------------------------------------o
o-----------------------------------------------------------o
q_175. (p (r))
q_175. (p (r))
</pre>
</pre>
|-
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<pre>
<pre>
o-----------------------------------------------------------o
o-----------------------------------------------------------o
q_139. (p (q))(q (r))
q_139. (p (q))(q (r))
</pre>
</pre>
|}
Among other things, these images make it visually obvious that the constraint on the three boolean variables ''p'', ''q'', ''r'' that we indicate by asserting either of the forms "(p (q))(q (r))" or "''p'' ≤ ''q'' ≤ ''r''" is one that implies a constraint on the two boolean variables ''p'', ''r'' that we indicate by either of the forms "(p (r))" or "''p'' ≤ ''r''", but that it imposes additional constraints on these variables that are not captured by the illative conclusion.
Among other things, these images make it visually obvious that the constraint on the three boolean variables ''p'', ''q'', ''r'' that we indicate by asserting either of the forms "(p (q))(q (r))" or "''p'' ≤ ''q'' ≤ ''r''" is one that implies a constraint on the two boolean variables ''p'', ''r'' that we indicate by either of the forms "(p (r))" or "''p'' ≤ ''r''", but that it imposes additional constraints on these variables that are not captured by the illative conclusion.
Thus we obtain the following four relational data tables for the propositions that we are looking at in Example 2.
Thus we obtain the following four relational data tables for the propositions that we are looking at in Example 2.
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<pre>
<pre>
[| q_207 |] = [| p =< q |]
[| q_207 |] = [| p =< q |]
o-----------------------------o
o-----------------------------o
</pre>
</pre>
|-
|
<pre>
<pre>
[| q_187 |] = [| q =< r |]
[| q_187 |] = [| q =< r |]
o-----------------------------o
o-----------------------------o
</pre>
</pre>
|-
|
<pre>
<pre>
[| q_175 |] = [| p =< r |]
[| q_175 |] = [| p =< r |]
o-----------------------------o
o-----------------------------o
</pre>
</pre>
|-
|
<pre>
<pre>
[| q_139 |] = [| p =< q =< r |]
[| q_139 |] = [| p =< q =< r |]
o-----------------------------o
o-----------------------------o
</pre>
</pre>
|}
In the medium of these unassuming examples, we begin to see the activities of logical inference and methodical inquiry as ''information clarifying operations'' (ICO's).
In the medium of these unassuming examples, we begin to see the activities of logical inference and methodical inquiry as ''information clarifying operations'' (ICO's).
For ease of reference during the rest of this discussion, let us refer to the propositional form ''f'' : '''B'''<sup>3</sup> → '''B''' such that ''f''(''p'', ''q'', ''r'') = ''q''<sub>139</sub>(''p'', ''q'', ''r'') = (p (q))(q (r)) as the ''syllogism mapping'', written as ''syll'' : '''B'''<sup>3</sup> → '''B''', and let us refer to the fiber ''syll''<sup>−1</sup>(1) ⊆ '''B'''<sup>3</sup> as the ''syllogism relation'', written as ''Syll'' ⊆ '''B'''<sup>3</sup>. Table 25-a shows ''Syll'' as a relational dataset.
For ease of reference during the rest of this discussion, let us refer to the propositional form ''f'' : '''B'''<sup>3</sup> → '''B''' such that ''f''(''p'', ''q'', ''r'') = ''q''<sub>139</sub>(''p'', ''q'', ''r'') = (p (q))(q (r)) as the ''syllogism mapping'', written as ''syll'' : '''B'''<sup>3</sup> → '''B''', and let us refer to the fiber ''syll''<sup>−1</sup>(1) ⊆ '''B'''<sup>3</sup> as the ''syllogism relation'', written as ''Syll'' ⊆ '''B'''<sup>3</sup>. Table 25-a shows ''Syll'' as a relational dataset.
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<pre>
<pre>
Table 25-a. Syllogism Relation
Table 25-a. Syllogism Relation
o-----------------------------o
o-----------------------------o
</pre>
</pre>
|}
One of the first questions that we might ask about a 3-adic relation, in this case ''Syll'', is whether it is ''determined by'' its 2-adic projections. I will illustrate what this means in the present case.
One of the first questions that we might ask about a 3-adic relation, in this case ''Syll'', is whether it is ''determined by'' its 2-adic projections. I will illustrate what this means in the present case.
Table 25-b repeats the relation ''Syll'' in the first column, listing its 3-tuples in bit-string form, followed by the 2-adic or ''planar'' projections of ''Syll'' in the next three columns. For instance, ''Syll''<sub>''pq''</sub> is the 2-adic projection of ''Syll'' on the ''pq'' plane that is arrived at by deleting the ''r'' column and counting each 2-tuple that results just one time. Likewise, ''Syll''<sub>''pr''</sub> is obtained by deleting the ''q'' column and ''Syll''<sub>''qr''</sub> is derived by deleting the p column, ignoring whatever duplicate pairs may result. The final row of the right three columns gives the propositions of the form ''f'' : '''B'''<sup>2</sup> → '''B''' that indicate the 2-adic relations that result from these projections.
Table 25-b repeats the relation ''Syll'' in the first column, listing its 3-tuples in bit-string form, followed by the 2-adic or ''planar'' projections of ''Syll'' in the next three columns. For instance, ''Syll''<sub>''pq''</sub> is the 2-adic projection of ''Syll'' on the ''pq'' plane that is arrived at by deleting the ''r'' column and counting each 2-tuple that results just one time. Likewise, ''Syll''<sub>''pr''</sub> is obtained by deleting the ''q'' column and ''Syll''<sub>''qr''</sub> is derived by deleting the p column, ignoring whatever duplicate pairs may result. The final row of the right three columns gives the propositions of the form ''f'' : '''B'''<sup>2</sup> → '''B''' that indicate the 2-adic relations that result from these projections.
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<pre>
<pre>
Table 25-b. Dyadic Projections of the Syllogism Relation
Table 25-b. Dyadic Projections of the Syllogism Relation
o-------------o-------------o-------------o-------------o
o-------------o-------------o-------------o-------------o
</pre>
</pre>
|}
Let us make the simple observation that taking a projection, in our framework, deleting a column from a relational table, is like taking a derivative in differential calculus. What it means is that our attempt to return to the integral from whence the derivative was derived will in general encounter an indefinite variation on account of the circumstance that real information may have been destroyed by the derivation.
Let us make the simple observation that taking a projection, in our framework, deleting a column from a relational table, is like taking a derivative in differential calculus. What it means is that our attempt to return to the integral from whence the derivative was derived will in general encounter an indefinite variation on account of the circumstance that real information may have been destroyed by the derivation.
Figure 28-a shows the familiar picture of a boolean 3-cube, wherein the points of '''B'''<sup>3</sup> are coordinated as bit strings of length three. Looking at the functions ''f'' : '''B'''<sup>3</sup> → '''B''' and the relations ''L'' ⊆ '''B'''<sup>3</sup> on this pattern, one views the construction of either type of object as a matter of coloring the nodes of the 3-cube with choices from a pair of colors that stipulate which points are in the relation ''L'' = <nowiki>[|</nowiki>''f''<nowiki>|]</nowiki> and which points are out of it. Bowing to common convention, we may use the color "1" for points that are "in" a given relation and the color "0" for points that are "out" of that same relation. However, it will be more convenient here to indicate the former case by writing the coordinates in the place of the node and to indicate the latter case by plotting the point as an unlabeled node "o".
Figure 28-a shows the familiar picture of a boolean 3-cube, wherein the points of '''B'''<sup>3</sup> are coordinated as bit strings of length three. Looking at the functions ''f'' : '''B'''<sup>3</sup> → '''B''' and the relations ''L'' ⊆ '''B'''<sup>3</sup> on this pattern, one views the construction of either type of object as a matter of coloring the nodes of the 3-cube with choices from a pair of colors that stipulate which points are in the relation ''L'' = <nowiki>[|</nowiki>''f''<nowiki>|]</nowiki> and which points are out of it. Bowing to common convention, we may use the color "1" for points that are "in" a given relation and the color "0" for points that are "out" of that same relation. However, it will be more convenient here to indicate the former case by writing the coordinates in the place of the node and to indicate the latter case by plotting the point as an unlabeled node "o".
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<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
Figure 28-a. Boolean 3-Cube B^3
Figure 28-a. Boolean 3-Cube B^3
</pre>
</pre>
|}
Table 28-b shows the 3-adic relation ''Syll'' ⊆ '''B'''<sup>3</sup> again, and Figure 28-c shows it plotted on a 3-cube template.
Table 28-b shows the 3-adic relation ''Syll'' ⊆ '''B'''<sup>3</sup> again, and Figure 28-c shows it plotted on a 3-cube template.
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<pre>
<pre>
Table 28-b. Syll c B^3
Table 28-b. Syll c B^3
o-----------------------o
o-----------------------o
</pre>
</pre>
|-
|
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
Figure 28-c. Triadic Relation Syll c B^3
Figure 28-c. Triadic Relation Syll c B^3
</pre>
</pre>
|}
We return once more to the plane projections of ''Syll'' ⊆ '''B'''<sup>3</sup>.
We return once more to the plane projections of ''Syll'' ⊆ '''B'''<sup>3</sup>.
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<pre>
<pre>
Table 29-a. Syll c B^3
Table 29-a. Syll c B^3
o-----------------------o
o-----------------------o
</pre>
</pre>
|-
|
<pre>
<pre>
Table 29-b. Dyadic Projections of Syll
Table 29-b. Dyadic Projections of Syll
o-----------o o-----------o o-----------o
o-----------o o-----------o o-----------o
</pre>
</pre>
|}
In showing the 2-adic projections of a 3-adic relation ''L'' ⊆ '''B'''<sup>3</sup>, I will translate the coordinates of the points in each relation to the plane of the projection, there dotting out with a dot "." the bit of the bit string that is out of place on that plane.
In showing the 2-adic projections of a 3-adic relation ''L'' ⊆ '''B'''<sup>3</sup>, I will translate the coordinates of the points in each relation to the plane of the projection, there dotting out with a dot "." the bit of the bit string that is out of place on that plane.
Figure 29-c shows ''Syll'' and its three 2-adic projections:
Figure 29-c shows ''Syll'' and its three 2-adic projections:
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<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
Figure 29-c. Syll c B^3 and its Dyadic Projections
Figure 29-c. Syll c B^3 and its Dyadic Projections
</pre>
</pre>
|}
We now compute the tacit extensions of the 2-adic projections of ''Syll'', alias ''q''<sub>139</sub>, and this makes manifest its relationship to the other functions and fibers, namely, ''q''<sub>175</sub>, ''q''<sub>187</sub>, ''q''<sub>207</sub>.
We now compute the tacit extensions of the 2-adic projections of ''Syll'', alias ''q''<sub>139</sub>, and this makes manifest its relationship to the other functions and fibers, namely, ''q''<sub>175</sub>, ''q''<sub>187</sub>, ''q''<sub>207</sub>.
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<pre>
<pre>
Table 30-a. Syll c B^3
Table 30-a. Syll c B^3
o-----------------------o
o-----------------------o
</pre>
</pre>
|-
|
<pre>
<pre>
Table 30-b. Dyadic Projections of Syll
Table 30-b. Dyadic Projections of Syll
o-----------o o-----------o o-----------o
o-----------o o-----------o o-----------o
</pre>
</pre>
|-
|
<pre>
<pre>
Table 30-c. Tacit Extensions of Projections of Syll
Table 30-c. Tacit Extensions of Projections of Syll
o---------------o o---------------o o---------------o
o---------------o o---------------o o---------------o
</pre>
</pre>
|-
|
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
Figure 30-d. Tacit Extension TE_12_3 (Syll_12)
Figure 30-d. Tacit Extension TE_12_3 (Syll_12)
</pre>
</pre>
|-
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<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
Figure 30-e. Tacit Extension TE_13_2 (Syll_13)
Figure 30-e. Tacit Extension TE_13_2 (Syll_13)
</pre>
</pre>
|-
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<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
Figure 30-f. Tacit Extension TE_23_1 (Syll_23)
Figure 30-f. Tacit Extension TE_23_1 (Syll_23)
</pre>
</pre>
|}
The reader may wish to contemplate Figure 31 and use it to verify the following two facts:
The reader may wish to contemplate Figure 31 and use it to verify the following two facts:
: ''Syll''<sub>13</sub> = ''Syll''<sub>12</sub> ο ''Syll''<sub>23</sub>
: ''Syll''<sub>13</sub> = ''Syll''<sub>12</sub> ο ''Syll''<sub>23</sub>
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<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
Figure 31. Syll = TE(Syll_12) |^| TE(Syll_23)
Figure 31. Syll = TE(Syll_12) |^| TE(Syll_23)
</pre>
</pre>
|}
I don't know about you, but I am still puzzled by all of thus stuff, that is to say, by the entanglements of composition and projection and their relationship to the information processing properties of logical inference rules. What I lack is a single picture that could show me all of the pieces and make the pattern of their informational relationships clear.
I don't know about you, but I am still puzzled by all of thus stuff, that is to say, by the entanglements of composition and projection and their relationship to the information processing properties of logical inference rules. What I lack is a single picture that could show me all of the pieces and make the pattern of their informational relationships clear.
