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Directory:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives (view source)
Revision as of 03:04, 24 October 2007
, 03:04, 24 October 2007→Commentary Note 10.3: markup
===Commentary Note 10.3===
===Commentary Note 10.3===
I will devote some time to drawing out the relationships that exist among the different pictures of relations and relative terms that were shown above, or as redrawn here:
<pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| |
o-------------------------------------------------o
o-------------------------------------------------o
Figure 1. Lover of a Servant of a Woman
Figure 1. Lover of a Servant of a Woman
</pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| |
o-------------------------------------------------o
o-------------------------------------------------o
Figure 2. Giver of a Horse to a Lover of a Woman
Figure 2. Giver of a Horse to a Lover of a Woman
</pre>
<pre>
Table 3. Relational Composition
Table 3. Relational Composition
o---------o---------o---------o---------o
o---------o---------o---------o---------o
| L o S # X | | Z |
| L o S # X | | Z |
o---------o---------o---------o---------o
o---------o---------o---------o---------o
</pre>
<pre>
o-------------------------------------------------o
o-------------------------------------------------o
| |
| |
o-------------------------------------------------o
o-------------------------------------------------o
Figure 4. Relational Composition
Figure 4. Relational Composition
</pre>
Figures 1 and 2 exhibit examples of relative multiplication
Figures 1 and 2 exhibit examples of relative multiplication in one of Peirce's styles of syntax, to which I subtended lines of identity to mark the anaphora of the correlates. These pictures are adapted to showing the anatomy of the relative terms, while the forms of analysis illustrated in Table 3 and Figure 4 are designed to highlight the structures of the objective relations themselves.
in one of Peirce's styles of syntax, to which I subtended
lines of identity to mark the anaphora of the correlates.
These pictures are adapted to showing the anatomy of the
relative terms, while the forms of analysis illustrated
in Table 3 and Figure 4 are designed to highlight the
structures of the objective relations themselves.
There are many ways that Peirce might have gotten from his 1870 Notation
There are many ways that Peirce might have gotten from his 1870 Notation for the Logic of Relatives to his more evolved systems of Logical Graphs. For my part, I find it interesting to speculate on how the metamorphosis might have been accomplished by way of transformations that act on these nascent forms of syntax and that take place not too far from the pale of its means, that is, as nearly as possible according to the rules and the permissions of the initial system itself.
for the Logic of Relatives to his more evolved systems of Logical Graphs.
For my part, I find it interesting to speculate on how the metamorphosis
might have been accomplished by way of transformations that act on these
nascent forms of syntax and that take place not too far from the pale of
its means, that is, as nearly as possible according to the rules and the
permissions of the initial system itself.
In Existential Graphs, a relation is represented by a node
In Existential Graphs, a relation is represented by a node whose degree is the adicity of that relation, and which is adjacent via lines of identity to the nodes that represent its correlative relations, including as a special case any of its terminal individual arguments.
whose degree is the adicity of that relation, and which is
adjacent via lines of identity to the nodes that represent
its correlative relations, including as a special case any
of its terminal individual arguments.
In the 1870 Logic of Relatives, implicit lines of identity are invoked by
In the 1870 Logic of Relatives, implicit lines of identity are invoked by the subjacent numbers and marks of reference only when a correlate of some relation is the relate of some relation. Thus, the principal relate, which is not a correlate of any explicit relation, is not singled out in this way.
the subjacent numbers and marks of reference only when a correlate of some
relation is the relate of some relation. Thus, the principal relate, which
is not a correlate of any explicit relation, is not singled out in this way.
Remarkably enough, the comma modifier itself provides us with a mechanism
Remarkably enough, the comma modifier itself provides us with a mechanism to abstract the logic of relations from the logic of relatives, and thus to forge a possible link between the syntax of relative terms and the more graphical depiction of the objective relations themselves.
to abstract the logic of relations from the logic of relatives, and thus
to forge a possible link between the syntax of relative terms and the
more graphical depiction of the objective relations themselves.
Figure 5 demonstrates this possibility, posing a transitional case between
Figure 5 demonstrates this possibility, posing a transitional case between the style of syntax in Figure 1 and the picture of composition in Figure 4.
the style of syntax in Figure 1 and the picture of composition in Figure 4.
<pre>
o-----------------------------------------------------------o
o-----------------------------------------------------------o
| |
| |
o-----------------------------------------------------------o
o-----------------------------------------------------------o
Figure 5. Anything that is a Lover of a Servant of Anything
Figure 5. Anything that is a Lover of a Servant of Anything
</pre>
In this composite sketch, the diagonal extension of the universe 1
In this composite sketch, the diagonal extension of the universe 1 is invoked up front to anchor an explicit line of identity for the leading relate of the composition, while the terminal argument ''w'' has been generalized to the whole universe 1, in effect, executing an act of abstraction. This type of universal bracketing isolates the composing of the relations ''L'' and ''S'' to form the composite ''L'' o ''S''. The three relational domains ''X'', ''Y'', ''Z'' may be distinguished from one another, or else rolled up into a single universe of discourse, as one prefers.
is invoked up front to anchor an explicit line of identity for the
leading relate of the composition, while the terminal argument "w"
has been generalized to the whole universe 1, in effect, executing
an act of abstraction. This type of universal bracketing isolates
the composing of the relations L and S to form the composite L o S.
The three relational domains X, Y, Z may be distinguished from one
another, or else rolled up into a single universe of discourse, as
one prefers.
===Commentary Note 10.4===
===Commentary Note 10.4===