Peirce’s 1870 “Logic Of Relatives” • Comment 10.9

The use of the concepts of identity and teridentity is not to identify a thing-in-itself with itself, much less twice or thrice over — there is no need and therefore no utility in that. I can imagine Peirce asking, on Kantian principles if not entirely on Kantian premisses, Where is the manifold to be unified? The manifold that demands unification does not reside in the object but in the phenomena, that is, in the appearances that might have been appearances of different objects but that happen to be constrained by these identities to being just so many aspects, facets, parts, roles, or signs of one and the same object.

For example, notice how the various identity concepts actually functioned in the last example, where they had the opportunity to show their behavior in something like their natural habitat.

The use of the teridentity concept in the case of the “giver of a horse to a taker of it” is to say that the thing appearing with respect to its quality under an absolute term, a horse, the thing appearing with respect to its existence as the correlate of a dyadic relative, a potential possession, and the thing appearing with respect to its synthesis as the correlate of a triadic relative, a gift, are one and the same thing.

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Peirce’s 1870 “Logic Of Relatives” • Comment 10.8

There’s a critical transition point in sight of Peirce’s 1870 Logic of Relatives and it’s a point that turns on the teridentity relation.

In taking up the next example of relational composition, let’s substitute the relation \mathit{t} = \text{taker of}\, \underline{~~~~} for Peirce’s relation \mathit{o} = \text{owner of}\, \underline{~~~~}, simply for the sake of avoiding conflicts in the symbols we use. In this way, Figure 17 is transformed into Figure 22.


LOR 1870 Figure 22
(22)

The hypergraph picture of the abstract composition is given in Figure 23.


LOR 1870 Figure 23
(23)

If we analyze this in accord with the spreadsheet model of relational composition, the core of it is a particular way of composing a triadic ”giving” relation G \subseteq X \times Y \times Z with a dyadic ”taking” relation T \subseteq Y \times Z in such a way as to determine a certain dyadic relation (G \circ T) \subseteq X \times Z. Table 24 schematizes the associated constraints on tuples.

\text{Table 24.} ~~ \text{Another Brand of Composition}
  \mathit{1} \mathit{1} \mathit{1}
G X Y Z
T   Y Z
G \circ T X   Z

So we see that the notorious teridentity relation, which I have left equivocally denoted by the same symbol as the identity relation \mathit{1}, is already implicit in Peirce’s discussion at this point.

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Peirce’s 1870 “Logic Of Relatives” • Comment 10.7

Here is what I get when I analyze Peirce’s “giver of a horse to a lover of a woman” example along the same lines as the dyadic compositions.

We may begin with the mark-up shown in Figure 19.


LOR 1870 Figure 8
(19)

If we analyze this in accord with the spreadsheet model of relational composition, the core of it is a particular way of composing a triadic giving relation G \subseteq T \times U \times V with a dyadic loving relation L \subseteq U \times W so as to obtain a specialized type of triadic relation (G \circ L) \subseteq T \times V \times W. The applicable constraints on tuples are shown in Table 20.

\text{Table 20.} ~~ \text{Composite of Triadic and Dyadic Relations}
  \mathit{1} \mathit{1} \mathit{1} \mathit{1}
G T U V  
L   U   W
G \circ L T   V W

The hypergraph picture of the abstract composition is given in Figure 21.


LOR 1870 Figure 21
(21)
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Peirce’s 1870 “Logic Of Relatives” • Comment 10.6

As Peirce observes, it is not possible to work with relations in general without eventually abandoning all of one’s algebraic principles, in due time the associative law and maybe even the distributive law, just as we already gave up the commutative law. It cannot be helped, as we cannot reflect on a law unless from a perspective outside it, in any case, virtually so.

This could be done from the standpoint of the combinator calculus, and there are places where Peirce verges on systems that are very similar, but here we are making a deliberate effort to stay within the syntactic neighborhood of Peirce’s 1870 Logic of Relatives. Not too coincidentally, it is for the sake of making smoother transitions between narrower and wider regimes of algebraic law that we have been developing the paradigm of Figures and Tables indicated above.

For the next few episodes, then, I will examine the examples that Peirce gives at the next level of complication in the multiplication of relative terms, for example, the three that are repeated below.


LOR 1870 Figure 8
(16)

LOR 1870 Figure 17
(17)

LOR 1870 Figure 18
(18)
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Peirce’s 1870 “Logic Of Relatives” • Comment 10.5

We have sufficiently covered the application of the comma functor to absolute terms, so let us return to where we were in working our way through CP 3.73 and see whether we can validate Peirce’s statements about the commafications of dyadic relative terms and the corresponding diagonal extensions to triadic relations.

But not only may any absolute term be thus regarded as a relative term, but any relative term may in the same way be regarded as a relative with one correlate more.  It is convenient to take this additional correlate as the first one.

Then:

\mathit{l},\!\mathit{s}\mathrm{w}

will denote a lover of a woman that is a servant of that woman.

The comma here after \mathit{l} should not be considered as altering at all the meaning of \mathit{l}\,, but as only a subjacent sign, serving to alter the arrangement of the correlates.

(Peirce, CP 3.73)

Just to plant our feet on a more solid stage, let us apply this idea to the Othello example. For this performance only, just to make the example more interesting, let us assume that \mathrm{Jeste\, (J)} is secretly in love with \mathrm{Desdemona\, (D)}.

Then we begin with the modified data set:

\begin{array}{*{15}{c}}  \mathrm{w} & = &  \mathrm{B} & +\!\!, &  \mathrm{D} & +\!\!, &  \mathrm{E}  \\[6pt]  \mathit{l} & = &  \mathrm{B} \!:\! \mathrm{C} & +\!\!, &  \mathrm{C} \!:\! \mathrm{B} & +\!\!, &  \mathrm{D} \!:\! \mathrm{O} & +\!\!, &  \mathrm{E} \!:\! \mathrm{I} & +\!\!, &  \mathrm{I} \!:\! \mathrm{E} & +\!\!, &  \mathrm{J} \!:\! \mathrm{D} & +\!\!, &  \mathrm{O} \!:\! \mathrm{D}  \\[6pt]  \mathit{s} & = &  \mathrm{C} \!:\! \mathrm{O} & +\!\!, &  \mathrm{E} \!:\! \mathrm{D} & +\!\!, &  \mathrm{I} \!:\! \mathrm{O} & +\!\!, &  \mathrm{J} \!:\! \mathrm{D} & +\!\!, &  \mathrm{J} \!:\! \mathrm{O}  \end{array}

And we next derive the following results:

\begin{array}{lll}  \mathit{l},  & = &  \text{lover that is}\, \underline{~~~~}\, \text{of}\, \underline{~~~~}  \\[6pt]  & = &  (\mathrm{B\!:\!B\!:\!C} ~+\!\!,~ \mathrm{C\!:\!C\!:\!B} ~+\!\!,~ \mathrm{D\!:\!D\!:\!O} ~+\!\!,~ \mathrm{E\!:\!E\!:\!I} ~+\!\!,~ \mathrm{I\!:\!I\!:\!E} ~+\!\!,~ \mathrm{J\!:\!J\!:\!D} ~+\!\!,~ \mathrm{O\!:\!O\!:\!D})  \\[12pt]  \mathit{l},\!\mathit{s}\mathrm{w}  & = &  (\mathrm{B\!:\!B\!:\!C} ~+\!\!,~ \mathrm{C\!:\!C\!:\!B} ~+\!\!,~ \mathrm{D\!:\!D\!:\!O} ~+\!\!,~ \mathrm{E\!:\!E\!:\!I} ~+\!\!,~ \mathrm{I\!:\!I\!:\!E} ~+\!\!,~ \mathrm{J\!:\!J\!:\!D} ~+\!\!,~ \mathrm{O\!:\!O\!:\!D})  \\  & & \times  \\  & &  (\mathrm{C\!:\!O} ~+\!\!,~ \mathrm{E\!:\!D} ~+\!\!,~ \mathrm{I\!:\!O} ~+\!\!,~ \mathrm{J\!:\!D} ~+\!\!,~ \mathrm{J\!:\!O})  \\  & & \times  \\  & &  (\mathrm{B} ~+\!\!,~ \mathrm{D} ~+\!\!,~ \mathrm{E})  \end{array}

Now what are we to make of that?

If we operate in accordance with Peirce’s example of \mathfrak{g}\mathit{o}\mathrm{h} as the “giver of a horse to an owner of that horse”, then we may assume that the associative law and the distributive law are in force, allowing us to derive this equation:

\begin{array}{lll}  \mathit{l},\!\mathit{s}\mathrm{w}  & = &  \mathit{l},\!\mathit{s}(\mathrm{B} ~~+\!\!,~~ \mathrm{D} ~~+\!\!,~~ \mathrm{E})  \\[6pt]  & = &  \mathit{l},\!\mathit{s}\mathrm{B} ~~+\!\!,~~ \mathit{l},\!\mathit{s}\mathrm{D} ~~+\!\!,~~ \mathit{l},\!\mathit{s}\mathrm{E}  \end{array}

Evidently what Peirce means by the associative principle, as it applies to this type of product, is that a product of elementary relatives having the form (\mathrm{R\!:\!S\!:\!T})(\mathrm{S\!:\!T})(\mathrm{T}) is equal to \mathrm{R} but that no other form of product yields a non-null result. Scanning the implied terms of the triple product tells us that only the case (\mathrm{J\!:\!J\!:\!D})(\mathrm{J\!:\!D})(\mathrm{D}) = \mathrm{J} is non-null.

It follows that:

\begin{array}{lll}  \mathit{l},\!\mathit{s}\mathrm{w}  & = &  \text{lover and servant of a woman}  \\[6pt]  & = &  \text{lover that is a servant of a woman}  \\[6pt]  & = &  \text{lover of a woman that is a servant of that woman}  \\[6pt]  & = &  \mathrm{J}  \end{array}

And so what Peirce says makes sense in this case.

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Peirce’s 1870 “Logic Of Relatives” • Comment 10.4

From now on I will use the forms of analysis exemplified in the last set of Figures and Tables as a routine bridge between the logic of relative terms and the logic of their extended relations. For future reference, we may think of Table 13 as illustrating the spreadsheet model of relational composition, while Figure 14 may be thought of as making a start toward a hypergraph model of generalized compositions. I will explain the hypergraph model in some detail at a later point. The transitional form of analysis represented by Figure 15 may be called the universal bracketing of relatives as relations.

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Peirce’s 1870 “Logic Of Relatives” • Comment 10.3

We’ve been using several different styles of picture to illustrate relative terms and the relations they denote. Let us now examine the relationships that exist among the variety of visual schemes. Two examples of relative multiplication that we considered before are diagrammed again in Figures 11 and 12.


LOR 1870 Figure 7
(11)

LOR 1870 Figure 8
(12)

Figures 11 and 12 employ one style of syntax Peirce used for relative multiplication, to which I added lines of identity to connect the corresponding marks of reference. These forms are adapted to showing the anatomy of relative terms themselves, while the forms of analysis in Table 13 and Figure 14 are designed to highlight the structures of the objective relations they denote.

\text{Table 13.} ~~ \text{Relational Composition}
  \mathit{1} \mathit{1} \mathit{1}
L X Y  
S   Y Z
L \circ S X   Z


LOR 1870 Figure 14
(14)

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. It is 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 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 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 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 15 demonstrates this possibility, posing a transitional case between the style of syntax in Figure 11 and the picture of composition in Figure 14.


LOR 1870 Figure 15
(15)

In this composite sketch the diagonal extension \mathit{1} of the universe \mathbf{1} is invoked up front to anchor an explicit line of identity for the leading relate of the composition, while the terminal argument \mathrm{w} has been generalized to the whole universe \mathbf{1}. Doing this amounts to an act of abstraction from the particular application to \mathrm{w}. This form of universal bracketing isolates the serial composition of the relations L and S to form the composite L \circ S.

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