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
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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
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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
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LOR 1870 Figure 17
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LOR 1870 Figure 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
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LOR 1870 Figure 8
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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
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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
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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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Peirce’s 1870 “Logic Of Relatives” • Comment 10.2

To say that a relative term “imparts a relation” is to say that it conveys information about the space of tuples in a cartesian product, that is, it determines a particular subset of that space.  When we study the combinations of relative terms, from the most elementary forms of composition to the most complex patterns of correlation, we are considering the ways that these constraints, determinations, and informations, as imparted by relative terms, are compounded in the formation of syntax.

Let us go back and look more carefully at just how it happens that Peirce’s adjacent terms and subjacent indices manage to impart their respective measures of information about relations.  Consider the examples shown in Figures 7 and 8, where connecting lines of identity have been drawn between the corresponding occurrences of the subjacent marks of reference:  \dagger\, \ddagger\, \parallel\, \S\, \P.


LOR 1870 Figure 7
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LOR 1870 Figure 8
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One way to approach the problem of “information fusion” in Peirce’s syntax is to soften the distinction between adjacent terms and subjacent signs and treat the types of constraints they separately signify more on a par with each other.  To that purpose, let us consider a way of thinking about relational composition that emphasizes the set-theoretic constraints involved in the construction of a composite relation.

For example, given the relations L \subseteq X \times Y and M \subseteq Y \times Z, Table 9 and Figure 10 present two ways of picturing the constraints that are involved in constructing the relational composition L \circ M \subseteq X \times Z.

\text{Table 9.} ~~ \text{Relational Composition}
  \mathit{1} \mathit{1} \mathit{1}
L X Y  
M   Y Z
L \circ M X   Z

The way to read Table 9 is to imagine that you are playing a game that involves placing tokens on the squares of a board that is marked in just this way.  The rules are that you have to place a single token on each marked square in the middle of the board in such a way that all of the indicated constraints are satisfied.  That is, you have to place a token whose denomination is a value in the set X on each of the squares marked X, and similarly for the squares marked Y and Z, meanwhile leaving all of the blank squares empty.  Furthermore, the tokens placed in each row and column have to obey the relational constraints that are indicated at the heads of the corresponding row and column.  Thus, the two tokens from X have to denote the very same value from X, and likewise for Y and Z, while the pairs of tokens on the rows marked L and M are required to denote elements that are in the relations L and M, respectively.  The upshot is that when just this much is done, that is, when the L, M, and \mathit{1} relations are satisfied, then the row marked L \circ M will automatically bear the tokens of a pair of elements in the composite relation L \circ M.

Figure 10 shows a different way of viewing the same situation.


LOR 1870 Figure 10
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Peirce’s 1870 “Logic Of Relatives” • Comment 10.1

What Peirce is attempting to do in CP 3.75 is absolutely amazing and I personally did not see anything on a par with it again until I began to study the application of mathematical category theory to computation and logic, back in the mid 1980s. To completely evaluate the success of this attempt we would have to return to Peirce’s earlier paper “Upon the Logic of Mathematics” (1867) to pick up some of the ideas about arithmetic that he set out there.

Another branch of the investigation would require that we examine more carefully the entire syntactic mechanics of subjacent signs that Peirce uses to establish linkages among relational domains. It is important to note that these types of indices constitute a diacritical, interpretive, syntactic category under which Peirce also places the comma functor.

The way that I would currently approach both of these branches of the investigation would be to open up a wider context for the study of relational compositions, attempting to get at the essence of what is going on when we relate relations, possibly complex, to other relations, possibly simple.

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