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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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
(7)

LOR 1870 Figure 8
(8)

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
(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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Peirce’s 1870 “Logic Of Relatives” • Selection 10

We continue with §3. Application of the Algebraic Signs to Logic.

The Signs for Multiplication (cont.)

The sum x + x generally denotes no logical term.  But {x,}_\infty + \, {x,}_\infty may be considered as denoting some two x’s.

It is natural to write:

  x ~+~ x ~=~ \mathit{2}.x ~,   
and {x,}_\infty + \, {x,}_\infty ~=~ \mathit{2}.{x,}_\infty ~,   

where the dot shows that this multiplication is invertible.

We may also use the antique figures so that:

  \mathit{2}.{x,}_\infty ~=~ \mathfrak{2}x ~,   
just as \mathit{1}_\infty ~=~ \mathfrak{1} ~.   

Then \mathfrak{2} alone will denote some two things.

But this multiplication is not in general commutative, and only becomes so when it affects a relative which imparts a relation such that a thing only bears it to one thing, and one thing alone bears it to a thing.

For instance, the lovers of two women are not the same as two lovers of women, that is:

  \mathit{l}\mathfrak{2}.\mathrm{w} ~\text{and}~ \mathfrak{2}.\mathit{l}\mathrm{w}   

are unequal;  but the husbands of two women are the same as two husbands of women, that is:

  \mathit{h}\mathfrak{2}.\mathrm{w} ~=~ \mathfrak{2}.\mathit{h}\mathrm{w} ~,   
and in general; x,\!\mathfrak{2}.y ~=~ \mathfrak{2}.x,\!y ~.   

(Peirce, CP 3.75)

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Peirce’s 1870 “Logic Of Relatives” • Comment 9.7

From this point forward we may think of idempotents, selectives, and zero-one diagonal matrices as being roughly equivalent notions. The only reason I say roughly is that we are comparing ideas at different levels of abstraction in proposing these connections.

We have covered the way that Peirce uses his invention of the comma modifier to assimilate boolean multiplication, logical conjunction, and what we may think of as serial selection under his more general account of relative multiplication.

But the comma functor has its application to relative terms of any arity, not just the zeroth arity of absolute terms, and so there will be a lot more to explore on this point. But now I must return to the anchorage of Peirce’s text and hopefully get a chance to revisit this topic later.

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Peirce’s 1870 “Logic Of Relatives” • Comment 9.6

By way of fixing the current array of relational concepts in our minds, let us work through a sample of products from our relational multiplication table that will serve to illustrate the application of a comma relative to an absolute term, presented in both matrix and bigraph pictures.

Example 1

Example 2

Example 3

Example 4

Example 5

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Peirce’s 1870 “Logic Of Relatives” • Comment 9.5

Peirce’s comma operation, in its application to an absolute term, is tantamount to the representation of that term’s denotation as an idempotent transformation, which is commonly represented as a diagonal matrix. Hence the alternate name, diagonal extension.

An idempotent element x is given by the abstract condition that xx = x, but elements like these are commonly encountered in more concrete circumstances, acting as operators or transformations on other sets or spaces, and in that action they will often be represented as matrices of coefficients.

Let’s see how this looks in the matrix and graph pictures of absolute and relative terms.

Absolute Terms

\begin{array}{*{17}{l}}  \mathbf{1} & = & \text{anything} & = &  \mathrm{B} & +\!\!, &  \mathrm{C} & +\!\!, &  \mathrm{D} & +\!\!, &  \mathrm{E} & +\!\!, &  \mathrm{I} & +\!\!, &  \mathrm{J} & +\!\!, &  \mathrm{O}  \\[6pt]  \mathrm{m} & = & \text{man} & = &  \mathrm{C} & +\!\!, &  \mathrm{I} & +\!\!, &  \mathrm{J} & +\!\!, &  \mathrm{O}  \\[6pt]  \mathrm{n} & = & \text{noble} & = &  \mathrm{C} & +\!\!, &  \mathrm{D} & +\!\!, &  \mathrm{O}  \\[6pt]  \mathrm{w} & = & \text{woman} & = &  \mathrm{B} & +\!\!, &  \mathrm{D} & +\!\!, &  \mathrm{E}  \end{array}

Previously, we represented absolute terms as column arrays. The above four terms are given by the columns of the following table:

\begin{array}{c|cccc}  & \mathbf{1} & \mathrm{m} & \mathrm{n} & \mathrm{w} \\  \hline  \mathrm{B} & 1 & 0 & 0 & 1 \\  \mathrm{C} & 1 & 1 & 1 & 0 \\  \mathrm{D} & 1 & 0 & 1 & 1 \\  \mathrm{E} & 1 & 0 & 0 & 1 \\  \mathrm{I} & 1 & 1 & 0 & 0 \\  \mathrm{J} & 1 & 1 & 0 & 0 \\  \mathrm{O} & 1 & 1 & 1 & 0  \end{array}

The types of graphs known as bigraphs or bipartite graphs can be used to picture simple relative terms, dyadic relations, and their corresponding logical matrices. One way to bring absolute terms and their corresponding sets of individuals into the bigraph picture is to mark the nodes in some way, for example, hollow nodes for non-members and filled nodes for members of the indicated set, as shown below:


LOR 1870 Figure 4.1
(4.1)

LOR 1870 Figure 4.2
(4.2)

LOR 1870 Figure 4.3
(4.3)

LOR 1870 Figure 4.4
(4.4)

Diagonal Extensions

\begin{array}{lll}  \mathbf{1,} & = & \text{anything that is}\, \underline{~~~~}  \\[6pt]  & = & \mathrm{B\!:\!B} ~+\!\!,~ \mathrm{C\!:\!C} ~+\!\!,~ \mathrm{D\!:\!D} ~+\!\!,~ \mathrm{E\!:\!E} ~+\!\!,~ \mathrm{I\!:\!I} ~+\!\!,~ \mathrm{J\!:\!J} ~+\!\!,~ \mathrm{O\!:\!O}  \\[9pt]  \mathrm{m,} & = & \text{man that is}\, \underline{~~~~}  \\[6pt]  & = & \mathrm{C\!:\!C} ~+\!\!,~ \mathrm{I\!:\!I} ~+\!\!,~ \mathrm{J\!:\!J} ~+\!\!,~ \mathrm{O\!:\!O}  \\[9pt]  \mathrm{n,} & = & \text{noble that is}\, \underline{~~~~}  \\[6pt]  & = & \mathrm{C\!:\!C} ~+\!\!,~ \mathrm{D\!:\!D} ~+\!\!,~ \mathrm{O\!:\!O}  \\[9pt]  \mathrm{w,} & = & \text{woman that is}\, \underline{~~~~}  \\[6pt]  & = & \mathrm{B\!:\!B} ~+\!\!,~ \mathrm{D\!:\!D} ~+\!\!,~ \mathrm{E\!:\!E}  \end{array}

Naturally enough, the diagonal extensions are represented by diagonal matrices:

\begin{array}{c|*{7}{c}}  \mathbf{1,} & \mathrm{B} & \mathrm{C} & \mathrm{D} &  \mathrm{E}  & \mathrm{I} & \mathrm{J} & \mathrm{O} \\  \hline  \mathrm{B} & 1 &   &   &   &   &   &   \\  \mathrm{C} &   & 1 &   &   &   &   &   \\  \mathrm{D} &   &   & 1 &   &   &   &   \\  \mathrm{E} &   &   &   & 1 &   &   &   \\  \mathrm{I} &   &   &   &   & 1 &   &   \\  \mathrm{J} &   &   &   &   &   & 1 &   \\  \mathrm{O} &   &   &   &   &   &   & 1  \end{array}

\begin{array}{c|*{7}{c}}  \mathrm{m,} & \mathrm{B} & \mathrm{C} & \mathrm{D} &  \mathrm{E}  & \mathrm{I} & \mathrm{J} & \mathrm{O} \\  \hline  \mathrm{B} & 0 &   &   &   &   &   &   \\  \mathrm{C} &   & 1 &   &   &   &   &   \\  \mathrm{D} &   &   & 0 &   &   &   &   \\  \mathrm{E} &   &   &   & 0 &   &   &   \\  \mathrm{I} &   &   &   &   & 1 &   &   \\  \mathrm{J} &   &   &   &   &   & 1 &   \\  \mathrm{O} &   &   &   &   &   &   & 1  \end{array}

\begin{array}{c|*{7}{c}}  \mathrm{n,} & \mathrm{B} & \mathrm{C} & \mathrm{D} &  \mathrm{E}  & \mathrm{I} & \mathrm{J} & \mathrm{O} \\  \hline  \mathrm{B} & 0 &   &   &   &   &   &   \\  \mathrm{C} &   & 1 &   &   &   &   &   \\  \mathrm{D} &   &   & 1 &   &   &   &   \\  \mathrm{E} &   &   &   & 0 &   &   &   \\  \mathrm{I} &   &   &   &   & 0 &   &   \\  \mathrm{J} &   &   &   &   &   & 0 &   \\  \mathrm{O} &   &   &   &   &   &   & 1  \end{array}

\begin{array}{c|*{7}{c}}  \mathrm{w,} & \mathrm{B} & \mathrm{C} & \mathrm{D} &  \mathrm{E}  & \mathrm{I} & \mathrm{J} & \mathrm{O} \\  \hline  \mathrm{B} & 1 &   &   &   &   &   &   \\  \mathrm{C} &   & 0 &   &   &   &   &   \\  \mathrm{D} &   &   & 1 &   &   &   &   \\  \mathrm{E} &   &   &   & 1 &   &   &   \\  \mathrm{I} &   &   &   &   & 0 &   &   \\  \mathrm{J} &   &   &   &   &   & 0 &   \\  \mathrm{O} &   &   &   &   &   &   & 0  \end{array}

Cast into the bigraph picture of dyadic relations, the diagonal extension of an absolute term takes on a very distinctive sort of “straight-laced” character:


LOR 1870 Figure 5.1
(5.1)

LOR 1870 Figure 5.2
(5.2)

LOR 1870 Figure 5.3
(5.3)

LOR 1870 Figure 5.4
(5.4)
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