Inquiry Into Inquiry

Peirce’s 1870 “Logic of Relatives” • Comment 10.10

The last of Peirce’s three examples involving the composition of triadic relatives with dyadic relatives is shown again in Figure 25.

The hypergraph picture of the abstract composition is given in Figure 26.

This example illustrates the way Peirce analyzes the logical conjunction, we might even say the parallel conjunction, of a pair of dyadic relatives in terms of the comma extension and the same style of composition we saw in the last example, that is, according to a pattern of anaphora invoking the teridentity relation.

Laying out the above analysis of logical conjunction on the spreadsheet model of relational composition, the gist of it is the diagonal extension of a dyadic loving relation to a triadic being and loving relation which is then composed with a dyadic serving relation so as to determine a dyadic relation   Table 27 schematizes the associated constraints on tuples.

Resources

• Peirce’s 1870 Logic of Relatives • Part 1 • Part 2 • Part 3 • References

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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 thus 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 requiring unification does not reside in the object but in the phenomena — in the appearances which might have been appearances of different objects but are bound by the indicated identities to be just so many aspects, facets, parts, roles, or signs of one and the same object.

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 “giver of a horse to an owner of it” is to say 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.

Resources

• Peirce’s 1870 Logic of Relatives • Part 1 • Part 2 • Part 3 • References

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

Our progress through the 1870 Logic of Relatives brings us in sight of a critical transition point, one which turns on the teridentity relation.

The markup for Peirce’s “giver of a horse to an owner of it” is shown again in Figure 22.

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

If we analyze this in accord with the spreadsheet model of relational composition then the core of it is a particular way of composing a triadic giving relation with a dyadic owning relation in such a way as to determine a specialized dyadic relation   Table 24 schematizes the associated constraints on tuples.

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

Resources

• Peirce’s 1870 Logic of Relatives • Part 1 • Part 2 • Part 3 • References

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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.

If we analyze this in accord with the spreadsheet model of relational composition then the core of it is a particular way of composing a triadic giving relation with a dyadic loving relation so as to obtain a specialized type of triadic relation   The applicable constraints on tuples are shown in Table 20.

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

Resources

• Peirce’s 1870 Logic of Relatives • Part 1 • Part 2 • Part 3 • References

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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 the more usual algebraic principles, in due time the associative law and even the distributive law, just as we already gave up the commutative law.  It cannot be helped, as we cannot reflect on a law except from a perspective outside it, in any case, virtually so.

This could be done in the framework of the combinator calculus, and there are places where Peirce verges on systems of a comparable character, 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 realms of algebraic law that we have been developing the paradigm of Figures and Tables indicated above.

In the next several episodes, then, I’ll examine the cases Peirce uses to illustrate the next level of complexity in the multiplication of relative terms, as shown in the Figures below.

Resources

• Peirce’s 1870 Logic of Relatives • Part 1 • Part 2 • Part 3 • References

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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.

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 is secretly in love with

Then we begin with the modified data set:

And next we derive the following results:

Now what are we to make of that?

If we operate in accordance with Peirce’s example of as the “giver of a horse to an owner of that horse” then we may assume the associative and distributive laws remain in force, allowing us to derive the following equation.

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 is equal to 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 is non‑null.

It follows that:

And so what Peirce says makes sense in this case.

Resources

• Peirce’s 1870 Logic of Relatives • Part 1 • Part 2 • Part 3 • References

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

From now on the forms of analysis exemplified in the last set of Figures and Tables will serve as a convenient bridge between the logic of relative terms and the mathematics of relations themselves.  We may think of Table 13 as illustrating a spreadsheet model of relational composition while Figure 14 may be thought of as making a start toward a hypergraph model of generalized compositions.  I’ll explain the hypergraph model in more detail at a later point.  The transitional form of analysis represented by Figure 15 may be called the universal bracketing of relatives as relations.

Resources

• Peirce’s 1870 Logic of Relatives • Part 1 • Part 2 • Part 3 • References

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

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

Figures 11 and 12 employ one of the styles of syntax Peirce used for relative multiplication, to which I added lines of identity to connect the corresponding marks of reference.  Forms like these show the anatomy of the relative terms themselves, while the forms in Table 13 and Figure 14 are adapted to show the structures of the objective relations they denote.

There are many ways 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 acting on these nascent forms of syntax and taking place not too far from the pale of its means, that is, as nearly as possible according to the rules and 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 representing 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 rèlate of some relation.  Thus, the principal rèlate, 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.

In this composite sketch the diagonal extension of the universe is invoked up front to anchor an explicit line of identity for the leading rèlate of the composition, while the terminal argument is generalized to the whole universe   Doing this amounts to an act of abstraction from the particular application to   This form of universal bracketing isolates the serial composition of the relations and to form the composite

Resources

• Peirce’s 1870 Logic of Relatives • Part 1 • Part 2 • Part 3 • References

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Peirce’s 1870 “Logic of Relatives” • Comment 10.2

To say a relative term “imparts a relation” is to say 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 those 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: 

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 and Table 9 and Figure 10 present two ways of picturing the constraints involved in constructing the relational composition

The way to read Table 9 is to imagine you are playing a game which involves placing tokens on the squares of a board marked in just that way.  The rules are you have to place a single token on each marked square in the middle of the board in such a way that all the indicated constraints are satisfied.  That is, you have to place a token whose denomination is a value in the set on each of the squares marked and similarly for the squares marked and meanwhile leaving all the blank squares empty.

Furthermore, the tokens placed in each row and column have to obey the relational constraints indicated at the heads of the corresponding row and column.  Thus, the two tokens from have to denote the very same value from and likewise for and while the pairs of tokens on the rows marked and are required to denote elements in the relations and respectively.

The upshot is, when all that has been done, when the and relations are satisfied, then the row marked will automatically bear the tokens of a pair of elements in the composite relation

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

Resources

• Peirce’s 1870 Logic of Relatives • Part 1 • Part 2 • Part 3 • References

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cc: FB | Peirce Matters • Laws of Form • Peirce List (1) (2) (3) (4) (5) (6) (7)

Peirce’s 1870 “Logic of Relatives” • Comment 10.1

What Peirce is attempting to do at CP 3.75 is absolutely amazing.  I did not run across anything on a par with it again until the mid 1980s when I began studying the application of mathematical category theory to computation and logic.  Gauging the success of Peirce’s attempt would take a return to his earlier paper “Upon the Logic of Mathematics” (1867) to pick up the ideas about arithmetic he sets out there.

Another branch of the investigation would require us to examine the syntactic mechanics of subjacent signs Peirce uses to establish linkages among relational domains.  The indices employed for this purpose amount to a category of diacritical and interpretive signs which includes, among other things, the comma functor we have just been discussing.

Combining the two branches of this investigation opens a wider context for the study of relational compositions, distilling the essence of what it takes to relate relations, possibly complex, to other relations, possibly simple.

Resources

• Peirce’s 1870 Logic of Relatives • Part 1 • Part 2 • Part 3 • References

cc: Cybernetics • Ontolog Forum • Structural Modeling • Systems Science
cc: FB | Peirce Matters • Laws of Form • Peirce List (1) (2) (3) (4) (5) (6) (7)