Peirce’s 1870 “Logic of Relatives” • Comment 11.11

Posted on May 9, 2014 by Jon Awbrey

Peirce’s 1870 “Logic of Relatives” • Comment 11.11

The preceding exercises were intended to beef-up our “functional literacy” skills to the point where we can read our functional alphabets backwards and forwards and recognize the local functionalities immanent in relative terms no matter where they reside among the domains of relations.  These skills will serve us in good stead as we work to build a catwalk from Peirce’s platform of 1870 to contemporary scenes on the logic of relatives, and back again.

By way of extending a few very tentative planks, let us experiment with the following definitions.

  • A relative term ``p" and the corresponding relation P \subseteq X \times Y are both called functional on rèlates if and only if P is a function at X.  We write this in symbols as P : X \to Y.
  • A relative term ``p" and the corresponding relation P \subseteq X \times Y are both called functional on correlates if and only if P is a function at Y.  We write this in symbols as P : X \gets Y.

When a relation happens to be a function, it may be excusable to use the same name for it in both applications, writing out explicit type markers like P : X \times Y,   P : X \to Y, and P : X \gets Y, as the case may be, when and if it serves to clarify matters.

From this current, perhaps transient, perspective, it appears our next task is to examine how the known properties of relations are modified when aspects of functionality are spied in the mix.  Let us then return to our various ways of looking at relational composition and see what changes and what stays the same when the relations in question happen to be functions of various kinds at some of their domains.  Here is one generic picture of relational composition, cast in a style that hews pretty close to the line of potentials inherent in Peirce’s syntax of this period.

Universal Bracket P ◦ Q
\text{Figure 44. Anything that is a}~ p ~\text{of a}~ q ~\text{of Anything}

From this we extract the hypergraph picture of relational composition.

Relational Composition Figure P ◦ Q
\text{Figure 45. Relational Composition}~ P \circ Q

All the information contained in these Figures can be expressed in the form of a constraint satisfaction table, or spreadsheet picture of relational composition.

\text{Table 46. Relational Composition}~ P \circ Q

Relational Composition Table P ◦ Q

The following plan of study then presents itself, to see what easy mileage we can get in our exploration of functions by adopting the above templates as the primers of a paradigm.

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Peirce’s 1870 “Logic of Relatives” • Comment 11.10

Posted on May 7, 2014 by Jon Awbrey

Peirce’s 1870 “Logic of Relatives” • Comment 11.10

A dyadic relation F \subseteq X \times Y which qualifies as a function f : X \to Y may then enjoy a number of further distinctions.

Definitions

For example, the function f : X \to Y shown below is neither total nor tubular at its codomain Y so it can enjoy none of the properties of being surjective, injective, or bijective.

Function f : X → Y
\text{Figure 40. Function}~ f : X \to Y

An easy way to extract a surjective function from any function is to reset its codomain to its range.  For example, the range of the function f above is Y^\prime = \{ 0, 2, 5, 6, 7, 8, 9 \}.  If we form a new function g : X \to Y^\prime that looks just like f on the domain X but is assigned the codomain Y^\prime, then g is surjective, and is described as a mapping onto Y^\prime.

Function g : X → Y'
\text{Figure 41. Function}~ g : X \to Y'

The function h : Y' \to Y is injective.

Function h : Y' → Y
\text{Figure 42. Function}~ h : Y' \to Y

The function m : X \to Y is bijective.

Function m : X → Y
\text{Figure 43. Function}~ m : X \to Y

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Peirce’s 1870 “Logic of Relatives” • Comment 11.9

Posted on May 7, 2014 by Jon Awbrey

Peirce’s 1870 “Logic of Relatives” • Comment 11.9

Among the variety of regularities affecting dyadic relations we pay special attention to the c-regularity conditions where c is equal to 1.

Let P \subseteq X \times Y be an arbitrary dyadic relation.  The following properties can be defined.

Definitions 1

We previously examined dyadic relations exemplifying each of these regularity conditions.  Then we introduced a few bits of terminology and special-purpose notations for working with tubular relations.

Definitions 2

We arrive by way of this winding stair at the special cases of dyadic relations P \subseteq X \times Y variously described as 1-regular, total and tubular, or total prefunctions on specified domains, X or Y or both, and which are more often celebrated as functions on those domains.

If P is a pre-function P : X \rightharpoonup Y that happens to be total at X, then P is known as a function from X to Y, typically indicated as P : X \to Y.

To say that a relation P \subseteq X \times Y is total and tubular at X is to say that P is 1-regular at X.  Thus, we may formalize the following definitions.

Definitions 3

For example, let X = Y = \{ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 \} and let F \subseteq X \times Y be the dyadic relation depicted in the bigraph below.

Dyadic Relation F
\text{Figure 39. Dyadic Relation}~ F

We observe that F is a function at Y and we record this fact in either of the manners F : X \leftarrow Y or F : Y \to X.

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Peirce’s 1870 “Logic of Relatives” • Comment 11.8

Posted on May 6, 2014 by Jon Awbrey

Peirce’s 1870 “Logic of Relatives” • Comment 11.8

Let’s take a closer look at the numerical incidence properties of relations, concentrating on the assorted regularity conditions defined in the article on Relation Theory.

For example, L has the property of being c\text{-regular at}~ j if and only if the cardinality of the local flag L_{x @ j} is equal to c for all x in X_j, coded in symbols, if and only if |L_{x @ j}| = c for all x in X_j.

In like fashion, one may define the numerical incidence properties (< c)\text{-regular at}~ j, (> c)\text{-regular at}~ j, and so on.  For ease of reference, a number of such definitions are recorded below.

Definitions

Clearly, if any relation is (\le c)\text{-regular} on one of its domains X_j and also (\ge c)\text{-regular} on the same domain, then it must be (= c)\text{-regular} on that domain, in short, c\text{-regular} at j.

For example, let G = \{ r, s, t \} and H = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \} and consider the dyadic relation F \subseteq G \times H bigraphed below.

Dyadic Relation F
\text{Figure 38. Dyadic Relation}~ F

We observe that F is 3-regular at G and 1-regular at H.

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Peirce’s 1870 “Logic of Relatives” • Comment 11.7

Posted on May 5, 2014 by Jon Awbrey

Peirce’s 1870 “Logic of Relatives” • Comment 11.7

We come now to the special cases of dyadic relations known as functions.  It will serve a dual purpose in the present exposition to take the class of functions as a source of object examples for clarifying the more abstruse concepts of Relation Theory.

To begin, let us recall the definition of a local flag L_{a @ j} of a k-adic relation L.

Display 1

For a dyadic relation L \subseteq X \times Y the notation for local flags can be simplified in two ways.  First, the local flags L_{u @ 1} and L_{v @ 2} are often more conveniently notated as L_{u @ X} and L_{v @ Y}, respectively.  Second, the notation may be streamlined even further by making the following definitions.

Display 2

In light of these conventions, the local flags of a dyadic relation L \subseteq X \times Y may be comprehended under the following descriptions.

Display 3

The following definitions are also useful.

Display 4

A sufficient illustration is supplied by the earlier example E.

Dyadic Relation E
\text{Figure 35. Dyadic Relation}~ E

Figure 36 shows the local flag E_{3 @ X} of E.

Local Flag E_{3 @ X}
\text{Figure 36. Local Flag}~ E_{3 @ X}

Figure 37 shows the local flag E_{2 @ Y} of E.

Local Flag E_{2 @ Y}
\text{Figure 37. Local Flag}~ E_{2 @ Y}

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Peirce’s 1870 “Logic of Relatives” • Comment 11.6

Posted on May 4, 2014 by Jon Awbrey

Peirce’s 1870 “Logic of Relatives” • Comment 11.6

Let’s continue working our way through the above definitions, constructing appropriate examples as we go.

Relation E_1 \subseteq X \times Y exemplifies the quality of totality at X.

Dyadic Relation E₁
\text{Dyadic Relation}~ E_1

Relation E_2 \subseteq X \times Y exemplifies the quality of totality at Y.

Dyadic Relation E₂
\text{Dyadic Relation}~ E_2

Relation E_3 \subseteq X \times Y exemplifies the quality of tubularity at X.

Dyadic Relation E₃
\text{Dyadic Relation}~ E_3

Relation E_4 \subseteq X \times Y exemplifies the quality of tubularity at Y.

Dyadic Relation E₄
\text{Dyadic Relation}~ E_4

So E_3 is a pre-function e_3 : X \rightharpoonup Y and E_4 is a pre-function e_4 : X \leftharpoonup Y.

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Peirce’s 1870 “Logic of Relatives” • Comment 11.5

Posted on May 2, 2014 by Jon Awbrey

Peirce’s 1870 “Logic of Relatives” • Comment 11.5

Everyone knows the right sort of diagram can be a great aid in rendering complex matters comprehensible.  With that in mind, let’s extract what we need from the Relation Theory article to illuminate Peirce’s 1870 Logic of Relatives and use it to fashion what icons we can within the current frame of discussion.

For the immediate present, we may begin with dyadic relations and describe the most frequently encountered species of relations and functions in terms of their local and numerical incidence properties.

Let P \subseteq X \times Y be an arbitrary dyadic relation.  The following properties of P can then be defined.

Display 1

If P \subseteq X \times Y is tubular at X, then P is known as a partial function or a pre-function from X to Y, frequently signalized by renaming P with an alternate lower case name, say {}^{\backprime\backprime} p {}^{\prime\prime}, and writing p : X \rightharpoonup Y.

Just by way of formalizing the definition:

Display 2

To illustrate these properties, let us fashion a generic enough example of a dyadic relation, E \subseteq X \times Y, where X = Y = \{ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 \}, and where the bigraph picture of E is shown in Figure 30.

Dyadic Relation E
\text{Figure 30. Dyadic Relation}~ E

If we scan along the X dimension from 0 to 9 we see that the incidence degrees of the X nodes with the Y domain are 0, 1, 2, 3, 1, 1, 1, 2, 0, 0 in that order.

If we scan along the Y dimension from 0 to 9 we see that the incidence degrees of the Y nodes with the X domain are 0, 0, 3, 2, 1, 1, 2, 1, 1, 0 in that order.

Thus, E is not total at either X or Y since there are nodes in both X and Y having incidence degrees less than 1.

Also, E is not tubular at either X or Y since there are nodes in both X and Y having incidence degrees greater than 1.

Clearly then the relation E cannot qualify as a pre-function, much less as a function, on either of its relational domains.

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Peirce’s 1870 “Logic of Relatives” • Comment 11.4

Posted on May 1, 2014 by Jon Awbrey

Peirce’s 1870 “Logic of Relatives” • Comment 11.4

The task before us is to clarify the relationships among relative terms, relations, and the special cases of relations given by equivalence relations, functions, and so on.

The first obstacle to get past is the order convention Peirce’s orientation to relative terms causes him to use for functions.  To focus on a concrete example of immediate use in this discussion, let’s take the “number of” function Peirce denotes by means of square brackets and re-formulate it as a dyadic relative term v in the following way.

v(t) ~:=~ [t] ~=~ \text{the number of the term}~ t.

To set the dyadic relative term v within a suitable context of interpretation, let’s suppose v corresponds to a relation V \subseteq \mathbb{R} \times S where \mathbb{R} is the set of real numbers and S is a suitable syntactic domain, here described as a set of terms.  The dyadic relation V is at first sight a function from S to \mathbb{R}.  It is, however, not always possible to assign a number to every term in whatever syntactic domain S we happen to pick, so we may eventually be forced to treat the dyadic relation V as a partial function from S to \mathbb{R}.  All things considered, then, let’s try the following budget of strategies and compromises.

First, let’s adapt the arrow notation for functions in such a way as to allow detaching the functional orientation from the order in which the names of domains are written on the page.  Second, let’s change the notation for partial functions, or pre-functions, to mark more clearly their distinction from functions.  This produces the following scheme.

q : X \to Y means q is functional at X.

q : X \leftarrow Y means q is functional at Y.

q : X \rightharpoonup Y means q is pre-functional at X.

q : X \leftharpoonup Y means q is pre-functional at Y.

Until it becomes necessary to stipulate otherwise, let’s assume v is a function in \mathbb{R} of S, written v : \mathbb{R} \leftarrow S, amounting to a functional alias of the dyadic relation V \subseteq \mathbb{R} \times S and associated with the dyadic relative term v whose rèlate lies in the set \mathbb{R} of real numbers and whose correlate lies in the set S of syntactic terms.

Note.  Please refer to the article on Relation Theory for the definitions of functions and pre‑functions used in the above discussion.

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Peirce’s 1870 “Logic of Relatives” • Comment 11.3

Posted on April 30, 2014 by Jon Awbrey

Peirce’s 1870 “Logic of Relatives” • Comment 11.3

Before I can discuss Peirce’s “number of” function in greater detail I will need to deal with an expositional difficulty I have been carefully dancing around all this time, but one which will no longer abide its assigned place under the rug.

Functions have long been understood, from well before Peirce’s time to ours, as special cases of dyadic relations, so the “number of” function is already to be numbered among the class of dyadic relatives we’ve been dealing with all this time.  But Peirce’s manner of representing a dyadic relative term mentions the “rèlate” first and the “correlate” second, a convention going over into functional terms as making the functional value first and the functional argument second.  The problem is, almost anyone brought up in our present time frame is accustomed to thinking of a function as a set of ordered pairs where the order in each pair lists the functional argument first and the functional value second.

Syntactic wrinkles of this sort can be ironed out smoothly enough in a framework of flexible interpretive conventions, but not without introducing an order of anachronism into Peirce’s text I want to avoid as much as possible.  This will require me to experiment with various styles of compromise.  Among other things, the interpretation of Peirce’s 1870 “Logic of Relatives” can be facilitated by introducing a few items of background material on relations in general, as regarded from a combinatorial point of view.

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Peirce’s 1870 “Logic of Relatives” • Comment 11.2

Posted on April 30, 2014 by Jon Awbrey

Peirce’s 1870 “Logic of Relatives” • Comment 11.2

NOF Said …

Let’s bring together the various things Peirce has said about the number of function up to this point in the paper.

NOF 1

I propose to assign to all logical terms, numbers;  to an absolute term, the number of individuals it denotes;  to a relative term, the average number of things so related to one individual.  Thus in a universe of perfect men (men), the number of “tooth of” would be 32.  The number of a relative with two correlates would be the average number of things so related to a pair of individuals;  and so on for relatives of higher numbers of correlates.  I propose to denote the number of a logical term by enclosing the term in square brackets, thus, [t].

(Peirce, CP 3.65)

NOF 2

But not only do the significations of  =  and  <  here adopted fulfill all absolute requirements, but they have the supererogatory virtue of being very nearly the same as the common significations.  Equality is, in fact, nothing but the identity of two numbers;  numbers that are equal are those which are predicable of the same collections, just as terms that are identical are those which are predicable of the same classes.

So, to write 5 < 7 is to say that 5 is part of 7, just as to write \mathrm{f} < \mathrm{m} is to say that Frenchmen are part of men.  Indeed, if \mathrm{f} < \mathrm{m}, then the number of Frenchmen is less than the number of men, and if \mathrm{v} = \mathrm{p}, then the number of Vice-Presidents is equal to the number of Presidents of the Senate;  so that the numbers may always be substituted for the terms themselves, in case no signs of operation occur in the equations or inequalities.

(Peirce, CP 3.66)

NOF 3

It is plain that both the regular non-invertible addition and the invertible addition satisfy the absolute conditions.  But the notation has other recommendations.  The conception of taking together involved in these processes is strongly analogous to that of summation, the sum of 2 and 5, for example, being the number of a collection which consists of a collection of two and a collection of five.  Any logical equation or inequality in which no operation but addition is involved may be converted into a numerical equation or inequality by substituting the numbers of the several terms for the terms themselves — provided all the terms summed are mutually exclusive.

Addition being taken in this sense, nothing is to be denoted by zero, for then

x ~+\!\!,~ 0 ~=~ x

whatever is denoted by x;  and this is the definition of zero.  This interpretation is given by Boole, and is very neat, on account of the resemblance between the ordinary conception of zero and that of nothing, and because we shall thus have

[0] ~=~ 0.

(Peirce, CP 3.67)

NOF 4

The conception of multiplication we have adopted is that of the application of one relation to another.  …

Even ordinary numerical multiplication involves the same idea, for 2 \times 3 is a pair of triplets, and 3 \times 2 is a triplet of pairs, where “triplet of” and “pair of” are evidently relatives.

If we have an equation of the form

xy ~=~ z,

and there are just as many x’s per y as there are, per things, things of the universe, then we have also the arithmetical equation,

[x][y] ~=~ [z].

For instance, if our universe is perfect men, and there are as many teeth to a Frenchman (perfect understood) as there are to any one of the universe, then

[\mathit{t}][\mathrm{f}] ~=~ [\mathit{t}\mathrm{f}]

holds arithmetically.

So if men are just as apt to be black as things in general,

[\mathrm{m,}][\mathrm{b}] ~=~ [\mathrm{m,}\mathrm{b}],

where the difference between [\mathrm{m}] and [\mathrm{m,}] must not be overlooked.

It is to be observed that

[\mathit{1}] ~=~ \mathfrak{1}.

Boole was the first to show this connection between logic and probabilities.  He was restricted, however, to absolute terms.  I do not remember having seen any extension of probability to relatives, except the ordinary theory of expectation.

Our logical multiplication, then, satisfies the essential conditions of multiplication, has a unity, has a conception similar to that of admitted multiplications, and contains numerical multiplication as a case under it.

(Peirce, CP 3.76)

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