Peirce’s 1870 “Logic of Relatives” • Comment 11.17

Posted on May 27, 2014 by Jon Awbrey

Peirce’s 1870 “Logic of Relatives” • Comment 11.17

I think the reader is beginning to get an inkling of the crucial importance of the “number of” function in Peirce’s way of looking at logic.  It is one plank in the bridge from logic to the theories of probability, statistics, and information, in which setting logic forms but a limiting case at one scenic turnout on the expanding vista.  It is one of the ways Peirce forges a link between the eternal, logical, or rational realm and the secular, empirical, or real domain.

With that note of encouragement and exhortation, let us return to the details of the text.

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)

Peirce observes that the measure \mathit{v} on logical terms preserves the relations of implication or inclusion which impose an ordering on those terms.  Here Peirce uses a single symbol ``\!<\!" to denote the linear ordering on numbers, but also what amounts to the implication ordering on logical terms and the inclusion ordering on classes.  Later he will introduce distinctive symbols for the logical orderings.  The links among terms, sets, and numbers can be pursued in all directions and Peirce has already indicated in an earlier paper how he would construct the integers from sets, that is, from the aggregate denotations of terms.  I will try to get back to that another time.

We have a statement of the following form.

If \mathrm{f} < \mathrm{m} then the number of Frenchmen is less than the number of men.

This goes into symbolic form as follows.

\begin{matrix} \mathrm{f} < \mathrm{m} & \Rightarrow & [\mathrm{f}] < [\mathrm{m}]. \end{matrix}

In this setting the ``\!<\!" on the left is a logical ordering on syntactic terms while the ``\!<\!" on the right is an arithmetic ordering on real numbers.

The question that arises in this case is whether a map between two ordered sets is order-preserving.  In order to formulate the question in more general terms, we may begin with the following set-up.

Let X_1 be a set with the ordering <_1.

Let X_2 be a set with the ordering <_2.

An order relation is typically defined by a set of axioms that determines its properties.  Since we have frequent occasion to view the same set in the light of several different order relations, we often resort to explicit specifications like (X, <_1),\ (X, <_2), and so on to indicate a set with a given ordering.

A map F : (X_1, <_1) \to (X_2, <_2) is order-preserving if and only if a statement of a particular form holds for all x and y in (X_1, <_1), namely, the following.

\begin{matrix} x <_1 y & \Rightarrow & F(x) <_2 F(y). \end{matrix}

The “number of” map v : (S, <_1) \to (\mathbb{R}, <_2) has just this character, as exemplified in the case at hand.

\begin{matrix} \mathrm{f} & < & \mathrm{m} & \Rightarrow & [\mathrm{f}] & < & [\mathrm{m}] \\[6pt] \mathrm{f} & < & \mathrm{m} & \Rightarrow & v(\mathrm{f}) & < & v(\mathrm{m}) \end{matrix}

The ``\!<\!" on the left is read as proper inclusion, in other words, subset of but not equal to, while the ``\!<\!" on the right is read as the usual less than relation.

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Peirce’s 1870 “Logic of Relatives” • Comment 11.16

Posted on May 21, 2014 by Jon Awbrey

Peirce’s 1870 “Logic of Relatives” • Comment 11.16

We now have enough material on morphisms to go back and cast a more studied eye on what Peirce is doing with that “number of” function, whose application to a logical term \mathit{t} is indicated by writing the term in square brackets, as [\mathit{t}].  It is convenient to have a prefix notation for the function mapping a term \mathit{t} to a number [\mathit{t}] but Peirce previously reserved the letter ``n" for logical \text{not}, so let’s use v(\mathit{t}) as a variant for [\mathit{t}].

My plan will be nothing less plodding than to work through the statements Peirce made in defining and explaining the “number of” function up to our present place in the paper, namely, the budget of points collected in Comment 11.2.

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 [\mathit{t}].

(Peirce, CP 3.65)

The role of the “number of” function may be formalized by assigning it a name and a type, in the present discussion v : S \to \mathbb{R}, where S is a suitable set of signs, a syntactic domain, containing all the logical terms whose numbers we need to evaluate in a given context, and where \mathbb{R} is the set of real numbers.

Transcribing Peirce’s example:

\begin{array}{ll} \text{Let} & \mathrm{m} ~=~ \text{man} \\[8pt] \text{and} & \mathit{t} ~=~ \text{tooth of}\,\underline{~~~~}. \\[8pt] \text{Then} & v(\mathit{t}) ~=~ [\mathit{t}] ~=~ \displaystyle\frac{[\mathit{t}\mathrm{m}]}{[\mathrm{m}]}. \end{array}

To spell it out in words, the number of the relative term ``\text{tooth of}\,\underline{~~~~}" in a universe of perfect human dentition is equal to the number of teeth of humans divided by the number of humans, that is, 32.

The dyadic relative term \mathit{t} determines a dyadic relation T \subseteq X \times Y, where X and Y contain all the teeth and all the people, respectively, under discussion.

A rough indication of the bigraph for T might be drawn as follows, showing just the first few items in the toothy part of X and the peoply part of Y.

Dyadic Relation T
\text{Figure 51. Dyadic Relation}~ T \subseteq X \times Y

Notice that the “number of” function v : S \to \mathbb{R} needs the data represented by the entire bigraph for T in order to compute the value [\mathit{t}].

Finally, one observes this component of T is a function in the direction T : X \to Y, since we are counting only teeth which occupy exactly one mouth of a tooth-bearing creature.

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Peirce’s 1870 “Logic of Relatives” • Comment 11.15

Posted on May 15, 2014 by Jon Awbrey

Peirce’s 1870 “Logic of Relatives” • Comment 11.15

I’m going to elaborate a little further on the subject of arrows, morphisms, or structure-preserving mappings, as a modest amount of extra work at this point will repay ample dividends when it comes time to revisit Peirce’s “number of” function on logical terms.

The structure preserved by a structure-preserving map is just the structure we all know and love as a triadic relation.  Very typically, it will be the type of triadic relation that defines the type of binary operation that obeys the rules of a mathematical structure known as a group, that is, a structure satisfying the axioms for closure, associativity, identities, and inverses.

For example, in the case of the logarithm map J we have the following data.

\begin{array}{lcccll} J & : & \mathbb{R} & \gets & \mathbb{R} & \text{(properly restricted)} \\[6pt] K & : & \mathbb{R} & \gets & \mathbb{R} \times \mathbb{R} & \text{where}~ K(r, s) = r + s \\[6pt] L & : & \mathbb{R} & \gets & \mathbb{R} \times \mathbb{R} & \text{where}~ L(u, v) = u \cdot v \end{array}

Real number addition and real number multiplication (suitably restricted) are examples of group operations.  If we write the sign of each operation in brackets as a name for the triadic relation that defines the corresponding group, we have the following set-up.

\begin{matrix} J & : & [+] \gets [\,\cdot\,] \\[6pt] [+] & \subseteq & \mathbb{R} \times \mathbb{R} \times \mathbb{R} \\[6pt] [\,\cdot\,] & \subseteq & \mathbb{R} \times \mathbb{R} \times \mathbb{R} \end{matrix}

It often happens that both group operations are indicated by the same sign, usually one from the set \{ \cdot, *, + \} or simple concatenation, but they remain in general distinct whether considered as operations or as relations, no matter what signs of operation are used.  In such a setting, our chiasmatic theme may run a bit like one of the following two variants.

\textit{The image of the sum is the sum of the images.}

\textit{The image of the product is the sum of the images.}

Figure 50 presents a generic picture for groups G and H.

Group Homomorphism J : G ← H
\text{Figure 50. Group Homomorphism}~ J : G \gets H

In a setting where both groups are written with a plus sign, perhaps even constituting the same group, the defining formula of a morphism, J(L(u, v)) = K(Ju, Jv), takes on the shape J(u + v) = Ju + Jv, which looks analogous to the distributive multiplication of a factor J over a sum (u + v).  That is why morphisms are regarded as generalizations of linear functions and are frequently referred to in those terms.

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Peirce’s 1870 “Logic of Relatives” • Comment 11.14

Posted on May 15, 2014 by Jon Awbrey

Peirce’s 1870 “Logic of Relatives” • Comment 11.14

Let’s now look at a concrete example of a morphism J, say, one of the mappings of reals into reals commonly known as logarithm functions, where you get to pick your favorite base.

Here we have K(r, s) = r + s and L(u, v) = u \cdot v and the formula J(L(u, v)) = K(Ju, Jv) becomes J(u \cdot v) = J(u) + J(v), where ordinary multiplication and addition are indicated by a dot (\cdot) and a plus sign (+) respectively.

Figure 49 shows how the multiplication, addition, and logarithm operations fit together.

Logarithm Arrow J : {+} ← {⋅}
\text{Figure 49. Logarithm Arrow}~ J : \{ + \} \gets \{ \cdot \}

In short, where the image operation J is the logarithm map, the source operation is the numerical product, and the target operation is the numerical sum, we have the following rule of thumb.

The image of the product is the sum of the images.

\begin{array}{lll} J(u \cdot v) & = & J(u) + J(v) \end{array}

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Peirce’s 1870 “Logic of Relatives” • Comment 11.13

Posted on May 14, 2014 by Jon Awbrey

Peirce’s 1870 “Logic of Relatives” • Comment 11.13

As we make our way toward the foothills of Peirce’s 1870 Logic of Relatives there are several pieces of equipment we must not leave the plains without, namely, the utilities variously known as arrows, morphisms, homomorphisms, structure-preserving maps, among other names, depending on the altitude of abstraction we happen to be traversing at the moment in question.  As a moderate to middling but not too beaten track, let’s examine a few ways of defining morphisms that will serve us in the present discussion.

Suppose we are given three functions J, K, L satisfying the following conditions.

\begin{array}{lcccl} J & : & X & \gets & Y \\[6pt] K & : & X & \gets & X \times X \\[6pt] L & : & Y & \gets & Y \times Y \end{array}

\begin{array}{lll} J(L(u, v)) & = & K(Ju, Jv) \end{array}

Our sagittarian leitmotif can be rubricized in the following slogan.

The J-image of the L-product is the K-product of the J-images.

Figure 47 presents us with a picture of the situation in question.

Structure Preserving Transformation J : K ← L
\text{Figure 47. Structure Preserving Transformation}~ J : K \gets L

Table 48 gives the constraint matrix version of the same thing.

\text{Table 48. Structure Preserving Transformation}~ J : K \gets L
Structure Preserving Transformation J : K ← L

One way to read the Table is in terms of the informational redundancies it summarizes.  For example, one way to read it says that satisfying the constraint in the L row along with all the constraints in the J columns automatically satisfies the constraint in the K row.  Quite by design, that is one way to understand the equation J(L(u, v)) = K(Ju, Jv).

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Peirce’s 1870 “Logic of Relatives” • Comment 11.12

Posted on May 12, 2014 by Jon Awbrey

Peirce’s 1870 “Logic of Relatives” • Comment 11.12

Since functions are special cases of dyadic relations and since the space of dyadic relations is closed under relational composition — that is, the composition of two dyadic relations is again a dyadic relation — we know the relational composition of two functions has to be a dyadic relation.  If the relational composition of two functions is necessarily a function, too, then we would be justified in speaking of functional composition and also in saying the space of functions is closed under this functional form of composition.

Just for novelty’s sake, let’s try to prove this for relations that are functional on correlates.

The task is this — We are given a pair of dyadic relations:

\begin{array}{lll} P \subseteq X \times Y & \text{and} & Q \subseteq Y \times Z \end{array}

The dyadic relations P and Q are assumed to be functional on correlates, a premiss we express as follows.

\begin{array}{lll} P : X \gets Y & \text{and} & Q : Y \gets Z \end{array}

We are charged with deciding whether the relational composition P \circ Q \subseteq X \times Z is also functional on correlates, in symbols, whether P \circ Q : X \gets Z.

It always helps to begin by recalling the pertinent definitions.

For a dyadic relation L \subseteq X \times Y, we have the following equivalence.

\begin{array}{lll} L ~\text{is a function}~ L : X \gets Y & \iff & L ~\text{is}~ 1\text{-regular at}~ Y. \end{array}

As for the definition of relational composition, it is enough to consider the coefficient of the composite relation on an arbitrary ordered pair, i\!:\!j.  For that we have the following formula, where the summation indicated is logical disjunction.

(P \circ Q)_{ij} ~=~ \sum_k P_{ik} Q_{kj}

So let’s begin.

  • P : X \gets Y, or the fact that P ~\text{is}~ 1\text{-regular at}~ Y, means there is exactly one ordered pair i\!:\!k \in P for each k \in Y.
  • Q : Y \gets Z, or the fact that Q ~\text{is}~ 1\text{-regular at}~ Z, means there is exactly one ordered pair k\!:\!j \in Q for each j \in Z.
  • As a result, there is exactly one ordered pair i\!:\!j \in P \circ Q for each j \in Z, which means P \circ Q ~\text{is}~ 1\text{-regular at}~ Z, and so we have the function P \circ Q : X \gets Z.

And we are done.

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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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