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Peirce's 1870 Logic Of Relatives • Part 2
Author: Jon Awbrey
• Overview • Part 1 • Part 2 • Part 3 • References • Resources • Blog Series • Document History •
Contents
 1 Selection 11
 1.1 The Signs for Multiplication (concl.)
 1.2 Commentary Note 11.1
 1.3 Commentary Note 11.2
 1.4 Commentary Note 11.3
 1.5 Commentary Note 11.4
 1.6 Commentary Note 11.5
 1.7 Commentary Note 11.6
 1.8 Commentary Note 11.7
 1.9 Commentary Note 11.8
 1.10 Commentary Note 11.9
 1.11 Commentary Note 11.10
 1.12 Commentary Note 11.11
 1.13 Commentary Note 11.12
 1.14 Commentary Note 11.13
 1.15 Commentary Note 11.14
 1.16 Commentary Note 11.15
 1.17 Commentary Note 11.16
 1.18 Commentary Note 11.17
 1.19 Commentary Note 11.18
 1.20 Commentary Note 11.19
 1.21 Commentary Note 11.20
 1.22 Commentary Note 11.21
 1.23 Commentary Note 11.22
 1.24 Commentary Note 11.23
 1.25 Commentary Note 11.24
 2 Selection 12
 3 Selection 13
Selection 11
The Signs for Multiplication (concl.)
The conception of multiplication we have adopted is that of the application of one relation to another. So, a quaternion being the relation of one vector to another, the multiplication of quaternions is the application of one such relation to a second. Even ordinary numerical multiplication involves the same idea, for is a pair of triplets, and is a triplet of pairs, where "triplet of" and "pair of" are evidently relatives. If we have an equation of the form: 
and there are just as many 's per as there are, per things, things of the universe, then we have also the arithmetical equation: 
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: 
holds arithmetically. So if men are just as apt to be black as things in general: 
where the difference between and must not be overlooked. It is to be observed that: 
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). 
Commentary Note 11.1
We have reached a suitable place to pause in our reading of Peirce's text — actually, it's more like a place to run as fast as we can along a parallel track — where I can pay off a few of the expository IOUs I've been using to pave the way to this point.
The more pressing debts that come to mind are concerned with the matter of Peirce's “number of” function that maps a term into a number and with my justification for calling a certain style of illustration the hypergraph picture of relational composition. As it happens, there is a thematic relation between these topics, and so I can make my way forward by addressing them together.
At this point we have two good pictures of how to compute the relational compositions of arbitrary dyadic relations, namely, the bigraph representation and the matrix representation, each of which has its differential advantages in different types of situations.
But we do not have a comparable picture of how to compute the richer variety of relational compositions that involve triadic or any higher adicity relations. As a matter of fact, we run into a nontrivial classification problem simply to enumerate the different types of compositions that arise in these cases.
Therefore, let us inaugurate a systematic study of relational composition, general enough to articulate the “generative potency” of Peirce's 1870 Logic of Relatives.
Commentary Note 11.2
Let's bring together the various things that 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 (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 is to say that is part of , just as to write is to say that Frenchmen are part of men. Indeed, if , then the number of Frenchmen is less than the number of men, and if , then the number of VicePresidents 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 noninvertible 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 
whatever is denoted by ; 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 
(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 is a pair of triplets, and is a triplet of pairs, where “triplet of” and “pair of” are evidently relatives. If we have an equation of the form: 
and there are just as many 's per as there are, per things, things of the universe, then we have also the arithmetical equation: 
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: 
holds arithmetically. So if men are just as apt to be black as things in general: 
where the difference between and must not be overlooked. It is to be observed that: 
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). 
Commentary Note 11.3
Before I can discuss Peirce's “number of” function in greater detail I will need to deal with an expositional difficulty that I have been very carefully dancing around all this time, but one that 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 itself is already to be numbered among the types of dyadic relatives that we've been explicitly mentioning and implicitly using all this time. But Peirce's way of talking about a dyadic relative term is to list the “relate” first and the “correlate” second, a convention that goes over into functional terms as making the functional value first and the functional argument second, whereas almost anyone brought up in our present time frame has difficulty thinking of a function any other way than as a set of ordered pairs where the order in each pair lists the functional argument first and the functional value second.
All of these syntactic wrinkles can be ironed out in a very smooth way, given a sufficiently general context of flexible enough interpretive conventions, but not without introducing an order of anachronism into Peirce's presentation that I am presently trying to avoid as much as possible. Thus, I will need to experiment with various styles of compromise formation.
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.
Commentary Note 11.4
The task before us is to clarify the relationships among relative terms, relations, and the special cases of relations that are given by equivalence relations, functions, and so on.
The first obstacle to get past is the order convention that 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 that Peirce denotes by means of square brackets and reformulate it as a dyadic relative term as follows:
To set the dyadic relative term within a suitable context of interpretation, let us suppose that corresponds to a relation where is the set of real numbers and is a suitable syntactic domain, here described as a set of terms. The dyadic relation is at first sight a function from to There is, however, a very great likelihood that we cannot always assign numbers to every term in whatever syntactic domain we happen to choose, so we may eventually be forced to treat the dyadic relation as a partial function from to All things considered, then, let me try out the following impedimentaria of strategies and compromises.
First, I adapt the functional arrow notation so that it allows us to detach the functional orientation from the order in which the names of domains are written on the page. Second, I change the notation for partial functions, or prefunctions, to one that is less likely to be confounded. This gives the scheme:
means that is functional at 
means that is functional at 
means that is prefunctional at 
means that is prefunctional at 
Until it becomes necessary to stipulate otherwise, let us assume that is a function in of written amounting to the functional alias of the dyadic relation and associated with the dyadic relative term whose relate lies in the set of real numbers and whose correlate lies in the set of syntactic terms.
Note. See the article Relation Theory for the definitions of functions and prefunctions used in this section.
Commentary Note 11.5
The right form of diagram can be a great aid in rendering complex matters comprehensible, so let's extract the overly compressed bits of the “Relation Theory” article that we need to illuminate Peirce's 1870 “Logic Of Relatives” and draw what icons we can within the current frame.
For the immediate present, we may start with dyadic relations and describe the customary species of relations and functions in terms of their local and numerical incidence properties.
Let be an arbitrary dyadic relation. The following properties of can be defined:

If is tubular at then is known as a partial function or a prefunction from to frequently signalized by renaming with an alternate lower case name, say and writing
Just by way of formalizing the definition:

To illustrate these properties, let us fashion a generic enough example of a dyadic relation, where and where the bigraph picture of looks like this:
(30) 
If we scan along the dimension from to we see that the incidence degrees of the nodes with the domain are in that order.
If we scan along the dimension from to we see that the incidence degrees of the nodes with the domain are in that order.
Thus, is not total at either or since there are nodes in both and having incidence degrees less than
Also, is not tubular at either or since there are nodes in both and having incidence degrees greater than
Clearly, then, the relation cannot qualify as a prefunction, much less as a function on either of its relational domains.
Commentary Note 11.6
Let's continue working our way through the above definitions, constructing appropriate examples as we go.
exemplifies the quality of totality at
(31) 
exemplifies the quality of totality at
(32) 
exemplifies the quality of tubularity at
(33) 
exemplifies the quality of tubularity at
(34) 
So is a prefunction and is a prefunction
Commentary Note 11.7
We come now to the very special cases of dyadic relations that are known as functions. It will serve a dual purpose on behalf of the present exposition if we take the class of functions as a source of object examples to clarify the more abstruse concepts in the Relation Theory material.
To begin, let's recall the definition of a local flag:
In the case of a dyadic relation it is possible to simplify the notation for local flags in a couple of ways. First, it is often easier in the dyadic case to refer to as and as Second, the notation may be streamlined even further by writing as and as
In light of these considerations, the local flags of a dyadic relation may be formulated as follows:

The following definitions are also useful:

A sufficient illustration is supplied by the earlier example
(35) 
The local flag is displayed here:
(36) 
The local flag is displayed here:
(37) 
Commentary Note 11.8
Next let's reexamine the numerical incidence properties of relations, concentrating on the definitions of the assorted regularity conditions.
For example, is said to be if and only if the cardinality of the local flag is equal to for all coded in symbols, if and only if for all
In a similar fashion, it is possible to define the numerical incidence properties and so on. For ease of reference, a few of these definitions are recorded below.

Clearly, if any relation is on one of its domains and also on the same domain, then it must be on that domain, in effect, at
For example, let and and consider the dyadic relation that is bigraphed here:
(38) 
We observe that is 3regular at and 1regular at
Commentary Note 11.9
Among the variety of conceivable regularities affecting dyadic relations we pay special attention to the regularity conditions where is equal to 1.
Let be an arbitrary dyadic relation. The following properties of can be defined:

We have already looked at dyadic relations that separately exemplify each of these regularities. We also introduced a few bits of additional terminology and specialpurpose notations for working with tubular relations:

We arrive by way of this winding stair at the special stamps of dyadic relations that are variously described as 1regular, total and tubular, or total prefunctions on specified domains, either or or both, and that are more often celebrated as functions on those domains.
If is a prefunction that happens to be total at then is known as a function from to typically indicated as
To say that a relation is totally tubular at is to say that is 1regular at Thus, we may formalize the following definitions:

For example, let and let be the dyadic relation depicted in the bigraph below:
(39) 
We observe that is a function at and we record this fact in either of the manners or
Commentary Note 11.10
In the case of a dyadic relation that has the qualifications of a function there are a number of further differentia that arise:

For example, the function depicted below is neither total at nor tubular at and so it cannot enjoy any of the properties of being surjective, injective, or bijective.
(40) 
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 above is Thus, if we form a new function that looks just like on the domain but is assigned the codomain then is surjective, and is described as mapping onto
(41) 
The function is injective.
(42) 
The function is bijective.
(43) 
Commentary Note 11.11
The preceding exercises were intended to beefup our “functional” literacy skills to the point where we can read our functional alphabets backwards and forwards and recognize the local functionalities that may be immanent in relative terms no matter where they locate themselves within 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 and the corresponding relation are both called functional on relates if and only if is a function at in symbols, 
A relative term and the corresponding relation are both called functional on correlates if and only if is a function at in symbols, 
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 as the case may be, when and if it serves to clarify matters.
From this current, perhaps transient, perspective, it appears that our next task is to examine how the known properties of relations are modified when an aspect of functionality is 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 different 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.
(44) 
From this we extract the hypergraph picture of relational composition:
(45) 
All of the relevant information of these Figures can be compressed into the form of a spreadsheet, or constraint satisfaction table:
So the following presents itself as a reasonable plan of study: Let's see how much easy mileage we can get in our exploration of functions by adopting the above templates as a paradigm.
Commentary Note 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 that 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 that 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:
and are assumed to be functional on correlates, a premiss that we express as follows:
We are charged with deciding whether the relational composition is also functional on correlates, in symbols, whether
It always helps to begin by recalling the pertinent definitions.
For a dyadic relation we have:

As for the definition of relational composition, it is enough to consider the coefficient of the composite relation on an arbitrary ordered pair, For that, we have the following formula, where the summation indicated is logical disjunction:
So let's begin.
or the fact that means that there is exactly one ordered pair for each 
or the fact that means that there is exactly one ordered pair for each 
As a result, there is exactly one ordered pair for each which means that and so we have the function 
And we are done.
Commentary Note 11.13
As we make our way toward the foothills of Peirce's 1870 Logic of Relatives, there are several pieces of equipment that we must not leave the plains without, namely, the utilities variously known as arrows, morphisms, homomorphisms, structurepreserving 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 that satisfy the following conditions:


Our sagittarian leitmotif can be rubricized in the following slogan:
(Where is the image, is the compound, and is the ligature.) 
Figure 47 presents us with a picture of the situation in question.
(47) 
Table 48 gives the constraint matrix version of the same thing.
One way to read this Table is in terms of the informational redundancies that it schematizes. In particular, it can be read to say that when one satisfies the constraint in the row, along with all the constraints in the columns, then the constraint in the row is automatically true. That is one way of understanding the equation:
Commentary Note 11.14
Now, as promised, let's look at a more homely example of a morphism, say, any one of the mappings (roughly speaking) that are commonly known as logarithm functions, where you get to pick your favorite base. In this case, and and the defining formula comes out looking like writing a dot and a plus sign for the ordinary binary operations of arithmetical multiplication and arithmetical summation, respectively.
(49) 
Thus, where the image is the logarithm map, the compound is the numerical sum, and the ligature is the numerical product, one has the following rule of thumb:


Commentary Note 11.15
I'm going to elaborate a little further on the subject of arrows, morphisms, or structurepreserving maps, 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 that is preserved by a structurepreserving map is just the structure that 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 that is known as a group, that is, a structure that satisfies the axioms for closure, associativity, identities, and inverses.
For example, in the previous case of the logarithm map we have the data:

Real number addition and real number multiplication (suitably restricted) are examples of group operations. If we write the sign of each operation in braces as a name for the triadic relation that constitutes or defines the corresponding group, then we have the following setup:

In many cases, one finds that both group operations are indicated by the same sign, typically , , , 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 these two variants:


Figure 50 presents a generic picture for groups and
(50) 
In a setting where both groups are written with a plus sign, perhaps even constituting the very same group, the defining formula of a morphism, takes on the shape which looks very analogous to the distributive multiplication of a sum by a factor Hence another popular name for a morphism: a linear map.
Commentary Note 11.16
We have enough material on morphisms now 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 is indicated by writing the term in square brackets, as It is convenient to have a prefix notation for the function that maps a term to a number but Peirce has previously reserved for the logical so let's use as a variant for
My plan will be nothing less plodding than to work through the statements that 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 Section 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 (Peirce, CP 3.65). 
The role of the “number of” function may be formalized by assigning it a name and a type as where is a suitable set of signs, a >syntactic domain, containing all the logical terms whose numbers we need to evaluate in a given discussion, and where is the set of real numbers.
Transcribing Peirce's example:
Let  
and  
Then 
Thus, in a universe of perfect human dentition, the number of the relative term is equal to the number of teeth of humans divided by the number of humans, that is,
The dyadic relative term determines a dyadic relation where contains all the teeth and contains all the people that happen to be under discussion.
A rough indication of the bigraph for might be drawn as follows, showing just the first few items in the toothy part of and the peoply part of
(51) 
Notice that the “number of” function needs the data that is represented by this entire bigraph for in order to compute the value
Finally, one observes that this component of is a function in the direction since we are counting only teeth that occupy exactly one mouth of a toothbearing creature.
Commentary Note 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. Among other things it is one of the planks 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, as a matter of necessity and a matter of fact, practically speaking at any rate, one way that Peirce forges a link between the eternal, logical, or rational realm and the secular, empirical, or real domain.
With that little bit of encouragement and exhortation, let us return to the nitty gritty 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 is to say that is part of , just as to write is to say that Frenchmen are part of men. Indeed, if , then the number of Frenchmen is less than the number of men, and if , then the number of VicePresidents 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 is here remarking on the principle that the measure on terms preserves or respects the prevailing implication, inclusion, or subsumption relations that impose an ordering on those terms. In these initiatory passages of the text, Peirce is using a single symbol to denote the usual linear ordering on numbers, but also what amounts to the implication ordering on logical terms and the inclusion ordering on classes. Later, of course, he will introduce distinctive symbols for logical orders. 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 then the number of Frenchmen is less than the number of men. 
This goes into symbolic form as follows:

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 orderpreserving. In order to formulate the question in more general terms, we may begin with the following setup:
Let be a set with the ordering 
Let be a set with the ordering 
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 and so on, to indicate a set with a given ordering.
A map is orderpreserving if and only if a statement of a particular form holds for all and in namely, the following:

The “number of” map has just this character, as exemplified in the case at hand:

Here, 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 ordinary less than relation.
Commentary Note 11.18
An orderpreserving map is a special case of a structure preserving map, and the idea of preserving structure, as used in mathematics, always means preserving some but not necessarily all the structure of the source domain in question. People sometimes express this by speaking of structure preservation in measure, the implication being that any property that is amenable to being qualified in manner is potentially amenable to being quantified in degree, perhaps in such a way as to answer questions like “How structurepreserving is it?”
Let's see how this remark applies to the orderpreserving property of the “number of” mapping For any pair of absolute terms and in the syntactic domain we have the following implications, where denotes the logical subsumption relation on terms and denotes the less than or equal to relation on the real number domain

Equivalently:

Nowhere near the number of logical distinctions that exist on the left hand side of the implication arrow can be preserved as one passes to the linear ordering of real numbers on the right hand side of the implication arrow, but that is not required in order to call the map orderpreserving, or what is known as an order morphism.
Commentary Note 11.19
Up to this point in the 1870 Logic of Relatives, Peirce has introduced the “number of” function on logical terms and discussed the extent to which its use as a measure, such that satisfies the relevant measuretheoretic principles, for starters, these two:
1.  The “number of” map exhibits a certain type of uniformity property, whereby the value of the measure on a uniformly qualified population is in fact actualized by each member of the population. 
2.  The “number of” map satisfies an order morphism principle, whereby the illative partial ordering of logical terms is reflected up to a partial extent by the arithmetical linear ordering of their measures. 
Peirce next takes up the action of the “number of” map on the two types of, loosely speaking, additive operations that we normally consider in logic.
NOF 3.1
It is plain that both the regular noninvertible addition and the invertible addition satisfy the absolute conditions. (Peirce, CP 3.67). 
The sign denotes what Peirce calls “the regular noninvertible addition”, corresponding to the inclusive disjunction of logical terms or the union of their extensions as sets.
The sign denotes what Peirce calls “the invertible addition”, corresponding to the exclusive disjunction of logical terms or the symmetric difference of their extensions as sets.
NOF 3.2
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 and for example, being the number of a collection which consists of a collection of two and a collection of five. (Peirce, CP 3.67). 
A full interpretation of this remark will require us to pick up the precise technical sense in which Peirce is using the word collection, and that will take us back to his logical reconstruction of certain aspects of number theory, all of which I am putting off to another time, but it is still possible to get a rough sense of what he's saying relative to the present frame of discussion.
The “number of” map evidently induces some sort of morphism with respect to logical sums. If this were straightforwardly true, we could write:

Equivalently:

Of course, things are not quite that simple when it comes to inclusive disjunctions and settheoretic unions, so it is usual to introduce the concept of a subadditive measure to describe the principle that does hold here, namely, the following:

Equivalently:

This is why Peirce trims his discussion of this point with the following hedge:
NOF 3.3
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. (Peirce, CP 3.67). 
Finally, a morphism with respect to addition, even a contingently qualified one, must do the right stuff on behalf of the additive identity:
NOF 3.4
Addition being taken in this sense, nothing is to be denoted by zero, for then 
whatever is denoted by ; 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 
(Peirce, CP 3.67). 
With respect to the nullity in and the number in we have:
In sum, therefore, it can be said: It also serves that only preserves a due respect for the function of a vacuum in nature.
Commentary Note 11.20
We arrive at the last of Peirce's statements about the “number of” map that we singled out above:
NOF 4.1
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 is a pair of triplets, and is a triplet of pairs, where “triplet of” and “pair of” are evidently relatives. If we have an equation of the form: 
and there are just as many 's per as there are per things, things of the universe, then we have also the arithmetical equation: 
(Peirce, CP 3.76). 
Peirce is here observing what we might call a contingent morphism. Provided that a certain condition, to be named in short order, happens to be satisfied, we would find it holding that the “number of” map such that serves to preserve the multiplication of relative terms, that is to say, the composition of relations, in the form: So let us try to uncross Peirce's manifestly chiasmatic encryption of the condition that is called on in support of this preservation.
The proviso for the equation to hold is this:
There are just as many 's per as there are per things, things of the universe. (Peirce, CP 3.76). 
Returning to the example that Peirce gives:
NOF 4.2
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: 
holds arithmetically. (Peirce, CP 3.76). 
Now that is something that we can sink our teeth into and trace the bigraph representation of the situation. It will help to recall our first examination of the “tooth of” relation and to adjust the picture we sketched of it on that occasion.
Transcribing Peirce's example:
Let  
and  
Then 
That is to say, the number of the relative term is equal to the number of teeth of humans divided by the number of humans. In a universe of perfect human dentition this gives a quotient of
The dyadic relative term determines a dyadic relation where contains all the teeth and contains all the people that happen to be under discussion.
To make the case as simple as possible and still cover the point, suppose there are just four people in our universe of discourse and just two of them are French. The bigraphical composition below shows the pertinent facts of the case.
(52) 
In this picture the order of relational composition flows down the page. For convenience in composing relations, the absolute term is inflected by the comma functor to form the dyadic relative term which in turn determines the idempotent representation of Frenchmen as a subset of mankind,
By way of a legend for the figure, we have the following data:

Now let's see if we can use this picture to make sense of the following statement:
NOF 4.3
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: 
holds arithmetically. (Peirce, CP 3.76). 
In statistical terms, Peirce is saying this: If the population of Frenchmen is a fair sample of the general population with regard to the factor of dentition, then the morphic equation,
whose transpose gives the equation,
is every bit as true as the defining equation in this circumstance, namely,
Commentary Note 11.21
One more example and one more general observation, and then we will be all caught up with our homework on Peirce's “number of” function.
NOF 4.4
So if men are just as apt to be black as things in general, 
where the difference between and must not be overlooked. (Peirce, CP 3.76). 
The protasis, “men are just as apt to be black as things in general”, is elliptic in structure, and presents us with a potential ambiguity. If we had no further clue to its meaning, it might be read as either of the following:
1.  Men are just as apt to be black as things in general are apt to be black. 
2.  Men are just as apt to be black as men are apt to be things in general. 
The second interpretation, if grammatical, is pointless to state, since it equates a proper contingency with an absolute certainty. So I think it is safe to assume this paraphrase of what Peirce intends:
Men are just as likely to be black as things in general are likely to be black. 
Stated in terms of the conditional probability:
From the definition of conditional probability:
Equivalently:
Taking everything together, we obtain the following result:
This, of course, is the definition of independent events, as applied to the event of being Black and the event of being a Man. It seems to be the most likely guess that this is the meaning of Peirce's statement about frequencies:
The terms of this equation can be normalized to produce the corresponding statement about probabilities:
Let's see if this checks out.
Let be the number of things in general. In terms of Peirce's “number of” function, then, we have the equation On the assumption that and are associated with independent events, we obtain the following sequence of equations:

As a result, we have to interpret = “the average number of men per things in general” as = “the probability of a thing in general being a man”. This seems to make sense.
Commentary Note 11.22
Let's look at that last example from a different angle.
NOF 4.4
So if men are just as apt to be black as things in general, 
where the difference between and must not be overlooked. (Peirce, CP 3.76). 
In different lights the formula presents itself as an aimed arrow, fair sample, or stochastic independence condition.
The example apparently assumes a universe of things in general, encompassing among other things the denotations of the absolute terms and That suggests to me that we might well illustrate this case in relief, by returning to our earlier staging of Othello and seeing how well that universe of dramatic discourse observes the premiss that “men are just as apt to be black as things in general”.
Here are the relevant data:

The fair sampling condition is tantamount to this: “Men are just as apt to be black as things in general are apt to be black”. In other words, men are a fair sample of things in general with respect to the factor of being black.
Should this hold, the consequence would be:
When is not zero, we obtain the result:
As before, it is convenient to represent the absolute term by means of the corresponding idempotent term
Consider the bigraph for the composition:
This is represented below in the equivalent form:
(53) 
Thus we observe one of the more factitious facts affecting this very special universe of discourse, namely:
This is equivalent to the implication that Peirce would have written in the form
That is enough to puncture any notion that and are statistically independent, but let us continue to develop the plot a bit more. Putting all the general formulas and particular facts together, we arrive at the following summation of the situation in the Othello case:
If the fair sampling condition were true, it would have the following consequence:
On the contrary, we have the following fact:
In sum, it is not the case in the Othello example that “men are just as apt to be black as things in general”.
Expressed in terms of probabilities: and
If these were independent terms we would have:
In point of fact, however, we have:
Another way to see it is to observe that: while
Commentary Note 11.23
Peirce's description of logical conjunction and conditional probability via the logic of relatives and the mathematics of relations is critical to understanding the relationship between logic and measurement, in effect, the qualitative and quantitative aspects of inquiry. To ground this connection firmly in mind, I will try to sum up as succinctly as possible, in more current notation, the lesson we ought to take away from Peirce's last “number of” example, since I know the account I have given so far may appear to have wandered widely.
NOF 4.4
So if men are just as apt to be black as things in general, 
where the difference between and must not be overlooked. (Peirce, CP 3.76). 
In different lights the formula presents itself as an aimed arrow, fair sampling, or statistical independence condition. The concept of independence was illustrated above by means of a case where independence fails. The details of that counterexample are summarized below.
(54) 
The condition that “men are just as apt to be black as things in general” is expressed in terms of conditional probabilities as which means that the probability of the event given the event is equal to the unconditional probability of the event
In the Othello example, it is enough to observe that while in order to recognize the bias or dependency of the sampling map.
The reduction of a conditional probability to an absolute probability, as is one of the ways we come to recognize the condition of independence, via the definition of conditional probability,
To recall the derivation, the definition of conditional probability plus the independence condition yields in short,
As Hamlet discovered, there's a lot to be learned from turning a crank.
Commentary Note 11.24
We come to the end of the “number of” examples that we found on our agenda at this point in the text:
NOF 4.5
It is to be observed that 
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 and CE 2, 376). 
There are problems with the printing of the text at this point. Let us first recall the conventions we are using in this transcription, in particular, for the italic 1 that signifies the dyadic identity relation and for the “antique figure one” that Peirce defines as
CP 3 gives which I cannot make sense of. CE 2 gives the 1's in different styles of italics, but reading the equation as makes the best sense if the “1” on the right hand side is read as the numeral “1” that denotes the natural number 1, and not as the absolute term “1” that denotes the universe of discourse. Read this way, is the average number of things related by the identity relation to one individual, and so it makes sense that where is the set of nonnegative integers
With respect to the relative term in the syntactic domain and the number in the nonnegative integers we have:
And so the “number of” mapping has another one of the properties that would be required of an arrow
The manner in which these arrows and qualified arrows help us to construct a suspension bridge that unifies logic, semiotics, statistics, stochastics, and information theory will be one of the main themes I aim to elaborate throughout the rest of this inquiry.
Selection 12
The Sign of Involution
I shall take involution in such a sense that will denote everything which is an for every individual of Thus will be a lover of every woman. Then will denote whatever stands to every woman in the relation of servant of every lover of hers; and will denote whatever is a servant of everything that is lover of a woman. So that 
(Peirce, CP 3.77). 
Commentary Note 12.1
To get a better sense of why the above formulas mean what they do, and to prepare the ground for understanding more complex relational expressions, it will help to assemble the following materials and definitions:
is a set singled out in a particular discussion as the universe of discourse. 
is the 1adic relation, or set, whose elements fall under the absolute term The elements of are sometimes referred to as the denotation or the settheoretic extension of the term 
is the 2adic relation associated with the relative term 
is the 2adic relation associated with the relative term 
is the 1dimensional matrix representation of the set and the term 
is the 2dimensional matrix representation of the relation and the relative term 
is the 2dimensional matrix representation of the relation and the relative term 
Recalling a few definitions, the local flags of the relation are given as follows:

The applications of the relation are defined as follows:

Commentary Note 12.2
Let us make a few preliminary observations about the operation of logical involution, as Peirce introduces it here:
I shall take involution in such a sense that will denote everything which is an for every individual of Thus will be a lover of every woman. (Peirce, CP 3.77). 
In ordinary arithmetic the involution or the exponentiation of to the power of is the repeated application of the multiplier for as many times as there are ones making up the exponent
In analogous fashion, the logical involution is the repeated application of the term for as many times as there are individuals under the term According to Peirce's interpretive rules, the repeated applications of the base term are distributed across the individuals of the exponent term In particular, the base term is not applied successively in the manner that would give something like “a lover of a lover of … a lover of a woman”.
For example, suppose that a universe of discourse numbers among its contents just three women, This could be expressed in Peirce's notation by writing:
Under these circumstances the following equation would hold:
This says that a lover of every woman in the given universe of discourse is a lover of that is a lover of that is a lover of In other words, a lover of every woman in this context is a lover of and a lover of and a lover of
The denotation of the term is a subset of that can be obtained as follows: For each flag of the form with collect the elements that appear as the first components of these ordered pairs, and then take the intersection of all these subsets. Putting it all together:
It is very instructive to examine the matrix representation of at this point, not the least because it effectively dispels the mystery of the name involution. First, let us make the following observation. To say that is a lover of every woman is to say that loves if is a woman. This can be rendered in symbols as follows:
Reading the formula as “ loves if is a woman” highlights the operation of converse implication inherent in it, and this in turn reveals the analogy between implication and involution that accounts for the aptness of the latter name.
The operations defined by the formulas and for are tabulated below:

It is clear that these operations are isomorphic, amounting to the same operation of type All that remains is to see how this operation on coefficient values in induces the corresponding operations on sets and terms.
The term determines a selection of individuals from the universe of discourse that may be computed by means of the corresponding operation on coefficient matrices. If the terms and are represented by the matrices and respectively, then the operation on terms that produces the term must be represented by a corresponding operation on matrices, say, that produces the matrix In other words, the involution operation on matrices must be defined in such a way that the following equations hold:
The fact that denotes the elements of a subset of means that the matrix is a 1dimensional array of coefficients in that is indexed by the elements of The value of the matrix at the index is written and computed as follows:
Commentary Note 12.3
We now have two ways of computing a logical involution that raises a dyadic relative term to the power of a monadic absolute term, for example, for “lover of every woman”.
The first method operates in the medium of set theory, expressing the denotation of the term as the intersection of a set of relational applications:
The second method operates in the matrix representation, expressing the value of the matrix with respect to an argument as a product of coefficient powers:
Abstract formulas like these are more easily grasped with the aid of a concrete example and a picture of the relations involved.
Example 6
Consider a universe of discourse that is subject to the following data:

Figure 55 shows the placement of within and the placement of within
(55) 
To highlight the role of more clearly, the Figure represents the absolute term by means of the relative term that conveys the same information.
Computing the denotation of by way of the settheoretic formula, we can show our work as follows:
With the above Figure in mind, we can visualize the computation of as follows:
1.  Pick a specific in the bottom row of the Figure. 
2.  Pan across the elements in the middle row of the Figure. 
3.  If links to then otherwise 
4.  If in the middle row links to in the top row then otherwise 
5.  Compute the value for each in the middle row. 
6.  If any of the values is then the product is otherwise it is 
As a general observation, we know that the value of goes to just as soon as we find a such that and in other words, such that but If there is no such then
Running through the program for each the only case that produces a nonzero result is That portion of the work can be sketched as follows:
Commentary Note 12.4
Peirce next considers a pair of compound involutions, stating an equation between them that is analogous to a law of exponents in ordinary arithmetic, namely,
Then will denote whatever stands to every woman in the relation of servant of every lover of hers; and will denote whatever is a servant of everything that is lover of a woman. So that 
(Peirce, CP 3.77). 
Articulating the compound relative term in settheoretic terms is fairly immediate:
On the other hand, translating the compound relative term into a settheoretic equivalent is less immediate, the hangup being that we have yet to define the case of logical involution that raises a dyadic relative term to the power of a dyadic relative term. As a result, it looks easier to proceed through the matrix representation, drawing once again on the inspection of a concrete example.
Example 7

(56) 
There is a “servant of every lover of” link between and if and only if But the vacuous inclusions, that is, the cases where have the effect of adding nonintuitive links to the mix.
The computational requirements are evidently met by the following formula:
In other words, if and only if there exists a such that and
Commentary Note 12.5
The equation can be verified by establishing the corresponding equation in matrices:
If and are two 1dimensional matrices over the same index set then if and only if for every Thus, a routine way to check the validity of is to check whether the following equation holds for arbitrary
Taking both ends toward the middle, we proceed as follows:

The products commute, so the equation holds. In essence, the matrix identity turns on the fact that the law of exponents in ordinary arithmetic holds when the values are restricted to the boolean domain Interpreted as a logical statement, the law of exponents amounts to a theorem of propositional calculus that is otherwise expressed in the following ways:

Commentary Note 12.6
Note. May need more explanation here.
Selection 13
The Sign of Involution (cont.)
A servant of every man and woman will be denoted by and will denote a servant of every man that is a servant of every woman. So that 
(Peirce, CP 3.77). 
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