## Survey of Inquiry Driven Systems • 1

This is a Survey of previous blog and wiki posts on Inquiry Driven Systems, material that I plan to refine toward a more compact and systematic treatment of the subject.

## Survey of Differential Logic • 1

This is a Survey of previous blog and wiki posts on Differential Logic, material that I plan to develop toward a more compact and systematic account.

## Survey of Animated Logical Graphs • 1

This is one of several Survey posts I’ll be drafting from time to time, starting with minimal stubs and collecting links to the better variations on persistent themes I’ve worked on over the years.  After that I’ll look to organizing and revising the assembled material with an eye toward developing more polished articles.

## Peirce’s 1880 “Algebra Of Logic” Chapter 3 • Comment 7.5

Suppose we add another individual to our initial universe of discourse, arriving at a three-point universe $\{ \mathrm{I}, \mathrm{J}, \mathrm{K} \}.$

It might be thought that adding one more element to the universe of discourse would allow slightly more complicated relations to be compounded from its basic ingredients, but the truth is that crossing the threshold from a two-point universe to a three-point universe occasions a steep ascent in the complexity of relations generated.

Looking back from the ascent we see that the two-point universe $\{ \mathrm{I}, \mathrm{J} \}$ manifests a type of formal degeneracy (loss of generality) compared with the three-point universe $\{ \mathrm{I}, \mathrm{J}, \mathrm{K} \}.$  This is due to the circumstance that the number of “diagonal” pairs, those of the form $\mathrm{A\!:\!A},$ equals the number of “off-diagonal” pairs, those of the form $\mathrm{A\!:\!B},$ so the two-point case exhibits symmetries that will be broken as soon as one adds another element to the universe.

A universe of three individuals $\mathrm{I, J, K}$ yields exactly nine individual dual relatives or ordered pairs of universe elements:

$\begin{matrix} \mathrm{I\!:\!I}, & \mathrm{I\!:\!J}, & \mathrm{I\!:\!K}, & \mathrm{J\!:\!I}, & \mathrm{J\!:\!J}, & \mathrm{J\!:\!K}, & \mathrm{K\!:\!I}, & \mathrm{K\!:\!J}, & \mathrm{K\!:\!K}. \end{matrix}$

It is convenient arrange the pairs in a square array:

$\left( \begin{matrix} \mathrm{I\!:\!I} & \mathrm{I\!:\!J} & \mathrm{I\!:\!K} \\[4pt] \mathrm{J\!:\!I} & \mathrm{J\!:\!J} & \mathrm{J\!:\!K} \\[4pt] \mathrm{K\!:\!I} & \mathrm{K\!:\!J} & \mathrm{K\!:\!K} \end{matrix} \right)$

There are $2^9 = 512$ dual relatives over this universe of discourse, since each one is formed by choosing a subset of the nine ordered pairs and then “aggregating” them, forming their logical sum, or simply regarding them as a subset.  Taking the square array of ordered pairs as a backdrop, any one of the $512$ dual relatives may be represented by a square matrix of binary values, a value of $1$ occupying the place of each ordered pair that belongs to the subset and a value of $0$ occupying the place of each ordered pair that does not belong to the subset in question.

The matrix representations of the $512$ dual relatives or dyadic relations over the universe $\{ \mathrm{I}, \mathrm{J}, \mathrm{K} \}$ are tabulated below according to the following plan:

• Since the diagonal and off-diagonal components of the matrices vary independently of each other, the whole set of matrices factors as a product of two smaller sets, making the break along the lines of the corresponding cardinalities, $2^9 = 2^3 \times 2^6.$
• The eight diagonal matrices are shown in the first row of the display, omitting the off-diagonal zeroes for ease of reading and pattern recognition.
• The $64$ off-diagonal matrices are shown in the rest of the display, arranged in rank order by increasing numbers of $\text{1's}$ and decreasing numbers of $\text{0's},$ suppressing the fixed zeroes along the diagonals to make the changing patterns of $\text{0's}$ and $\text{1's}$ easier to follow.
• The number of off-diagonal matrices of rank $k$ is equal to the binomial coefficient $\tbinom{6}{k}$ or $\mathrm{C}(6, k).$  The values of $\mathrm{C}(6, k)$  are given by the row of Pascal’s Triangle that contains the sequence ${1, 6, 15, 20, 15, 6, 1}.$

$\begin{pmatrix} 0 & ~ & ~ \\ ~ & 0 & ~ \\ ~ & ~ & 0 \end{pmatrix} \, \begin{pmatrix} 1 & ~ & ~ \\ ~ & 0 & ~ \\ ~ & ~ & 0 \end{pmatrix} \, \begin{pmatrix} 0 & ~ & ~ \\ ~ & 1 & ~ \\ ~ & ~ & 0 \end{pmatrix} \, \begin{pmatrix} 0 & ~ & ~ \\ ~ & 0 & ~ \\ ~ & ~ & 1 \end{pmatrix} \, \begin{pmatrix} 1 & ~ & ~ \\ ~ & 1 & ~ \\ ~ & ~ & 0 \end{pmatrix} \, \begin{pmatrix} 1 & ~ & ~ \\ ~ & 0 & ~ \\ ~ & ~ & 1 \end{pmatrix} \, \begin{pmatrix} 0 & ~ & ~ \\ ~ & 1 & ~ \\ ~ & ~ & 1 \end{pmatrix} \, \begin{pmatrix} 1 & ~ & ~ \\ ~ & 1 & ~ \\ ~ & ~ & 1 \end{pmatrix}$

$\times$

$\begin{pmatrix} ~ & 0 & 0 \\ 0 & ~ & 0 \\ 0 & 0 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 1 & 0 \\ 0 & ~ & 0 \\ 0 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 1 \\ 0 & ~ & 0 \\ 0 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 0 \\ 1 & ~ & 0 \\ 0 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 0 \\ 0 & ~ & 1 \\ 0 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 0 \\ 0 & ~ & 0 \\ 1 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 0 \\ 0 & ~ & 0 \\ 0 & 1 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 1 & 1 \\ 0 & ~ & 0 \\ 0 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 0 \\ 1 & ~ & 0 \\ 0 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 0 \\ 0 & ~ & 1 \\ 0 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 0 \\ 0 & ~ & 0 \\ 1 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 0 \\ 0 & ~ & 0 \\ 0 & 1 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 0 & 1 \\ 1 & ~ & 0 \\ 0 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 1 \\ 0 & ~ & 1 \\ 0 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 1 \\ 0 & ~ & 0 \\ 1 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 1 \\ 0 & ~ & 0 \\ 0 & 1 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 0 & 0 \\ 1 & ~ & 1 \\ 0 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 0 \\ 1 & ~ & 0 \\ 1 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 0 \\ 1 & ~ & 0 \\ 0 & 1 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 0 & 0 \\ 0 & ~ & 1 \\ 1 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 0 \\ 0 & ~ & 1 \\ 0 & 1 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 0 & 0 \\ 0 & ~ & 0 \\ 1 & 1 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 1 & 1 \\ 1 & ~ & 0 \\ 0 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 1 \\ 0 & ~ & 1 \\ 0 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 1 \\ 0 & ~ & 0 \\ 1 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 1 \\ 0 & ~ & 0 \\ 0 & 1 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 1 & 0 \\ 1 & ~ & 1 \\ 0 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 0 \\ 1 & ~ & 0 \\ 1 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 0 \\ 1 & ~ & 0 \\ 0 & 1 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 1 & 0 \\ 0 & ~ & 1 \\ 1 & 0 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 0 \\ 0 & ~ & 1 \\ 0 & 1 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 1 & 0 \\ 0 & ~ & 0 \\ 1 & 1 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 0 & 1 \\ 1 & ~ & 1 \\ 0 & 0 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 0 & 1 \\ 1 & ~ & 0 \\ 0 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 1 \\ 1 & ~ & 0 \\ 1 & 0 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 0 & 1 \\ 0 & ~ & 0 \\ 1 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 1 \\ 0 & ~ & 1 \\ 0 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 1 \\ 0 & ~ & 1 \\ 1 & 0 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 0 & 0 \\ 0 & ~ & 1 \\ 1 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 0 \\ 1 & ~ & 0 \\ 1 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 0 \\ 1 & ~ & 1 \\ 0 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 0 \\ 1 & ~ & 1 \\ 1 & 0 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 1 & 1 \\ 1 & ~ & 1 \\ 0 & 0 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 1 & 1 \\ 1 & ~ & 0 \\ 0 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 1 \\ 1 & ~ & 0 \\ 1 & 0 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 1 & 1 \\ 0 & ~ & 0 \\ 1 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 1 \\ 0 & ~ & 1 \\ 0 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 1 \\ 0 & ~ & 1 \\ 1 & 0 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 1 & 0 \\ 0 & ~ & 1 \\ 1 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 0 \\ 1 & ~ & 0 \\ 1 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 0 \\ 1 & ~ & 1 \\ 0 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 0 \\ 1 & ~ & 1 \\ 1 & 0 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 0 & 0 \\ 1 & ~ & 1 \\ 1 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 1 \\ 0 & ~ & 1 \\ 1 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 1 \\ 1 & ~ & 0 \\ 1 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 1 \\ 1 & ~ & 1 \\ 0 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 0 & 1 \\ 1 & ~ & 1 \\ 1 & 0 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 0 & 1 \\ 1 & ~ & 1 \\ 1 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 0 \\ 1 & ~ & 1 \\ 1 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 1 \\ 0 & ~ & 1 \\ 1 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 1 \\ 1 & ~ & 0 \\ 1 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 1 \\ 1 & ~ & 1 \\ 0 & 1 & ~ \end{pmatrix} \, \begin{pmatrix} ~ & 1 & 1 \\ 1 & ~ & 1 \\ 1 & 0 & ~ \end{pmatrix}$

$\begin{pmatrix} ~ & 1 & 1 \\ 1 & ~ & 1 \\ 1 & 1 & ~ \end{pmatrix}$

### References

• Peirce, C.S. (1880), “On the Algebra of Logic”, American Journal of Mathematics 3, 15–57.  Collected Papers (CP 3.154–251), Chronological Edition (CE 4, 163–209).
• Peirce, C.S., Collected Papers of Charles Sanders Peirce, vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958.  Volume 3 : Exact Logic, 1933.
• Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1981–.  Volume 4 (1879–1884), 1986.

## Peirce’s 1880 “Algebra Of Logic” Chapter 3 • Comment 7.4

Dyadic relations enjoy yet another form of graph-theoretic representation as labeled bipartite graphs or labeled bigraphs.  I’ll just call them bigraphs here, letting the labels be understood in this logical context.

The figure below shows the bigraphs of the 16 dyadic relations on two points, adopting the same arrangement as the previous displays of binary matrices and loopy digraphs.

### References

• Peirce, C.S. (1880), “On the Algebra of Logic”, American Journal of Mathematics 3, 15–57.  Collected Papers (CP 3.154–251), Chronological Edition (CE 4, 163–209).
• Peirce, C.S., Collected Papers of Charles Sanders Peirce, vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958.  Volume 3 : Exact Logic, 1933.
• Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1981–.  Volume 4 (1879–1884), 1986.

## Peirce’s 1880 “Algebra Of Logic” Chapter 3 • Comment 7.3

Dyadic relations have graph-theoretic representations as labeled directed graphs with loops, also known as labeled pseudo-digraphs in some schools of graph theory.  I’ll just call them digraphs here, letting the labels and loops be understood in this logical context.

The figure below shows the digraphs of the 16 dyadic relations on two points, adopting the same arrangement as the previous display of binary matrices.

### References

• Peirce, C.S. (1880), “On the Algebra of Logic”, American Journal of Mathematics 3, 15–57.  Collected Papers (CP 3.154–251), Chronological Edition (CE 4, 163–209).
• Peirce, C.S., Collected Papers of Charles Sanders Peirce, vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958.  Volume 3 : Exact Logic, 1933.
• Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1981–.  Volume 4 (1879–1884), 1986.

## Peirce’s 1880 “Algebra Of Logic” Chapter 3 • Comment 7.2

Because it can sometimes be difficult to reconnect abstractions with their concrete instances, especially after the abstract types have become autonomous and taken on a life of their own, let us resort to a simple concrete case and examine the implications of what Peirce is saying about the relation between general relatives and individual relatives.

Suppose our initial universe of discourse has exactly two individuals, $\mathrm{I}$ and $\mathrm{J}.$  Then there are exactly four individual dual relatives or ordered pairs of universe elements:

$\begin{matrix} \mathrm{I\!:\!I}, & \mathrm{I\!:\!J}, & \mathrm{J\!:\!I}, & \mathrm{J\!:\!J}. \end{matrix}$

It is convenient arrange these in a square array:

$\left( \begin{array}{rr} \mathrm{I\!:\!I} & \mathrm{I\!:\!J} \\[4pt] \mathrm{J\!:\!I} & \mathrm{J\!:\!J} \end{array} \right)$

There are $2^4 = 16$ dual relatives in general over this universe of discourse, since each one is formed by choosing a subset of the four ordered pairs and then “aggregating” them, forming their logical sum, or simply regarding them as a subset.  Taking the square array of ordered pairs as a backdrop, any one of the $16$ dual relatives may be represented by a square matrix of binary values, a value of $1$ occupying the place of each ordered pair that belongs to the subset and a value of $0$ occupying the place of each ordered pair that does not belong to the subset in question.  The matrix representations of the $16$ dual relatives or dyadic relations over the universe $\{ \mathrm{I}, \mathrm{J} \}$ are displayed below:

$\begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \quad \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \quad \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \quad \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$

$\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} \quad \begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix} \quad \begin{pmatrix} 0 & 1 \\ 0 & 1 \end{pmatrix} \quad \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$

$\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} \quad \begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix} \quad \begin{pmatrix} 0 & 0 \\ 1 & 1 \end{pmatrix} \quad \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$

$\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \quad \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix} \quad \begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix} \quad \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$

Relative to the universe $\{ \mathrm{I}, \mathrm{J} \}$, the individual dual relatives of the form $\mathrm{A\!:\!A}$ are $\mathrm{I\!:\!I}$ and $\mathrm{J\!:\!J}$ while the individual dual relatives of the form $\mathrm{A\!:\!B}$ are $\mathrm{I\!:\!J}$ and $\mathrm{J\!:\!I}.$

Peirce assigns the name concurrents to dual relatives all whose individual aggregants are of the form $\mathrm{A\!:\!A}.$  There are exactly $4$ of these and their matrices are shown in the top row of the above display.  All the rest are called opponents and their matrices are listed in the bottom three rows.

Peirce gives the name alio-relatives to dual relatives all whose individual aggregants are of the form $\mathrm{A\!:\!B}.$  There are exactly $4$ of these and their matrices are shown in the first column of the above display.  All the rest are called self-relatives and their matrices are listed in the right hand three columns.

Notice that the relative ${0},$ represented by the matrix with all ${0}$ entries, falls under the definitions of both a concurrent and an alio-relative.

### References

• Peirce, C.S. (1880), “On the Algebra of Logic”, American Journal of Mathematics 3, 15–57.  Collected Papers (CP 3.154–251), Chronological Edition (CE 4, 163–209).
• Peirce, C.S., Collected Papers of Charles Sanders Peirce, vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958.  Volume 3 : Exact Logic, 1933.
• Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1981–.  Volume 4 (1879–1884), 1986.