Survey of Precursors Of Category Theory • 1

Posted on May 15, 2015 by Jon Awbrey

A few years ago I began a sketch on the “Precursors of Category Theory”, aiming to trace the continuities of the category concept from Aristotle, to Kant and Peirce, through Hilbert and Ackermann, to contemporary mathematical practice.  A Survey of blog and wiki notes on the subject is given below, still very rough and incomplete, but perhaps a few will find it of use.

Wiki Notes

Blog Posts

  • Notes On Categories • (1)
  • Precursors Of Category Theory • (1) • (2) • (3)

Categories à la Peirce

Posted in Abstraction, Ackermann, Analogy, Aristotle, C.S. Peirce, Carnap, Category Theory, Diagrams, Dyadic Relations, Equational Inference, Form, Foundations of Mathematics, Functional Logic, Hilbert, History of Mathematics, Hypostatic Abstraction, Kant, Logic, Logic of Relatives, Mathematics, Peirce, Propositions As Types Analogy, Relation Theory, Saunders Mac Lane, Semiotics, Sign Relations, Surveys, Triadic Relations, Type Theory, Universals | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | 20 Comments

Survey of Inquiry Driven Systems • 1

Posted on May 14, 2015 by Jon Awbrey

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

An inquiry driven system is a system having among its state variables some representing its state of information with respect to various topics of interest, for example, its own state and the states of any potential object systems.  Thus it has a component of state tracing a trajectory though an information state space.

Elements

Developments

Applications

Posted in Abduction, Action, Adaptive Systems, Aristotle, Artificial Intelligence, Automated Research Tools, Change, Cognitive Science, Communication, Cybernetics, Deduction, Descartes, Dewey, Discovery, Doubt, Education, Educational Systems Design, Educational Technology, Fixation of Belief, Induction, Information, Information Theory, Inquiry, Inquiry Driven Systems, Inquiry Into Inquiry, Intelligent Systems, Interpretation, Invention, Kant, Knowledge, Learning, Learning Theory, Logic, Logic of Science, Mathematics, Mental Models, Peirce, Pragmatic Maxim, Pragmatism, Process Thinking, Scientific Inquiry, Semiotics, Sign Relations, Surveys, Teaching, Triadic Relations, Visualization | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

Survey of Differential Logic • 1

Posted on May 11, 2015 by Jon Awbrey

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

Elements

Architectonics

Applications

Explorations

Posted in Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Category Theory, Change, Cybernetics, Differential Analytic Turing Automata, Differential Calculus, Differential Logic, Discrete Dynamics, Equational Inference, Frankl Conjecture, Functional Logic, Gradient Descent, Graph Theory, Hologrammautomaton, Indicator Functions, Inquiry Driven Systems, Leibniz, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Propositional Calculus, Surveys, Time, Topology, Visualization, Zeroth Order Logic | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | 1 Comment

Survey of Animated Logical Graphs • 1

Posted on May 10, 2015 by Jon Awbrey

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.

Elements

Excursions

Applications

Blog Dialogs

Posted in Abstraction, Amphecks, Animata, Boole, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Computational Complexity, Constraint Satisfaction Problems, Equational Inference, Graph Theory, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Propositional Calculus, Propositional Equation Reasoning Systems, Spencer Brown, Surveys, Theorem Proving, Visualization, Zeroth Order Logic | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , | 2 Comments

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

Posted on May 1, 2015 by Jon Awbrey

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.

Resources

Posted in Dyadic Relations, Graph Theory, Logic, Logic of Relatives, Mathematics, Matrix Theory, Peirce, Relation Theory, Semiotics, Sign Relations, Triadic Relations | Tagged , , , , , , , , , , | 10 Comments

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

Posted on April 24, 2015 by Jon Awbrey

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.

Dyadic Relation Bigraphs 2 Points

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.

Resources

Posted in Dyadic Relations, Graph Theory, Logic, Logic of Relatives, Mathematics, Matrix Theory, Peirce, Relation Theory, Semiotics, Sign Relations, Triadic Relations | Tagged , , , , , , , , , , | 9 Comments

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

Posted on April 23, 2015 by Jon Awbrey

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.

Dyadic Relations 2 Points

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.

Resources

Posted in Dyadic Relations, Graph Theory, Logic, Logic of Relatives, Mathematics, Matrix Theory, Peirce, Relation Theory, Semiotics, Sign Relations, Triadic Relations | Tagged , , , , , , , , , , | 10 Comments

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

Posted on April 19, 2015 by Jon Awbrey

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.

Resources

Posted in Dyadic Relations, Logic, Logic of Relatives, Mathematics, Matrix Theory, Peirce, Relation Theory, Semiotics, Sign Relations, Triadic Relations | Tagged , , , , , , , , , | 11 Comments

Peirce’s 1880 “Algebra Of Logic” Chapter 3 • Comment 7.1

Posted on April 13, 2015 by Jon Awbrey

I wanted to call attention to a very important statement from Selection 7 (CP 3.225–226).  Peirce enumerates the fundamental forms of individual dual relatives in the following terms:

225.   Individual relatives are of one or other of the two forms

\begin{array}{lll} \mathrm{A : A} & \qquad & \mathrm{A : B}, \end{array}

and simple relatives are negatives of one or other of these two forms.

And then he makes the following observation:

226.   The forms of general relatives are of infinite variety, but the following may be particularly noticed.

Relatives may be divided into those all whose individual aggregants are of the form \mathrm{A : A} and those which contain individuals of the form \mathrm{A : B}.  The former may be called concurrents, the latter opponents.

This tells us that Peirce understands the distinction between general dual relatives and individual dual relatives, the individuals being “aggregated” or logically summed to form the generals, and that singling out special cases of general relatives in the way he does next is but a first rough cut toward a complete classification.  This needs to be born in mind as we proceed toward the enumeration of triadic relations and beyond, especially as it affects the classification of triadic sign relations.

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.

Resources

Posted in Dyadic Relations, Logic, Logic of Relatives, Mathematics, Peirce, Relation Theory, Semiotics, Sign Relations, Triadic Relations | Tagged , , , , , , , , | 9 Comments

Objective Frameworks • Properties and Instances 1

Posted on April 9, 2015 by Jon Awbrey

Dealing with sign relations containing many types of signs — icons, indices, symbols, and more complex varieties — calls for a flexible and powerful organizational framework, one with the ability to grow and develop over time.  This is one of those applications where I found it useful to consider a “relative membership” relation, adding a parameter for the interpreter to the ordinary set-theoretic membership.

I laid out the details of a formalization in the following paragraphs:

It begins as follows:

In accounting for the special characters of icons and indices that arose in previous discussions, it was necessary to open the domain of objects coming under formal consideration to include unspecified numbers of properties and instances of whatever objects were initially set down.  This is a general phenomenon, affecting every motion toward explanation whether pursued by analytic or synthetic means.  What it calls for in practice is a way of organizing growing domains of objects, without having to specify in advance all the objects there are.

Posted in C.S. Peirce, Icon Index Symbol, Inquiry, Interpretive Frameworks, Logic, Logic of Relatives, Mathematics, Objective Frameworks, Peirce, Relation Theory, Relative Membership, Semiotics, Set Theory, Sign Relations | Tagged , , , , , , , , , , , , , | 2 Comments