Survey of Definition and Determination • 4

Posted on May 3, 2025 by Jon Awbrey

In the early 1990s, “in the middle of life’s journey” as the saying goes, I returned to grad school in a systems engineering program with the idea of taking a more systems-theoretic approach to my development of Peircean themes, from signs and scientific inquiry to logic and information theory.

Two of the first questions calling for fresh examination were the closely related concepts of definition and determination, not only as Peirce used them in his logic and semiotics but as researchers in areas as diverse as computer science, cybernetics, physics, and systems science would find themselves forced to reconsider the concepts in later years.  That led me to collect a sample of texts where Peirce and a few other writers discuss the issues of definition and determination.  There are copies of those selections at the following sites.

What follows is a Survey of blog and wiki posts on Definition and Determination, with a focus on the part they play in Peirce’s interlinked theories of signs, information, and inquiry.  In classical logical traditions the concepts of definition and determination are closely related and their bond acquires all the more force when we view the overarching concept of constraint from an information-theoretic point of view, as Peirce did beginning in the 1860s.

Blog Dialogs

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Posted in C.S. Peirce, Comprehension, Constraint, Definition, Determination, Extension, Form, Indication, Information, Information = Comprehension × Extension, Inquiry Driven Systems, Logic, Mathematics, Scientific Method, Semiotics, Sign Relations, Structure, Systems Theory, Visualization | Tagged , , , , , , , , , , , , , , , , , , | Leave a comment

Survey of Cybernetics • 5

Posted on May 2, 2025 by Jon Awbrey

Again, in a ship, if a man were at liberty to do what he chose, but were devoid of mind and excellence in navigation (αρετης κυβερνητικης), do you perceive what must happen to him and his fellow sailors?

— Plato • Alcibiades • 135 A

This is a Survey of blog posts relating to Cybernetics.  It includes the selections from Ashby’s Introduction and the comment on them I’ve posted so far, plus two series of reflections on the governance of social systems in light of cybernetic and semiotic principles.

Anthem

Ashby’s Introduction to Cybernetics

  • Chapter 11 • Requisite Variety

Blog Series

  • Theory and Therapy of Representations • (1) • (2) • (3) • (4) • (5)

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Posted in Abduction, C.S. Peirce, Communication, Control, Cybernetics, Deduction, Determination, Discovery, Doubt, Epistemology, Fixation of Belief, Induction, Information, Information = Comprehension × Extension, Information Theory, Inquiry, Inquiry Driven Systems, Inquiry Into Inquiry, Interpretation, Invention, Knowledge, Learning Theory, Logic, Logic of Relatives, Logic of Science, Mathematics, Peirce, Philosophy, Philosophy of Science, Pragmatic Information, Probable Reasoning, Process Thinking, Relation Theory, Scientific Inquiry, Scientific Method, Semeiosis, Semiosis, Semiotic Information, Semiotics, Sign Relational Manifolds, Sign Relations, Surveys, Triadic Relations, Uncertainty | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | 1 Comment

Survey of Animated Logical Graphs • 8

Posted on May 2, 2025 by Jon Awbrey

This is a Survey of blog and wiki posts on Logical Graphs, encompassing several families of graph‑theoretic structures originally developed by Charles S. Peirce as graphical formal languages or visual styles of syntax amenable to interpretation for logical applications.

Beginnings

Elements

Examples

Blog Series

  • Logical Graphs • Interpretive Duality • (1) • (2) • (3) • (4)
  • Logical Graphs, Iconicity, Interpretation • (1) • (2)
  • Genus, Species, Pie Charts, Radio Buttons • (1)

Excursions

Applications

Anamnesis

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Posted in Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Computational Complexity, Constraint Satisfaction Problems, Differential Logic, Equational Inference, Graph Theory, Group Theory, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Model Theory, Painted Cacti, Peirce, Proof Theory, Propositional Calculus, Propositional Equation Reasoning Systems, Spencer Brown, Theorem Proving, Visualization | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , | 114 Comments

Survey of Abduction, Deduction, Induction, Analogy, Inquiry • 5

Posted on May 1, 2025 by Jon Awbrey

This is a Survey of blog and wiki posts on three elementary forms of inference, as recognized by a logical tradition extending from Aristotle through Charles S. Peirce.  Particular attention is paid to the way the inferential rudiments combine to form the more complex patterns of analogy and inquiry.

Anthem

Blog Dialogs

Blog Series

Blog Surveys

OEIS Wiki

Ontolog Forum

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Posted in Abduction, Aristotle, C.S. Peirce, Deduction, Dewey, Discovery, Doubt, Fixation of Belief, Functional Logic, Icon Index Symbol, Induction, Inference, Information, Inquiry, Invention, Logic, Logic of Science, Mathematics, Morphism, Paradigmata, Paradigms, Pattern Recognition, Peirce, Philosophy, Pragmatic Maxim, Pragmatism, Scientific Inquiry, Scientific Method, Semiotics, Sign Relations, Surveys, Syllogism, Triadic Relations, Visualization | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

Cactus Language • Preliminaries 9

Posted on April 25, 2025 by Jon Awbrey

We now have the materials in place to formulate a definition of our subject.

The painted cactus language with paints in the set \mathfrak{P} = \{ p_j : j \in J \} is the formal language \mathfrak{L} = \mathfrak{C} (\mathfrak{P}) \subseteq \mathfrak{A}^* = (\mathfrak{M} \cup \mathfrak{P})^* defined as follows.

\begin{array}{ll} \text{PC 1.} & \text{The blank symbol}~ m_1 ~\text{is a sentence.} \\ \text{PC 2.} & \text{The paint}~ p_j ~\text{is a sentence for each}~ j ~\text{in}~ J. \\ \text{PC 3.} & \mathrm{Conc}^0 ~\text{and}~ \mathrm{Surc}^0 ~\text{are sentences.} \\ \text{PC 4.} & \text{For each positive integer}~ n, \\ & \text{if}~ s_1, \ldots, s_n ~\text{are sentences} \\ & \text{then}~ \mathrm{Conc}_{k=1}^n s_k ~\text{is a sentence} \\ & \text{and}~ \mathrm{Surc}_{k=1}^n s_k ~\text{is a sentence.} \end{array}

In the idiom of formal language theory, a string s is called a sentence of \mathfrak{L} if and only if it belongs to \mathfrak{L}, or simply a sentence if the language \mathfrak{L} is understood.  A sentence of \mathfrak{C} (\mathfrak{P}) is referred to as a painted and rooted cactus expression on the palette \mathfrak{P}, or a cactus expression for short.

Resources

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Cactus Language • Preliminaries 8

Posted on April 22, 2025 by Jon Awbrey

Defining the basic operations of concatenation and surcatenation on arbitrary strings gives them operational meaning for the all‑inclusive language \mathfrak{L} = \mathfrak{A}^*.  With that in hand it is time to adjoin the notion of a more discriminating grammaticality, in other words, a more properly restrictive concept of a sentence.

If \mathfrak{L} is an arbitrary formal language over an alphabet of the type we have been discussing, that is, an alphabet of the form \mathfrak{A} = \mathfrak{M} \cup \mathfrak{P}, then there are a number of basic structural relations which can be defined on the strings of \mathfrak{L}.

Concatenation

s is the concatenation of s_1 and s_2 in \mathfrak{L}
if and only if
s_1 is a sentence of \mathfrak{L}, s_2 is a sentence of \mathfrak{L},
and
s = s_1 \cdot s_2

s is the concatenation of the k strings s_1, \ldots, s_k in \mathfrak{L}
if and only if
s_j is a sentence of \mathfrak{L} for all j = 1 \ldots k
and
s = \mathrm{Conc}_{j=1}^k s_j = s_1 \cdot \ldots \cdot s_k

Discatenation

s is the discatenation of s_1 by t
if and only if
s_1 is a sentence of \mathfrak{L}, t is an element of \mathfrak{A},
and
s_1 = s \cdot t
in which case we more commonly write
s = s_1 \cdot t^{-1}

Subclause

s is a subclause of \mathfrak{L}
if and only if
s is a sentence of \mathfrak{L}
and
s ends with a ``\text{)}"

Subcatenation

s is the subcatenation of s_1 by s_2
if and only if
s_1 is a subclause of \mathfrak{L}, s_2 is a sentence of \mathfrak{L},
and
s = s_1 \cdot (``\text{)}")^{-1} \cdot ``\text{,}" \cdot s_2 \cdot ``\text{)}"

Surcatenation

s is the surcatenation of the k strings s_1, \ldots, s_k in \mathfrak{L}
if and only if
s_j is a sentence of \mathfrak{L} for all {j = 1 \ldots k}
and
s = \mathrm{Surc}_{j=1}^k s_j = ``\text{(}" \cdot s_1 \cdot ``\text{,}" \cdot \ldots \cdot ``\text{,}" \cdot s_k \cdot ``\text{)}"

The converses of the above decomposition relations amount to the corresponding composition operations.  As complementary forms of analysis and synthesis they make it possible to articulate the structures of strings and sentences in two directions.

Resources

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Posted in Automata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Differential Logic, Equational Inference, Formal Grammars, Formal Languages, Graph Theory, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Propositional Calculus, Visualization | Tagged , , , , , , , , , , , , , , , , | 3 Comments

Cactus Language • Preliminaries 7

Posted on April 19, 2025 by Jon Awbrey

The array of syntactic operators may be put in more organized form by making a few additional conventions and auxiliary definitions.

Concatenation

The conception of concatenation permits extension to its natural prequel, the corresponding operator on zero operands.

\mathrm{Conc}^0 = ``" = \text{the empty string.}

From that beginning the operation of concatenation may be broken into stages by means of the following conceptions.

The precatenation \mathrm{Prec}(s_1, s_2) of two strings s_1, s_2 is defined as follows.

\mathrm{Prec} (s_1, s_2) = s_1 \cdot s_2.

The concatenation of n strings s_1, \ldots, s_n may now be given a new definition as the iterated precatenation of n+1 strings beginning with s_0 = \mathrm{Conc}^0 = ``" and continuing through the remaining n strings.

\text{For}~ n = 0, ~\mathrm{Conc}_{k=0}^n s_k = \mathrm{Conc}^0 = ``".

\text{For}~ n > 0, ~\mathrm{Conc}_{k=1}^n s_k = \mathrm{Prec}(\mathrm{Conc}_{k=0}^{n-1} s_k, s_n).

Surcatenation

The conception of surcatenation permits extension to its natural prequel, the corresponding operator on zero operands.

\mathrm{Surc}^0 = ``()".

From that beginning the operation of surcatenation may be broken into stages by means of the following conceptions.

A subclause in \mathfrak{A}^* is a string ending with ``)".

The subcatenation \mathrm{Subc} (s_1, s_2) of a subclause s_1 by a string s_2 is defined as follows.

\mathrm{Subc} (s_1, s_2) = s_1 \cdot (``)")^{-1} \cdot ``," \cdot s_2 \cdot ``)".

The surcatenation of n strings s_1, \ldots, s_n may now be given a new definition as the iterated subcatenation of n+1 strings beginning with s_0 = \mathrm{Surc}^0 = ``()" and continuing through the remaining n strings.

\text{For}~ n = 0, ~\mathrm{Surc}_{k=0}^n s_k = \mathrm{Surc}^0 = ``()".

\text{For}~ n > 0, ~\mathrm{Surc}_{k=1}^n s_k = \mathrm{Subc}(\mathrm{Surc}_{k=0}^{n-1} s_k, s_n).

Notice that the expressions \mathrm{Conc}_{k=0}^0 s_k and \mathrm{Surc}_{k=0}^0 s_k are defined in such a way that the respective operators \mathrm{Conc}^0 and \mathrm{Surc}^0 simply ignore, in the manner of constants, whatever sequences of strings s_k may be listed as their ostensible arguments.

Resources

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Posted in Automata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Differential Logic, Equational Inference, Formal Grammars, Formal Languages, Graph Theory, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Propositional Calculus, Visualization | Tagged , , , , , , , , , , , , , , , , | 3 Comments

Cactus Language • Preliminaries 6

Posted on April 16, 2025 by Jon Awbrey

The definitions of the syntactic connectives can be made a little more succinct by defining the following pair of generic operators on strings.

Concatenation

The concatenation \mathrm{Conc}_{k=1}^n of the sequence of n strings (s_k)_{k=1}^n is defined recursively as follows.

\text{For}~ n = 1, ~\mathrm{Conc}_{k=1}^n s_k = s_1.

\text{For}~ n > 1, ~\mathrm{Conc}_{k=1}^n s_k = \mathrm{Conc}_{k=1}^{n-1} s_k \cdot s_n.

Surcatenation

The surcatenation \mathrm{Surc}_{k=1}^n of the sequence of n strings (s_k)_{k=1}^n is defined recursively as follows.

\text{For}~ n = 1, ~\mathrm{Surc}_{k=1}^n s_k = ``\text{(}" \cdot s_1 \cdot ``\text{)}".

\text{For}~ n > 1, ~\mathrm{Surc}_{k=1}^n s_k = \mathrm{Surc}_{k=1}^{n-1} s_k \cdot (``\text{)}")^{-1} \cdot ``\text{,}" \cdot s_n \cdot ``\text{)}".

Resources

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Posted in Automata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Differential Logic, Equational Inference, Formal Grammars, Formal Languages, Graph Theory, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Propositional Calculus, Visualization | Tagged , , , , , , , , , , , , , , , , | 4 Comments

Cactus Language • Preliminaries 5

Posted on April 13, 2025 by Jon Awbrey

The easiest way to define the language \mathfrak{C}(\mathfrak{P}) is to indicate the general run of operations required to construct the greater share of its sentences from the designated few which require a special election.

To do that we introduce a family of operations called syntactic connectives on the strings of \mathfrak{A}^*.  If the strings on which they operate are already sentences of \mathfrak{C}(\mathfrak{P}) then the operations amount to sentential connectives.  If the syntactic sentences, viewed as abstract strings of uninterpreted signs, are provided with a semantics where they denote propositions, in other words, indicator functions on a universe of discourse, then the operations amount to propositional connectives.

Rather than presenting the most concise description of cactus languages right from the beginning, it aids comprehension to develop a picture of their forms in gradual stages, starting with the most natural ways of viewing their elements, if somewhat at a distance, and working through the most easily grasped impressions of their structures, if not always the sharpest acquaintances with their details.

We begin by defining two sets of basic operations on strings of \mathfrak{A}^*.

Concatenation

The concatenation of one string s_1 is the string s_1.

The concatenation of two strings s_1, s_2 is the string {s_1 \cdot s_2}.

The concatenation of k strings (s_j)_{j = 1}^k is the string {s_1 \cdot \ldots \cdot s_k}.

Surcatenation

The surcatenation of one string s_1 is the string ``\text{(}" \cdot s_1 \cdot ``\text{)}".

The surcatenation of two strings s_1, s_2 is the string ``\text{(}" \cdot s_1 \cdot ``\text{,}" \cdot s_2 \cdot ``\text{)}".

The surcatenation of k strings (s_j)_{j = 1}^k is the string ``\text{(}" \cdot s_1 \cdot ``\text{,}" \cdot \ldots \cdot ``\text{,}" \cdot s_k \cdot ``\text{)}".

Resources

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Posted in Automata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Differential Logic, Equational Inference, Formal Grammars, Formal Languages, Graph Theory, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Propositional Calculus, Visualization | Tagged , , , , , , , , , , , , , , , , | 3 Comments

Cactus Language • Preliminaries 4

Posted on April 11, 2025 by Jon Awbrey

The informal mechanisms illustrated in the preceding discussion equip us with a description of cactus language adequate to providing conceptual and computational representations for the minimal formal logical system variously known as propositional logic or sentential calculus.

The painted cactus language \mathfrak{C} is actually a parameterized family of languages, consisting of one language \mathfrak{C}(\mathfrak{P}) for each set \mathfrak{P} of paints.

The alphabet \mathfrak{A} = \mathfrak{M} \cup \mathfrak{P} is the disjoint union of the following two sets of symbols.

\mathfrak{M} is the alphabet of markers, the set of punctuation marks, or the collection of syntactic constants common to all the languages \mathfrak{C}(\mathfrak{P}).  Various ways of representing the elements of \mathfrak{M} are shown in the following display.

Cactus Language Display 2

\mathfrak{P} is the palette, the alphabet of paints, or the collection of syntactic variables peculiar to the language \mathfrak{C}(\mathfrak{P}).  The set of signs in \mathfrak{P} may be enumerated as follows.

\mathfrak{P} = \{ \mathfrak{p}_j : j \in J \}.

Resources

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Posted in Automata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Differential Logic, Equational Inference, Formal Grammars, Formal Languages, Graph Theory, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Propositional Calculus, Visualization | Tagged , , , , , , , , , , , , , , , , | 5 Comments