Cf: Theme One Program • Exposition 2
https://inquiryintoinquiry.com/2022/06/16/theme-one-program-exposition-2-2/
Re: Theme One Program • Exposition 1
https://inquiryintoinquiry.com/2022/06/15/theme-one-program-exposition-1-2/
All,
The previous post described the elementary data structure
used to represent nodes of graphs in the Theme One program.
This post describes the specific family of graphs employed
by the program.
Figure 1 shows a typical example of a “painted and rooted cactus”.
Figure 1. Painted And Rooted Cactus
https://inquiryintoinquiry.files.wordpress.com/2022/06/theme-exposition-pai…
The graph itself is a mathematical object and does not inhabit the
page or other medium before our eyes, and it must not be confused
with any picture or other representation of it, anymore than we’d
want someone to confuse us with a picture of ourselves, but it’s
a fair enough picture, once we understand the conventions of
representation involved.
Let V(G) be the set of nodes in a graph G and let L be a set of identifiers.
We often find ourselves in situations where we have to consider many different
ways of associating the nodes of G with the identifiers in L. Various manners
of associating nodes with identifiers have been given conventional names by
different schools of graph theorists. I will give one way of describing
a few of the most common patterns of association.
• A graph is “painted” if there is a relation between its node set
and a set of identifiers, in which case the relation is called
a “painting” and the identifiers are called “paints”.
• A graph is “colored” if there is a function from its node set
to a set of identifiers, in which case the function is called
a “coloring” and the identifiers are called “colors”.
• A graph is “labeled” if there is a one-to-one mapping between
its node set and a set of identifiers, in which case the mapping
is called a “labeling” and the identifiers are called “labels”.
• A graph is said to be “rooted” if it has a unique distinguished node,
in which case the distinguished node is called the “root” of the graph.
The graph in Figure 1 has a root node marked by the “at” sign or amphora
symbol “@”.
The graph in Figure 1 has eight nodes plus the five paints in the
set {a, b, c, d, e}. The painting of nodes is indicated by drawing
the paints of each node next to the node they paint. Observe that
some nodes may be painted with an empty set of paints.
The structure of a painted and rooted cactus may be encoded in the form of
a character string called a “painted and rooted cactus expression”. For the
remainder of this discussion the terms “cactus” and “cactus expression” will
be used to mean the painted and rooted varieties. A cactus expression is
formed on an alphabet consisting of the relevant set of identifiers, the
“paints”, together with three punctuation marks: the left parenthesis,
the comma, and the right parenthesis.
Regards,
Jon

Cf: Theme One Program • Exposition 1
https://inquiryintoinquiry.com/2018/06/08/theme-one-program-exposition-1/
All,
Theme One is a program for building and transforming a particular species
of graph-theoretic data structures, forms designed to support a variety of
fundamental learning and reasoning tasks.
The program evolved over the course of an exploration into the integration of
contrasting types of activities involved in learning and reasoning, especially
the types of algorithms and data structures capable of supporting a range of
inquiry processes, from everyday problem solving to scientific investigation.
In its current state, Theme One integrates over a common data structure
fundamental algorithms for one type of inductive learning and one type
of deductive reasoning.
We begin by describing the class of graph-theoretic data structures
used by the program, as determined by their local and global aspects.
It will be the usual practice to shift around and to view these graphs
at many different levels of detail, from their abstract definition to
their concrete implementation, and many points in between.
The main work of the Theme One program is achieved by building and
transforming a single species of graph-theoretic data structures.
In their abstract form these structures are closely related to the
graphs that are called “cacti” and “conifers” in graph theory,
so we'll generally refer to them under those names.
The graph-theoretic data structures used by the program are built up from
a basic data structure called an “idea-form flag”. That structure is defined
as a pair of Pascal data types by means of the following specifications.
Box 1. Type Idea = ^Form
https://inquiryintoinquiry.files.wordpress.com/2022/06/theme-exposition-typ…
An “idea” is a pointer to a “form”.
A “form” is a record consisting of:
A “sign” of type char;
Four pointers, as, up, on, by, of type idea;
A “code” of type numb, that is, an integer in [0, max integer].
Represented in terms of “digraphs”, or directed graphs, the combination
of an “idea” pointer and a “form” record is most easily pictured as an arc,
or directed edge, leading to a node that is labeled with the other data,
in this case, a letter and a number.
At the roughest but quickest level of detail, an idea of a form can be drawn like this.
Box 2. Idea^Form Node
https://inquiryintoinquiry.files.wordpress.com/2022/06/theme-exposition-ide…
When it is necessary to fill in more detail, the following schematic pattern can be used.
Box 3. Idea^Form Flag
https://inquiryintoinquiry.files.wordpress.com/2022/06/theme-exposition-ide…
The idea-form type definition determines the local structure of
the whole host of graphs used by the program, including a motley
array of ephemeral buffers, temporary scratch lists, and other
graph-theoretic data structures used for their transient utilities
at specific points in the program.
I will put off discussing these more incidental graph structures
until the points where they actually arise, focusing here on the
particular varieties and the specific variants of cactoid graphs
that constitute the main formal media of the program's operation.
Regards,
Jon

Cf: Functional Logic • Inquiry and Analogy • 9
https://inquiryintoinquiry.com/2022/05/02/functional-logic-inquiry-and-anal…
Inquiry and Analogy • Dewey’s “Sign of Rain” • An Example of Inquiry
https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#Dewey_…
Inquiry and Inference
https://oeis.org/wiki/Functional_Logic_%E2%80%A2_Inquiry_and_Analogy#Dewey_…
If we follow Dewey’s “Sign of Rain” example far enough to consider
the import of thought for action, we realize the subsequent conduct
of the interpreter, progressing up through the natural conclusion of
the episode — the quickening steps, seeking shelter in time to escape
the rain — all those acts form a series of further interpretants,
contingent on the active causes of the individual, for the originally
recognized signs of rain and the first impressions of the actual case.
Just as critical reflection develops the associated and alternative
signs which gather about an idea, pragmatic interpretation explores
the consequential and contrasting actions which give effective and
testable meaning to a person’s belief in it.
Figure 10 charts the progress of inquiry in Dewey’s Sign of Rain example
according to the stages of reasoning identified by Peirce, focusing on
the compound or mixed form of inference formed by the first two steps.
Figure 10. Cycle of Inquiry
https://inquiryintoinquiry.files.wordpress.com/2022/04/cycle-of-inquiry-gra…
Step 1 is Abductive,
abstracting a Case from the consideration of a Fact and a Rule.
• Fact : C ⇒ A, In the Current situation the Air is cool.
• Rule : B ⇒ A, Just Before it rains, the Air is cool.
• Case : C ⇒ B, The Current situation is just Before it rains.
Step 2 is Deductive,
admitting the Case to another Rule and arriving at a novel Fact.
• Case : C ⇒ B, The Current situation is just Before it rains.
• Rule : B ⇒ D, Just Before it rains, a Dark cloud will appear.
• Fact : C ⇒ D, In the Current situation, a Dark cloud will appear.
What precedes is nowhere near a complete analysis of Dewey’s example,
even so far as it might be carried out within the constraints of the
syllogistic framework, and it covers only the first two steps of the
inquiry process, but perhaps it will do for a start.
Regards,
Jon