Inquiry Into Inquiry

Pragmatic Truth • 2

Posted on July 10, 2024 by Jon Awbrey

Truth as the Good of Logic

Pragmatic theories of truth enter on a stage set by the philosophies of former ages, with special reference to the Ancient Greeks, the Scholastics, and Immanuel Kant. Recalling a few elements of that background can provide valuable insight into the play of ideas as they have developed up through our time. Because pragmatic ideas about truth are often confused with a number of quite distinct notions it is useful say a few words about those other theories and to highlight the points of significant contrast.

In one classical formulation, truth is defined as the good of logic, where logic is classed as a normative science, in other words, an inquiry into a good or value which seeks to arrive at knowledge of it and the means to achieve it. In that view, truth cannot be discussed to much effect outside the context of inquiry, knowledge, and logic, all very broadly conceived.

Resources

cc: FB | Inquiry Driven Systems • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Posted in Aristotle, C.S. Peirce, Coherence, Concordance, Congruence, Consensus, Convergence, Correspondence, Dewey, Fixation of Belief, Information, Inquiry, John Dewey, Kant, Logic, Logic of Science, Method, Peirce, Philosophy, Pragmatic Maxim, Pragmatism, Semiotics, Sign Relations, Triadic Relations, Truth, Truth Theory, William James | Tagged Aristotle, C.S. Peirce, Coherence, Concordance, Congruence, Consensus, Convergence, Correspondence, Dewey, Fixation of Belief, Information, Inquiry, John Dewey, Kant, Logic, Logic of Science, Method, Peirce, Philosophy, Pragmatic Maxim, Pragmatism, Semiotics, Sign Relations, Triadic Relations, Truth, Truth Theory, William James | 6 Comments

Pragmatic Truth • 1

Posted on July 8, 2024 by Jon Awbrey

Questions about the pragmatic conception of truth have broken out in several quarters, asking in effect, “What conceptions of truth arise most naturally from and are best suited to pragmatic ways of thinking?” My best thoughts on that score were written out quite a few years ago, in an article I originally wrote for Wikipedia. I haven’t dared look at what’s become of it on that site — linked below is my current fork on another wiki.

Pragmatic Theory Of Truth

It begins as follows …

Pragmatic theory of truth refers to those accounts, definitions, and theories of the concept truth distinguishing the philosophies of pragmatism and pragmaticism. The conception of truth in question varies along lines reflecting the influence of several thinkers, initially and notably, Charles Sanders Peirce, William James, and John Dewey, but a number of common features can be identified.

The most characteristic features are (1) a reliance on the pragmatic maxim as a means of clarifying the meanings of difficult concepts, truth in particular, and (2) an emphasis on the fact that the product variously branded as belief, certainty, knowledge, or truth is the result of a process, namely, inquiry.

Et sic deinceps …

Resources

cc: FB | Inquiry Driven Systems • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Constraints and Indications • 2

Posted on July 4, 2024 by Jon Awbrey

Re: Constraints and Indications • 1
Re: Ontolog Forum • Joseph Simpson

Coping with collaboration, communication, context, integration, interoperability, perspective, purpose, and the reality of the information dimension demands a transition from conceptual environments bounded by dyadic relations to those informed by triadic relations, especially the variety of triadic sign relations employed by pragmatic semiotics.

Along the lines of my first post on this topic I am presently concerned with the logical and mathematical requirements of dealing with constraints but when it comes to the constraints involved in communicating across cultural and disciplinary barriers I could recommend a paper Susan Awbrey and I wrote for a conference devoted to those very issues.

Conference Presentation

Awbrey, S.M., and Awbrey, J.L. (1999), “Organizations of Learning or Learning Organizations : The Challenge of Creating Integrative Universities for the Next Century”, Second International Conference of the Journal ‘Organization’, Re‑Organizing Knowledge, Trans‑Forming Institutions : Knowing, Knowledge, and the University in the 21st Century, University of Massachusetts, Amherst, MA. Online.

Published Paper

Awbrey, S.M., and Awbrey, J.L. (2001), “Conceptual Barriers to Creating Integrative Universities”, Organization : The Interdisciplinary Journal of Organization, Theory, and Society 8(2), Sage Publications, London, UK, 269–284. Abstract. Online.

Resources

cc: FB | Inquiry Driven Systems • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Posted in Adaptive Systems, Artificial Intelligence, Ashby, C.S. Peirce, Constraint, Control, Cybernetics, Determination, Error-Controlled Regulation, Feedback, Indication, Indicator Functions, Information, Inquiry, Inquiry Driven Systems, Intelligent Systems, Intentionality, Learning Theory, Semiotic Information, Semiotics, Systems Theory, Uncertainty | Tagged Adaptive Systems, Artificial Intelligence, Ashby, C.S. Peirce, Constraint, Control, Cybernetics, Determination, Error-Controlled Regulation, Feedback, Indication, Indicator Functions, Information, Inquiry, Inquiry Driven Systems, Intelligent Systems, Intentionality, Learning Theory, Semiotic Information, Semiotics, Systems Theory, Uncertainty | Leave a comment

Constraints and Indications • 1

Posted on July 2, 2024 by Jon Awbrey

Re: Peirce List • Kaina Stoicheia and the Symbol Grounding Problem
Re: Jerry Chandler • Christophe Menant • Jon Awbrey • Christophe Menant

The system‑theoretic concept of constraint is one that unifies a manifold of other notions — definition, determination, habit, information, law, predicate, regularity, and so on. Indeed, it is often the best way to understand the entire complex of concepts.

Entwined with the concept of constraint is the concept of information, the power signs bear to reduce uncertainty and advance inquiry. Asking what consequences those ideas have for Peirce’s theory of triadic sign relations led me some years ago to the thoughts recorded on the following page.

Pragmatic Semiotic Information

Here I am thinking of the concept of constraint that constitutes one of the fundamental ideas of classical cybernetics and mathematical systems theory.

For example, here is how W. Ross Ashby introduces the concept of constraint in his Introduction to Cybernetics (1956).

A most important concept, with which we shall be much concerned later, is that of constraint. It is a relation between two sets, and occurs when the variety that exists under one condition is less than the variety that exists under another. Thus, the variety of the human sexes is 1 bit; if a certain school takes only boys, the variety in the sexes within the school is zero; so as 0 is less than 1, constraint exists. (1964 ed., p. 127).

At its simplest, then, constraint is an aspect of the subset relation.

The objective of an agent, organism, or similar regulator is to keep within its viable region, a particular subset of its possible state space. That is the constraint of primary interest to the agent.

Reference

Ashby, W.R. (1956), Introduction to Cybernetics, Methuen, London, UK.

Resources

cc: FB | Inquiry Driven Systems • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Theme One Program • Jets and Sharks 3

Posted on June 23, 2024 by Jon Awbrey

Re: Theme One Program • Jets and Sharks • (1) • (2)

Example 5. Jets and Sharks (cont.)

Given a representation of the Jets and Sharks universe in computer memory, we naturally want to see if the memory serves to supply the facts a well-constructed data base should.

In their PDP Handbook presentation of the Jets and Sharks example, McClelland and Rumelhart suggest several exercises for the reader to explore the performance of their neural pool memory model on the tasks of retrieval and generalization (Exercise 2.1).

Using cactus graphs or minimal negations to implement pools of mutually inhibitory neurons lends itself to neural architectures on a substantially different foundation from the garden variety connectionist models. At a high level of abstraction, however, there is enough homology between the two orders to compare their performance on many of the same tasks. With that in mind, I tried Theme One on a number of examples like the ones suggested by McClelland and Rumelhart.

What follows is a brief discussion of two examples as given in the original User Guide. Next time I’ll fill in more details about the examples and discuss their bearing on the larger issues at hand.

With a query on the name “ken” we obtain the following output, giving all the features associated with Ken.

$\text{Jets and Sharks} \stackrel{_\bullet}{} \text{Query 1}$

With a query on the two features “college” and “sharks” we obtain the following outline of all features satisfying those constraints.

$\text{Jets and Sharks} \stackrel{_\bullet}{} \text{Query 2}$

From this we discover all college Sharks are 30‑something and married. Further, we have a complete listing of their names broken down by occupation.

References

McClelland, J.L. (2015), Explorations in Parallel Distributed Processing : A Handbook of Models, Programs, and Exercises, 2nd ed. (draft), Stanford Parallel Distributed Processing Lab. Online, Section 2.3, Figure 2.1.
McClelland, J.L., and Rumelhart, D.E. (1988), Explorations in Parallel Distributed Processing : A Handbook of Models, Programs, and Exercises, MIT Press, Cambridge, MA. “Figure 1. Characteristics of a number of individuals belonging to two gangs, the Jets and the Sharks”, p. 39, from McClelland (1981).
McClelland, J.L. (1981), “Retrieving General and Specific Knowledge From Stored Knowledge of Specifics”, Proceedings of the Third Annual Conference of the Cognitive Science Society, Berkeley, CA.

Resources

cc: FB | Theme One Program • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Posted in Algorithms, Animata, Artificial Intelligence, Boolean Functions, C.S. Peirce, Cactus Graphs, Cognition, Computation, Constraint Satisfaction Problems, Data Structures, Differential Logic, Equational Inference, Formal Languages, Graph Theory, Inquiry Driven Systems, Laws of Form, Learning Theory, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Propositional Calculus, Semiotics, Spencer Brown, Visualization | Tagged Algorithms, Animata, Artificial Intelligence, Boolean Functions, C.S. Peirce, Cactus Graphs, Cognition, Computation, Constraint Satisfaction Problems, Data Structures, Differential Logic, Equational Inference, Formal Languages, Graph Theory, Inquiry Driven Systems, Laws of Form, Learning Theory, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Propositional Calculus, Semiotics, Spencer Brown, Visualization | 3 Comments

Theme One Program • Jets and Sharks 2

Posted on June 22, 2024 by Jon Awbrey

Example 5. Jets and Sharks (cont.)

As we saw last time, Theme One reads the text file shown below and constructs a cactus graph data structure in computer memory. The cactus graph represents a single logical formula in propositional calculus and that proposition embodies the logical constraints defining the Jets and Sharks data base.

$\text{Jets and Sharks} \stackrel{_\bullet}{} \text{Log File}$

Our cactus graph incorporates a vocabulary of 41 logical terms, each of which represents a boolean variable, so the proposition in question, call it $``q",$ is a boolean function of the form $q : \mathbb{B}^{41} \to \mathbb{B}.$ Given $2^{41} = 2,199,023,255,552$ we know a truth table for $q$ takes over two trillion rows and a venn diagram for $q$ takes the same number of cells. Topping it off, there are $2^{2^{41}}$ boolean functions of the form $f : \mathbb{B}^{41} \to \mathbb{B}$ and $q$ is just one of them.

Measures of strategy are clearly needed to negotiate patches of cacti like those.

References

McClelland, J.L. (2015), Explorations in Parallel Distributed Processing : A Handbook of Models, Programs, and Exercises, 2nd ed. (draft), Stanford Parallel Distributed Processing Lab. Online, Section 2.3, Figure 2.1.
McClelland, J.L., and Rumelhart, D.E. (1988), Explorations in Parallel Distributed Processing : A Handbook of Models, Programs, and Exercises, MIT Press, Cambridge, MA. “Figure 1. Characteristics of a number of individuals belonging to two gangs, the Jets and the Sharks”, p. 39, from McClelland (1981).
McClelland, J.L. (1981), “Retrieving General and Specific Knowledge From Stored Knowledge of Specifics”, Proceedings of the Third Annual Conference of the Cognitive Science Society, Berkeley, CA.

Resources

cc: FB | Theme One Program • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Theme One Program • Jets and Sharks 1

Posted on June 20, 2024 by Jon Awbrey

It is easy to spend a long time on the rudiments of learning and logic before getting down to practical applications — but I think we’ve circled square one long enough to expand our scope and see what the category of programs envisioned in Theme One can do with more substantial examples and exercises.

During the development of the Theme One program I tested successive implementations of its Reasoning Module or Logical Modeler on appropriate examples of logical problems current in the literature of the day. The PDP Handbook of McClelland and Rumelhart set one of the wittiest gems ever to whet one’s app‑titude so I could hardly help but take it on. The following text is a light revision of the way I set it up in the program’s User Guide.

Example 5. Jets and Sharks

The propositional calculus based on the minimal negation operator can be interpreted in a way resembling the logic of activation states and competition constraints in one class of neural network models. One way to do this is to interpret the blank or unmarked state as the resting state of a neural pool, the bound or marked state as its activated state, and to represent a mutually inhibitory pool of neurons $A, B, C$ by the proposition $\texttt{(} A \texttt{,} B \texttt{,} C \texttt{)}.$ The manner of representation may be illustrated by transcribing a well-known example from the parallel distributed processing literature (McClelland and Rumelhart 1988) and working through a couple of the associated exercises as translated into logical graphs.

Displayed below is the text expression of a traversal string which Theme One parses into a cactus graph data structure in computer memory. The cactus graph represents a single logical formula in propositional calculus and this proposition embodies all the logical constraints defining the Jets and Sharks data base.

$\text{Jets and Sharks} \stackrel{_\bullet}{} \text{Log File}$

References

McClelland, J.L. (2015), Explorations in Parallel Distributed Processing : A Handbook of Models, Programs, and Exercises, 2nd ed. (draft), Stanford Parallel Distributed Processing Lab. Online, Section 2.3, Figure 2.1.
McClelland, J.L., and Rumelhart, D.E. (1988), Explorations in Parallel Distributed Processing : A Handbook of Models, Programs, and Exercises, MIT Press, Cambridge, MA. “Figure 1. Characteristics of a number of individuals belonging to two gangs, the Jets and the Sharks”, p. 39, from McClelland (1981).
McClelland, J.L. (1981), “Retrieving General and Specific Knowledge From Stored Knowledge of Specifics”, Proceedings of the Third Annual Conference of the Cognitive Science Society, Berkeley, CA.

Resources

cc: FB | Theme One Program • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Theme One Program • Exposition 9

Posted on June 19, 2024 by Jon Awbrey

Transformation Rules and Equivalence Classes

The abstract character of the cactus language relative to its logical interpretations makes it possible to give abstract rules of equivalence for transforming cacti among themselves and partitioning the space of cacti into formal equivalence classes. The transformation rules and equivalence classes are “purely formal” in the sense of being indifferent to the logical interpretation, entitative or existential, one happens to choose.

Two definitions are useful here:

A reduction is a transformation which preserves equivalence classes and reduces the level of graphical complexity.
A basic reduction is a reduction which applies to a basic connective, either a node connective or a lobe connective.

The two kinds of basic reductions are described as follows.

A node reduction is permitted if and only if every component cactus joined to a node itself reduces to a node.

A lobe reduction is permitted if and only if exactly one component cactus listed in a lobe reduces to an edge.

That is roughly the gist of the rules. More formal definitions can wait for the day when we need to explain their use to a computer.

Resources

cc: FB | Theme One Program • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Theme One Program • Exposition 8

Posted on June 18, 2024 by Jon Awbrey

Mathematical Structure and Logical Interpretation

The main things to take away from the previous post are the following two ideas, one syntactic and one semantic.

Syntax. The compositional structures of cactus graphs and cactus expressions are constructed from two kinds of connective operations.
Semantics. There are two ways of mapping the compositional structures of syntax into the compositional structures of propositional sentences.

The two kinds of connective operations are described as follows.

The node connective joins a number of component cacti $C_1, \ldots, C_k$ to a node, as shown below.

The lobe connective joins a number of component cacti $C_1, \ldots, C_k$ to a lobe, as shown below.

The two ways of mapping cactus structures to logical meanings are summarized in Table 3, which compares the entitative and existential interpretations of the basic cactus structures, in effect, the graphical constants and connectives.

$\text{Table 3. Logical Interpretations of Cactus Structures}$

Resources

cc: FB | Theme One Program • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

Theme One Program • Exposition 7

Posted on June 17, 2024 by Jon Awbrey

Logical Cacti

Up till now we’ve been working to hammer out a two‑edged sword of syntax, honing the syntax of cactus graphs and cactus expressions and turning it to use in taming the syntax of two‑level formal languages.

But the purpose of a logical syntax is to support a logical semantics, which means, for starters, to bear interpretation as sentential signs capable of denoting objective propositions about a universe of objects.

One of the difficulties we face is that the words interpretation, meaning, semantics, and their ilk take on so many different meanings from one moment to the next of their use. A dedicated neologician might be able to think up distinctive names for all the aspects of meaning and all the approaches to them that concern us, but I will do the best I can with the common lot of ambiguous terms, leaving it to context and intelligent interpreters to sort it out as much as possible.

The formal language of cacti is formed at such a high level of abstraction that its graphs bear at least two distinct interpretations as logical propositions. The two interpretations concerning us here are descended from the ones C.S. Peirce called the entitative and the existential interpretations of his systems of graphical logics.

Existential Interpretation

Table 1 illustrates the existential interpretation of cactus graphs and cactus expressions by providing English translations for a few of the most basic and commonly occurring forms.

$\text{Table 1. Existential Interpretation}$

Entitative Interpretation

Table 2 illustrates the entitative interpretation of cactus graphs and cactus expressions by providing English translations for a few of the most basic and commonly occurring forms.

$\text{Table 2. Entitative Interpretation}$

Resources

cc: FB | Theme One Program • Laws of Form • Ma^thstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

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Truth as the Good of Logic

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Example 5. Jets and Sharks (cont.)

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Example 5. Jets and Sharks (cont.)

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Example 5. Jets and Sharks

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Transformation Rules and Equivalence Classes

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Mathematical Structure and Logical Interpretation

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Logical Cacti

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