Inquiry Into Inquiry

An unexpected benefit of cleaning out our basement and putting our belongings in storage was finding a trove of work I thought I’d lost, in a stash of 3½ inch floppies, no less.  I uploaded a sample to a couple of folders on Google Drive.

• Theme One Program
• Theme One Guide

The first contains a minimal set of files for running the program.
The second contains the example files mentioned in the User Guide.

Apologies in advance for Theme One being a bare prototype, a “test of concept” sort of program.  The user interface is pre-mouse and very finicky but I, its mother, was able to nurse it along far enough to learn a lot from it and many are lessons of still timely pertinence to our perennial issues.

cc: Ontolog Forum • Peirce List (1) (2) (3) (4)

As an alternative to piling generalities on generalities, not that there’s anything wrong with that, it also helps to look at issues as they arise in concrete applications.

One of the most concrete applications I ever attempted was the program I worked on all through the 1980s designed to integrate a basic form of inductive (data‑driven) learning with a fundamental form of deductive (concept‑driven) reasoning.  Having recently begun a fresh attempt to essay all that on my blog I think it might serve our ends to share it here.

• Survey of Theme One Program

cc: Ontolog Forum

Logical Cacti (cont.)

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 an equivalence transformation which applies in the direction of decreasing 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 have to explain all this to a computer.

Resources

• Theme One Program • Overview
• Theme One Program • Exposition
• Theme One Program • User Guide

cc: Peirce List • (1) • (2) • (3)

Logical Cacti (cont.)

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

• The compositional structures of cactus graphs and cactus expressions are constructed from two kinds of connective operations.
• There are two ways of mapping these compositional structures 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 to a node:

• The lobe connective joins a number of component cacti to a lobe:

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.

Resources

• Theme One Program • Overview
• Theme One Program • Exposition
• Theme One Program • User Guide

cc: Peirce List • (1) • (2) • (3)

The portions of exposition just skipped over covered the use of cactus graphs in the program’s learning module to learn sequences of characters called “words” or “strings” and sequences of words called “sentences” or “strands”.  Leaving the matter of grammar to another time we turn to the use of cactus graphs in the program’s reasoning module to represent logical propositions on the order of Peirce’s alpha graphs and Spencer Brown’s calculus of indications.

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.

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.

Resources

• Theme One Program • Overview
• Theme One Program • Exposition
• Theme One Program • User Guide

cc: Peirce List • (1) • (2)

Re: Laws Of Form • Armahedi Mahzar

AM:  Why do you need XOR in your inquiry system?

Clearly we need a way to represent exclusive disjunction, along with its dual, logical equivalence, in any calculus capable of covering propositional logic, so I assume this is a question about why I chose to represent those two operations more compactly with cactus graphs instead of using trees and defining them in terms of conjunctions and negations.

The generalization from trees to cacti presented itself at the point where multiple lines of problem-solving effort converged.  Some of the problems were conceptual, arising from a desire to include the types of operator-variables Peirce considered.  Other problems were computational, provoked by a need to avoid combinatorial explosions in the evaluation of logical formulas.

But, as I remarked earlier, “the genesis of that generalization is a tale worth telling another time”, after we’ve gotten a better handle on the basic logical issues.

Resources

• Theme One Program • Overview
• Theme One Program • Exposition
• Survey of Theme One Program

cc: Cybernetics • Ontolog Forum (1) (2) • Systems Science (1) (2)
cc: Peirce List (12-12) (18-02) (18-03) (20-09) (20-10) (21-10)
cc: FB | Theme One Program • Laws of Form (1) (2)