Directory talk:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6

Deletions

6.38. Considering the Source

There is one remaining form of useful continuity that can be established between these newly formalized inventions and the ordinary conventions of common practice that are customary to apply in the informal context. Conforming to the ascriptions made above, I revive an old usage for framing interjections and enunciate the quotation \({}^{\backprime\backprime} \text{X} {}^{\prime\prime\text{I}}\!\) as \({}^{\backprime\backprime} \text{X} {}^{\prime\prime} ~ \text{quotha}.\!\) Readers who find this custom too curious for words might consider the twofold origins of inquiry and interpretation, one in the virtue of addressing uncertainty and another in the acknowledgment of surprise.

Fragments

6.19. Examples of Self-Reference

In previous work I developed a version of propositional calculus based on C.S. Peirce's existential graphs and implemented this calculus in computational form as a sentential calculus interpreter. Taking this calculus as a point of departure, I devised a theory of differential extensions for propositional domains that can be used, figuratively speaking, to put universes of discourse “in motion”, in other words, to provide qualitative descriptions of processes taking place in logical spaces. See (Awbrey, 1989 and 1994) for an account of this calculus, documentation of its computer program, and a detailed treatment of differential extensions.

In previous work (Awbrey, 1989) I described a system of notation for propositional calculus based on C.S. Peirce's existential graphs, documented a computer implementation of this formalism, and showed how to provide this calculus with a differential extension that can be used to describe changing universes of discourse. In subsequent work (Awbrey, 1994) the resulting system of differential logic was applied to give qualitative descriptions of change in discrete dynamical systems. This section draws on that earlier work, summarizing the conceptions that are needed to give logical representations of sign relations and recording a few changes of a minor nature in the typographical conventions used.

Abstractly, a domain of propositions is known by the axioms it satisfies. Concretely, one thinks of a proposition as applying to the objects it is true of.

Logically, a domain of properties or propositions is known by the axioms it is subject to. Concretely, a property or proposition is known by the things or situations it is true of. Typically, the signs of properties and propositions are called terms and sentences, respectively.

6.23. Intensional Representations of Sign Relations

In the formalized examples of IRs to be presented in this work, I will keep to the level of logical reasoning that is usually referred to as propositional calculus or sentential logic.

The contrast between ERs and IRs is strongly correlated with another dimension of interest in the study of inquiry, namely, the tension between empirical and rational modes of inquiry.

This section begins the explicit discussion of ERs by taking a second look at the sign relations \(L(\text{A})\!\) and \(L(\text{B}).\!\) Since the form of these examples no longer presents any novelty, this second presentation of \(L(\text{A})\!\) and \(L(\text{B})\!\) provides a first opportunity to introduce some new material. In the process of reviewing this material, it is useful to anticipate a number of incidental issues that are on the point of becoming critical, and to begin introducing the generic types of technical devices that are needed to deal with them.

Therefore, the easiest way to begin an explicit treatment of ERs is by recollecting the Tables of the sign relations \(L(\text{A})\!\) and \(L(\text{B})\!\) and by finishing the corresponding Tables of their dyadic components. Since the form of the sign relations \(L(\text{A})\!\) and \(L(\text{B})\!\) no longer presents any novelty, I can use the second presentation of these examples as a first opportunity to examine a selection of their finer points, previously overlooked.

Starting from this standpoint, the easiest way to begin developing an explicit treatment of ERs is to gather the relevant materials in the forms already presented, to fill out their missing details and expand the abbreviated contents of these forms, and to review their full structures in a more formal light.

Because of the perfect parallelism that the literal coding contrives between individual signs and grammatical categories, this arrangement illustrates not so much a code transformation as a re-interpretation of the original signs under different headings.

6.33. Sign Relational Complexes

I would like to record here, in what is topically the appropriate place, notice of a number of open questions that will have to be addressed if anyone desires to make a consistent calculus out of this link notation. Perhaps it is only because the franker forms of liaison involved in the couple \(a \widehat{~} b\!\) are more subject to the vagaries of syntactic elision than the corresponding bindings of the anglish ligature \((a, b),\!\) but for some reason or other the circumflex character of these diacritical notices are much more liable to suggest various forms of elaboration, including higher order generalizations and information-theoretic partializations of the very idea of \(n\!\)-tuples and sequences.

One way to deal with the problems of partial information …

Relational Complex?

\(L ~=~ L^{(1)} \cup \ldots \cup L^{(k)}\!\)

Sign Relational Complex?

\(L ~=~ L^{(1)} \cup L^{(2)} \cup L^{(3)}\!\)

Linkages can be chained together to form sequences of indications or \(n\!\)-tuples, without worrying too much about the order of collecting terms in the corresponding angle brackets.

\(\begin{matrix} a \widehat{~} b \widehat{~} c & = & (a, b, c) & = & (a, (b, c)) & = & ((a, b), c). \end{matrix}\)

These equivalences depend on the existence of natural isomorphisms between different ways of constructing \(n\!\)-place product spaces, that is, on the associativity of pairwise products, a not altogether trivial result (Mac Lane, CatWorkMath, ch. 7).

Higher Order Indications (HOIs)?

\(\begin{matrix} \widehat{~} x & = & (~, x) & ? \'"`UNIQ-MathJax1-QINU`"'. In contrast, the SER for interpreter \(\text{B}\!\) yields the semiotic equations:   \([{}^{\backprime\backprime} \text{A} {}^{\prime\prime}]_\text{B}\!\) \(=\!\) \([{}^{\backprime\backprime} \text{u} {}^{\prime\prime}]_\text{B}\!\)   \([{}^{\backprime\backprime} \text{B} {}^{\prime\prime}]_\text{B}\!\) \(=\!\) \([{}^{\backprime\backprime} \text{i} {}^{\prime\prime}]_\text{B}\!\) or  \({}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!\) \(=_\text{B}\!\)  \({}^{\backprime\backprime} \text{u} {}^{\prime\prime}\!\)    \({}^{\backprime\backprime} \text{B} {}^{\prime\prime}\!\) \(=_\text{B}\!\)  \({}^{\backprime\backprime} \text{i} {}^{\prime\prime}\!\) and the semiotic partition\[\{ \{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \} , \{ {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \} \}\!\]. 6.38. Considering the Source Attributed Sign Relation \(\begin{array}{ccl} O & = & \{ \text{A}, \text{B} \} \'"`UNIQ-MathJax3-QINU`"'. In contrast, the SER for interpreter \(\text{B}\!\) yields the semiotic equations:   \([{}^{\backprime\backprime} \text{A} {}^{\prime\prime}]_\text{B}\!\) \(=\!\) \([{}^{\backprime\backprime} \text{u} {}^{\prime\prime}]_\text{B}\!\)   \([{}^{\backprime\backprime} \text{B} {}^{\prime\prime}]_\text{B}\!\) \(=\!\) \([{}^{\backprime\backprime} \text{i} {}^{\prime\prime}]_\text{B}\!\) or  \({}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!\) \(=_\text{B}\!\)  \({}^{\backprime\backprime} \text{u} {}^{\prime\prime}\!\)    \({}^{\backprime\backprime} \text{B} {}^{\prime\prime}\!\) \(=_\text{B}\!\)  \({}^{\backprime\backprime} \text{i} {}^{\prime\prime}\!\) and the semiotic partition\[\{ \{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \} , \{ {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \} \}\!\]. Table 86. Confounded Sign Relation C Object Sign Interpretant A "A" "A" A "A" "i" A "A" "u" A "i" "A" A "i" "i" A "u" "A" A "u" "u" B "B" "B" B "B" "i" B "B" "u" B "i" "B" B "i" "i" B "u" "B" B "u" "u" Table 87. Disjointed Sign Relation D Object Sign Interpretant AA "A"A "A"A AA "A"A "i"A AA "i"A "A"A AA "i"A "i"A AB "A"B "A"B AB "A"B "u"B AB "u"B "A"B AB "u"B "u"B BA "B"A "B"A BA "B"A "u"A BA "u"A "B"A BA "u"A "u"A BB "B"B "B"B BB "B"B "i"B BB "i"B "B"B BB "i"B "i"B