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Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 6 (view source)
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§ 30. Connected, Integrated, Reflective Symbols
§ 30. Connected, Integrated, Reflective Symbols
The next seven sections (§§ 31–37) are designed to motivate the idea that a language as simple as propositional calculus can be used to articulate significant properties of <math>n\!</math>-place relations. The course of the discussion will proceed as follows:
The next seven sections (§§ 31–37) are designed to incrementally motivate the idea that a language as simple as propositional calculus, remarkably enough, can be used to articulate significant properties of n place relations. The course of the discussion will proceed as follows:
§ 31. First, I introduce concepts and notation designed to expand and generalize the orders of relations that are available to be discussed in an adequate fashion.
§ 31. First, I introduce concepts and notation designed to expand and generalize the orders of relations that are available to be discussed in an adequate fashion.
§ 32. Second, I elaborate a particular mode of abstraction, that is, a systematic strategy for generalizing the collections of formal objects that are initially given to discussion. This dimension of abstraction or direction of generalization will be described under the thematic heading of "partiality".
§ 32. Second, I elaborate a particular mode of abstraction, that is, a systematic strategy for generalizing the collections of formal objects that are initially given to discussion. This dimension of abstraction or direction of generalization will be described under the thematic heading of ''partiality''.
§ 33. Third, I present an alternative approach to the issue of "degenerate", "defective", or "fragmentary" n place relations, proceeding by way of generalized objects known as "n place relational complexes". Illustrating these ideas with respect to their bearing on sign relations the discussion arrives at a notion of "sign relational complexes", or "sign complexes".
§ 33. Third, I present an alternative approach to the issue of ''defective'', ''degenerate'', or ''fragmentary'' <math>n\!</math>-place relations, proceeding by way of generalized objects known as ''<math>n\!</math>-place relational complexes''. Illustrating these ideas with respect to their bearing on sign relations the discussion arrives at a notion of ''sign-relational complexes'', or ''sign complexes''.
In the next three sections (§§ 34 36) I consider a collection of "identification tasks" for n place relations. Of particular interest is the extent to which the determination of an n place relation is constrained by a particular type of data, namely, by the specification of lower arity relations that occur as its projections. This topic is often treated as a question about a relation's "reducibility" or "irreduciblity" with respect to its projections. For instance, if the identity of an n place relation is completely determined by the data of its k place projections, then R is said to be "identifiable by", "reducible to", or "reconstructible from" its k place components, otherwise R is said to be "irreducible" with respect to its k place projections.
In the next three sections (§§ 34–36) I consider a collection of ''identification tasks'' for <math>n\!</math>-place relations. Of particular interest is the extent to which the determination of an <math>n\!</math>-place relation is constrained by a particular type of data, namely, by the specification of lower arity relations that occur as its projections. This topic is often treated as a question about a relation's ''reducibility'' or ''irreduciblity'' with respect to its projections. For instance, if the identity of an <math>n\!</math>-place relation <math>L\!</math> is completely determined by the data of its <math>k\!</math>-place projections, then <math>L\!</math> is said to be ''identifiable by'', ''reducible to'', or ''reconstructible from'' its <math>k\!</math>-place components, otherwise <math>L\!</math> is said to be ''irreducible'' with respect to its <math>k\!</math>-place projections.
§ 34. First, I consider a number of set theoretic operations that can be utilized in discussing these "identification", "reducibility", or "reconstruction" questions. Once a level of general discussion has been surveyed enough to make a start, these tools can be specialized and applied to concrete examples in the realm of sign relations and also applied in the neighborhood of closely associated triadic relations.
§ 34. First, I consider a number of set-theoretic operations that can be utilized in discussing these ''identification'', ''reducibility'', or ''reconstruction'' questions. Once a level of general discussion has been surveyed enough to make a start, these tools can be specialized and applied to concrete examples in the realm of sign relations and also applied in the neighborhood of closely associated triadic relations.
§ 35. This section considers the positive case of reducibility, presenting examples of triadic relations that can be reconstructed from their dyadic projections. In fact, it happens that the sign relations A and B fall into this category of dyadically reducible triadic relations.
§ 35. This section considers the positive case of reducibility, presenting examples of triadic relations that can be reconstructed from their dyadic projections. In fact, it happens that the sign relations <math>L(A)\!</math> and <math>L(B)\!</math> fall into this category of dyadically reducible triadic relations.
§ 36. This section considers the negative case of reduciblity, presenting examples of "irreducibly triadic relations", or triadic relations that cannot be reconstructed from their lower dimensional projections or "faces".
§ 36. This section considers the negative case of reducibility, presenting examples of ''irreducibly triadic relations'', or triadic relations that cannot be reconstructed from their lower dimensional projections or ''faces''.
§ 37. Finally, the discussion culminates in an exposition of the so called "propositions as types" (PAT) analogy, outlining a formal system of "type expressions" or "type formulas" that bears a strong resemblance to propositional calculus. Properly interpreted, the resulting "calculus of propositional types" (COPT) can be used as a language for talking about well formed types of n place relations.
§ 37. Finally, the discussion culminates in an exposition of the so called ''propositions as types'' (PAT) analogy, outlining a formal system of ''type expressions'' or ''type formulas'' that bears a strong resemblance to propositional calculus. Properly interpreted, the resulting ''calculus of propositional types'' (COPT) can be used as a language for talking about well-formed types of <math>k\!</math>-place relations.
§ 38. Considering the Source
§ 38. Considering the Source
§ 39. Prospective Indices : Pointers to Future Work
§ 39. Prospective Indices : Pointers to Future Work
§ 40. Interlaced with the structural and reflective developments that go into the OF and the IF is a conceptual arrangement called the "dynamic evaluative framework" (DEF). This utility works to isolate the aspects of process and purpose that are observable on either side of the objective interpretive divide and helps to organize the graded notions of directed change that can be actualized in the RIF.
§ 40. Interlaced with the structural and reflective developments that go into the OF and the IF is a conceptual arrangement called the ''dynamic evaluative framework'' (DEF). This utility works to isolate the aspects of process and purpose that are observable on either side of the objective interpretive divide and helps to organize the graded notions of directed change that can be actualized in the RIF.
§ 41. Elective and Motive Forces
§ 41. Elective and Motive Forces
§ 44. Reflections on Closure
§ 44. Reflections on Closure
§ 45. Intelligence => Critical Reflection
§ 45. Intelligence <math>\Rightarrow</math> Critical Reflection
§ 46. Looking Ahead : The Meta Issue
§ 46. Looking Ahead : The “Meta” Issue
§ 47. Mutually Intelligible Codes
§ 47. Mutually Intelligible Codes
§ 50. Revisiting the Source
§ 50. Revisiting the Source
===6.1. The Phenomenology of Reflection===
===6.1. The Phenomenology of Reflection===