System E - A New Language that Reveals New Distinctions in Laws of Form's Notational Space

摘要

In the book Laws of Form, the cross,2 that is , does not indicate the precise region of notational space it occupies. In resolving this ambiguity, a language of concepts and terms called System E reveals new distinctions. We initially take the notation in the laws of form (LoF) systemwithout regard to its values (i.e., not taken as indicators), nor to its transformations—we call this arrangement space. To properly parse and see arrangement space, we make continence properties visually explicit, and we introduce the concept of line boundaries whose thickness is void,and formally adjoin them to the space. These void-line-boundaries are the immediate- boundary of all units, including the entire space as well as both the inside and outside of the right- angle mark , and they thereby partition any given space into its units: units whose shape is either nestor(enclosing a distinct unit) or nonnestor. The elementary nestor units are called covers. Covers have a shade rule, and each cover has an n-arity. The LoF expression-formation operations of forming the image and (what I call) juxtaposition and are then expressed as outward adjoinings of elementaryunits. We also define open, closed, and clopen forms. The unwritten cross of Laws of Form (p. 7) may be translated as: a void-line-boundary covering a clopen form. Moreover, a void-line-boundary immediately inside a right-angle mark can function in notation (!) as the boundary of the firstdistinction in prenotation. For these and other reasons, void-line-boundaries, and not the LoF cross, become the fundamental boundary in notation. Keywords: G. Spencer-Brown, laws of form, pre-notation, notation, space, extension, System E, ndimensionality, extendedness, unmarked, void, spacefor,shade, boundary, void-line-boundary, scope, parse, partition, natural units, shape, nestors, covers, n-arity, elementary adjoinment units, formation, image, opus, unwritten cross, clopen. Typographical Notes: I use a pair of single-quote marks ' ' to show a LoF arrangement spacein typographical space: e.g., I show a void LoF arrangement space in typographical space by ''. I sometimes use a hyphen to bind consecutive words into a unit-meaning: For example, for nouns such as void-line-boundary. Their intent is to allow the reader to properly parse a sentenceon its initial reading (first pass), thereby avoiding a double take of re-reading to ponder the possible meaningful parsings. Notes on Citations: Reference citations in parentheses, like (p. 1), (section 'Instruction') or (chapter 5), refer to Spencer-Brown's 1969 Laws of Form.Brackets are used to indicate direction to a section in this paper, for example: [see VIII.9].