Changes

States of knowledge about the location of a system or about the distribution of a population of systems in a state space <math>X = \mathbb{R}^n</math> can now be expressed by taking the set <math>\underline\mathcal{X} = \{\underline{x}_i\}</math> as a basis of logical features.  In picturesque terms, one may think of the underscore and the subscript as combining to form a subtextual spelling for the <math>i^\operatorname{th}\!</math> threshold map.  This can help to remind us that the ''threshold operator'' <math>(\underline{~})_i</math> acts on <math>\mathbf{x}</math> by setting up a kind of a "hurdle" for it.  In this interpretation the coordinate proposition <math>\underline{x}_i</math> asserts that the representative point <math>\mathbf{x}</math> resides ''above'' the <math>i^\operatorname{th}\!</math> threshold.

States of knowledge about the location of a system or about the distribution of a population of systems in a state space <math>X = \mathbb{R}^n</math> can now be expressed by taking the set <math>\underline\mathcal{X} = \{\underline{x}_i\}</math> as a basis of logical features.  In picturesque terms, one may think of the underscore and the subscript as combining to form a subtextual spelling for the <math>i^\operatorname{th}\!</math> threshold map.  This can help to remind us that the ''threshold operator'' <math>(\underline{~})_i</math> acts on <math>\mathbf{x}</math> by setting up a kind of a "hurdle" for it.  In this interpretation the coordinate proposition <math>\underline{x}_i</math> asserts that the representative point <math>\mathbf{x}</math> resides ''above'' the <math>i^\operatorname{th}\!</math> threshold.

Primitive assertions of the form <math>\underline{x}_i (\mathbf{x})</math> may then be negated and joined by means of propositional connectives in the usual ways to provide information about the state <math>\mathbf{x}</math> of a contemplated system or a statistical ensemble of systems.  Parentheses <math>(\ldots)</math> may be used to indicate logical negation.  Eventually one discovers the usefulness of the <math>k\!</math>-ary ''just one false'' operators of the form <math>(a_1, \ldots, a_k)</math>, as treated in earlier reports.  This much tackle generates a space of points (cells, interpretations), <math>\underline{X} = \langle \underline\mathcal{X} \rangle \cong \mathbb{B}^n,</math> and a space of functions (regions, propositions), <math>\underline{X}^\uparrow \cong (\mathbb{B}^n \to \mathbb{B}).</math>  Together these form a new universe of discourse <u>''X''</u>^&nbsp;&bull; = [<font face="lucida calligraphy"><u>X</u></font>] of the type ('''B'''^''n'',&nbsp;('''B'''^''n''&nbsp;&rarr;&nbsp;'''B''')), which we may abbreviate as '''B'''^''n''&nbsp;+&rarr;&nbsp;'''B''', or most succinctly as ['''B'''^''n''].

Primitive assertions of the form <math>\underline{x}_i (\mathbf{x})</math> may then be negated and joined by means of propositional connectives in the usual ways to provide information about the state <math>\mathbf{x}</math> of a contemplated system or a statistical ensemble of systems.  Parentheses <math>(\ldots)</math> may be used to indicate logical negation.  Eventually one discovers the usefulness of the <math>k\!</math>-ary ''just one false'' operators of the form <math>(a_1, \ldots, a_k)</math>, as treated in earlier reports.  This much tackle generates a space of points (cells, interpretations), <math>\underline{X} \cong \mathbb{B}^n,</math> and a space of functions (regions, propositions), <math>\underline{X}^\uparrow \cong (\mathbb{B}^n \to \mathbb{B}).</math>  Together these form a new universe of discourse <math>\underline{X}^\circ</math> of the type <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})),</math> which we may abbreviate as <math>\mathbb{B}^n\ +\!\to \mathbb{B}</math> or most succinctly as <math>[\mathbb{B}^n].</math>

The square brackets have been chosen to recall the rectangular frame of a venn diagram.  In thinking about a universe of discourse it is a good idea to keep this picture in mind, where we constantly think of the elementary cells <u>''x''</u>, the defining features <u>''x''</u>_''i'', and the potential shadings ''f''&nbsp;:&nbsp;<u>''X''</u>&nbsp;&rarr;&nbsp;'''B''', all at the same time, remaining aware of the arbitrariness of the way that we choose to inscribe our distinctions in the medium of a continuous space.

The square brackets have been chosen to recall the rectangular frame of a venn diagram.  In thinking about a universe of discourse it is a good idea to keep this picture in mind, where we constantly think of the elementary cells <u>''x''</u>, the defining features <u>''x''</u>_''i'', and the potential shadings ''f''&nbsp;:&nbsp;<u>''X''</u>&nbsp;&rarr;&nbsp;'''B''', all at the same time, remaining aware of the arbitrariness of the way that we choose to inscribe our distinctions in the medium of a continuous space.