Changes

|-

|-

| <math>\begin{matrix}\mathcal{X} & = & \{ x_1, \ldots, x_n \}\end{matrix}</math>

| <math>\begin{matrix}\mathcal{X} & = & \{ x_1, \ldots, x_n \}\end{matrix}</math>

| <math>\begin{matrix}\underline\mathcal{X} & = & \{ \underline{x}_1, \ldots, \underline{x}_n \}\end{matrix}</math>

| <math>\begin{matrix}\underline{\mathcal{X}} & = & \{ \underline{x}_1, \ldots, \underline{x}_n \}\end{matrix}</math>

| <math>\begin{matrix}\mathcal{A} & = & \{ a_1, \ldots, a_n \}\end{matrix}</math>

| <math>\begin{matrix}\mathcal{A} & = & \{ a_1, \ldots, a_n \}\end{matrix}</math>

|-

|-

\underline{X}

\underline{X}

\\

\\

= & \langle \underline\mathcal{X} \rangle

= & \langle \underline{\mathcal{X}} \rangle

\\

\\

= & \langle \underline{x}_1, \ldots, \underline{x}_n \rangle

= & \langle \underline{x}_1, \ldots, \underline{x}_n \rangle

\underline{X}^\bullet

\underline{X}^\bullet

\\

\\

= & [\underline\mathcal{X}]

= & [\underline{\mathcal{X}}]

\\

\\

= & [\underline{x}_1, \ldots, \underline{x}_n]

= & [\underline{x}_1, \ldots, \underline{x}_n]

Notice that, as defined here, there need be no actual relation between the <math>n\!</math>-dimensional subsets <math>\{L_i\}\!</math> and the coordinate axes corresponding to <math>\{x_i\},\!</math> aside from the circumstance that the two sets have the same cardinality.  In concrete cases, though, one usually has some reason for associating these "volumes" with these "lines", for instance, <math>L_i\!</math> is bounded by some hyperplane that intersects the <math>i^\text{th}\!</math> axis at a unique threshold value <math>r_i \in \mathbb{R}.\!</math>  Often, the hyperplane is chosen normal to the axis.  In recognition of this motive, let us make the following convention.  When the set <math>L_i\!</math> has points on the <math>i^\text{th}\!</math> axis, that is, points of the form <math>(0, \ldots, 0, r_i, 0, \ldots, 0)</math> where only the <math>x_i\!</math> coordinate is possibly non-zero, we may pick any one of these coordinate values as a parametric index of the relation.  In this case we say that the indexing is ''real'', otherwise the indexing is ''imaginary''.  For a knowledge based system <math>X,\!</math> this should serve once again to mark the distinction between ''acquaintance'' and ''opinion''.

Notice that, as defined here, there need be no actual relation between the <math>n\!</math>-dimensional subsets <math>\{L_i\}\!</math> and the coordinate axes corresponding to <math>\{x_i\},\!</math> aside from the circumstance that the two sets have the same cardinality.  In concrete cases, though, one usually has some reason for associating these "volumes" with these "lines", for instance, <math>L_i\!</math> is bounded by some hyperplane that intersects the <math>i^\text{th}\!</math> axis at a unique threshold value <math>r_i \in \mathbb{R}.\!</math>  Often, the hyperplane is chosen normal to the axis.  In recognition of this motive, let us make the following convention.  When the set <math>L_i\!</math> has points on the <math>i^\text{th}\!</math> axis, that is, points of the form <math>(0, \ldots, 0, r_i, 0, \ldots, 0)</math> where only the <math>x_i\!</math> coordinate is possibly non-zero, we may pick any one of these coordinate values as a parametric index of the relation.  In this case we say that the indexing is ''real'', otherwise the indexing is ''imaginary''.  For a knowledge based system <math>X,\!</math> this should serve once again to mark the distinction between ''acquaintance'' and ''opinion''.

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^\text{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 &ldquo;hurdle&rdquo; 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^\mathrm{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^\text{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 &ldquo;hurdle&rdquo; 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^\mathrm{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>\texttt{(} \ldots \texttt{)}\!</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>\texttt{(} a_1 \texttt{,} \ldots \texttt{,} a_k \texttt{)},\!</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}^\bullet</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>

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>\texttt{(} \ldots \texttt{)}\!</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>\texttt{(} a_1 \texttt{,} \ldots \texttt{,} a_k \texttt{)},\!</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}^\bullet</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, graphically illustrating the links among the elementary cells <math>\underline\mathbf{x},</math> the defining features <math>\underline{x}_i,</math> and the potential shadings <math>f : \underline{X} \to \mathbb{B}</math> all at the same time, not to mention the arbitrariness of the way 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, graphically illustrating the links among the elementary cells <math>\underline{\mathbf{x}},</math> the defining features <math>\underline{x}_i,</math> and the potential shadings <math>f : \underline{X} \to \mathbb{B}</math> all at the same time, not to mention the arbitrariness of the way we choose to inscribe our distinctions in the medium of a continuous space.

Finally, let <math>X^*\!</math> denote the space of linear functions, <math>(\ell : X \to \mathbb{K}),</math> which has in the finite case the same dimensionality as <math>X,\!</math> and let the same notation be extended across the Table.

Finally, let <math>X^*\!</math> denote the space of linear functions, <math>(\ell : X \to \mathbb{K}),</math> which has in the finite case the same dimensionality as <math>X,\!</math> and let the same notation be extended across the Table.