Changes

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^\operatorname{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^\operatorname{th}\!</math> axis, that is, points of the form '''&lsaquo;'''&nbsp;<math>0, \ldots, 0, r_i, 0, \ldots, 0</math>&nbsp;'''&rsaquo;''' 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^\operatorname{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^\operatorname{th}\!</math> axis, that is, points of the form '''&lsaquo;'''&nbsp;<math>0, \ldots, 0, r_i, 0, \ldots, 0</math>&nbsp;'''&rsaquo;''' 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 ''X'' = '''R'''^''n'' can now be expressed by taking the set <font face="lucida calligraphy"><u>X</u></font>&nbsp;=&nbsp;{<u>''x''</u>_''i''} 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 ''i''^th threshold map.  This can

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.

help to remind us that the ''threshold operator'' <u>&nbsp;</u>)_''i'' acts on ''x'' by setting up a kind of a "hurdle" for it.  In this interpretation, the coordinate proposition <u>''x''</u>_''i'' asserts that the representative point ''x'' resides ''above'' the ''i''^th threshold.

Primitive assertions of the form <u>''x''</u>_''i''(''x'') can then be negated and joined by means of propositional connectives in the usual ways to provide information about the state ''x'' of a contemplated system or a statistical ensemble of systems.  Parentheses "(&nbsp;)" may be used to indicate negation.  Eventually one discovers the usefulness of the ''k''-ary ''just one false'' operators of the form "(&nbsp;,&nbsp;,&nbsp;,&nbsp;)", as treated in earlier reports.  This much tackle generates a space of points (cells, interpretations), <u>''X''</u>&nbsp;=&nbsp;〈<font face="lucida calligraphy"><u>X</u></font>〉&nbsp;<math>\cong</math>&nbsp;'''B'''^''n'', and

Primitive assertions of the form <u>''x''</u>_''i''(''x'') can then be negated and joined by means of propositional connectives in the usual ways to provide information about the state ''x'' of a contemplated system or a statistical ensemble of systems.  Parentheses "(&nbsp;)" may be used to indicate negation.  Eventually one discovers the usefulness of the ''k''-ary ''just one false'' operators of the form "(&nbsp;,&nbsp;,&nbsp;,&nbsp;)", as treated in earlier reports.  This much tackle generates a space of points (cells, interpretations), <u>''X''</u>&nbsp;=&nbsp;〈<font face="lucida calligraphy"><u>X</u></font>〉&nbsp;<math>\cong</math>&nbsp;'''B'''^''n'', and