Changes

\end{matrix}</math>

\end{matrix}</math>

Notice that, as defined here, there need be no actual relation between the <math>n\!</math>-dimensional subsets {''L''_''i''} and the coordinate axes corresponding to {''x''_''i''}, 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, ''L''_''i'' is bounded by some hyperplane that intersects the ''i''^th axis at a unique threshold value ''r''_''i'' &isin; '''R'''. Often, the hyperplane is chosen normal to the axis.  In recognition of this motive, let us make the following convention.  When the set ''L''_''i'' has points on the ''i''^th axis, that is, points of the form ‹0,&nbsp;&hellip;,&nbsp;0,&nbsp;''r''_''i'',&nbsp;0,&nbsp;&hellip;,&nbsp;0› where only the ''x''_''i'' 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 ''X'', 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 ''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