Changes

Notice that, as defined here, there need be no actual relation between the ''n''-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'' ∈ '''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, …, 0, ''r''_''i'', 0, …, 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 ''n''-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'' ∈ '''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, …, 0, ''r''_''i'', 0, …, 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''.

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 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.

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