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Finally, two things are important to keep in mind with regard to the simplicity, linearity, positivity, and singularity of propositions.

Finally, two things are important to keep in mind with regard to the simplicity, linearity, positivity, and singularity of propositions.

First, all of these properties are relative to a particular basis.  For example, a singular proposition with respect to a basis <font face="lucida calligraphy">A</font> will not remain singular if <font face="lucida calligraphy">A</font> is extended by a number of new and independent features.  Even if we stick to the original set of pairwise options {''a''_''i''} &cup; {(''a''_''i'')} to select a new basis, the sets of linear and positive propositions are determined by the choice of simple propositions, and this determination is tantamount to the conventional choice of a cell as origin.

First, all of these properties are relative to a particular basis.  For example, a singular proposition with respect to a basis <math>\mathcal{A}</math> will not remain singular if <math>\mathcal{A}</math> is extended by a number of new and independent features.  Even if we stick to the original set of pairwise options <math>\{a_i\} \cup \{(a_i)\}</math> to select a new basis, the sets of linear and positive propositions are determined by the choice of simple propositions, and this determination is tantamount to the conventional choice of a cell as origin.

Second, the singular propositions '''B'''^''n''&nbsp;<font face=symbol>'''××>'''</font>&nbsp;'''B''', picking out as they do a single cell or a coordinate tuple of '''B'''^''n'', become the carriers or the vehicles of a certain type-ambiguity that vacillates between the dual forms '''B'''^''n'' and ('''B'''^''n''&nbsp;<font face=symbol>'''××>'''</font>&nbsp;'''B''') and infects the whole hierarchy of types built on them.  In plainer language, the terms that signify the interpretations ''x''&nbsp;:&nbsp;'''B'''^''n'' and the singular propositions ''x''&nbsp;:&nbsp;'''B'''^''n''&nbsp;<font face=symbol>'''××>'''</font>&nbsp;'''B''' are fully equivalent in information, and this means that every

Second, the singular propositions '''B'''^''n''&nbsp;<font face=symbol>'''××>'''</font>&nbsp;'''B''', picking out as they do a single cell or a coordinate tuple of '''B'''^''n'', become the carriers or the vehicles of a certain type-ambiguity that vacillates between the dual forms '''B'''^''n'' and ('''B'''^''n''&nbsp;<font face=symbol>'''××>'''</font>&nbsp;'''B''') and infects the whole hierarchy of types built on them.  In plainer language, the terms that signify the interpretations ''x''&nbsp;:&nbsp;'''B'''^''n'' and the singular propositions ''x''&nbsp;:&nbsp;'''B'''^''n''&nbsp;<font face=symbol>'''××>'''</font>&nbsp;'''B''' are fully equivalent in information, and this means that every