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\section{Transitional remarks}
 
\section{Transitional remarks}
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Up to this point we have been treating the universe of discourse $X,$ the quality $q,$ and the symbol $``q"$ as all of one piece, almost as if the entire context marked by $X$ and $q$ and $``q"$ amounted to the only way of viewing $X.$  That is clearly not the case, but the fact is that people often use the term ``universe of discourse" to cover a particular set of distinctions drawn in the space $X$ and even sometimes a particular calculus or language for discussing the elements of $X.$  If it were possible to coin a new phrase in this realm one might distinguish these latter components as the ``discursive universe", but there is probably no escape from simply recognizing the equivocal senses of the terms already in use and trying to clarify the senses intended in context.
      
$\ldots$
 
$\ldots$
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The basic propositions $a_i : \mathbb{B}^n \to \mathbb{B}$ are both linear and positive.  So these two kinds of propositions, the linear and the positive, may be viewed as two different ways of generalizing the class of basic propositions.
 
The basic propositions $a_i : \mathbb{B}^n \to \mathbb{B}$ are both linear and positive.  So these two kinds of propositions, the linear and the positive, may be viewed as two different ways of generalizing the class of basic propositions.
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Finally, it is important to note that all of the above distinctions are relative to the choice of a particular logical basis $\mathcal{A} = \{ a_1, \ldots, a_n \}.$  For example, a singular proposition with respect to the basis $\mathcal{A}$ will not remain singular if $\mathcal{A}$ is extended by a number of new and independent features.  Even if one keeps to the original set of pairwise options $\{ a_i \} \cup \{ (a_i) \}$ to pick out a new basis, the sets of linear propositions and positive propositions are both determined by the choice of basic propositions, and this whole determination is tantamount to the purely conventional choice of a cell as origin.
    
\subsection{Differential extensions}
 
\subsection{Differential extensions}
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