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These concepts and notations can now be explained in greater detail.  In order to begin as simply as possible, I distinguish two levels of analysis and set out initially on the easier path.  On the first level of analysis, I take spaces like '''B''', '''B'''^''n'', and ('''B'''^''n'' → '''B''') at face value and treat them as the primary objects of interest.  On the second level of analysis, I use these spaces as coordinate charts for talking about points and functions in more fundamental spaces.

These concepts and notations may now be explained in greater detail.  In order to begin as simply as possible, let us distinguish two levels of analysis and set out initially on the easier path.  On the first level of analysis we take spaces like <math>\mathbb{B},</math> <math>\mathbb{B}^n,</math> and <math>(\mathbb{B}^n \to \mathbb{B})</math> at face value and treat them as the primary objects of interest.  On the second level of analysis we use these spaces as coordinate charts for talking about points and functions in more fundamental spaces.

A pair of spaces, of types '''B'''^''n'' and ('''B'''^''n''&nbsp;&rarr;&nbsp;'''B'''), give typical expression to everything that we commonly associate with the ordinary picture of a venn diagram.  The dimension, ''n'', counts the number of "circles" or simple closed curves that are inscribed in the universe of discourse, corresponding to its relevant logical features or basic propositions.  Elements of type '''B'''^''n'' correspond to what are often called propositional

A pair of spaces, of types <math>\mathbb{B}^n</math> and <math>(\mathbb{B}^n \to \mathbb{B}),</math> give typical expression to everything that we commonly associate with the ordinary picture of a venn diagram.  The dimension, <math>n,\!</math> counts the number of "circles" or simple closed curves that are inscribed in the universe of discourse, corresponding to its relevant logical features or basic propositions.  Elements of type <math>\mathbb{B}^n</math> correspond to what are often called propositional ''interpretations'' in logic, that is, the different assignments of truth values to sentence letters.  Relative to a given universe of discourse, these interpretations are visualized as its ''cells'', in other words, the smallest enclosed areas or undivided regions of the venn diagram.  The functions <math>f : \mathbb{B}^n \to \mathbb{B}</math> correspond to the different ways of shading the venn diagram to indicate arbitrary propositions, regions, or sets.  Regions included under a shading indicate the ''models'', and regions excluded represent the ''non-models'' of a proposition.  To recognize and formalize the natural cohesion of these two layers of concepts into a single universe of discourse, we introduce the type notations <math>[\mathbb{B}^n] = \mathbb{B}^n\ +\!\to \mathbb{B}</math> to stand for the pair of types <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})).</math> The resulting "stereotype" serves to frame the universe of discourse as a unified categorical object, and makes it subject to prescribed sets of evaluations and transformations (categorical morphisms or ''arrows'') that affect the universe of discourse as an integrated whole.

''interpretations'' in logic, that is, the different assignments of truth values to sentence letters.  Relative to a given universe of discourse, these interpretations are visualized as its ''cells'', in other words, the smallest enclosed areas or undivided regions of the venn diagram.  The functions ''f''&nbsp;:&nbsp;'''B'''^''n''&nbsp;&rarr;&nbsp;'''B''' correspond to the different ways of shading

the venn diagram to indicate arbitrary propositions, regions, or sets.  Regions included under a shading indicate the ''models'', and regions excluded represent the ''non-models'' of a proposition.  To recognize and formalize the natural cohesion of these two layers of concepts into a single universe of discourse, I introduce the type notations ['''B'''^''n''] = '''B'''^''n''&nbsp;+&rarr;&nbsp;'''B''' to stand for the pair of types ('''B'''^''n'', ('''B'''^''n''&nbsp;&rarr;&nbsp;'''B''')).  The resulting "stereotype" serves to frame the universe of discourse as a unified categorical object, and makes it subject to prescribed sets of evaluations and transformations (categorical morphisms or ''arrows'') that affect the universe of discourse as an integrated whole.

Most of the time we can serve the algebraic, geometric, and logical interests of our study without worrying about their occasional conflicts and incidental divergences.  The conventions and definitions already set down will continue to cover most of the algebraic and functional aspects of our discussion, but to handle the logical and qualitative aspects we will need to add a few more.  In general, abstract sets may be denoted by gothic, greek, or script capital variants of ''A'', ''B'', ''C'', and so on, with elements denoted by a corresponding set of subscripted letters in plain lower case, for example, <font face="lucida calligraphy">A</font> = {''a''_''i''}.  Most

Most of the time we can serve the algebraic, geometric, and logical interests of our study without worrying about their occasional conflicts and incidental divergences.  The conventions and definitions already set down will continue to cover most of the algebraic and functional aspects of our discussion, but to handle the logical and qualitative aspects we will need to add a few more.  In general, abstract sets may be denoted by gothic, greek, or script capital variants of ''A'', ''B'', ''C'', and so on, with elements denoted by a corresponding set of subscripted letters in plain lower case, for example, <font face="lucida calligraphy">A</font> = {''a''_''i''}.  Most