Changes

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.

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.

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 <math>A, B, C,\!</math> and so on, with their elements being denoted by a corresponding set of subscripted letters in plain lower case, for example, <math>\mathcal{A} = \{a_i\}.</math>  Most

of the time, a set such as <font face="lucida calligraphy">A</font> = {''a''_''i''} will be employed as the ''alphabet'' of

of the time, a set such as <math>\mathcal{A} = \{a_i\}</math> will be employed as the ''alphabet'' of a [[formal language]].  These alphabet letters serve to name the logical features (properties or propositions) that generate a particular universe of discourse.  When we want to discuss the particular features of a universe of discourse, beyond the abstract designation of a type like <math>(\mathbb{B}^n\ +\!\to \mathbb{B}),</math> then we may use the following notations.  If <math>\mathcal{A} = \{a_1, \ldots, a_n\}</math> is an alphabet of logical features, then <math>A = \langle \mathcal{A} \rangle = \langle a_1, \ldots, a_n \rangle</math> is the set of interpretations, <math>A^\uparrow = (A \to \mathbb{B})</math> is the set of propositions, and <math>A^\circ = [\mathcal{A}] = [a_1, \ldots, a_n]</math> is the combination of these interpretations and propositions into the universe of discourse that is based on the features <math>\{a_1, \ldots, a_n\}.</math>

a [[formal language]].  These alphabet letters serve to name the logical features (properties or propositions) that generate a particular universe of discourse.  When we want to discuss the particular features of a universe of discourse,

beyond the abstract designation of a type like ('''B'''^''n''&nbsp;+&rarr;&nbsp;'''B'''), then we may use the following notations.  If <font face="lucida calligraphy">A</font> = {''a''₁,&nbsp;&hellip;,&nbsp;''a''_''n''} is an alphabet of logical

features, then ''A'' = 〈<font face="lucida calligraphy">A</font>〉 = 〈''a''₁,&nbsp;&hellip;,&nbsp;''a''_''n''〉 is the set of interpretations, ''A''^ = (''A'' &rarr; '''B''') is the set of propositions, and ''A''^&nbsp;&bull; = [<font face="lucida calligraphy">A</font>] = [''a''₁,&nbsp;&hellip;,&nbsp;''a''_''n''] is

the combination of these interpretations and propositions into the universe of discourse that is based on the features {''a''₁,&nbsp;&hellip;,&nbsp;''a''_''n''}.

As always, especially in concrete examples, these rules may be dropped whenever necessary, reverting to a free assortment of feature labels.  However, when we need to talk about the logical aspects of a space that is already named as a vector space, it will be necessary to make special provisions.  At any rate, these elaborations can be deferred until actually needed.

As always, especially in concrete examples, these rules may be dropped whenever necessary, reverting to a free assortment of feature labels.  However, when we need to talk about the logical aspects of a space that is already named as a vector space, it will be necessary to make special provisions.  At any rate, these elaborations can be deferred until actually needed.