Changes

The set of logical features <math>\{a_1, \ldots, a_n\}</math> provides a basis for generating an <math>n\!</math>-dimensional ''[[universe of discourse]]'' that I denote as <math>[a_1, \ldots, a_n].</math>  It is useful to consider each universe of discourse as a unified categorical object that incorporates both the set of points <math>\langle a_1, \ldots, a_n \rangle</math> and the set of propositions <math>f : \langle a_1, \ldots, a_n \rangle \to \mathbb{B}</math> that are implicit with the ordinary picture of a venn diagram on <math>n\!</math> features.  Thus, we may regard the universe of discourse <math>[a_1, \ldots, a_n]</math> as an ordered pair having the type <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}),</math> and we may abbreviate this last type designation as <math>\mathbb{B}^n\ +\!\to \mathbb{B},</math> or even more succinctly as <math>[\mathbb{B}^n].</math>  (Used this way, the angle brackets <math>\langle\ldots\rangle</math> are referred to as ''generator brackets''.)

The set of logical features <math>\{a_1, \ldots, a_n\}</math> provides a basis for generating an <math>n\!</math>-dimensional ''[[universe of discourse]]'' that I denote as <math>[a_1, \ldots, a_n].</math>  It is useful to consider each universe of discourse as a unified categorical object that incorporates both the set of points <math>\langle a_1, \ldots, a_n \rangle</math> and the set of propositions <math>f : \langle a_1, \ldots, a_n \rangle \to \mathbb{B}</math> that are implicit with the ordinary picture of a venn diagram on <math>n\!</math> features.  Thus, we may regard the universe of discourse <math>[a_1, \ldots, a_n]</math> as an ordered pair having the type <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}),</math> and we may abbreviate this last type designation as <math>\mathbb{B}^n\ +\!\to \mathbb{B},</math> or even more succinctly as <math>[\mathbb{B}^n].</math>  (Used this way, the angle brackets <math>\langle\ldots\rangle</math> are referred to as ''generator brackets''.)

Table 2 exhibits the scheme of notation I use to formalize the domain of propositional calculus, corresponding to the logical content of truth tables and venn diagrams.  Although it overworks the square brackets a bit, I also use either one of the equivalent notations [''n''] or '''''n''''' to denote the data type of a finite set on n elements.

Table 2 exhibits the scheme of notation I use to formalize the domain of propositional calculus, corresponding to the logical content of truth tables and venn diagrams.  Although it overworks the square brackets a bit, I also use either one of the equivalent notations <math>[n]\!</math> or <math>\mathbf{n}</math> to denote the data type of a finite set on <math>n\!</math> elements.

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:96%"

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:96%"

|+ '''Table 2.  Fundamental Notations for Propositional Calculus'''

|+ '''Table 2.  Propositional Calculus : Basic Notation'''

|- style="background:ghostwhite"

|- style="background:ghostwhite"

! Symbol

! Symbol

! Type

! Type

|-

|-

| <font face="lucida calligraphy">A<font>

| <math>\mathfrak{A}</math>

| {''a''₁, &hellip;, ''a''_''n''}

| <math>\lbrace\!</math>&nbsp;“<math>a_1\!</math>”&nbsp;<math>, \ldots,\!</math>&nbsp;“<math>a_n\!</math>”&nbsp;<math>\rbrace\!</math>

| Alphabet

| Alphabet

| [''n''] = '''n'''

| <math>[n] = \mathbf{n}</math>

|-

|-

| ''A''_''i''

| <math>\mathcal{A}</math>

| {(''a''_''i''), ''a''_''i''}

| <math>\{ a_1, \ldots, a_n \}</math>

| <math>[n] = \mathbf{n}</math>

|-

|-

| ''A''

| <math>A_i\!</math>

|

| <math>\{ \overline{a_i}, a_i \}\!</math>

〈<font face="lucida calligraphy">A</font>〉<br>

| Dimension <math>i\!</math>

〈''a''₁, &hellip;, ''a''_''n''〉<br>

| <math>\mathbb{B}</math>

{‹''a''₁, &hellip;, ''a''_''n''›}<br>

''A''₁ &times; &hellip; &times; ''A''_''n''<br>

| <math>A\!</math>

&prod;_''i'' ''A''_''i''

| <math>\langle \mathcal{A} \rangle</math><br>

|

<math>\langle a_1, \ldots, a_n \rangle</math><br>

Set of cells,<br>

<math>\{ (a_1, \ldots, a_n) \}\!</math>

<math>A_1 \times \ldots \times A_n</math><br>

<math>\textstyle \prod_i A_i\!</math>

| Set of cells,<br>

coordinate tuples,<br>

coordinate tuples,<br>

points, or vectors<br>

points, or vectors<br>

in the universe<br>

in the universe<br>

of discourse

of discourse

| '''B'''^''n''

| <math>\mathbb{B}^n</math>

|-

|-

| ''A''*

| <math>A^*\!</math>

| (hom : ''A'' &rarr; '''B''')

| <math>(\operatorname{hom} : A \to \mathbb{B})</math>

| Linear functions

| Linear functions

| ('''B'''^''n'')* = '''B'''^''n''

| <math>(\mathbb{B}^n)^* \cong \mathbb{B}^n</math>

|-

|-

| ''A''^

| <math>A^\uparrow</math>

| (''A'' &rarr; '''B''')

| <math>(A \to \mathbb{B})</math>

| Boolean functions

| Boolean functions

| '''B'''^''n'' &rarr; '''B'''

| <math>\mathbb{B}^n \to \mathbb{B}</math>

|-

|-

| ''A''^&bull;

| <math>A^\circ</math>

|

| <math>[ \mathcal{A} ]</math><br>

[<font face="lucida calligraphy">A</font>]<br>

<math>(A, A^\uparrow)</math><br>

(''A'', ''A''^)<br>

<math>(A\ +\!\to \mathbb{B})</math><br>

(''A'' +&rarr; '''B''')<br>

<math>(A, (A \to \mathbb{B}))</math><br>

(''A'', (''A'' &rarr; '''B'''))<br>

<math>[ a_1, \ldots, a_n ]</math>

[''a''₁, &hellip;, ''a''_''n'']

| Universe of discourse<br>

|

Universe of discourse<br>

based on the features<br>

based on the features<br>

{''a''₁, &hellip;, ''a''_''n''}

<math>\{ a_1, \ldots, a_n \}</math>

|

| <math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))</math><br>

('''B'''^''n'', ('''B'''^''n'' &rarr; '''B'''))<br>

<math>(\mathbb{B}^n\ +\!\to \mathbb{B})</math><br>

('''B'''^''n'' +&rarr; '''B''')<br>

<math>[\mathbb{B}^n]</math>

['''B'''^''n'']

|}

|}

</font><br>

<br>

===Qualitative Logic and Quantitative Analogy===

===Qualitative Logic and Quantitative Analogy===