Changes

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

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

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

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

|}

|}<br>

<br>

===Qualitative Logic and Quantitative Analogy===

===Qualitative Logic and Quantitative Analogy===

<p><math>\mbox{Transformation}\!</math></p>

<p><math>\mbox{Transformation}\!</math></p>

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

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

|}

|}<br>

The Table exhibits a sample of likely parallels between the real and boolean domains.  The central column gives a selection of terminology that is borrowed from differential geometry and extended in its meaning to the logical side of the Table.  These are the varieties of spaces that come up when we turn to analyzing the dynamics of processes that pursue their courses through the states of an arbitrary space <math>X.\!</math>  Moreover, when it becomes necessary to approach situations of overwhelming dynamic complexity in a succession of qualitative reaches, then the methods of logic that are afforded by the boolean domains, with their declarative means of synthesis and deductive modes of analysis, supply a natural battery of tools for the task.

The Table exhibits a sample of likely parallels between the real and boolean domains.  The central column gives a selection of terminology that is borrowed from differential geometry and extended in its meaning to the logical side of the Table.  These are the varieties of spaces that come up when we turn to analyzing the dynamics of processes that pursue their courses through the states of an arbitrary space <math>X.\!</math>  Moreover, when it becomes necessary to approach situations of overwhelming dynamic complexity in a succession of qualitative reaches, then the methods of logic that are afforded by the boolean domains, with their declarative means of synthesis and deductive modes of analysis, supply a natural battery of tools for the task.

<p><math>\mbox{Transformation}\!</math></p>

<p><math>\mbox{Transformation}\!</math></p>

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

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

|}

|}<br>

<br>

First, observe that the type of a ''tangent vector at a point'', also known as a ''directional derivative'' at that point, has the form <math>(\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K},</math> where <math>\mathbb{K}</math> is the chosen ground field, in the present case either <math>\mathbb{R}</math> or <math>\mathbb{B}.</math>  At a point in a space of type <math>\mathbb{K}^n,</math> a directional derivative operator <math>\vartheta\!</math> takes a function on that space, an <math>f\!</math> of type <math>(\mathbb{K}^n \to \mathbb{K}),</math> and maps it to a ground field value of type <math>\mathbb{K}.</math>  This value is known as the ''derivative'' of <math>f\!</math> in the direction <math>\vartheta\!</math> [Che46, 76&ndash;77].  In the boolean case <math>\vartheta  : (\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math> has the form of a proposition about propositions, in other words, a proposition of the next higher type.

First, observe that the type of a ''tangent vector at a point'', also known as a ''directional derivative'' at that point, has the form <math>(\mathbb{K}^n \to \mathbb{K}) \to \mathbb{K},</math> where <math>\mathbb{K}</math> is the chosen ground field, in the present case either <math>\mathbb{R}</math> or <math>\mathbb{B}.</math>  At a point in a space of type <math>\mathbb{K}^n,</math> a directional derivative operator <math>\vartheta\!</math> takes a function on that space, an <math>f\!</math> of type <math>(\mathbb{K}^n \to \mathbb{K}),</math> and maps it to a ground field value of type <math>\mathbb{K}.</math>  This value is known as the ''derivative'' of <math>f\!</math> in the direction <math>\vartheta\!</math> [Che46, 76&ndash;77].  In the boolean case <math>\vartheta  : (\mathbb{B}^n \to \mathbb{B}) \to \mathbb{B}</math> has the form of a proposition about propositions, in other words, a proposition of the next higher type.

| <math>\mbox{Derivation}\!</math>

| <math>\mbox{Derivation}\!</math>

| <math>(\mathbb{K}^n \to \mathbb{K}) \to (\mathbb{K}^n \to \mathbb{K})</math>

| <math>(\mathbb{K}^n \to \mathbb{K}) \to (\mathbb{K}^n \to \mathbb{K})</math>

|}

|}<br>

<br>

===Reality at the Threshold of Logic===

===Reality at the Threshold of Logic===

Table 5 accumulates an array of notation that I hope will not be too distracting.  Some of it is rarely needed, but has been filled in for the sake of completeness.  Its purpose is simple, to give literal expression to the visual intuitions that come with venn diagrams, and to help build a bridge between our qualitative and quantitative outlooks on dynamic systems.

Table 5 accumulates an array of notation that I hope will not be too distracting.  Some of it is rarely needed, but has been filled in for the sake of completeness.  Its purpose is simple, to give literal expression to the visual intuitions that come with venn diagrams, and to help build a bridge between our qualitative and quantitative outlooks on dynamic systems.

{| 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 5.  A Bridge Over Troubled Waters'''

|+ '''Table 5.  A Bridge Over Troubled Waters'''

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

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

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

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

|}

|}<br>

</font><br>

The left side of the Table collects mostly standard notation for an ''n''-dimensional vector space over a field '''K'''.  The right side of the table repeats the first elements of a notation that I sketched above, to be used in further developments of propositional calculus.  (I plan to use this notation in the logical analysis of neural network systems.)  The middle column of the table is designed as a transitional step from the case of an arbitrary field '''K''', with a special interest in the continuous line '''R''', to the qualitative and discrete situations that are instanced and typified by '''B'''.

The left side of the Table collects mostly standard notation for an ''n''-dimensional vector space over a field '''K'''.  The right side of the table repeats the first elements of a notation that I sketched above, to be used in further developments of propositional calculus.  (I plan to use this notation in the logical analysis of neural network systems.)  The middle column of the table is designed as a transitional step from the case of an arbitrary field '''K''', with a special interest in the continuous line '''R''', to the qualitative and discrete situations that are instanced and typified by '''B'''.

| true

| true

| 1

| 1

|}

|}<br>

<br>

Propositional forms on two variables correspond to boolean functions ''f''&nbsp;:&nbsp;'''B'''²&nbsp;&rarr;&nbsp;'''B'''.  In Table 7 each function ''f''_''i'' is indexed by the values that it takes on the points of the universe ''X''^&nbsp;&bull;&nbsp;=&nbsp;[''x'',&nbsp;''y'']&nbsp;<math>\cong</math>&nbsp;'''B'''².  Converting the binary index thus generated to a decimal equivalent, we obtain the functional nicknames that are listed in the first column.  The 2² points of the universe ''X''^&nbsp;&bull; are coordinated as a space of type '''B'''², as indicated under the heading of the Table, where the coordinate projections ''x'' and ''y'' run through the various combinations of their values in '''B'''.

Propositional forms on two variables correspond to boolean functions ''f''&nbsp;:&nbsp;'''B'''²&nbsp;&rarr;&nbsp;'''B'''.  In Table 7 each function ''f''_''i'' is indexed by the values that it takes on the points of the universe ''X''^&nbsp;&bull;&nbsp;=&nbsp;[''x'',&nbsp;''y'']&nbsp;<math>\cong</math>&nbsp;'''B'''².  Converting the binary index thus generated to a decimal equivalent, we obtain the functional nicknames that are listed in the first column.  The 2² points of the universe ''X''^&nbsp;&bull; are coordinated as a space of type '''B'''², as indicated under the heading of the Table, where the coordinate projections ''x'' and ''y'' run through the various combinations of their values in '''B'''.

|-

|-

| f₁₅ || f₁₁₁₁ || 1 1 1 1 || ((&nbsp;)) || true || 1

| f₁₅ || f₁₁₁₁ || 1 1 1 1 || ((&nbsp;)) || true || 1

|}

|}<br>

<br>

==A Differential Extension of Propositional Calculus==

==A Differential Extension of Propositional Calculus==