Changes

===Qualitative Logic and Quantitative Analogy===

===Qualitative Logic and Quantitative Analogy===

<p>Logical, however, is used in a third sense, which is at once more vital and more practical;  to denote, namely, the systematic care, negative and positive, taken to safeguard reflection so that it may yield the best results under the given conditions.</p>

 

<p>[[John Dewey]], ''[[How We Think]]'', [Dew, 56]</p>

''Logical'', however, is used in a third sense, which is at once more vital and more practical;  to denote, namely, the systematic care, negative and positive, taken to safeguard reflection so that it may yield the best results under the given conditions.

| align="right" colspan="3" | &mdash; John Dewey, ''How We Think'', [Dew, 56]

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''&nbsp;&rarr;&nbsp;'''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 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''&nbsp;&rarr;&nbsp;'''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.

===Philosophy of Notation : Formal Terms and Flexible Types===

===Philosophy of Notation : Formal Terms and Flexible Types===

<p>Where number is irrelevant, regimented mathematical technique has hitherto tended to be lacking.  Thus it is that the progress of natural science has depended so largely upon the discernment of measurable quantity of one sort or another.</p>

 

<p>W.V. Quine, ''Mathematical Logic'', [Qui, 7]</p>

Where number is irrelevant, regimented mathematical technique has hitherto tended to be lacking.  Thus it is that the progress of natural science has depended so largely upon the discernment of measurable quantity of one sort or another.

| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]

For much of our discussion propositions and boolean functions are treated as the same formal objects, or as different interpretations of the same formal calculus.  This rule of interpretation has exceptions, though.  There is a distinctively logical interest in the use of propositional calculus that is not exhausted by its functional interpretation.  It is part of our task in this study to deal with these uniquely logical characteristics as they present themselves both in our subject matter and in our formal calculus.  Just to provide a hint of what's at stake:  In logic, as opposed to the more imaginative realms of mathematics, we consider it a good thing to always know what we are talking about.  Where mathematics encourages tolerance for uninterpreted symbols as intermediate terms, logic exerts a keener effort to interpret directly each oblique carrier of meaning, no matter how slight, and to unfold the complicities of every indirection in the flow of information.  Translated into functional terms, this means that we want to maintain a continual, immediate, and persistent sense of both the converse relation ''f''^&ndash;1 &sube; '''B'''&nbsp;&times;&nbsp;'''B'''^''n'', or what is the same thing, ''f''^&ndash;1 : '''B''' &rarr; ''Pow''('''B'''^''n''), and the ''fibers'' or inverse images ''f''^&ndash;1(0) and ''f''^&ndash;1(1), associated with each boolean function ''f'' : '''B'''^''n'' &rarr; '''B''' that we use.  In practical terms, the desired implementation of a propositional interpreter should incorporate our intuitive recognition that the induced partition of the functional domain into level sets ''f''^&ndash;1(''b''), for ''b'' &isin; '''B''', is part and parcel of understanding the denotative uses of each propositional function ''f''.

For much of our discussion propositions and boolean functions are treated as the same formal objects, or as different interpretations of the same formal calculus.  This rule of interpretation has exceptions, though.  There is a distinctively logical interest in the use of propositional calculus that is not exhausted by its functional interpretation.  It is part of our task in this study to deal with these uniquely logical characteristics as they present themselves both in our subject matter and in our formal calculus.  Just to provide a hint of what's at stake:  In logic, as opposed to the more imaginative realms of mathematics, we consider it a good thing to always know what we are talking about.  Where mathematics encourages tolerance for uninterpreted symbols as intermediate terms, logic exerts a keener effort to interpret directly each oblique carrier of meaning, no matter how slight, and to unfold the complicities of every indirection in the flow of information.  Translated into functional terms, this means that we want to maintain a continual, immediate, and persistent sense of both the converse relation ''f''^&ndash;1 &sube; '''B'''&nbsp;&times;&nbsp;'''B'''^''n'', or what is the same thing, ''f''^&ndash;1 : '''B''' &rarr; ''Pow''('''B'''^''n''), and the ''fibers'' or inverse images ''f''^&ndash;1(0) and ''f''^&ndash;1(1), associated with each boolean function ''f'' : '''B'''^''n'' &rarr; '''B''' that we use.  In practical terms, the desired implementation of a propositional interpreter should incorporate our intuitive recognition that the induced partition of the functional domain into level sets ''f''^&ndash;1(''b''), for ''b'' &isin; '''B''', is part and parcel of understanding the denotative uses of each propositional function ''f''.

===The Analogy Between Real and Boolean Types===

===The Analogy Between Real and Boolean Types===

<p>Measurement consists in correlating our subject matter with the series of real numbers;  and such correlations are desirable because, once they are set up, all the well-worked theory of numerical mathematics lies ready at hand as a tool for our further reasoning.</p>

 

<p>W.V. Quine, ''Mathematical Logic'', [Qui, 7]</p>

Measurement consists in correlating our subject matter with the series of real numbers;  and such correlations are desirable because, once they are set up, all the well-worked theory of numerical mathematics lies ready at hand as a tool for our further reasoning.

| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]

There are two further reasons why I am spending so much time on a careful treatment of types, and they both have to do with our being able to take full computational advantage of certain dimensions of flexibility in the types that apply to terms.  First, the domains of differential geometry and logic programming are connected by analogies between real and boolean types of the same pattern.  Second, the types involved in these patterns have important isomorphisms connecting them that apply on both the real and the boolean sides of the picture.

There are two further reasons why I am spending so much time on a careful treatment of types, and they both have to do with our being able to take full computational advantage of certain dimensions of flexibility in the types that apply to terms.  First, the domains of differential geometry and logic programming are connected by analogies between real and boolean types of the same pattern.  Second, the types involved in these patterns have important isomorphisms connecting them that apply on both the real and the boolean sides of the picture.

===Theory of Control and Control of Theory===

===Theory of Control and Control of Theory===

<p>You will hardly know who I am or what I mean,<br>

You will hardly know who I am or what I mean,<br>

But I shall be good health to you nevertheless,<br>

But I shall be good health to you nevertheless,<br>

And filter and fibre your blood.</p>

And filter and fibre your blood.

 

<p>Walt Whitman, ''Leaves of Grass'', [Whi, 88]</p>

| align="right" | &mdash; Walt Whitman, ''Leaves of Grass'', [Whi, 88]

In the boolean context, a function ''f''&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;'''B''' is tantamount to a ''proposition'' about elements of ''X'', and the elements of ''X'' constitute the ''interpretations'' of that proposition.  The fiber ''f''^&ndash;1(1) comprises the set of ''models'' of ''f'', or examples of elements in ''X'' satisfying the proposition ''f''.  The fiber ''f''^&ndash;1(0) collects the complementary set of ''anti-models'', or the exceptions to the proposition ''f'' that exist in ''X''.  Of course, the space of functions (''X''&nbsp;&rarr;&nbsp;'''B''') is isomorphic to the set of all subsets of X, called the ''power set'' of ''X'' and often denoted as <font face="lucida calligraphy">Pow</font>(''X'') or 2^''X''.

In the boolean context, a function ''f''&nbsp;:&nbsp;''X''&nbsp;&rarr;&nbsp;'''B''' is tantamount to a ''proposition'' about elements of ''X'', and the elements of ''X'' constitute the ''interpretations'' of that proposition.  The fiber ''f''^&ndash;1(1) comprises the set of ''models'' of ''f'', or examples of elements in ''X'' satisfying the proposition ''f''.  The fiber ''f''^&ndash;1(0) collects the complementary set of ''anti-models'', or the exceptions to the proposition ''f'' that exist in ''X''.  Of course, the space of functions (''X''&nbsp;&rarr;&nbsp;'''B''') is isomorphic to the set of all subsets of X, called the ''power set'' of ''X'' and often denoted as <font face="lucida calligraphy">Pow</font>(''X'') or 2^''X''.

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

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

<p>But no science can rest entirely on measurement, and many scientific investigations are quite out of reach of that device.  To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope.</p>

 

<p>W.V. Quine, ''Mathematical Logic'', [Qui, 7]</p>

But no science can rest entirely on measurement, and many scientific investigations are quite out of reach of that device.  To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope.

| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7]

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.

===Tables of Propositional Forms===

===Tables of Propositional Forms===

<p>To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope.  It provides explicit techniques for manipulating the most basic ingredients of discourse.</p>

 

<p>W.V. Quine, ''Mathematical Logic'', [Qui, 7-8]</p>

To the scientist longing for non-quantitative techniques, then, mathematical logic brings hope.  It provides explicit techniques for manipulating the most basic ingredients of discourse.

| align="right" colspan="3" | &mdash; W.V. Quine, ''Mathematical Logic'', [Qui, 7&ndash;8]

To prepare for the next phase of discussion, Tables 6 and 7 collect and summarize all of the propositional forms on one and two variables.  These propositional forms are represented over bases of boolean variables as complete sets of boolean-valued functions.  Adjacent to their names and specifications are listed what are roughly the simplest expressions in the ''[[cactus language]]'', the particular syntax for propositional calculus that I use in formal and computational contexts.  For the sake of orientation, the English paraphrases and the more common notations are listed in the last two columns.  As simple and circumscribed as these low-dimensional universes may appear to be, a careful exploration of their differential extensions will involve us in complexities sufficient to demand our attention for some time to come.

To prepare for the next phase of discussion, Tables 6 and 7 collect and summarize all of the propositional forms on one and two variables.  These propositional forms are represented over bases of boolean variables as complete sets of boolean-valued functions.  Adjacent to their names and specifications are listed what are roughly the simplest expressions in the ''[[cactus language]]'', the particular syntax for propositional calculus that I use in formal and computational contexts.  For the sake of orientation, the English paraphrases and the more common notations are listed in the last two columns.  As simple and circumscribed as these low-dimensional universes may appear to be, a careful exploration of their differential extensions will involve us in complexities sufficient to demand our attention for some time to come.