Changes

==Differential Logic : Series B==

==Differential Logic : Series B==

===Note 1===

===Linear Topics : The Differential Theory of Qualitative Equations===

 

'''Linear Topics : The Differential Theory of Qualitative Equations'''

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That's the basic idea.  The next order of business is to develop the logical side of the analogy a bit more fully, and to take up the elaboration of some moderately simple applications of these ideas to a selection of relatively concrete examples.

That's the basic idea.  The next order of business is to develop the logical side of the analogy a bit more fully, and to take up the elaboration of some moderately simple applications of these ideas to a selection of relatively concrete examples.

===Note 2===

===Example 1.  A Polymorphous Concept===

 

'''Example 1.  A Polymorphous Concept'''

I start with an example that is simple enough that it will allow us to compare the representations of propositions by venn diagrams, truth tables, and my own favorite version of the syntax for propositional calculus all in a relatively short space.  To enliven the exercise, I borrow an example from a book with several independent dimensions of interest, ''Topobiology'' by Gerald Edelman.  One finds discussed there the notion of a "polymorphous set".  Such a set is defined in a universe of discourse whose elements can be described in terms of a fixed number <math>k\!</math> of logical features.  A "polymorphous set" is one that can be defined in terms of sets whose elements have a fixed number <math>j\!</math> of the <math>k\!</math> features.

I start with an example that is simple enough that it will allow us to compare the representations of propositions by venn diagrams, truth tables, and my own favorite version of the syntax for propositional calculus all in a relatively short space.  To enliven the exercise, I borrow an example from a book with several independent dimensions of interest, ''Topobiology'' by Gerald Edelman.  One finds discussed there the notion of a "polymorphous set".  Such a set is defined in a universe of discourse whose elements can be described in terms of a fixed number <math>k\!</math> of logical features.  A "polymorphous set" is one that can be defined in terms of sets whose elements have a fixed number <math>j\!</math> of the <math>k\!</math> features.

For the most threadbare kind of logical system that we find residing in propositional calculus, this notion of model is almost too simple to deserve the name, yet it can be of service to fashion some form of continuity between the simple and the complex.

For the most threadbare kind of logical system that we find residing in propositional calculus, this notion of model is almost too simple to deserve the name, yet it can be of service to fashion some form of continuity between the simple and the complex.

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This tells us that changing any two or more of the features <math>u, v, w\!</math> will take us from the center cell to a cell outside the shaded region for the set <math>Q.\!</math>

This tells us that changing any two or more of the features <math>u, v, w\!</math> will take us from the center cell to a cell outside the shaded region for the set <math>Q.\!</math>

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<blockquote>

Next I will discuss several applications of logical differentials, developing along the way their logical and practical implications.

Next I will discuss several applications of logical differentials, developing along the way their logical and practical implications.

We have come to the point of making a connection, at a very primitive level, between propositional logic and the classes of mathematical structures that are employed in mathematical systems theory to model dynamical systems of very general sorts.

We have come to the point of making a connection, at a very primitive level, between propositional logic and the classes of mathematical structures that are employed in mathematical systems theory to model dynamical systems of very general sorts.

In our initial consideration of the proposition <math>q\!</math>, we naturally interpret it as a function of the three variables that it wears on its sleeve, as it were, namely, those that we find contained in the basis <math>\{ u, v, w \}</math>.  As we begin to regard this proposition from the standpoint of a differential analysis, however, we may need to regard it as ''tacitly embedded'' in any number of higher dimensional spaces.  Just by way of starting out, our immediate interest is with the ''first order differential analysis'' (FODA), and this requires us to regard all of the propositions in sight as functions of the variables in the first order extended basis, specifically, those in the set <math>\{ u, v, w, \operatorname{d}u, \operatorname{d}v, \operatorname{d}w \}</math>.  Now this does not change the expression of any proposition, like <math>q\!</math>, that does not mention the extra variables, only changing how it gets interpreted as a function.  A level of interpretive flexibility of this order is very useful, and it is quite common throughout mathematics.  In this discussion, I will invoke its application under the name of the ''[[tacit extension]]'' of a proposition to any universe of discourse based on a superset of its original basis.

In our initial consideration of the proposition <math>q\!</math>, we naturally interpret it as a function of the three variables that it wears on its sleeve, as it were, namely, those that we find contained in the basis <math>\{ u, v, w \}</math>.  As we begin to regard this proposition from the standpoint of a differential analysis, however, we may need to regard it as ''tacitly embedded'' in any number of higher dimensional spaces.  Just by way of starting out, our immediate interest is with the ''first order differential analysis'' (FODA), and this requires us to regard all of the propositions in sight as functions of the variables in the first order extended basis, specifically, those in the set <math>\{ u, v, w, \operatorname{d}u, \operatorname{d}v, \operatorname{d}w \}</math>.  Now this does not change the expression of any proposition, like <math>q\!</math>, that does not mention the extra variables, only changing how it gets interpreted as a function.  A level of interpretive flexibility of this order is very useful, and it is quite common throughout mathematics.  In this discussion, I will invoke its application under the name of the ''[[tacit extension]]'' of a proposition to any universe of discourse based on a superset of its original basis.

I think that we finally have enough of the preliminary set-ups and warm-ups out of the way that we can begin to tackle the differential analysis proper of the sample proposition, the truth-function <math>q(u, v, w)\!</math> that is given by the following expression:

I think that we finally have enough of the preliminary set-ups and warm-ups out of the way that we can begin to tackle the differential analysis proper of the sample proposition, the truth-function <math>q(u, v, w)\!</math> that is given by the following expression:

           On the Proposition q, Evaluated at c

           On the Proposition q, Evaluated at c

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One more piece of notation will save us a few bytes in the length of many of our schematic formulations.

One more piece of notation will save us a few bytes in the length of many of our schematic formulations.

That was pretty tedious, and I know that it all appears to be totally trivial, which is precisely why we usually just leave it "tacit" in the first place, but hard experience, and a real acquaintance with the confusion that can beset us when we do not render these implicit grounds explicit, have taught me that it will ultimately be necessary to get clear about it, and by this ''clear'' to say ''marked'', not merely ''transparent''.

That was pretty tedious, and I know that it all appears to be totally trivial, which is precisely why we usually just leave it "tacit" in the first place, but hard experience, and a real acquaintance with the confusion that can beset us when we do not render these implicit grounds explicit, have taught me that it will ultimately be necessary to get clear about it, and by this ''clear'' to say ''marked'', not merely ''transparent''.

Before going on, it would probably be a good idea to remind ourselves of just why we are going through with this exercise.  It is to unify the world of change, for which aspect or regime of the world I occasionally evoke the eponymous figures of Prometheus and Heraclitus, and the world of logic, for which facet or realm of the world I periodically recur to the prototypical shades of Epimetheus and Parmenides, at least, that is, to state it more carefully, to encompass the antics and the escapades of these all too manifestly strife-born twins within the scopes of our thoughts and within the charts of our theories, as it is most likely the only places where ever they will, for the moment and as long as it lasts, be seen or be heard together.

Before going on, it would probably be a good idea to remind ourselves of just why we are going through with this exercise.  It is to unify the world of change, for which aspect or regime of the world I occasionally evoke the eponymous figures of Prometheus and Heraclitus, and the world of logic, for which facet or realm of the world I periodically recur to the prototypical shades of Epimetheus and Parmenides, at least, that is, to state it more carefully, to encompass the antics and the escapades of these all too manifestly strife-born twins within the scopes of our thoughts and within the charts of our theories, as it is most likely the only places where ever they will, for the moment and as long as it lasts, be seen or be heard together.

That sums up, but rather more carefully, the material that I ran through just a bit too quickly the first time around.  Next time, I will begin to develop an alternative style of diagram for depicting these types of differential settings.

That sums up, but rather more carefully, the material that I ran through just a bit too quickly the first time around.  Next time, I will begin to develop an alternative style of diagram for depicting these types of differential settings.

Another way of looking at this situation is by letting the (first order) differential features <math>\operatorname{d}u, \operatorname{d}v, \operatorname{d}w</math> be viewed as the features of another universe of discourse, called the ''tangent universe'' to <math>X\!</math> with respect to the interpretation <math>c\!</math> and represented as <math>\operatorname{d}X \cdot c</math>&nbsp;.  In this setting, <math>\operatorname{D}q \cdot c</math>&nbsp;, the ''difference proposition'' of <math>q\!</math> at the interpretation <math>c\!</math>&nbsp;, where <math>c = u\ v\ w</math>&nbsp;, is marked by the shaded region in Figure&nbsp;4.

Another way of looking at this situation is by letting the (first order) differential features <math>\operatorname{d}u, \operatorname{d}v, \operatorname{d}w</math> be viewed as the features of another universe of discourse, called the ''tangent universe'' to <math>X\!</math> with respect to the interpretation <math>c\!</math> and represented as <math>\operatorname{d}X \cdot c</math>&nbsp;.  In this setting, <math>\operatorname{D}q \cdot c</math>&nbsp;, the ''difference proposition'' of <math>q\!</math> at the interpretation <math>c\!</math>&nbsp;, where <math>c = u\ v\ w</math>&nbsp;, is marked by the shaded region in Figure&nbsp;4.

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===Note 10===

===From "Theme One : A Program of Inquiry" by Jon Awbrey and Susan Awbrey, 09 Aug 1989===

 

'''From "Theme One : A Program of Inquiry" by Jon Awbrey and Susan Awbrey, 09 Aug 1989'''

'''Example 5.  Jets and Sharks'''

'''Example 5.  Jets and Sharks'''

Be at liberty to sing out of it.

Be at liberty to sing out of it.

===Note 11===

===Interlude===

 

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But that was then, this is now, so let me try to say it planar.

But that was then, this is now, so let me try to say it planar.

===Note 12===

===Notes on Cactus Language===

 

'''Notes on Cactus Language'''

I happened on the graphical syntax for propositional calculus that I now call the ''cactus language'' while exploring the confluence of three streams of thought.  There was C.S. Peirce's use of operator variables in logical forms and the operational representations of logical concepts, there was George Spencer Brown's explanation of a variable as the contemplated presence or absence of a constant, and then there was the graph theory and group theory that I had been picking up, bit by bit, since I first encountered them in tandem in Frank Harary's foundations of math course, ''c.'' 1970.

I happened on the graphical syntax for propositional calculus that I now call the ''cactus language'' while exploring the confluence of three streams of thought.  There was C.S. Peirce's use of operator variables in logical forms and the operational representations of logical concepts, there was George Spencer Brown's explanation of a variable as the contemplated presence or absence of a constant, and then there was the graph theory and group theory that I had been picking up, bit by bit, since I first encountered them in tandem in Frank Harary's foundations of math course, ''c.'' 1970.

According to my earlier, if somewhat sketchy interpretive suggestions, we are supposed to picture a quasi-neural pool that contains a couple of quasi-neural agents or ''units'', that between the two of them stand for the logical variables ''jets'' and ''sharks'', respectively.  Further, we imagine these agents to be mutually inhibitory, so that settlement of the dynamic between them achieves equilibrium when just one of the two is ''active'' or ''changing'' and the other is ''stable''or ''enduring''.

According to my earlier, if somewhat sketchy interpretive suggestions, we are supposed to picture a quasi-neural pool that contains a couple of quasi-neural agents or ''units'', that between the two of them stand for the logical variables ''jets'' and ''sharks'', respectively.  Further, we imagine these agents to be mutually inhibitory, so that settlement of the dynamic between them achieves equilibrium when just one of the two is ''active'' or ''changing'' and the other is ''stable''or ''enduring''.

We were focussing on a particular figure of syntax, presented here in both graph and string renditions:

We were focussing on a particular figure of syntax, presented here in both graph and string renditions:

I'm going to let that settle a while.

I'm going to let that settle a while.

Table&nbsp;5 sums up the facts of the physical situation at equilibrium.  If we let <math>\mathbf{B} = \{ \mathrm{charged}, \mathrm{resting} \} = \{ \mathrm{moving}, \mathrm{steady} \} = \{ \mathrm{note}, \mathrm{rest} \},</math> or whatever candidates you pick for the 2-membered set in question, the Table shows a function <math>f : \mathbf{B} \times \mathbf{B} \to \mathbf{B},</math> where <math>f(x, y) = (x, y) = \operatorname{XOR}(x, y).\!</math>

Table&nbsp;5 sums up the facts of the physical situation at equilibrium.  If we let <math>\mathbf{B} = \{ \mathrm{charged}, \mathrm{resting} \} = \{ \mathrm{moving}, \mathrm{steady} \} = \{ \mathrm{note}, \mathrm{rest} \},</math> or whatever candidates you pick for the 2-membered set in question, the Table shows a function <math>f : \mathbf{B} \times \mathbf{B} \to \mathbf{B},</math> where <math>f(x, y) = (x, y) = \operatorname{XOR}(x, y).\!</math>

Although the syntax of the cactus language modifies the syntax of Peirce's graphical formalisms to some extent, the first interpretation corresponds to what he called the ''entitative graphs'' and the second interpretation corresponds to what he called the ''existential graphs''.  In working through the present example, I have chosen the existential interpretation of cactus expressions, and so the form "<code>(jets , sharks)</code>" is interpreted as saying that everything in the universe of discourse is either a Jet or a Shark, but never both at once.

Although the syntax of the cactus language modifies the syntax of Peirce's graphical formalisms to some extent, the first interpretation corresponds to what he called the ''entitative graphs'' and the second interpretation corresponds to what he called the ''existential graphs''.  In working through the present example, I have chosen the existential interpretation of cactus expressions, and so the form "<code>(jets , sharks)</code>" is interpreted as saying that everything in the universe of discourse is either a Jet or a Shark, but never both at once.

Before we tangle with the rest of the Jets and Sharks example, let's look at a cactus expression that's next in the series we just considered, this time a lobe with three variables.  For instance, let's analyze the cactus form whose graph and string expressions are shown in the next display.

Before we tangle with the rest of the Jets and Sharks example, let's look at a cactus expression that's next in the series we just considered, this time a lobe with three variables.  For instance, let's analyze the cactus form whose graph and string expressions are shown in the next display.

o-----------------------------------o-----------o

o-----------------------------------o-----------o

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The cactus lobe operators <math>(\ ), (x_1), (x_1, x_2), (x_1, x_2, x_3), \ldots, (x_1, \ldots, x_k)\!</math> are often referred to as ''boundary operators'' and one of the reasons for this can be seen most easily in the venn diagram for the <math>k\!</math>-argument boundary operator <math>(x_1, \ldots, x_k).\!</math>  Figure&nbsp;10 shows the venn diagram for the 3-fold boundary form <math>(x, y, z).\!</math>

The cactus lobe operators <math>(\ ), (x_1), (x_1, x_2), (x_1, x_2, x_3), \ldots, (x_1, \ldots, x_k)\!</math> are often referred to as ''boundary operators'' and one of the reasons for this can be seen most easily in the venn diagram for the <math>k\!</math>-argument boundary operator <math>(x_1, \ldots, x_k).\!</math>  Figure&nbsp;10 shows the venn diagram for the 3-fold boundary form <math>(x, y, z).\!</math>

In relation to the central cell indicated by the conjunction <math>x\ y\ z,</math> the region indicated by "<math>(x, y, z)\!</math>" is composed of the ''adjacent'' or the ''bordering'' cells.  Thus they are the cells that are just across the boundary of the center cell, arrived at by taking all of Leibniz's ''minimal changes'' from the given point of departure.

In relation to the central cell indicated by the conjunction <math>x\ y\ z,</math> the region indicated by "<math>(x, y, z)\!</math>" is composed of the ''adjacent'' or the ''bordering'' cells.  Thus they are the cells that are just across the boundary of the center cell, arrived at by taking all of Leibniz's ''minimal changes'' from the given point of departure.

Any cell in a venn diagram has a well-defined set of nearest neighbors, and so we can apply a boundary operator of the appropriate rank to the list of signed features that conjoined would indicate the cell in view.

Any cell in a venn diagram has a well-defined set of nearest neighbors, and so we can apply a boundary operator of the appropriate rank to the list of signed features that conjoined would indicate the cell in view.

Figure 11.  Venn Diagram for ((x),(y),(z))

Figure 11.  Venn Diagram for ((x),(y),(z))

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Given the foregoing explanation of the ''k''-fold boundary operator, along with its use to express such forms of logical constraints as "just 1 of ''k'' is false" and "just 1 of ''k'' is true", there will be no trouble interpreting an expression of the following shape from the Jets and Sharks example:

Given the foregoing explanation of the ''k''-fold boundary operator, along with its use to express such forms of logical constraints as "just 1 of ''k'' is false" and "just 1 of ''k'' is true", there will be no trouble interpreting an expression of the following shape from the Jets and Sharks example:

We may note in passing that <math>(x, y) = ((x),(y)),\!</math> but a rule of this form holds only in the case of the 2-fold boundary operator.

We may note in passing that <math>(x, y) = ((x),(y)),\!</math> but a rule of this form holds only in the case of the 2-fold boundary operator.

Let's collect the various ways of representing the structure of a universe of discourse that is described by the following cactus form, verbalized as "just 1 of <math>x, y , z\!</math> is true".

Let's collect the various ways of representing the structure of a universe of discourse that is described by the following cactus form, verbalized as "just 1 of <math>x, y , z\!</math> is true".

Figure 15.  Quotient Structure Venn Diagram for ((x),(y),(z))

Figure 15.  Quotient Structure Venn Diagram for ((x),(y),(z))

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Let's now look at the last type of clause that we find in my transcription of the Jets and Sharks data base, for instance, as exemplified by the following couple of lobal expressions:

Let's now look at the last type of clause that we find in my transcription of the Jets and Sharks data base, for instance, as exemplified by the following couple of lobal expressions:

The same analysis applies to the generic form <math>(x, (x_1), \ldots, (x_k)),\!</math> specifying a pie-chart with a genus <math>X\!</math> and the <math>k\!</math> species <math>X_1, \ldots, X_k.\!</math>

The same analysis applies to the generic form <math>(x, (x_1), \ldots, (x_k)),\!</math> specifying a pie-chart with a genus <math>X\!</math> and the <math>k\!</math> species <math>X_1, \ldots, X_k.\!</math>

==Differential Logic : Series C==

==Differential Logic : Series C==