Changes

It is a classic catch, whose pattern was traced out long ago in the paradox of Plato's ''Meno''.  Discussion of this dialogue and of the task it sets for AI, cognitive science, education, including the design of intelligent tutoring systems, can be found in (H. Gardner, 1985), (Chomsky, 1965, '72, '75, '80, '86), (Fodor, 1975, 1983), (Piattelli-Palmarini, 1980), and in (Collins & Stevens, 1991).  Though it appears to mask a legion of diversions, this text will present itself at least twice more in the current engagement, both on the horizon and at the gates of the project to fathom and to build intelligent systems.  Therefore, it is worth recalling how this inquiry begins.  The interlocutor Meno asks:

It is a classic catch, whose pattern was traced out long ago in the paradox of Plato's ''Meno''.  Discussion of this dialogue and of the task it sets for AI, cognitive science, education, including the design of intelligent tutoring systems, can be found in (H. Gardner, 1985), (Chomsky, 1965, '72, '75, '80, '86), (Fodor, 1975, 1983), (Piattelli-Palmarini, 1980), and in (Collins & Stevens, 1991).  Though it appears to mask a legion of diversions, this text will present itself at least twice more in the current engagement, both on the horizon and at the gates of the project to fathom and to build intelligent systems.  Therefore, it is worth recalling how this inquiry begins.  The interlocutor Meno asks:

<blockquote>

{| align="center" cellpadding="8" width="90%" <!--QUOTE-->

Can you tell me, Socrates, whether virtue can be taught, or is acquired by practice, not teaching?  Or if neither by practice nor by learning, whether it comes to mankind by nature or in some other way?  (Plato, ''Meno'', p. 265).

</blockquote>

<p>Can you tell me, Socrates, whether virtue can be taught, or is acquired by practice, not teaching?  Or if neither by practice nor by learning, whether it comes to mankind by nature or in some other way?  (Plato, ''Meno'', p. 265).</p>

Whether the word "virtue" (''arete'') is interpreted to mean virtuosity in some special skill or a more general excellence of conduct, it is evidently easy, in the understandable rush to "knowledge", to forget or to ignore what the primary subject of this dialogue is.  Only when the difficulties of the original question, whether virtue is teachable, have been moderated by a tentative analysis does knowledge itself become a topic of the conversation.  This hypothetical mediation of the problem takes the following tack:  If virtue were a kind of knowledge, and if every kind of knowledge could be taught, would it not follow that virtue could be taught?

Whether the word "virtue" (''arete'') is interpreted to mean virtuosity in some special skill or a more general excellence of conduct, it is evidently easy, in the understandable rush to "knowledge", to forget or to ignore what the primary subject of this dialogue is.  Only when the difficulties of the original question, whether virtue is teachable, have been moderated by a tentative analysis does knowledge itself become a topic of the conversation.  This hypothetical mediation of the problem takes the following tack:  If virtue were a kind of knowledge, and if every kind of knowledge could be taught, would it not follow that virtue could be taught?

The primary factorization is typically only the first in a series of analytic decompositions that are needed to fully describe a complex domain of phenomena.  The question about proper factorization that this discussion has been at pains to point out becomes compounded into a question about the reality of all the various distinctions of analytic order.  Do the postulated levels really exist in nature, or do they arise only as the artifacts of our attempts to mine the ore of nature?  An early appreciation of the hypothetical character of these distinctions and the post hoc manner of their validation is recorded in (Chomsky, 1975, p. 100).

The primary factorization is typically only the first in a series of analytic decompositions that are needed to fully describe a complex domain of phenomena.  The question about proper factorization that this discussion has been at pains to point out becomes compounded into a question about the reality of all the various distinctions of analytic order.  Do the postulated levels really exist in nature, or do they arise only as the artifacts of our attempts to mine the ore of nature?  An early appreciation of the hypothetical character of these distinctions and the post hoc manner of their validation is recorded in (Chomsky, 1975, p. 100).

<blockquote>

{| align="center" cellpadding="8" width="90%" <!--QUOTE-->

<p>In linguistic theory, we face the problem of constructing this system of levels in an abstract manner, in such a way that a simple grammar will result when this complex of abstract structures is given an interpretation in actual linguistic material.</p>

<p>In linguistic theory, we face the problem of constructing this system of levels in an abstract manner, in such a way that a simple grammar will result when this complex of abstract structures is given an interpretation in actual linguistic material.</p>

<p>Since higher levels are not literally constructed out of lower ones, in this view, we are quite free to construct levels of a high degree of interdependence, i.e., with heavy conditions of compatibility between them, without the fear of circularity that has been so widely stressed in recent theoretical work in linguistics.</p>

<p>Since higher levels are not literally constructed out of lower ones, in this view, we are quite free to construct levels of a high degree of interdependence, i.e., with heavy conditions of compatibility between them, without the fear of circularity that has been so widely stressed in recent theoretical work in linguistics.</p>

To summarize the main points:  A system of analytic levels is a theoretical unity, to be judged as a whole for the insight it provides into a whole body of empirical data mediately gathered.  A level within such a system is really a perspective taken up by the beholder, not a cross-section slicing through the phenomenon itself.  Although there remains an ideal of locating natural articulations, the theory is an artificial device in relation to the nature it explains.  Facts are made, not born, and already a bit factitious in being grasped as facts.

To summarize the main points:  A system of analytic levels is a theoretical unity, to be judged as a whole for the insight it provides into a whole body of empirical data mediately gathered.  A level within such a system is really a perspective taken up by the beholder, not a cross-section slicing through the phenomenon itself.  Although there remains an ideal of locating natural articulations, the theory is an artificial device in relation to the nature it explains.  Facts are made, not born, and already a bit factitious in being grasped as facts.

One version of Peirce's sign definition is especially useful for the present purpose.  It establishes for signs a fundamental role in logic and is stated in terms of abstract relational properties that are flexible enough to be interpreted in the materials of dynamic systems.  Peirce gave this definition of signs in his 1902 "Application to the Carnegie Institution":

One version of Peirce's sign definition is especially useful for the present purpose.  It establishes for signs a fundamental role in logic and is stated in terms of abstract relational properties that are flexible enough to be interpreted in the materials of dynamic systems.  Peirce gave this definition of signs in his 1902 "Application to the Carnegie Institution":

<blockquote>

{| align="center" cellpadding="8" width="90%" <!--QUOTE-->

<p>Logic is ''formal semiotic''.  A sign is something, ''A'', which brings something, ''B'', its ''interpretant'' sign, determined or created by it, into the same sort of correspondence (or a lower implied sort) with something, ''C'', its ''object'', as that in which itself stands to ''C''.  This definition no more involves any reference to human thought than does the definition of a line as the place within which a particle lies during a lapse of time.  (Peirce, NEM 4, 54).</p>

<p>Logic is ''formal semiotic''.  A sign is something, ''A'', which brings something, ''B'', its ''interpretant'' sign, determined or created by it, into the same sort of correspondence (or a lower implied sort) with something, ''C'', its ''object'', as that in which itself stands to ''C''.  This definition no more involves any reference to human thought than does the definition of a line as the place within which a particle lies during a lapse of time.  (Peirce, NEM 4, 54).</p>

<p>It is from this definition, together with a definition of "formal", that I deduce mathematically the principles of logic.  I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non-psychological conception of logic has ''virtually'' been quite generally held, though not generally recognized.  (Peirce, NEM 4, 21).</p>

<p>It is from this definition, together with a definition of "formal", that I deduce mathematically the principles of logic.  I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non-psychological conception of logic has ''virtually'' been quite generally held, though not generally recognized.  (Peirce, NEM 4, 21).</p>

A placement and appreciation of this theory in a philosophical context that extends from Aristotle's early treatise ''On Interpretation'' through John Dewey's later elaborations and applications (Dewey, 1910, 1929, 1938) is the topic of (Awbrey & Awbrey, 1992).  Here, only a few features of this definition will be noted that are especially relevant to the goal of implementing intelligent interpreters.

A placement and appreciation of this theory in a philosophical context that extends from Aristotle's early treatise ''On Interpretation'' through John Dewey's later elaborations and applications (Dewey, 1910, 1929, 1938) is the topic of (Awbrey & Awbrey, 1992).  Here, only a few features of this definition will be noted that are especially relevant to the goal of implementing intelligent interpreters.

Why have I chosen differential geometry and logic programming to try jamming together?  A clue may be picked up in the quotation below.  When the foundations of that ingenious duplex, AI and cybernetics, were being poured, one who was present placed these words in a cornerstone of the structure (Ashby, 1956, p. 9).

Why have I chosen differential geometry and logic programming to try jamming together?  A clue may be picked up in the quotation below.  When the foundations of that ingenious duplex, AI and cybernetics, were being poured, one who was present placed these words in a cornerstone of the structure (Ashby, 1956, p. 9).

<blockquote>

{| align="center" cellpadding="8" width="90%" <!--QUOTE-->

The most fundamental concept in cybernetics is that of "difference", either that two things are recognisably different or that one thing has changed with time.

The most fundamental concept in cybernetics is that of "difference", either that two things are recognisably different or that one thing has changed with time.

A deliberate continuity of method extends from this use of difference in goal-seeking behavior to the baby steps of AI per se, namely, the use of difference-reduction methods in the form of what is variously described as means-ends analysis, goal regression, or general problem solving.

A deliberate continuity of method extends from this use of difference in goal-seeking behavior to the baby steps of AI per se, namely, the use of difference-reduction methods in the form of what is variously described as means-ends analysis, goal regression, or general problem solving.

Legend tells us that the primal twins of AI, the strife-born siblings of Goal-Seeking and Hill-Climbing, began to stumble and soon came to grief on certain notorious obstacles.  The typical scenario runs as follows.

Legend tells us that the primal twins of AI, the strife-born siblings of Goal-Seeking and Hill-Climbing, began to stumble and soon came to grief on certain notorious obstacles.  The typical scenario runs as follows.

<blockquote>

{| align="center" cellpadding="8" width="90%" <!--QUOTE-->

At any moment in time the following question is posed:<br>

<p>At any moment in time the following question is posed:<br>

In this problem space how ought one choose to operate<br>

In this problem space how ought one choose to operate<br>

in order to forge of one's current state a new update<br>

in order to forge of one's current state a new update<br>

that has hopes of being nearer to one's engoaled fate?

that has hopes of being nearer to one's engoaled fate?</p>

</blockquote>

But before Jack and Jill can start up the hill they will need a whole bucket of prior notions to prime the pump.  There must be an idea of distance, in short, a metric function defined on pairs of states in the problem space.  There must be an idea of direction, a longing toward a goal that informs the moment, that fixes a relation of oriented distances to transition operators on states.  Stated in linguistic terms the directive is a factor that commands and instructs.  It arranges a form of interpretation that endows disparities with a particular sense of operational meaning.

But before Jack and Jill can start up the hill they will need a whole bucket of prior notions to prime the pump.  There must be an idea of distance, in short, a metric function defined on pairs of states in the problem space.  There must be an idea of direction, a longing toward a goal that informs the moment, that fixes a relation of oriented distances to transition operators on states.  Stated in linguistic terms the directive is a factor that commands and instructs.  It arranges a form of interpretation that endows disparities with a particular sense of operational meaning.

A symbolic calculus is needed to assist our reasoning and computation in the realm of propositions.  With an eye toward efficiency of computing and ease of human use, while preserving both functional and declarative properties of propositions, I have implemented an interpreter and assorted utilities for one such calculus.  The original form of this particular calculus goes back to the logician C.S. Peirce, who is my personal favorite candidate for the grand-uncle of AI.  Among other things, Peirce discovered the logical importance of NAND/NNOR operators (CP 4.12 ff, 4.264 f), (NE 4, ch. 5), inspired early ideas about logic machines (Peirce, 1883), is credited with "the first known effort to apply Boolean algebra to the design of switching circuits" (M. Gardner, p. 116 n), and even speculated on the nature of abstract interpreters and other "Quasi-Minds" (Peirce, CP 4.536, 4.550 ff).

A symbolic calculus is needed to assist our reasoning and computation in the realm of propositions.  With an eye toward efficiency of computing and ease of human use, while preserving both functional and declarative properties of propositions, I have implemented an interpreter and assorted utilities for one such calculus.  The original form of this particular calculus goes back to the logician C.S. Peirce, who is my personal favorite candidate for the grand-uncle of AI.  Among other things, Peirce discovered the logical importance of NAND/NNOR operators (CP 4.12 ff, 4.264 f), (NE 4, ch. 5), inspired early ideas about logic machines (Peirce, 1883), is credited with "the first known effort to apply Boolean algebra to the design of switching circuits" (M. Gardner, p. 116 n), and even speculated on the nature of abstract interpreters and other "Quasi-Minds" (Peirce, CP 4.536, 4.550 ff).

<blockquote>

{| align="center" cellpadding="8" width="90%" <!--QUOTE-->

Thought is not necessarily connected with a brain.  It appears in the work of bees, of crystals, and throughout the purely physical world;  and one can no more deny that it is really there, than that the colors, the shapes, etc., of objects are really there.  (CP 4.551).

</blockquote>

<p>Thought is not necessarily connected with a brain.  It appears in the work of bees, of crystals, and throughout the purely physical world;  and one can no more deny that it is really there, than that the colors, the shapes, etc., of objects are really there.  (CP 4.551).</p>

One could hardly invent a better anthem for the work being done today in the AI/systems hybrid areas of cellular automata (Burks, 1970), (Ulam, ch. 12), (Nicolis & Prigogine, 1989), emergent computation (Forrest, 1991), and "society of mind" theories (Minsky, 1986).  I hope it will emerge that these workers achieve the same grade of well-honed insight regarding the mind's apical functions that Peirce was able to inspire, having once acquired a taste for it in the higher combines of logic's hive.

One could hardly invent a better anthem for the work being done today in the AI/systems hybrid areas of cellular automata (Burks, 1970), (Ulam, ch. 12), (Nicolis & Prigogine, 1989), emergent computation (Forrest, 1991), and "society of mind" theories (Minsky, 1986).  I hope it will emerge that these workers achieve the same grade of well-honed insight regarding the mind's apical functions that Peirce was able to inspire, having once acquired a taste for it in the higher combines of logic's hive.