A 4-Dimensional Bifurcation Map

II. Dynamical Systems Theory

A. Introduction to Dynamical Systems Theory [DST]

B. What is a Complex, Nonlinear System?

C. Tools of DST

1. Phase Space
2. Attractors
3. Bifurcation Diagrams
4. Fractal Geometry

D. Characteristics of Fractal Geometry

* Bifurcation Diagrams
* How a Bifurcation Diagram is Produced
* Four Characteristics of Dynamical Systems

III. Linguistic Categorization

A. Traditional Categorization

B. The Prototype System of Categorization

* How a Category May be Formed via the Prototype Theory

* Characteristics of the Prototype System

* Additional Characteristics of the Prototype System

IV. The Prototype System as a Dynamical System

A. Complex forms

B. High Sensitivity to Initial Conditions

C. Feedback Mechanisms and Nonlinearity

* The Prototype System Applied to the Category "OBSCENE"
* How Meaning Emerges in the Dynamic Prototype System of Categorization
* Modeling "OBSCENE" through the Attractor States
* How to Deal with the "OBSCENITY" Issue
* Category Formation and the Mandelbrot Set

V. Conclusion

VI. Endnotes

VII. Appendix

VIII. References

We will begin by introducing DST. This will be a discussion of the subject matter of DST that will define terms, and will introduce the tools of DST, including: "phase space;" "attractors;" the "bifurcation diagram;" and "fractal geometry." Four important characteristics of dynamical systems will then be introduced. Then, we will turn our attention to the subject of linguistic categorization.

We will discuss the traditional explanation of the system of categorization and outline its assumptions. The traditional approach, it will be argued, fails to account for the adaptive nature of categorization observed through experience with a language. The traditional system is found to be more rigid than it is adaptive. This will lead us into a discussion of the prototype theory of categorization.

It will be argued that the prototype system is a more accurate description of how we actually categorize the world. This is because it is found to be more adaptive than the traditional approach, while still maintaining stability. We will then begin an explication of the prototype system by first finding its roots in the work of the later Wittgenstein and his notion of "family resemblances." From there, we will explore the workings of the prototype system of categorization. Characteristics of this system will be outlined that contrast with those of the traditional system. It will be found that several characteristics are shared by both the prototype theory and dynamical systems by comparing the characteristics of each. This is where the information provided thus far will begin to come together.

From there, we will reiterate the characteristics of dynamical systems and explore by illustration and discussion how and why the prototype system can be considered a dynamical system. At this point, we will illustrate one example of how the tools of DST, when applied to the prototype theory of categorization, may influence descriptions of category formation in the criminal justice system. Using the tool of "attractors" from DST, the systemic development of a jury’s category construction is described in various qualitatively distinct states. Finally, we will conclude with a recapitulation of the important points covered throughout the paper.

Introduction to Dynamical Systems Theory

Dynamical systems theory [DST] is a burgeoning interdisciplinary effort that explores complex, non-linear systems. This "new science," as it has been called by James Gleick (1987) alternatively goes by the names "chaos theory," or "complex systems theory." These appellations are often used interchangeably. In this paper, I will primarily use the term "dynamical systems theory," for I think that this term best evokes the important components of the subject matter at hand; namely, a system of human categorization that is dynamic, as opposed to static.

DST’s origins are found primarily in physics, mathematics and biology, though advancement of the field has come from such diverse disciplines as meteorology, business management, and information systems, among many others. Though the applicability of DST is well proven in the natural sciences, application to the social sciences has been somewhat recalcitrant.

We might point out, in searching for reasons for this lag, that the social sciences are, in the main, driven by a linear paradigm that can be traced back to Newtonian physics. The social sciences have been developed with this paradigm embedded as its assumptions. We may therefore posit one possible reason for the social sciences dragging behind the natural sciences in nonlinear investigation; the social sciences have been caught in the inertia of hundreds of years of a linear paradigm and since it is only recently that the natural sciences have made nonlinear investigations manageable, the social sciences have not yet caught up.

Another possible reason for DST coming slow to the social sciences may be found in the difficulty in measuring the system being observed. Because of the extremely complex nature of social systems, locating the important variables contributing to the behavior of these systems has been difficult (Vallacher and Nowak, 1994; 279-8). While it is possible to simplify a model of a complex system to attain results, these models have little, if any, information to tell us about the original system.

Despite these difficulties, many social scientists have succeeded in using DST to aid in their research. This has been made possible primarily through the advancement of computer technology1. The tools of DST, made available through this technology, are applied by metaphor, or analogy by social scientists. They model social systems on those of the more easily observable and reproducible systems of the natural sciences and mathematics. This practice follows a precedent that has long been established among social scientists. Historically, some of the most remarkable work in the social sciences has been the result of such an approach. I need only point to a few examples.

In our modern, post-Newtonian period of philosophy, many efforts were made to utilize the Newtonian paradigm in the social sciences. Of particular note is David Hume, who paralleled Newton’s fundamental units of study; Newton’s atoms, molecules and compounds were paralleled with Hume’s "impressions" and "ideas" and Newton’s three laws of motion were paralleled with Hume’s own three laws of motion that governed the movement between these impressions and ideas (Hume 1977, 48-9). The force and success of the Newtonian paradigm influenced such thinkers as Hume to the benefit of generations to come. A second example of a social scientist following the lead of scientific advancement elsewhere (mathematics in this case) is found in the work of Spinoza. I refer to Spinoza’s ethics that were based directly on Euclidean methodologies (Russell 1972, 572). Spinoza deduced an entire system of ethics from some axiomatic principles as Euclid did with his geometry. These examples show not only the precedence of the application of the methodologies of natural sciences to the social sciences, they also remind us of the possible utility of such efforts. I will use DST through analogy in this paper to consider its applications to the human system of categorization. Let us now turn to examine DST, its subject matter, and its tools.

As was said of above, DST deals with complex, non-linear systems. What is a "complex non-linear system?" First, a "system" is any group of interacting elements that maintains some stability in their relations over time (Vallacher and Nowak 1994, 139). A linear system is one whose elements interact with strict causal regularity (Hayles 1990, 11). This means that, in a linear system, cause and effect are directly proportional; a large cause will produce a large effect. As an illustration, when I tap my pen on the table before me, the volume of sound produced increases as I increase the force with which I tap. There is a strict causal relationship between the force with which I tap and the volume of sound produced. As an example in the social sciences, if it were argued that a business contract was lost due to the tardiness of one member of a sales team at a business meeting, the thought process underlying this argument is clearly linear. Assumed in this argument is a strict causal relationship between the tardiness of the salesperson and the loss of the contract. This is in effect arguing that the salesperson’s tardiness was the "weak link" in securing the contract. Any time a "weak link" is said to have been discovered in an argument, or in an event for that matter, there is a linear thought process at work. The very metaphor from which the expression "weak link" arises relies on a conception of an argument as a linear chain of propositions arranged in some sequential, unidirectional ordering. Here is part of the linear picture.

A second part of the picture is that "linear" systems are defined as such due to the shape of their trajectories when plotted on a mathematical map. These trajectories consist in straight lines. This geometric regularity allows an exacting measurement to be taken of the trajectory. So, "linearity" may refer to either the type of causal relationship the elements are involved in, or it may refer to the geometry of the trajectory of the system when mathematically plotted. In sum, linear systems are systems whose elements maintain a strict causal relationship to one another, and whose trajectories in mathematical space are plotted as straight lines.

A non-linear system, on the other hand, is a group of interacting elements that have mutual influence on one another. For example, a basketball team, viewed as a system, can be described as possessing mutually interacting elements. If the center is injured on a play, the rest of the team will compensate. The one element (the center) effects all the others and this, in turn, allows the center to recuperate. So, not only is the center influencing the rest of the team, the team-members’ actions influence the center and each other as well. Indeed, characteristic of dynamic systems is the ability to produce adaptive behavior (Wegner and Tyler 1993, 21; Vallacher 1994, 139).

Also, in contrast to linear systems, nonlinear systems demonstrate a disproportionate relationship between cause and effect (Gleik 1988, 23). The changing of a seemingly minor element, such as the absence of our basketball team’s second-string center, may produce unpredictably drastic changes in the system, especially over time. The multiplying of an effect over time is due to "iteration" or feedback in the system. Briggs and Peat define iteration as "feedback involving the continual reabsorption or enfolding of what has come before..." (1989, 66). So, in contrast to linear systems, nonlinear systems display mutually interacting elements and a disproportionate relationship between cause and effect. In addition, nonlinear systems produce geometrically irregular trajectories that are difficult to quantify by traditional means.

So much for our explanations of the "nonlinear," and the "systems" parts of "complex, nonlinear systems;" what about the "complex" part? "Complexity" refers to the number of variables used to describe a system, and all of their possible relationships to one another. Different systems will have differing numbers of contributing elements and these elements will differ in their interrelations. Complexity is a matter of degrees. Because of the seemingly infinite number of variables affecting most dynamical systems, the effort to exactly quantify the degree of complexity at any given time will very likely be in vain. There are, however, methods of quantifying complexity in some arenas.

Fractal geometry, the invention of Benoit Mandelbrot2, is one method of doing so for spatial complexity. An irregular curve, such as the coastline of Britain, is given a numerical, fractal dimension (between integers). This number will refer to the spatial dimensions of an object. In classical, Euclidean geometry, an object may have zero, one, two or three dimensions; a point has zero dimensions, a line has one, a square has two dimensions, and a cube has three. Contrast this with fractal geometry where an object may have a fractional dimension . In our example, the coastline of Britain has been given a fractal dimension of 1.2 (Wegner and Tyler 1993, 17); the coastline has greater than one dimension, though less than two. Thus, the relative degree of complexity (the degree of irregularity) is quantified.

Now, having some general understanding of the subject matter of DST, we will canvass some of the tools that computer technology lends to the investigations of DST.

In this section, we will introduce the tools of DST including: "phase space," "attractors," "bifurcation diagrams" and "fractal geometry." The first tool we will consider is called "phase space."

Recall that "linearity" was described above in two parts and that the second part of this explanation said that the term "linearity" describes a system’s trajectory plotted on a mathematical map. This mathematical map is known as a "phase map." The phase map accommodates all possible movement of a system over time; the total possible movement of a system is known as its "phase space." This also refers to the mathematical space in which a system’s trajectory is plotted. The phase map is determined by coordinates that represent variables describing the system; a simple system, such as a pendulum can be described with two coordinates: position and velocity. For a more complex system, additional axes are added to the phase map for each additional variable necessary to describe the system.

Assume that our pendulum is of the most simple design. When pushed, the pendulum will swing back and forth until the friction on the system brings it to a halt. When plotted on a phase map, the trajectory of the pendulum will look like a spiral. The point at the center of the spiral locates the trajectory of the system at zero displacement and zero velocity. The pendulum seems to be attracted to this point in phase space. For this reason, this system, and ones with similar trajectories in phase space, are characterized as a "spiral point attractors," or more simply, "point attractors" (Peat 1991, 178). There are several types of "attractors" used to characterize systems, the spiral point being the most predictable. The concept of "attractors" is the second tool of DST that we will need to understand.

As was said above, the simple pendulum seems to be attracted to one particular value in phase space. An "attractor" is an area of phase space that are values that a system tends to over time (Vallacher and Nowak 1994, 140-1; Peat 1991, 179). Briggs and Peat describe an attractor a bit more poetically; "An attractor is a region of phase space which exerts a ‘magnetic’ appeal for a system, seemingly pulling the system toward it" (1989, 36). To create the image of an attractor, the movement of given system is plotted in phase space at various time intervals. On each occasion that the values of the system are plotted, a dot is installed on the phase map. For our simple pendulum, this dot represents the values of the two coordinates (position and velocity) at a given time. As a result of these plottings, an image begins to appear over time that is the composite of each plotting; this is the trajectory of the system. Attractors are useful for characterizing the behavior of a system; the same system can be characterized with different types of attractors at different times in its development. Attractors are also useful for comparing two or more systems’ behavior.

As was said previously, there are several other kinds of attractors. Now, having some understanding of what an attractor is and how one is produced, we will consider the "limit cycle," the "torus," and the "strange" attractors. We will begin with the "limit cycle" attractor.

Taking our simple pendulum and giving it a timed electrical kick would cancel out the dissipative effects of friction on the system. For this reason, the trajectory in phase space would no longer be a spiral, but rather, it would be a circle due to the now consistent values of both location and velocity. This pendulum’s trajectory is described as a "limit cycle" attractor because the trajectory is limited to the cycle of revolution around a circular form (Peat 1991, 179). As long as the electrical kick remains (providing other conditions haven’t changed) the trajectory will trace itself in phase space. Both the spiral point and the limit cycle attractors describe systems that are highly regular, and thus, predictable. The system that the spiral point describes will always come to a halt and that of the limit cycle will forever repeat its loop. Characteristic of the limit cycle attractor is the ability to resist change (Peat 1991, 181). A perturbation added to the system will cause the system to oscillate between values before quickly returning to a state of stability.

The torus attractor is the next attractor to consider moving up in complexity. This attractor requires three or more axes on the phase map. It can have three or more dimensions, each dimension being represented by another axis on the phase map. The torus looks like a donut in phase space. The system seems to be attracted to the surface area of the torus form in phase space. The torus attractor describes a system that has several degrees of freedom (possible directions). This system is globally predictable, though locally uncertain. Elsewhere, I have used the example of a mail-carrier to illustrate this point (Cislo1996, 21). The position of the mail-carrier, determined by plotting his or her mail-route over many days, would be globally predictable; we could predict that on a given day the mail-carrier would be in a certain area of town, but at any given time it would be difficult to predict where the mail-carrier would be.

The torus attractor may describe two limit cycles acting on one another. Depending on the period ratio of these two cycles (a period is one revolution around the loop) the torus attractor may become highly "sensitive to initial conditions." This results from the two cycles having a "periodic relationship." This means that their cycles meet up and as a result, a small perturbation may be fed back, or iterated back into the system, amplifying the perturbation over time. Just as in the example of the basketball team with the missing second-string center, this perturbation, iterated over time may produce drastically unpredictable effects. The result of this may be the torus attractor exploding in phase space (Peat 1991, 189); this is the origin of the "strange" attractor.

The "strange" attractor describes a system in phase space that seems to be attracted to the infinity of fragmentations left from the explosion of the torus. This unusual behavior is why the strange attractor is called "strange." Its shape in phase space defies any classical geometric assignment; it is a fractal. This attractor describes systems that are in a state of turbulence, such as a violent river or the wake of an aircraft. There still may be an overall pattern to the system, but there is little, if any, predictability.

Characteristics of Fractal Geometry

As was mentioned above, fractal objects or images are characteristically complex objects. Fractals, such as the Mandelbrot set, are also characteristically "self-similar" at all levels of magnification. This refers to an object’s repetitive pattern. A leaf, for example, may repeat the same patterns found in the entire plant from which it came (think of the fern plant). This description, so pervasive in objects of the natural world, is also relevant to mathematically generated images, such as the Mandelbrot set.

Without getting into too much detail, the Mandelbrot set (among other generated fractals) is created by running a non-linear equation and adding the product of that equation to a constant. This sum is fed back, or iterated, into the equation. After each iteration, the product of the equation is plotted on the complex plane. The axes of the complex plane are graded with coordinates that allow the determination of inclusion in or exclusion from the Mandelbrot set to made; the Mandelbrot set consists in the products of those equations that fall within the circumference of two on the complex plane after an arbitrary number of iterations. This work is carried out on the computer which assigns a color, or gradation of gray, given parameters, to each pixel on the computer screen where the products are plotted. The result is the mathematically generated fractal image known as the Mandelbrot set (Appendix A, Figure 1).

The term "self-similarity" in relation to the Mandelbrot set (our second characteristic of fractals under consideration) not only refers to the similarity noticed at first glance, as described with the fern plant, it also refers to a more deeply rooted structural similarity that is found by "zooming in" on the border area of the image (this is facilitated by use of the computer). Magnification of this border area reveals smaller or "baby Mandelbrots" that are equally complex; this pattern of similarity goes on ad infinitum.

Another characteristic of fractals is termed "high-sensitivity-to initial-conditions" [HSIC]. This term applies most conspicuously to mathematically generated fractals, being their genesis is understood as coming from the finite mathematical equation; thus, the (system’s) initial conditions are easily accessible. Briefly, this means that when running equations for the Mandelbrot set, for instance, two separate equations that are identical, though have slightly different input values, can have remarkably different results after iteration. The difference of even one digit in the thousandth decimal place can make one equation part of the set and the other a case of utter divergence after iteration. The initial conditions in this case are the two equations that we started with before iteration. High sensitivity to these initial conditions is the result of iteration; the difference between the two original equations is amplified due to each iteration.

So, we have described three characteristics of fractal geometry: (1) it describes irregular objects or images by an assignment of complexity, (2) the objects or images described are self-similar at all levels of magnification, and (3) these objects or images possess high-sensitivity-to-initial-conditions. Our next tool of DST to consider is called the "bifurcation diagram."

Like the phase map, the bifurcation diagram is a mathematical map on which the trajectory of a system is plotted (Appendix A, Figure 2). Where the phase map produces attractors that allow the behavior of a system to be viewed in some particular behavioral mode, e.g., spiral point, limit cycling, etc., the bifurcation diagram allows systemic behavior to be viewed still more globally, moving through the various attractor states. These various attractor states are preceded by what is called a "bifurcation." A "bifurcation" is a split in the trajectory of a system (Briggs and Peat 1989, 143; also see Gleik 1988, 71-8). With each bifurcation up to the strange attractor, the possibilities of the system doubles. Briggs and Peat have explained the bifurcation diagram using an example of a gypsy moth population (1989, 54-65). Being investigated, is the changing year to year population of these gypsy moths. For much of what is to immediately follow, I am indebted to Briggs’ and Peat’s illustration.

Secondly, there is need for a brief explanation of what the values on the horizontal axis represent. The values on our horizontal axis (that of birthrate) are numbers that represent the ratio between the population number and the number of progeny delivered. As an example, the value 3.5 would mean that the birthrate is 3.5 times the number of the previous population. If the starting population was one hundred moths, and the birthrate was 3.5 times the population number, then the birthrate would be 350, the ratio being 3.5:1.

One more piece of background information is needed before we demonstrate the use of the bifurcation diagram with our gypsy moth example. The "Verhulst equation" is a mathematical equation has been used to describe such varied phenomenon as population growth, how a rumor spreads, and theories of learning. What each of these systems have in common is that the future of each system depends nonlinearly on what came before. The Verhulst equation allows nonlinear, as opposed to exponential (linear) growth to be studied. The right hand side of the Verhulst growth equation [Xn+1=BXn(1-Xn)] has a self-balancing effect. When the Xn grows large, the (1-Xn) part grows smaller. To Multiply Xn by (1-Xn) is to multiply Xn by itself, this produces feedback and non-linearity. Of interest to note is that all systems in which the equation applies holds a potential for chaos (Briggs and Peat 1991, 56-7), or orderly disorder (the strange attractor).

Now, with that ground-clearing done, we can focus in on the bifurcation diagram. Assuming that some birth control, such as insecticide, has been administered to the system, and that the birthrate is no more than one, the population can be expected to decrease some every year.

By making "birthrate" the control parameter and manipulating the value of it, we can watch the behavior of the system change with each value. Raising the birthrate to 1.5 will cause the total population figure to reach a steady state of approximately sixty-six percent of the original population. Raising the control parameter still further to 2.5, and even to 2.98 will have the same result, the system will level out at approximately two thirds, or sixty-six percent of the original population. The value sixty-six percent seems to be an attractor for this system. But when the control parameter value is pushed to 3.0, something entirely different will happen; the sixty-six percent attractor becomes unstable and bifurcates, or splits in two. Now, the system will fluctuate between these two new values. Instead of one possibility for the system, there are now two.

Then, if we increase the control parameter beyond 3.4495, another bifurcation will occur, resulting in four possible courses for the system. Raising birthrate again past 3.56, still another bifurcation will occur, resulting in eight outcome basins. At 3.596 another bifurcation will occur, this time resulting in sixteen values that the system will equivocate between. And then, when the birthrate reaches 3.56999, the number of possibilities for the system explodes to infinity. At this point there is little, if any, predictability. The system at this point is characterized with the strange attractor.

Indeed, the bifurcation diagram corresponds to each of the attractors at different stages of systemic development. Before the first bifurcation, the system gravitates toward a point attractor. After the first bifurcation, the system gravitates toward the limit cycle attractor. After the second bifurcation, a torus attractor applies. The third and fourth bifurcations bring a torus attractor with increased dimensionality. And finally, after the last bifurcation at 3.56999, the strange attractor applies.

The bifurcation diagram not only displays this route to the strange attractor, it also shows several cycles of this progression repeating over and again. After the system reaches the point of the strange attractor, and is characterized by seemingly complete disorder, the system maintains this behavior for some time before spontaneously giving way to order once again. Order emerges and the entire cycle repeats itself. The point at which order begins to emerge once again is known as a "period of intermittency." This looks like a bar of relative inactivity on the bifurcation diagram. From disorder, the system organizes itself, and can be characterized once again with the point attractor, and then the limit cycle, the torus and the strange attractor. This is followed by another period of intermittency where the cycle repeats, and so on, and so forth.

What can we learn from this bifurcation diagram? Perhaps most notably, we learn that a dynamic system leading up to and through a state of chaos is not completely random. There is an ordered progression through the attractor states; the points at which the system bifurcates are universally valid. In 1975, Mitchell Feigenbaum made this discovery, and it has since been confirmed to be universal by various scientists working in a variety of disciplines with a variety of applications. The bifurcation points will always be at 3.0, 3.4495, 3.56, 3.596, and 3.5699.

Three other lessons are learned with the bifurcation diagram by noticing the way disorder spreads through phase space. First, there is a regular outline to the trajectory of the system looked at as a silhouette. Secondly, the way disorder spreads within the field is orderly. Notice the dark lines spreading out in parabolas. These lines represent places that the system is more likely to be found. Finally, the periods of intermittency show the disordered system spontaneously becoming stable and predictable again (Briggs and Peat 1989, 60-2).

These lessons have lead scientists to redefine the term "chaos." "Chaos" has long been thought to be void of information and so, nothing meaningful, it was thought, could be said about it. Now, with this new evidence, we see that there is order within disorder itself. Now, the distinction is made between randomness, which offers no information, and chaos, which as we have seen, we can say meaningful things about. This distinction and the lessons that brought it forth is what justifies Gleik calling chaos theory a "new science."

Chaos theory, or alternatively, dynamical systems theory, takes as its primary focus the strange attractor. This is because, through tools, such as the bifurcation diagram, we can now say meaningful things about a system in the strange attractor state that was previously thought to be complete disorder. What can be said about this most unusual of systemic attractors? Primarily, the strange attractor cannot be characterized using traditional Euclidean geometry. When a system jumps from the torus attractor to the strange attractor, a jump is also made from Euclidean geometry to fractal geometry. The strange attractor is of a fractal dimension. Here is what Peat has to say on the matter: "A strange attractor occupies a finite region of phase space, yet its surface area is infinite, its surface has no slope, and as a mathematical function, it cannot be differentiated because its detail is endless" (1991,189). This same description could be applied to the coastline of Britain. Both the coastline of Britain and the strange attractor are fractals. In addition to this, the bifurcation diagram, after the 3.56999 bifurcation, displays another image of the strange attractor (Briggs and Peat 1989, 60). Thus, we can apply the three descriptions given above about this region of the bifurcation diagram to the strange attractor.

Nonlinearity refers to the disproportionate causal relationship exhibited between elements of a system. A small cause can produce an unpredictably large effect over time. This is due to our second characteristic.

Feedback mechanisms produce a nonlinear causal relationship. "Feedback" refers to the recursive nature of the system. In mathematics this is referred to as "iteration." In an iterative system, the output of the system is fed back into it as an input. Thus, the system is critically dependent on what came before. Feedback, or iteration produces our third characteristic.

High sensitivity to initial conditions is the result of feedback mechanisms and nonlinearity. Because a small cause can produce a disproportionately large effect after iteration, any perturbation, no matter how small, cannot be thought of as inconsequential. This small effect may become magnified and exaggerated over time through iteration. This is sometimes called "the butterfly effect" in DST because it has been said by meteorologist Edward Lorenz that a butterfly flapping its wings in Peking can produce a rainstorm in New York City (Gleik 1988, 8).

Complex forms refers to the trajectory of a system in phase space - dynamical systems tend toward the strange attractor, a fractal in phase space. Fractals are self similar at all levels of magnification. That means that a replica of the fractal image can be seen again and again at increasing levels of magnification.

Now, having some understanding of DST, we can turn our attention to linguistic categorization.

In the traditional system, word meanings are built up from semantic primitives. These are the "atomic" structures of meaning; they cannot be further broken down into constituent parts. For Aristotle, these "semantic features," as they would later be called, constituted the "essence" of the subject. A word is defined by semantic features through the conjunction of "necessary and sufficient conditions." For example, the essence of "BACHELOR" is constituted by the combination of the semantic features [HUMAN], [MALE], [ADULT], and [NEVER MARRIED]. All are necessary for any case to be considered a member of the category "BACHELOR" and these features alone are enough, or sufficient, to consider any case part of the category.

Taylor points out eight assumptions of the classical approach (1995, 23-29)3.

Because linguistic categorization needs to have some stability in order to make language useful, each of these five assumptions were considered beneficial in the scholastic approach. The reductive attack in this system provides a comforting stability through its binary nature that is intended to avoid contradiction. However, a problem arises out of this very virtue. Can we, by this system, experience and communicate degrees of "justice," or "culpability?" Are there no degrees of "tall" or sensations of "red" to be spoken of? It seems clear that we actually do experience these things as a matters of degrees. An account of human categorization must be descriptive as opposed to prescriptive. There is no utility in prescribing how we ought to categorize the world (outside of ethics). Does the traditional approach meet the demands of a theory of categorization? Here is what Geeraerts calls for in a theory of cognition....

"Cognition should have a tendency toward structural stability: the categorial system can only work efficiently if it does not change drastically any time new data crop up. To prevent it from becoming chaotic, it should have a built-in tendency toward structural stability, but this stability should not become rigidity, lest the system stops being able to adapt itself to the ever-changing circumstances of the outside world..." (Taylor 1995, 53-54).

So, for Geeraerts, cognition, and therefore, our theories of cognition, must be adaptive in being able to accommodate new circumstances, while maintaining some structural stability. This seems intuitively correct. I submit (in agreement with Geeraerts) that because of "semantic primitives" and "necessary and sufficient conditions," and all that logically follows from them, the scholastic model is more rigid than it is adaptive and has difficulty in accommodating ambiguous cases; it does not permit degrees of category inclusion. The relevance of this shortcoming will become clear in the work of the later Wittgenstein, to which we will now turn our attention in opening a discussion of the prototype system of categorization.

The prototype theory of categorization is the logical conclusion to the work of the later Wittgenstein. In his "Philosophical Investigations," Wittgenstein asked a seemingly simple question; he asked his readers to define the word "game" (Wittgenstein 1953, sec. 66). One would begin this task by first listing examples of "games." One would include board-games, card-games, sporting-games, billiards, war-games, and even playing catch - anything that is spoken of as a game would be included. These examples would then be broken down in search of their necessary and sufficient semantic conditions. By calculating what semantic primitives are present in all cases one would attempt to determine the necessary and sufficient conditions for class inclusion in the category "GAME," thereby formulating a definition of "game."

The list of common features, one would imagine, would include such features as [THERE ARE WINNERS AND LOSERS] and [PROVIDES AN ELEMENT OF SATISFACTION AND/OR FUN]. But, are there always winners and losers in a game of catch? Is there always a feeling of satisfaction or fun when one participates in a highly competitive sporting event, for example? It becomes immediately apparent that there is no set of semantic features that is applicable to every example of the category "GAME," and some examples possess few and sometimes no features in common with the other examples. There seems to be no evident "necessary and sufficient conditions" with which to find class inclusion in the category "GAME." One could easily extend this to other categories, such as "OBSCENE," "INTENT" or "CONSENT."

By asking this seemingly simple question, Wittgenstein pointed out a major flaw in the classical approach; the boundaries of the category "GAME" are irregular. This led to his coining of the term "family resemblances" to describe the relationship that examples of a category share (Wittgenstein 1953, sec.67). This metaphor evokes the image of the family whose members may have many distinguishing features in common, yet possess some features that are unique to individuals, as a family may share the same color hair, yet have differing eye colors.

What then, if not necessary and sufficient conditions, determines how we categorize the world? Wittgenstein implicitly suggested, through his "family resemblances" metaphor, that there exists degrees of family, or categorical membership based on shared family resemblances. This, in turn would suggest that there is some method by which members of a family could be judged for level of class inclusion. This lead such scientists as Eleanor Rosch to suggest that we make judgments of class inclusion against a prototype, or exemplary member of the category4. Recent psychological and linguistic empirical evidence supports this position5. Prototypes seem to be the way by which we actually determine class inclusion.

The prototype is conceptualized as an ideal member that is actually nonexistent8, or alternatively, as an existing member that has the most amount of common features and least amount of distinctive features. Immediate family members have, as a group, the least amount of distinctive features and the most common features and so are considered part of the category. Members that fall within the irregular border area of this set are relatives; they may or may not be considered family members at a given time. And those that fall outside this border area all together are cases of utter categorical divergence; they are not related. Visualized as such, around a prototype, we can see how members of a category form a "family" that resemble each other in important ways, yet remain distinctive. Just like a human family. Let us now turn our attention to Figure 3 for further elaboration.

Figure 3 may represent any category. Here it represents the category "GAME"9. The letters in phase space represent various activities that we may call by the signifier "GAME." Letter "A" may be, for example, a variable for "baseball." In Figure 3, variables A, B, C, D, E, F, and G may be considered within the immediate family, while variables I, J, K, L, M, and H may be considered border cases, or relatives and O, P, R, Q and N may be cases of utter divergence. It is important to emphasize that the border area around the immediate family is best conceptualized as irregular and this border may change due to conditions, or context. The family is determined by category usage in a given society or culture at a particular point in time as interpreted by the subjective agent and thus becomes what might be considered a temporary area of definition. In some situations, the complexion of the entire family may change due to the use of metaphoric language, for instance.

The category "GAME" possesses entailments that determine class inclusion. The central members of the family "game" will (under normal conditions) likely possess the following entailments:

ENTAILMENTS OF GAME:

*activity

*not necessary for survival

*element of competition

*employs a strategy

*rules

*element of satisfaction, pleasure and/or fun

Just as the examples of "GAME" possess entailments that determine class inclusion, these entailments possess entailments as well. Our idea of "BASEBALL" includes certain entailments such as: activity, element of competition, rules, element of satisfaction, played with a bat, etc.. These entailments are not exclusively descriptors of our standard, American "BASEBALL," as they could be applied to what is commonly called "stick ball," being played by children in an alley, or a variety of other cases. There are no necessary and sufficient conditions in this system. The important entailments in "BASEBALL" are what is under consideration when determining if a particular activity may be referred to with a particular signifier. Again, what is considered important is determined by subjective agents embedded within a particular culture and context. At this time, in this culture, the employment of the standard baseball is generally considered an important entailment for an activity to be considered a member of the family "BASEBALL;" if we were performing a similar activity (common entailments) using a similar, though a larger and softer ball, we may refer to the activity as "SOFTBALL." In addition, these determinations are not homogenous. Different group associations, such as race, class, gender and subcultures, will come to different understandings as to what is important.

Some of the entailments for "GAME" listed above, will, no doubt, be present in things commonly referred to as "games," though they need not all be present. And, as mentioned above, this list is not exhaustive so, there will likely be other entailments not listed in common usage.

In sum, a determination of definition in the prototype system of categorization is based on entailments of the word/concept, and these are, in turn, considered in light of family resemblances to a prototype, this prototype "existing" as an exemplary member. The "meaning," or definition of a word is determined by subjective agents working within a societal or cultural system. It is this system that determines what entailments are important to consider a case part of the concept family, and to what degree, and creates a historically situated, contingent meaning.

Characteristics of the prototype system, it will be found, contrast with those of the traditional system.

1. Attributes, and metaphorical entailments (the entailment system) determine level of class inclusion. I.e., what is common and distinctive.

2. Class inclusion is a matter of degrees.

3. Categories have irregular boundaries. There can be ambiguous or borderline cases.

4. Members of a family do not have equal status. There are better and worse examples.

5. Attributes are not primitives. They possess attributes of their own that can be still further broken down into their attributes, and so on and so forth

So, the characteristics of the prototype system reveal a system that is adaptive, in that the categories are not rigid, and are, at the same time, stable by virtue of family resemblances. Let us now continue the quote from Geeraerts that we began earlier...

It will be clear that prototypical categories are eminently suited to fulfill the joint requirements of structural stability and flexible adaptability. On the one hand, the development of nuances within concepts indicates their dynamic ability to cope with changing conditions and changing expressive needs. On the other hand, the fact that marginally deviant concepts can be incorporated into existing categories as peripheral instantiations of the latter, proves that these categories have a tendency to maintain themselves as holistic entities, thus maintaining the overall structure of the categorical system (Geeraerts cited in Taylor 1995, 141).

It becomes clear from this discussion, that the prototype theory of categorization is better suited to represent linguistic categorization than is the classical approach. As Geeraerts has pointed out, the prototype system is both adaptive and stable. It allows for a central set of category members, borderline cases, and cases of categorical divergence. This would allow "baseball," and "chess" to be considered better examples of "GAME" than "gladiatorial events" under normal conditions.

It would be helpful to shift gears here to explore some further consequences of the prototype theory. First, it should be noted, that there is a nonlinear relationship between members of a category. There is no single direction of activity, rather, the members are interrelating. This is due to the process that determines a member’s level of class inclusion; the members are mutually dependent upon one another for their level of class inclusion through levels of commonality and distinctiveness. Also, in keeping with Rosch’s findings, some attributes will become more heavily weighted because of how we interact with the world, thus forming high sensitivity to context, or initial conditions. This is because an attribute may become amplified over time through the recursive effects inherent in the interrelated, nonlinear nature of the system.

From this discussion, we can begin to see that the prototype system of linguistic categorization seems to be a complex, nonlinear system that possesses HSIC, in that a small effect may be amplified over time due to feedback mechanisms, and interrelated components. It is also self-similar at all levels in that a category possesses entailments that possess entailments of their own and so forth.

As we have also seen, this is exactly the type of system that DST takes as its subject matter. DST has developed elaborate tools to analyze these systems such as attractors, bifurcation diagrams and fractal geometry.

Now, recall the characteristics of dynamical systems that we cited earlier:

1) Nonlinearity

2) Feedback mechanisms

3) High sensitivity to initial conditions

4) Complex forms

We will now turn to elaborate on how each of these characteristics can be seen to also be characteristic of the prototype system of categorization. For this discussion, we will reverse the order just listed. We will first explore the fourth listed, then the third and finally, we will explore the first and second characteristics jointly.

Complex Forms

Complex forms refers to the self-similar nature of fractals. To see whether or not the prototype system of categorization can be described as self-similar, we will need to return to our description of the prototype system above. The prototype system relies on entailments, metaphorical or otherwise. We said, "...a determination of definition is based on entailments of the category, and these are, in turn, considered in light of family resemblances to a prototype…"

As an example, the category "GAME," we saw, will likely possess the entailments: activity, not necessary for survival, element of competition, employs a strategy, rules, and an element of satisfaction, pleasure and/or fun. Examples of "GAME," such as baseball, tennis, chess, and poker will possess many these entailments. That is why each of these are considered "GAMES"; they have (varying degrees of ) family resemblances by way of entailments to the prototypical "game."

Having located the place of entailments within our language system, we will consider this entailment system more closely. Recall that entailments possess entailments which will, in turn, possess entailments, and so on. For example, one entailment of "BASEBALL" is; played with the standard American baseball. Each of the signifiers used in this entailment (most obviously: played, standard, American, and baseball) will possess entailments of their own. For example, the baseball (the object that is thrown, caught and hit) possesses the following entailments, among others: sphere (with certain weight and size specifications), formed by yarn wound around a small core of rubber, cork or combination of both, covered by two pieces of white horsehide or cowhide tightly stitched together (NCAA 1984, p.BA-9). We do not need to explore each of these entailments of (the) baseball to understand that we could do so, and again, we could list the entailments found there, and so on, ad infinitum. Just as the structure of the fern plant can be witnessed again in the branch and then again in the fronds, and just as the Mandelbrot set possesses levels of structural similarity, the structure of the language system, by way of the entailment system described, can be seen as structurally self-similar. It is this structural similarity that we will call "the self-similarity of language." The path to the self-similarity of language has been reductionist in nature. In this reduction, we continue in the classical tradition. We part with tradition however, because we are not satisfied that we could ever reduce a category to any necessary and sufficient conditions. Any attempt to find necessary and sufficient conditions faces the cycle of infinite regress and reduction to the point of absurdity. Without accepting necessary and sufficient conditions reduction yields no explanation for the creation of categories. We must posit some alternative exp

From this discussion, we conclude language can be thought of as self-similar and following this conclusion, complex in the sense that the strange attractor and the a Mandelbrot set are complex.

Now, recall what was said above about "important" entailments. We began the discussion with, "The important entailments in ‘BASEBALL’...etc.," and a discussion ensued that may be summed as: (A) important entailments of a phenomenon under consideration for class (signifier) inclusion determine class inclusion (or exclusion), and (B) importance is determined by subjective agents based on context. This "importance," it would seem, is dependent upon the determination of whether or not inclusion or exclusion of a particular entailment would utterly "break down" the "meaning" of the signifier under consideration. It is this determination that is situated within a particular context. It would seem then, that the context (which includes subjective agents) would count as initial conditions and the sensitivity to these initial conditions is evident in the reliance on context for the assignment of "meaning" to a given signifier. Perhaps an anecdote would serve clarification here.

A young woman told me recently of an exchange that she had with her mother. Their discussion centered around the preparations for her brother’s wedding ceremony and reception. In the course of conversation, the young woman hoped to inform her mother that John Doe, who would attend the festivities with her, was more than just a casual acquaintance. Not wanting to divert too much attention from the wedding, the young woman intended to hint to her mother of her romantic relationship with John in a way that would provoke questions on the subject. She chose to circumlocute by saying, "John will be my escort to the wedding" (emphasis mine). Much to the young woman’s dismay, her mother understood this as saying nothing beyond "John will be attending the wedding with me."

The mother’s interpretation of her daughter’s words seems quite natural considering the circumstances. The term "escort" is commonly used to refer to one’s guest at such events as weddings; this term generally carries no romantic connotations. If the circumstances (at the time of the utterance) were different, the mother’s response may have been closer to what the young woman hoped for - questions leading to the nature of her relationship with John. If the daughter were to have said to her mother, "John will be my escort to the movies Saturday" this would seem an odd occasion to use the term "escort" and would more likely lead to the desired line of questions. The obvious, though still instructive point here is that the societal understanding of the term "escort," coupled with the said circumstances, largely influenced message interpretation.

In the above anecdote, the consequences of message misinterpretation were somewhat inconsequential. Still, when the consequences for misinterpretation are potentially more severe, as in a criminal case, or in the case of the boy who cried wolf, the centrality of HSIC becomes all the more important. It seems a truism to say that our category formations possess a high sensitivity to context, or initial conditions - this much is obvious. Still, this obviousness is all the more reason to reveal these contingencies in our models of the language system.

In a social system, such as language, feedback mechanisms may work throughout various levels. Recall that feedback refers to the continual reabsorption of what came before in a system. In the system of categorization, the importance or weight assigned to a given entailment will be influenced not only by current conditions, it will also be influenced by past conditions. The idea of any category is the result of how the category has been employed in the past. For example, my idea of "consent" has been informed by each occurrence that this category has been used in various contexts. I am influenced by my parents’ usage of this term, as I am influenced by television, my friends, and billboards that have used this term. In this way, the prototype system allows for the changing of meaning of a particular signifier, as is common to language (think of how the meanings of contemporary English words have changed from their Old English usages). There is a recursive nature to the prototype system of categorization that acts like iteration in a mathematical equation. Because of this recursive nature, an influence may be magnified over time, thus changing the "definition" of the category. Because a small cause can produce a large effect over time, there is a disproportionate relationship between cause and effect. Thus, we can say that the prototype system is nonlinear due to feed back mechanisms.

Throughout this discussion we have seen why and how it seems appropriate to speak of the prototype system as a dynamical system. The characteristics of dynamical systems are also characteristics of the prototype system. Because of this correlation, there may be great utility in analyzing the prototype system as a dynamical system by employing the tools of and the lessons learned about dynamical systems in DST.

In the next section we will locate the dominant issues within the debate surrounding the category "OBSCENE" within our courts and then consider this category through the prototype theory. We will then discuss how DST might inform a theory of category construction.

The history of the category "OBSCENE" has been turbulent in our court system (see Gunther 1992, 766-792). The Supreme court has grappled on many occasions with the issues surrounding this subject. Some of the issues are concerned with what justification can be given to prohibit "obscene" materials in the first place. The questions here include:

1) is "obscene" material protected by the first amendment?
2) should non-consenting adults and children be protected from "obscene" material?
3) should violation of prohibitory laws be a criminal offense?
4) would prohibitory laws be designed here to control the thoughts of the citizenry, and if so, is this acceptable?

These are difficult questions to answer severally and this difficulty is compounded because these issues must be considered collectively. In addition, some of these questions already assume some definition of "obscene" exists, or is at least attainable.

A second major class of issues is concerned with the definition of the word "obscene." Several tests have been suggested as a tool to enable the trier of fact to decide whether or not a particular work can be considered "obscene." Most noteworthy are the "Hicklin Test" that questions whether or not the material would offend the sensibilities of the highly intolerant person and the "Average Person Test" that questions whether or not the "average person’s" sensibilities would be offended by the material. Perhaps the most influential test has come out of the Miller vs. California case of 1957. A three-part test was suggested that prohibited federal or state government from controlling the distribution of materials where these three guidelines (in some combination) were not satisfied with an affirmative response: a) whether "the average person, applying contemporary community standards" would find that the work, taken as a whole, appeals to the prurient interest; b) whether the work depicts or describes, in a patently offensive way, sexual conduct specifically defined by the applicable state law; and c) whether the work, taken as a whole, lacks serious literary, artistic, political, or scientific value (Gunther 1992, 780).

Some of the issues surrounding the debate about the definition of "obscenity" between 1954 and 1974 include: 1) whether or not the material is utterly without redeeming social importance; 2) whether such material would lead the consumer to anti-social acts; 3) where the line is between materials that possesses serious literary, artistic, political or scientific value and those that do not; 4) whether or not the material is degrading; 5) whether or not a law can be formulated such that a party can surmise that materials would be illegal to possess or distribute before the fact; 6) whether or not the law should distinguish between violent portrayals, degrading ones and ones that are merely sexually explicit; and 7) whether or not the existence of certain materials would threaten the "tone of society" (Gunther 1992, 766-792).

Clearly, these are difficult questions. This is evidenced by words of Justice Harlan, "The subject of obscenity has produced a variety of views among members of the Court unmatched in any other course of constitutional adjudication," and by the words of Justice Stewart, "I shall not today attempt further to define the kinds of material I understand to be embraced within that shorthand description; and perhaps I could never succeed in intelligibly doing so. But I know it [obscenity] when I see it…" (Gunther 1992, 767, 774).

The quote from Justice Stewart points to the heart of the Court’s difficulties with this issue. Everybody knows what "obscene" means; this is evidenced by the fact that it continues to be a productive word in our lexicon. How is it that we can use a word productively and yet be unable to define it such that all cases of it are included in the definition and all inappropriate cases are excluded? Adding to this difficulty, the Court is faced with the task of formulating the definition such that it excludes all that is not "obscene" in a fashion so as to be consistent with first amendment rights.

We faced a similar problem earlier when considering the category "GAME." At no point in reduction were we able to establish necessary and sufficient conditions such that all appropriate cases were included and all inappropriate cases were excluded in the category. The effects of our inability to define "GAME" (according to these conditions) are inconsequential compared with the effects that our inability to define the category "OBSCENE" may hold. Since the criteria of necessary and sufficient conditions fails us here, perhaps a different approach is in order. Let us now turn to an analysis of the category "OBSCENE" as understood through the prototype theory and Figure 3.

Here, Figure 3 represents the category "OBSCENE." The letters in phase space represent various activities that we may call by the signifier "OBSCENE." Letter "A" may be, for example, a variable for "swearing." In Figure 3, variables A, B, C, D, E, F, and G may be considered clear examples of the category, while variables H, I, J, K, L, and M may be considered border cases and N through R may be cases of utter divergence. The temporary area of definition that considers historical usage and all manner of context results in some parties referring to swearing ("A") and pornography ("R") by the signifier "OBSCENE" and the paintings of Carrivaggio ("O") by some other signifier.

The signifier "OBSCENE" possesses family members that that possess entailments that are judged for class inclusion. The central members of the family "OBSCENE" will likely possess the following entailments that determine the prototype and what "OBSCENE" is...

*Material depicts violent and sexually explicit content

*Material will lead consumer to anti-social acts

*Material is degrading

*Material affronts contemporary community standards as they relate to sex

*Material appeals to prurient interests

*Material lacks serious literary, artistic, political or scientific value

One’s idea of "pornography" may include certain entailments such as: depicts sexual activity, contains nudity, and appeals to prurient interest. These entailments are not exclusively descriptors of what is commonly referred to as "pornography," as they could be applied to what is commonly called "art," such as certain works of the great painters. The important entailments in "pornography" are what is under consideration when determining if a particular case may be referred to with that particular signifier. Again, what is considered important is societally or culturally determined within context; there is HSIC.

Some of the entailments of "OBSCENE" listed above, will, no doubt, be present in things commonly referred to as "pornography," though they need not all be present; these are not necessary and sufficient conditions for a case to be considered part of the class. And, as this list is not exhaustive, there will likely be other entailments not listed in common usage.

The immediate family of "OBSCENE" will include those examples of it that possess the most amount of common entailments with one another and the least amount of distinctive entailments. These members and their relatives are easy to pick out, as Justice Stewart knew well at the time of his statement, though they are difficult to define collectively. The reason for this is that the reductive, traditional system of categorization can never reduce a concept to necessary and sufficient conditions for class inclusion (however, there may be some exceptions with highly specialized and scientific jargon), and this is especially menacing in the case of the court defining the category "OBSCENE" because there multiple legal considerations when deciding what is to be included and excluded.

The prototype theory as a dynamical system informs our effort to define "OBSCENE" by reminding us that this system of categorization is 1) complex in that there are many levels of entailments in each of the examples of the class that are 2) mutually interdependent upon one another, 3) highly susceptible to initial conditions, or context and 4) that interact in a recursive, nonlinear manner. What this leads us to consider is how the prototype system of categorization can explain how "definition" is created for a category, making it useful.

As we reduce the examples of the category through the entailment system, the category "OBSCENE" becomes more complex. Just as the limit cycle attractor describes a system with more than one possible outcome, we now have more than one possible "definition" for the category "OBSCENE." We equivocate between the possibilities and ambiguity increases.

As we reduce the entailment system still further, ambiguity increases even more. Now, there may be many possible formulations of "OBSCENE." The system of categorization is in the torus attractor state. One or more formulations, in the state of the torus attractor, are acting upon one another in a periodic relationship.

As we continue to reduce the entailment system, at some point the system explodes and there are an infinite number of possibilities for the system. The system is in the strange attractor state. We have at this point realized that our efforts have been an exercise in futility, for they led us to reduction to the point of absurdity. There still may be some global definability for the category (i.e., some categories remain clearly outside of the system, perhaps the category "GOD" will remain divergent in our example), though there is very little ability to predict what will and what will not be considered a part of the category.

It may serve clarification to elaborate on the system of categorization moving through the attractor states. Recall that an "attractor" is an area of mathematical space that the trajectory of a system seems to gravitate toward. Recall also that there are several types of attractors including: the point; the limit cycle; the torus and the strange attractors. Each of these represent a nonlinear system in some qualitatively distinct behavioral state

We might imagine using these attractors to describe how the nonlinear system of linguistic categorization is behaving at a given time. We could paint in a context to make the use of these attractors more salient. To color this in, we will imagine that we are in court and the jury is out debating the definition of the word "obscene." In this case, the system consists in the members of the jury - as one member speaks, this effects the position of all the other members10.

The point attractor may describe the categorical system tending toward one value, or lexical item. Perhaps, at first, all jurors are in agreement that the definition of "obscene" is "appeals to prurient interest," or something of the like. Then, one of the jurors recalls hearing a definition of "obscene" during a prime-time television drama. She includes this in the conversation and this definition differs from that being agreed on previously. Some other jurors saw that show as well, and so now, there is disagreement as to what the definition should be.

The limit cycle may describe the system equivocating between the two items. The debate brings other considerations to the fore, and so now, more jurors are in disagreement with each other. The torus attractor may now describe the system in a state of global stability, though with local uncertainty. That is to say that the system oscillates between several family members.

And lastly, the strange attractor may describe the system in a chaotic state. There are an infinity of possible items that the system may gravitate toward (though this infinity is contained), and there is little or no predictability. The jurors are in complete disagreement and the nature of the discussion has become so complex, that there are virtually no limits to the possible definitions of the word "obscene."

Justice Stewart seems to have been in the strange attractor state at the time of his declaration. The Justice was certain that he could point to examples of the category "OBSCENE," yet, there was so much ambiguity when deriving a definition from the local level that he was left to throw up his hands. Order, or meaning, seems to have been present and not been present simultaneously. Order may exist on the global level, but at the local level, uncertainty prevails.

Using the finding of DST, we could suggest that, if we accept that the system of categorization is a dynamical system, order, or meaning, spontaneously emerges in the system that remains open to the ever-changing environment. Just as order is an emergent property in the system displayed in the bifurcation diagram, meaning may be thought of as an emergent property in the dynamic system of categorization. This is like saying that solidity is an emergent property in the system of neutrons, protons and electron clouds in the table before me. At the level of extreme reduction, at the atomic level, we do not experience solidity, but by virtue of the components interacting with one another, the experience of solidity emerges at the global level.

We get some sense of the category, such that under normal conditions, the ambiguity inherent at the local level is not menacing; we can talk about "obscenity" and "games." By virtue of family relations we are able to use a prototype as a sort of shorthand to mentally represent the category as a whole, though this ideal member does not actually exist. So, the effect is that when we try to pin it down, it always escapes our grip. There is no Aristotelian "essence" of the category to be spoken of, there is a set of examples that form family relations.

It is evident that the Justice had in mind a similar formulation of the problem. The prototype system as a dynamical system theory offers consolation to the justice in that it reveals the elements that come together to form the problem. So, while we are still unable to conquer the problem of defining "OBSCENE," we are able to understand the nature of the problem more clearly. This may be little consolation. Still, we may suggest a possible course of action through the lessons here.

Because each instance of categorization relies on a judgment of "important" entailments based on context, the best that the Court can do is examine each claim of "OBSCENITY" on a case by case basis. This would also be true of the category "GAME," or any other category, if the stakes were high enough. It is context that informs a judgment of categorization. Without context, we are left wandering at the local, indeterminate level of the strange attractor while all the while knowing that order exists at the global level. Under normal conditions, meaning; when we are not practicing Law, Linguistics, or Philosophy, we accept a certain degree of ambiguity and the categorical system works fine. But, when we are in extreme contexts, like the courtroom, reduction leads to our being overwhelmed by ambiguity. Disorder is the result of our attempts at order.

Let us keep in mind that the system of categorization that has been set forth here is descriptive in nature. In contrast to this, categorizations of such terms as "OBSCENE" necessarily contain a value judgment and so are at least implicitly prescriptive in nature. To say that some material is "obscene" is to imply that the material is unacceptable and that the material is negatively valued. This adds another element of difficulty because the inherent value judgment is necessarily subjective and, at the same time, it is the task of the Justices to prescribe some objective definition.

This is not to deny that value judgments can contain an element of objectivity. It is by virtue of a shared system by a language community that we are able to categorize anything at all. Even if I am not the person offended by the material that provokes me to exclaim "obscene," I still must recognize in choosing that category that some part of the language community will find the material offensive. Knowing this, the Court formulated the "Average Person Test" to avoid purely subjective interpretations. But, because of the amorphous nature of the language system as described here, no person can be expected to surmise what the "Average Person" is, no less how the "Average Person" categorizes the world.

The options for the Court are three: 1) they could, through political upheaval that disposes of the first amendment, make some final judgment as to what a material must possess in order to call it "obscene," such as censoring all nudity; 2) the court can consider each claim on a case-by-case basis; or 3) the court can rule that all expression whatever is protected under the first amendment. Option #1 seems unconscionable. Option #2 would likely result in the court hearing so many cases on this issue that the Court would be incapacitated. However, if there are to be prohibitions on expression, this seems the way to go. Perhaps institutional changes would be able to accommodate the added burden on the court system. Option #3, if opted for, would bring all the considerations of what justification can be given in the first place for censorship to the fore. In addition, we would have to question what exactly it means to offend the sensibilities through exposure to such material.

The author’s recommendation to the Court is to accept option #3. Clearly, we cannot formulate a definition for "OBSCENE" such that a citizen can know before the fact that the possession or distribution of certain materials would be illegal. Option #2 will result in ex post facto law. Since, I dismiss option #1 out of hand, the only reasonable option left is option #3.

The debate should be focused on what the justifications are for censorship first and foremost. This is an issue that is well represented on both sides. First amendment rights should be interpreted in their most liberal sense until some consensus can be reached. Only then, if censorship is well justified, can we implement option #2 and begin to make the necessary accommodations in the court system.

Through the tools of DST, such as attractors, bifurcation diagrams and fractal geometry, we may discuss categorization as a system that is fundamentally a complex, nonlinear system. This observation will serve to bring what is important in categorization

to the fore, namely HSIC, feedback mechanisms, nonlinearity, and complex, fractal forms. Nowhere is it more important to understand the system of categorization than in the courtroom.

In this section, we will expand on the model of the prototype system of categorization presented in Figure 3. When considering the category "GAME," we used two axes to determine placement of a given example with respect to all the other members. Now, to further explore the possibilities for modeling this system, we will recast the model on the complex plane (Appendix C, Figure 4).

The entire model represents the category "GAME." We may think of each quarter of the complex plane as the same sort of model as we saw in Figure 3. There are four systems like the one in figure 3 here. In this model, the axes will continue to represent the family member’s level of commonality and distinctiveness, but now, moving toward the point of intersection, or origin, increases the value of commonality and decreases the value of distinction for each of the quarters.

Each quarter represents some Basic-level category of "GAME;" the top-right section will represent "board games," the bottom-right will represent "gambling," the bottom-left will represent "sporting events," and the top left will represent "metaphorical usages of the category." In each quarter, variables have been plotted that represent different kinds of "games." Variables E and H may represent chess and Monopoly, respectively; variables D and F may represent poker and blackjack; variables C and G may represent Baseball and Soccer; and variables J and I may represent "war-games" and the "dating-game." Each of these variables were placed according to similarity judgments considering all the examples within context.

It will be noticed that the Mandelbrot set is superimposed over the complex plane. In doing this, we are comparing the graphical representation of a dynamical system with another graphical representation of another dynamical system. Interestingly, the Mandelbrot too is plotted on the complex plane. Recall that the Mandelbrot set consists of those points on the plane that do not fall outside of the circumference of two after an arbitrary number of iterations. In the Mandelbrot image, all points that fall within the pool of the Mandelbrot set are considered to be within the set. Under normal conditions, a given person, within a specific environment, at a given time, might hold a similar mental representation of the category "GAME."

The variables that fall within the boundaries of the Mandelbrot set are considered "GAMES." Those on the fractal border, may not be considered "GAMES," or they may be; there is uncertainty. And those outside of the fractal border altogether are not considered within the category. We might consider whether or not those on the fractal border might be considered "GAMES" if their entailments were judged for importance on some other "level" (there are infinite levels).

In this model, four Basic-level categories are represented, though there could be many more. Many systems in representation would require an additional spatial dimension (perhaps several more are conceivable). Each set of axes would need to be at a right angle. With this model imagined with a third dimension, depth is added.

Emphasized in this model is the fractal border. As we have seen, our categories do not seem to have clear and distinct borders, rather, they are irregular. This is the Fractal Nature of Language. Like the Mandelbrot set, our system of categorization displays HSIC, nonlinearity, recursive effects and complex forms. So, for these reasons, the Mandelbrot set seems well-suited to become a symbol for the dynamic system of linguistic categorization.

This preliminary investigation has set out to question the usefulness of the tools of DST in the analysis of category construction. As we have seen, the characteristics of the prototype system of linguistic categorization may be described in the same terms as those of complex, nonlinear systems. That is because the prototype system seems to be a dynamical system.

To arrive at this conclusion, we introduced DST, its subject matter and its tools of inquiry. Then we outlined four characteristics of dynamical systems. From there we explained the traditional and the prototype systems of categorization and examined the assumptions of each of these. It was found that the prototype system is a better representation of how we actually categorize the world. This, we saw, was due to its ability to adapt to changing circumstances while remaining stable, and its accommodation of borderline cases. Then we explored how the prototype system of categorization and dynamical systems can be seen to share the same characteristics. The prototype system of categorization, like the systems studied by DST, it was argued, is a nonlinear system that displays feedback mechanisms, HSIC, and complex, fractal forms that are self-similar at all levels of magnification.

For the reasons presented, it seems appropriate and even natural to speak of the prototype system of categorization as a dynamical system. By utilizing the recent advancements in the natural sciences, by way of the tools of DST, we may describe the prototype system in various qualitatively distinct states. We may describe the behavior of category construction the courtroom with attractors, for instance.

Nowhere is it more important than in the courtroom to understand the workings of category construction. Justice in our courts depends on agreement. We need to agree on the categories used to describe an event, their relationships to one another, and the weight attached to every part of this and to the whole as they represent an individual’s participation in society. In our courts, it is the job of witnesses, law enforcement officials, complainants and defendants to provide adequate information with which to form categories. It is the job of prosecutors and lawyers to categorically re-present that information in a fashion so as to reflect favorably on their respective clients. It is the job of judges to ensure that these categorical re-presentations are asserted in a fair and reasonable manner. And it is the job of the jurors to decide which is the most reasonable representation of events and to assign this a moral value. The vehicle of each of these categorical representations is language. Thus, we can understand the importance of linguistic categorization in the justice system.

There is evidence that the prototype theory of categorization is more consistent with how people actually use and create categories than is the traditional approach. With a move to the prototype theory, a system of categorization emerges whose description is in direct contrast to the traditional system. This new description is of a complex, nonlinear system of categorization. One that is adaptive to its environment, yet not at the expense of stability. One in which meaning is an emergent property. One that is eminently suited to become the study of dynamical systems theorists.

1. Systems in the natural sciences have traditionally been viewed through linear mathematical models. Nonlinear investigations have only recently come of age. What made these advancements possible? Many advances in DST have resulted from the advancement of computer technology, which resulted in (1) the ability to manage calculations that were previously unmanageable due to the sheer number of calculations involved and (2) graphical images of these mathematical functions expanded what was "interesting" to the investigator, they allowed a qualitative and intuitively graspable handle on systemic behavior that didn’t exist before (Smith 1995, 23). Of peripheral interest, we might note that the tools of DST that utilize computer technology, such as phase maps and attractors, made it possible for the layperson to understand phenomenon whose understanding was previously restricted to specialists (Smith 1995, 23; Loye and Eisler 1987, 58).

2. Foremost among the figures in fractal geometry is Benoit Mandelbrot, the inventor of this branch of geometry, and the namesake of the sovereign of all fractals, the Mandelbrot set.

3. Taylor’s last four assumptions come to semantics through phonology. It was my choice to omit the last three of these because of their inapplicability to my present purposes. However, the reader may be interested to see what the omitted assumptions were. It should be noticed that these three do not contradict any of my findings.

6) Features are universal. That is, features are drawn from a universal inventory by all languages.

7) Features are abstract. That is, the features are not contingent on what is actually the case in the world. This is an autonomous conception of the categorical system.

8) Features are innate. Since features are not acquired in the world (#7) they are seen to be an inherent property of the human mind.

* In contrast to Aristotelian-based categorical structures that depend on necessary and

sufficient conditions (and that result in clearly bounded, discrete categories), Rosch and Mervis set out in "Family Resemblances: Studies in the Internal Structure of Categories" (1975) to establish evidence for their theory that categories are formed around a prototypical member. These prototypes are surrounded by a "family" of instances of a category whose members vary in degree of category inclusion. The purpose of their article was to explore one structuring principle which "may govern the formation of the prototype structure of semantic categories." Rosch and Mervis set out to empirically investigate the defensibility of Wittgenstein’s notion of "family resemblances" to explain a structuring principle of category construction.

"Family resemblances" is somewhat akin to the mathematically concerned study of "cue validity." Cue validity seeks to quantify the attributes that are most distributed among members of a category and least distributed among other categories. Rosch and Mervis’ hypothesis was that those members that possess the most attributes in common with their own category and the least in common with other categories are likely to be used as prototypical members on which judgments of similarity will be based for determinations of category inclusion.

The authors report studies using three different types of categories. These are the "superordinate" semantic categories, the "basic-level" semantic categories and artificial categories formed of letter strings. Six experiments were reported with their results that, cumulatively taken, support the authors’ initial hypothesis. The results of the study support the position that family resemblances is a major factor in category formation. However, the authors are careful to include that family resemblances, while a major factor, is not the only factor involved. In addition, they argue that determinations of prototypicality are not merely a matter of quantifying the number of occasions that an attribute shows up - it is not merely quantitative, but must also be concerned with the "associated strength" with which attributes are related due to contextual factors. In the end, the authors argue for the importance of their findings with six points, each of which either solves a historically argued logical problem or settles disagreements regarding interpretations by linguists and/or cognitive psychologists.

* For example, in the October, 1995 issue of the Journal of Memory and Language,

James A. Hampton reports his experimental findings in his article "Testing the Prototype Theory of Concepts." Hampton set out to test two points regarding the Prototype theory. The first was whether or not the Prototype Theory could be distinguished from the Binary Model. For this, Hampton would need to find concepts with a Defining Feature (necessary for class inclusion) that also displayed Characteristic Features (features that would merely affect the degree of category inclusion). For his purposes, the term "Defining Features" would be reinterpreted to satisfy the requirements of each of the theories in question (they are not considered necessary conditions, as in the Classical explanation). The second point was to test one prevalent assumption in the literature on the Prototype theory. This assumption is that class inclusion is determined by adding features in a linear fashion (as opposed to nonlinear and therefore non-exponential); i.e., the relative "weight" assigned to a given characteristic is added to the existing weight of other features. The prediction was that the findings would display a nonlinear curve when plotted around a threshold (the area of mathematical space where class inclusion is most unpredictable, this is the "border area" of categories).

Four experiments were conducted on British-English speaking adults. These experiments were refined as difficulties arose. In each of them, the independent variable was a manipulation of the presence or absence of either a Defining Feature [DF], a Questionable Defining Feature [DF(?)], or a Characteristic Feature [CF]. Questions were asked of the subjects for the end of having them categorize through prompts.

Hampton found that it was extremely difficult to locate semantic examples to fit the criterion necessary for his first goal. As a consequence of this difficulty, he was led to conclude that there is no support for the Binary Model and that the Prototype Theory was sound in its basic assumptions. For his second prediction, Hampton (surprisingly) found that the effects of CF’s on categorization was greatest when there was already a high degree of class inclusion and that the presence of the CF bolstered the strength of category membership. Because of this finding, Hampton argues, the prototype theory will need to be refined.

6. I realize that there are many factors that this graphic mathematical model overlooks. It would be beyond the means of this preliminary investigation to attempt to formulate a model that is thoroughly mathematically sound. The model presented is intended to help the reader to understand the larger picture.

7. In my conception, both semantic features, redefined w/o necessary and sufficient conditions and called "entailments," and "metaphorical entailments," as per Lackoff and Johnson (1980), are considered for the judgment of distinctiveness and commonality and thus level of class inclusion. This allows metaphor to become a central part of semantics.

* Unlike Plato’s forms.

9. This category has been chosen so as to remain in the historical terms of the discussion for the moment.

10. Justice Stevens has been quoted, "Since obscenity is by no means a neutral subject, and since the ascertainment of the community standards is such a subjective task, the expression of the individual jurors’ sentiments will inevitably influence the perception of other jurors, particular those who would normally be in the minority" (Gunther 1992, 790, emphasis mine).

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