Computation and Cognition

by

Charles Wallis

Department of Philosophy

California State University, Long Beach

1250 Bellflower Boulevard

McIntosh Humanities Building (MHB) 917

Long Beach, CA 90840-2408

email:cwallis@csulb.edu

"When you understand the nature of a thing, you know what it is capable of." –Blade

I. Introduction. Since the 1980s a debate has raged within cognitive science concerning the exact nature of cognitive state transitions (arguably since 1958 when Rosenblatt proposed the perceptron). During the last twenty years three distinct positions have emerged within that debate; "classical" cognitive science, dynamic computationalism, and dynamic systems theory. Though I concern myself exclusively with the first two positions for the bulk of this paper, I will outline all three positions in the next section. In the third section I briefly discuss my reasons for excluding dynamic systems theory from the discussion in the remainder of the text. Following the exclusion of dynamic systems theory, I present and critique two arguments offered within the literature as to which notion of computation is the best model for cognitive state transitions in computational explanations in cognitive science. I conclude the paper by arguing that properly understood, all the types of arguments in the literature support dynamic computationalism.

II. Three Positions. As mentioned in the introduction, three distinct positions as to the nature of cognitive state transitions have emerged over the last twenty years. Each position has at least one iconic advocate. Jerry Fodor is the icon for what he terms "classical" cognitive science (CCS). Fodor aptly summarizes his view when he tells readers that:

It would not be unreasonable to describe Classical Cognitive Science as an extended attempt to apply the methods of proof theory to the modeling of thought.... (Fodor and Pylyshyn 1988, pp.29-30)

As Fodor suggests, the classic notion of cognitive state transitions is that of a series of recursive read-write operations on a string of quasi-linguistic symbols with a recursive combinatorial syntax and semantics. It is widely, though falsely, supposed by classicists as well as their critics within dynamic computationalism that "whatever can be calculated by a machine (working on finite data in accordance with a finite program of instructions) is Turing-machine-computable." (Copeland 2002) Hence, these theorists equated cognitive state transitions with the Turing-computable functions so that Turing's

results entail something remarkable, namely that a standard digital computer, given only the right program, a large enough memory and sufficient time, can compute any rule-governed input-output function. That is, it can display any systematic pattern of responses to the environment whatsoever. (Paul and Patricia Churchland 1990 p.26)

While I recognize the error inherent in the supposition, I do hold that the notion of state transitions essential to the classicist's position is one of recursive read-write operations on a string of quasi-linguistic symbols with a recursive combinatorial syntax and semantics.

One leading dynamic computationalist with a strong claim to iconoclastic status is Paul Smolensky. Smolensky describes dynamic computationalism as follows: "Subsymbols are not operated upon by symbol manipulation: They participate in numerical--not symbolic-- computation." (1988, p.3) In other words, computation occurs according to differential equations describing the numerical evolution of the activation patterns--not by means of the recursive application of a finite set of read/write rules for manipulating symbols. Though the dynamic computational model differs [or so I'll claim] dramatically from the classicist notion, it preserves the central insight of computation as operations on representations. Within dynamic computationalism, representations can and often do take the form of changes in activation patterns [not symbols with a recursive, combinatorial syntax and semantics] governed by physical laws [not recursive read/write operations].

Timothy van Gelder stands out as the icon for the third, and most recent position to emerge, that of dynamic systems theory (DST). On van Gelder's account, dynamic systems theorists explain behavior by supposing that the brain is a dynamic system operating in accordance with coupled nonlinear differential equations. The picture which emerges is one of a complex dynamic system imbedded in other complex dynamic systems. Such systems develop and interact with the environment continuously according to their governing equations. A cognitive process in DST becomes state-space evolution of a low-dimensional dynamical system governed by coupled nonlinear differential equations. Essential to van Gelder's view is that cognitive state transitions are not computational in nature.

Despite van Gelder's efforts, I believe that the dynamic systems hypothesis is the weakest of the three positions in the literature. In the next section I discuss my reason for omitting dynamic systems theory from the discussion in the remainder of the text.

III. Dynamic Systems Theory. Van Gelder offers two arguments in support of his contention that dynamic systems theory is both (a) non-computational and (b) the proper theoretical approach to understanding cognitive state transitions within explanations in cognitive science. In this section I review these arguments, claiming that they fail in ways that exclude dynamic systems theory from consideration as a theory of cognitive state transitions. Though my comments are necessarily brief, others have dealt with van Gelder's views in a facile and convincing manner elsewhere.1

Van Gelder intends the first of his two arguments to undermine the notion that cognitive state transitions are computational by arguing that such transitions are continuous, whereas computational state transitions are discrete. For van Gelder the necessary elements of computational state transitions (in this case as applied to a hypothetical steam valve governor) are that

...it literally completes the desired change in throttle valve by manipulating symbols according to a schedule of rules. Those symbols in, the context of the device and its situation, have meaning and the success of the governor in its task is owed to its symbol manipulations being in systematic accord with those meanings. The manipulations are discrete operations which necessarily occur in a determinate sequence;.... (van Gelder, 1995. p.350)

Likewise, van Gelder defines computers as:

Digital computers, in the sense that matters for cognitive science, are systems which carry

out effective computation over representations. That is, they are systems whose behaviors are algorithmically specified finite sequences of basic operations constituting manipulations of representations. (van Gelder, 1998. p.5)

In contrast to this computational picture, van Gelder asserts that a "...[cognitive] system's entire operation is smooth and continuous; there is no possibility of nonarbitrarily dividing its changes over time into distinct manipulatings, and no point in trying to do so." (van Gelder 1995, p. 354). In short, van Gelder alleges that computational processes are necessarily and essentially discrete whereas natural cognitive systems like other physical systems have state transitions that are necessarily and essentially continuous. While this argument may serve to distinguish cognitive state transitions as understood by dynamic systems from classicalist notions of computational state transitions, it fails as a general argument against the notion of understanding cognitive state transitions as computational. Van Gelder's argument offers readers a false dichotomy, suggesting that one must choose either classicalist notions of computation or dynamic systems theory. Dynamic computationalism constitutes one alternative van Gelder neither acknowledges nor excludes. Like dynamic systems theorists, dynamic computationalists allow for state changes best described by functions on the reals. However, dynamic computationalists have a well-defined understanding of these transitions as computations. Evidence of this flaw in van Gelder's argument can be seen in his later and much more conciliatory statement to the effect that:

...DH [the dynamic hypothesis] can embrace the idea that cognitive processes are computational, while preserving a contrast with the CH [the computational hypothesis]. This does not diminish but rather fortifies the DH, by allowing it incorporate computational ideas without inheriting orthodoxy's excess baggage. (van Gelder, 1998. p. 17)

In the second argument, van Gelder attacks computational approaches to cognition by attacking the applicability of the notion of representation to cognizers and in cognitive processes. Van Gelder tells readers that "a useful criterion of representation--a reliable way of telling whether a system contains them or not--is to ask whether there is any explanatory utility in describing the system in representational terms." (Van Gelder 1995, p. 352) When considering cognitive explanations, van Gelder concludes that

The real problem with describing the governor [or a cognizer] as a representational device, then, is that the relation of representing--something standing in for some other set of affairs--is too simple to capture the actual interaction between the governor and the engine [or the cognizer and its enviroment]. (Van Gelder 1995, p. 353)

Whatever the merits of the above argument, its conclusion excludes dynamic systems theory as an adequate approach within cognitive science. By definition cognitive science is a science of how systems come to know about the world and use that knowledge to interact with their distal environment. To cognize something is to come to know about it. However by eschewing representation van Gelder effectively removes from dynamic system's theory any means by which to connect system behavior to the world through knowledge. Indeed, Van Gelder lapses into a telling ambiguity when characterizing the explanandum within his vision of cognitive science. Most times van Gelder characterizes the explanandum as the performance of cognitive tasks. However, when challenged as to his appeal to "cognitive" and "cognition" in his characterizations, van Gelder falls back upon talk of behavior, telling readers that

it [Dynamic Systems Theory] requires that the resources of dynamics be developed and supplemented in order to provide explanations of those special kinds of behaviors [my italics]. Thus, dynamical cognitive science always incorporates considerations distinctive to particular kinds of cognition into dynamical frameworks to produce explanations that are fundamentally dynamical in form, but are nevertheless tailored to explain cognitive performances "as cognitive." (van Gelder, 1998, pp. 18-9)

Van Gelder never articulates the "special" nature of such behaviors in virtue of which they are cognitive. Therefore, I conclude that both arguments fail to establish the superiority fo dynamic systems theory as a conception of cognitive state transitions appropriate for explanations in cognitive science. Van Gelder in fact seems to illegitimately poach upon inapplicable notions from cognitive science in attempting to specify the explanandum and explanatory goals of dynamic systems theory.

IV. Function Arguments.	Within the cognitive science literature one finds two approaches to arguing for computational equivalence or non-equivalence. The quote with which the paper began expresses one approach, which I call the nature argument. Those who formulate the nature argument believe, like Blade, that "when you understand the nature of a thing, you know what it is capable of." According to this approach equivalence between the actual or theoretical operations and primitives that define a computational system is the measure by which one determines computational equivalence, or lack thereof, between systems. The nature approach to thinking about the relationships between computational mechanisms has languished in the shadow of the second type of argument consisting in (ironic given the vampire reference) the mirror of the sentiment expressed in the quote. The slogan for the functions argument to computational equivalence reads, "if you know what a thing is capable of, you understand its nature."

One reason for the ascendance of the functions argument is a popular misconception of the Church-Turing thesis. Pervasive throughout philosophy and the other disciplines in cognitive science is the false belief that Turing demonstrated that Turing machines can solve any problem that can be solved by any finite set of recursive instructions, explicitly stated rules, or procedures. It is widely supposed that Turing proved this result by showing that all notions (Turing, Church, Post-Markov) were capable of computing the same set of functions. In fact, Turing never offered such a proof, nor would he have accepted the above claim.

What I call "the myth of the Church-Turing thesis" has dramatically shaped the debate in cognitive science. Specifically, many theorists in the debate offer functions arguments in favor of their positions. However, I contend that they do not properly understand the nature of the arguments.

I'll call the version of the functions argument that establishes equivalence or non-
equivalence of two theoretically defined notions of computation based upon the identity of their respective classes of theoretically computable functions the strong functions argument. One can also formulate a weak functions argument. The weak functions argument can take several forms. It can argue that the computational complexity profile of two computational systems are identical, i.e., that the same functions have exactly (or even roughly) the same computational complexity for each system. It can also argue that within some set of conditions, two systems compute the same set of functions. This latter version receives the moniker "weak" because the result is conditional, which is perceived to contrast with the unconditional results of the strong argument.

The weak functions argument is rarely used in the debate regarding state transitions in cognitive science in all likelihood because it would not end the debate with seeming mathematical finality. For instance, functions arguments within classical cognitive science most often try to show that, in principle, the set of functions computable by connectionist systems is either coextensive with Turing-computable functions or worse still that cognitive functions are not connectionist-computable. In the next section I will look at these arguments in more detail. I will defend the view that the arguments often forwarded by classical theorists, though portrayed as strong, are actually weak functions arguments. Moreover, these arguments are not particularly apt versions of the weak functions argument. I will go on to argue that all versions of the functions argument, not just weak versions, are conditional in nature. Worse still, the assumptions of strong functions arguments are not as relevant to cognitive science as those often proposed by dynamic computationalists.

V. Systematicity. The systematicity argument forwarded by Fodor and Pylyshyn (1988) is framed as a strong functions argument. Fodor and McLaughlin express it as follows:

...cognitive capacities come in clumps. Thus, you don't find organisms that can learn to prefer the green triangle to the red square but can't learn to prefer the red triangle to the green square. You don't find organisms that can think the thought the girl loves John but can't think the thought that John loves the girl. You don't find organisms that can infer P from P&Q&R but can't infer P from P&Q. (F&M 1991, p.332)

In "Connectionism and Cognitive Architecture: A Critical Analysis," Fodor and Pylyshyn argue that connectionism is, in principle, an inadequate theory of the architecture for cognitive systems. They base their conclusion upon, among other things, their claim that connectionism cannot explain a strongly supported empirical fact that cognitive capacities are systematic. (F&P 1988, p. 41) To use Fodor and Pylyshyn's now famous example, connectionism cannot explain why a cognitive system is capable of thinking aRb iff it is capable of thinking bRa.

The sort of systematicity used by Fodor and others requires that one have the capacity to, for example, believe aRb iff one can has the capacity to believe bRa. It also requires that one can have the capacity to believe Pa iff one has the capacity to believe for some other object, b, Pb. In other words, systematicity requires that one be capable of believing that Arnold is a pig iff one is also capable of believing that Porky is a pig. Finally, systematicity requires that one can have the capacity to infer, for example, Pa from aRb and aRb Pa iff one has the general capacity to make conditional inferences, i.e., iff one has the capacity to infer aRb from Pa and Pa aRb .

Smolensky (1991 and 1991a) and others have responded to Fodor and Pylyshyn. (Chalmers 1990, Clark 1993, van Gelder 1990) Two important results are Smolensky's work on compositional structure in networks with distributed representations and Kolmogorov's Mapping Neural Network Existence Theorem. Kolmogorov's Theorem states that:

Given any continuous function ƒ:[0,1]n RM, ƒ(x) = y, ƒ can be implemented exactly by a three-layer feedfoward network have n fanout processing elements in the first (x - input) layer, (2n + 1) processing elements in the middle layer, and m processing elements in the top (y - output) layer. (Hecht-Nielsen 1991, p. 122)

Though a method for finding a function's Kolmogorov number remains undiscovered, the theorem proves important in that it states that for any Turing-computable function on the reals, a connectionist system exists that can compute that function.

On the other hand, Smolensky shows that

a true constituent can move around and fill any of a number of different roles. ...this [can] be done with vectors encoding distributed representations, and be done in a way that doesn't amount to simply implementing symbolic syntactic constituency.... (Smolensky 1991, p. 212)

Smolensky's (1991 and 1991a) papers have convinced Fodor that

...it is possible for Smolensky to wire a network so that is supports a vector that represents aRb if and only if it supports a vector that represents bRa; and perhaps it is possible for him to do that without making the imaginary units explicit.... (Fodor and McLaughlin 1991, pp. 348-9)

Thus, Fodor and McLaughlin (F&M) seem to accept that connectionist systems can support some systematic capacities without merely implementing classical architectures, forcing them to reformulate the systematicity argument in "Connectionism and the Problem of Systematicity." In this reformulation (perhaps all along) the systematicity argument takes on elements of a nature argument. F&M argue for their claims concerning which functions each system can compute by making claims of non-equivalence rooted in these system's primitive representations and operations. As shown below, these claims are actually false. Thus, in treating the systematicity argument as a functions argument I also address those elements of the argument that are more properly thought of as a nature argument.

Since F&M formulate the argument using only the aRb example, I restrict my initial discussion to that formulation. Readers should be cautioned that the argument I present here has never been full accepted by Fodor or McLaughlin. Nevertheless, I believe it to represent the argument in the literature and I try to indicate this by extensive quotations.

(FM1) It is nomologically necessary that cognitive capacities are systematic.

(FM2) (FM1) is true when and only when, if a system, S, is a cognitive system, then

it is nomologically necessary that S possesses a cognitive capacity with respect to aRb (e.x. the capacity to believe aRb) iff S has that capacity with respect to bRa.

(FM3) RTM and (FM2) are true when and only when, if S is a cognitive system, then

it is nomically necessary that with respect to a given cognitive capacity S is

able to represent aRb as part of that capacity iff S is able to represent bRa as

part of that capacity.

(FM4) Any adequate theory of cognitive architecture offers a nontrivial explanation

of the nomological necessity of systematicity as expressed in (FM3) [Hence,

(FM2) and (FM1) as well].

(FM5) Classical architectures compute over representations having a recursive

combinatorial syntactic and semantic structure.

(FM6) Given (FM5), "...in the Classical architecture, if you meet the conditions for

being able to represent aRb, YOU CANNOT BUT MEET THE

CONDITIONS FOR BEING ABLE TO REPRESENT bRa;...."

(F&M 1991, p.349)

(FM7) Therefore, the necessary systematicity of Classical architectures explains

(FM3).

(FM8) Since systematicity is a contingent feature of connectionist architectures they

cannot explain (FM3).

(FM9) Hence, Classical architectures are an adequate theory of cognitive architecture, while connectionist architectures prove inadequate.

At least as I understand it, the systematicity argument is clearly intended to be a strong functions argument establishing computational non-equivalence between connectionist systems and classical Turing-equivalent systems. As part of that argument F&M claim to establish the result that connectionism is, in principle, inadequate since connectionist systems cannot compute certain important cognitive functions. In Fodor's words,

...we assume without argument: (i) that cognitive capacities are generally systematic in this sense [the above quoted sense], both in humans and in many infrahuman organisms; (ii) that it is nomologically necessary (hence counterfactual supporting) that this is so; (iii) that there must therefore be some psychological mechanism in virtue of the functioning of which cognitive capacities are systematic; (iv) and that an adequate theory of cognitive architecture should exhibit this mechanism. (F&M 1991, p.332)

Later in the paper Fodor and McLaughlin state that

...the point of the problem that systematicity poses for connectionists,...is not to show that systematic cognitive capacities are possible given the assumptions of a connectionist architecture, but to explain how systematicity could be necessary--how it could be a law that cognitive capacities are systematic--given those assumptions. (F&M 1991, p.348)

I argue that F&M's (FM6), which one might call "the necessary systematicity of Classical architectures premise." is false. If I am correct, then the argument fails as a version of the strong functions argument. However, it might live on as a weak functions argument. I further claim that (FM1), which one might call "the nomic necessity of cognitive systematicity premise," also proves incorrect. Thus, the efficacy of the systematicity argument, I suggest is greatly diminished.

I begin by considering (FM6), the necessary systematicity of Classical architectures premise. F&M offer the following argument for the necessary systematicity of classical architectures premise:

...in the Classical architecture, if you meet the conditions for being able to represent aRb, YOU CANNOT BUT MEET THE CONDITIONS FOR BEING ABLE TO REPRESENT bRa; the architecture won't let you do so because (i) the representation of a, R and b are constituents of the representation of aRb, and (ii) you have to token the constituents of the representations that you token, so classical constituents can't just be imaginary. So then: it is built into the Classical picture that you can't think aRb unless you are able to think bRa (Ibid., p.349)

F&M reason as follows: In the classical picture, to think a thought is to represent it, and to represent it is to token it as a well-formed formula (wff) in one's language of thought, . Since in classical architectures has a recursive, combinatorial syntax and semantics, aRb will be a wff iff bRa is a wff. Hence, in the classical picture, one can think aRb iff one is able to think bRa.

F&M have erred. One can easily define the grammar of a language, c, having a recursive, combinatorial syntax and semantics and having a, R, and b as primitives in which aRb, but not bRa, is a wff. I conclude that the classical picture allows a person to think aRb, but not be able to think bRa, if, for instance, one's language of thought is c. I conclude that the necessary systematicity of classical architectures (FM6) is false. My example is a simple language, c, defined as follows:

The formulas of the language are strings (of finite length) of the following symbols:

The logical constants are the seven symbols: - v ( ) & .

The non-logical constants fall into two classes: the individual constants (a,c,d,e,...,t)

and the individual constants (b,c,d,e...,t).

The predicates consist entirely of the two-place predicate R.

An atomic formula consists of an individual constant from the set (a,c,d,e,...,t), followed by the two-place predicate R, followed by an individual constant from the set (b,c,d,e...,t).

A well-formed formula is either an atomic formula or else a formula built from one or more atomic formulas by a finite number of applications of the following rules:

(a) If is a wff, then - is a wff.

(b) If and are wff, then ( v ), ( ), ( & ), and ( ) are wffs.

Though nonstandard, c is a perfectly legitimate language with a recursive, combinatorial syntax and semantics. Moreover, since the common computer language Pascal has user defined data types, allowing one to define constants as done in c, one knows that classical architectures can support versions of c. Since classical architectures can support versions of c, one knows that there exists a Turing-equivalent machine that operates on wffs in c and only on wffs in c. The necessary systematicity of classical architectures premise, is false. One can expand the example to cover all of the putative types of systematicity discussed above. One can modify c to introduce unsystematic predicates:

The predicates consist entirely of the two-place predicate R and the one-place

predicate P.

An atomic formula consists of either

(a) an individual constant from the set (a,c,d,e,...,t), followed by the two-place

predicate R, followed by an individual constant from the set (b,c,d,e...,t).

(b) the one place predicate P followed by an individual constant from the set

(a,c,d,e,...,t).

The inferential capacity of a system depends upon the rules of inference. Nothing inherent to a language or rules of inference for a language requires the sort of systematicities to which Fodor refers. One can, therefore, define a system that has an unsystematic inference capacity by specifying unsystematic inference rules for the system. For example, one might formulate an arrow-out rule violating the systematicity example given above as follows:

For all wffs, and , where is an atomic sentence consisting of an individual

constant from the set (a,c,d,e,...,t), followed by the two-place predicate R,

followed by an individual constant from the set (b,c,d,e...,t), may be entered on

a line if and ( ) appear on earlier lines; as premise-numbers of the new

line take all premise numbers of those earlier lines.

To maintain the necessary systematicity of classical architectures premise, F&M must claim that it is impossible for a classical machine to have a language like c as its operating language and/or unsystematic rules of inference. I see no way to argue for this claim short of stipulating by definition that classical architectures must have systematic languages and inference rules. Since it is false that classical architectures are necessarily systematic, a classical architecture can explain the necessary systematicity of thought only with the additional assumption that cognitive architectures necessarily have systematic languages and inference rules. This stipulation, however, violates (FM4) and is equally available to the connectionist, who can claim that connectionist cognitive architectures are necessarily systematic. The result, I claim, is that classical cognitive science can explain the nomic necessity of the systematicity of cognitive capacities only on the assumption that cognitive capacities are necessarily systematic, i.e., the assumption that cognitive architectures are both classical and necessarily systematic.

Classicists troubled by their inability to explain the nomic necessity of systematicity ought to reexamine (FM1). The truth of the nomic necessity of cognitive systematicity premise, (FM1), requires that in all nomologically possible worlds in which one finds cognizers, their cognitive capacities are systematic in the manner discussed above. (Lewis 1973, 1986) Consider the following thought experiment designed to show that nothing about either classical cognitive science or our notion of a cognitive capacity provides evidence for (FM1): Alex starts off life as a systematic cognizer. Among his beliefs, he holds the belief that the empire state building is taller than himself, which is symbolized in Alex's language of thought as aRb. Before time t1 Alex has the capacity to believe that he is taller than the empire state building, which is symbolized bRa. At t1 an accident causes Alex's language of thought to become like the above defined c with respect to a, R, and b. Though Alex retains a wide range of cognitive abilities, he can no longer believe that bRa, because bRa is no longer a wff in his language of thought. When one says or writes "Alex is taller than the empire state building," Alex invariably blanks-out. In all other respects, Alex remains the same.

If (FM1), the nomic necessity of cognitive systematicity premise, were true, Alex would cease to be a cognitive creature and/or cease to have the cognitive capacity to form beliefs. I have strong intuitions that, despite his cognitive deficit, Alex is still a cognitive creature and he retains his belief-forming capacity. Moreover, one can explain Alex's deficit based upon one's classical theory of cognition, since it provides the mechanisms that could underlie Alex's deficit.

One possible response to Alex's case claims that it shows nothing because (FM1) requires only that Alex's cognitive capacities are mostly systematic. Alex's capacities are mostly systematic. Hence, (FM1) is not violated. I find this response to be of dubious cogency. Since Alex's cognitive deficit is consistent with the classical picture, Alex is unsystematic, and no other laws are operant, one cannot but wonder how the claim that "It is nomically necessary that cognitive capacities are mostly systematic," differs from the claim "It is nomically necessary that cognitive capacities are systematic, except when they are not."

Moreover, unsystematic cognitive capacities need not be restricted to isolated cases or come about as the result of an artificially induced cognitive deficit. Cognizers might gain strong selective advantages from unsystematic cognitive capacities and behavior. For example, in the case of the capacity to form beliefs systematicity requires the capacity to believe thoughts that always prove false or irrelevant in the animal's environment or in general. For example, it is always false for rats to believe that cats, hawks, etc. are friendly. The ability to believe such propositions might well decrease fitness by wasting cognitive resources or causing disastrously gregarious behavior towards hawks and cats. For this reason, it would help the overall fitness of the rat population if its members were not capable of thinking of any number of predators as friendly. In fact, rats cannot habituate to the smell of cat hair, and always exhibit a stress response in its presence.

Rats and many other animals can easily learn to prefer cheese to a substance that makes them sick on the basis of taste. They cannot learn to prefer the substance to cheese. On the Fodorian assumption of the LOT, preferences are sentences, like cPs, in a belief (or whatever) box, making rats capable of representing cPs, but not capable of representing sPc. One finds a similar ostensive lack of systematicity in the fact that rats can learn to associate illness with taste, but cannot learn to associate illness with noise or light. (Garcia and Koelling, 1966 and Garcia, Brett, and Rusiniak, 1989) Though such asymmetries in the capacity to form preferences suggest a lack of systematicity, no one labels rats "noncognitive," or denies that they have and can learn preferences.

Prima Facie unsystematic cognitive capacities are quite prominent in the cognitive dissociation literature, in which normal functioning is explored by studying damage to various brain areas and the resulting impairment of cognitive capacities. Such cases prove relevant for two reasons. First, the subject's cognitive capacities, though impaired, remain recognizable cognitive capacities, thus presenting actual cases of prima facie unsystematic cognitive capacities. Second, since such impairments are used to explore the particulars of normal functioning, any unsystematic impairments of a cognitive capacity present evidence strongly suggesting that normal functioning operates via unsystematic processing.

For example, humans suffering lesions to the right parietal lobe develop what psychologists call "contralateral neglect." They are, prima facie, unable to process visual, auditory, and/or somatosensory information localized to the left side of their body at the levels of conscious perception and planning despite the availability of the relevant precursor information. Hence, persons suffering from contralateral neglect have a prima facie unsystematic to ability to make the relevant inferences from available precursor sensory information. Additionally, the deficit is often accompanied by a lack of awareness of any problem. Though they still exhibit priming effects and illusions from the left visual field, such persons often reproduce only the right side of drawings, shave only half of their face, read only the right sides of words, etc. (Kolb and Whishaw 1990, pp. 424-6) Because their lesions are inexact, they can also exhibit unsystematic patterns of neglect. (Kolb and Whishaw 1990, p. 425)

Persons with "dorsal simultanagnosia," can detect objects in any area of their visual field, but cannot perceive the visual field as an integrated whole--resulting in unsystematic processing of visual scenes. For example,

When shown the [camel in front of pyramid] scene for two seconds, Tyler's subject said she saw "a mountain." When shown the figure for another two seconds, she said "a man," and did not indicate having seen the camel, desert, or pyramids, or realize that the man was related to what she had previously seen. When allowed thirty seconds to look at the picture, she eventually said, "it's a man looking at mountains." She never saw the "whole,".... (Farah 1990, p. 19).

Dorsal simultanagnosiacs cannot make the necessary inferences to systematically integrate visual wholes from available precursor information.

Anomic aphasics have tremendous problems generating the names of objects despite being able to comprehend, repeat, and produce meaningful speech. Kolb and Whishaw illustrate the anomic aphasic's prima facie unsystematic lexical access when noting that

...we saw a patient who when shown a picture of an ship anchor simply could not think of the name and finally said, "I know what it does . . . You use it to anchor a ship." Although he had actually used the word as a verb he was unable to use it as a noun. (1990, p.578)

Aphasias can also prevent access to particular subclasses of nouns; color words, words for animals, words for tools, etc.. (Damasio and Damasio 1992)

Lesions to the anterior perisylvian sector result in a prima facie unsystematic ability to grasp passive sentences because subjects

...do not always grasp reversible passive sentences such as the "The boy was kissed by the girl," in which boy and girl are equally likely to be the recipient of the action. Nevertheless, they can still assign the correct meaning to a nonreversible passive sentence such as "The apple was eaten by the boy".... (Damasio and Damasio 1992, p. 93)

Cases involving higher-order inferential ability and belief ascription are, in philosophical circles, more controversial. Nevertheless, conscientious adherents of the nomological necessity of cognitive systematicity ought to worry about cases involving sentences which seem to reflect grammatical systematicity, but are not possible beliefs since their appearance in a "belief-box" provides extremely strong evidence that the subject does not possess the requisite concept: One can believe, for instance, that the universe is finite or infinite. But, Dan Quayle not withstanding, one cannot believe that the universe is almost infinite. Nor can one believe that the liar is lying when he says that he is lying.

Fodor and Pylyshyn notice a similar problem when they point to and dismiss combinatorial failures in English. (1988, pp.41-6) "He kicked the bucket," should have a content related to buckets and kicking, and not related to death.

When arguing that humans are irrational, Stich (1985, 1988, 1990) has called upon the vast literature in cognitive science that suggests that human judgement abilities involving probabilities and some logical connectives are context-dependent and unsystematic under standard interpretations. For example, people who will pay $100 to reduce a risk from one in one million to zero will pay at most $1 to reduce the very same risk from two in one million to one in one million. (Piatelli-Palmarini 1991, p.33) Such reasoning, although widespread, looks prima facie unsystematic. That is, it treats the same objective risk differently in different cases.

Fodor and company might dispute the accepted interpretations of particular examples of seemingly unsystematic cognition in this extremely incomplete survey of the cognitive science literature. It seems unlikely that they could reinterpret so many different phenomena ranging across so many levels of analysis and cognitive tasks. Regardless, the burden of proof has shifted from the connectionists to the adherents of (FM1) on the empirical front. Classicists must establish systematicity as nomological necessity before making explanatory demands upon connectionists.

I conclude from the results of above-mentioned empirical and thought experiments that the nomic necessity of cognitive systematicity premise, (FM1), is doubtful, if not clearly false. Though possibly an interesting and widespread contingent empirical fact, systematicity is not a nomologically necessary truth. Fodor and McLaughlin must accept the existence of possible worlds with unsystematic cognition. The necessary systematicity of classical architectures premise, (FM6), is also false. Fodor et al, have not offered a sound argument from the necessary systematicity of cognition to the conclusion, (FM9), that classical architectures are an adequate theory of cognitive architecture, while connectionist architectures are inadequate.

Classicists might well acknowledge the above point and try to reformulate the systematicity argument as a weak functions argument. Unfortunately, there is no obvious way to specify systematicities in a way that does not presuppose any assumptions about how the mind represents. Worse still, as Cummins has aptly pointed out (Cummins, 1996 and Cummins et al., 2001) , if one supposes that the best (not ad hoc) explanations of systematicities make reference to structure, then (a) one will have to specify different structures for each non- isomorphic domain in which we are sensitive to systematicities (like vision and language), or (b) explain some systematicities in what one might consider an ad hoc or unprincipled fashion. Cummins has adroitly summarized much of this debate as follows:

You can make things look hard for (some) connectionists by (a) covertly relativizing systematicity to natural language, and (b) reminding them that they favor a nonclassical scheme of mental representation. If you can get them to accept this "challenge," they will then labor away.... Getting connectionists to accept this challenge is nice work if you can get it, and apparently you can. There may still be job openings in this area if you want employment in the confidence business. (Cummins, 1996 pp. 597-8)

VI. Limitations Arguments	The second family of arguments in the literature has a more

recent vintage. The idea behind the limitations arguments is that differences between connectionist and classical computational capacities are due to the fact that theoretical specifications of connectionist systems allow them to have activation values ranging over the reals. This presumption of infinite sensitivity to different activation values strikes the proponents of the limitations arguments as biologically unsound. For example, David Chalmers reasons as follows:

It is true that a system with unlimited precision might have cognitive capacities that no discrete systems could ever have. ... But the presence of noise implies that no biological process could reliably implement this system. ...so, human and animal brains must be limited to capacities that discrete systems can share. Beyond a certain key point..., uncontrollable fluctuations in background noise will wash out any further precision. ...if...we approximate the state of the system to this level of precision (perhaps...to 10-20...), then we will be doing as well as the system can reliably do. (Chalmers 1994 p. 393)

The "key point" assumption of the limitations argument acts to truncate representational capacity. Since under the key point assumption activation values become a finite (though large) subset of the reals, connectionist computational capacities are likewise limited to functions mapping between two finite (though large) subsets of the reals. As a result, it is possible to prove that some Turing machine can compute that function since it could do so by having a unique instruction for each input. Such a Turing machine would amount to a very elaborate, automated look-up table. Given the provable co-extension of the sets of computable functions, Chalmers and others conclude that classical and connectionist systems are equivalent notions of computability--at least for the purposes of cognitive science or for realistic systems (or similar qualifications).

Though this argument looks superficially like the strong functions argument, it's advocates fail to recognize some deep difficulties with this argument. First, the limitations argument conflates notions of theoretical equivalence with notions of implementational equivalence while also ignoring issues of computational complexity. In my view the strong functions argument only works in the context of theoretic definitions of computational systems. That is, strong functions arguments act to show what idealized systems satisfying only the computational definition could compute in a finite amount of time working at less than infinite speed. Such arguments are non-trivial. For example, Turing showed that Lambda-definability and computable by a Turing machine were computationally equivalent for computing functions on the positive integers.

The point of proving equivalence between computational systems is to show such systems to be interchangable within the specific context of interest. Turing's proof of equivalence occurred in the context of Church's proposal that one define "...an effectively calculable function of positive integers by identifying it with the notion of a recursive function of positive integers (or of a lambda-definable function of positive integers)." (1936) Thus, Turing showed that his notion of effective computability with regard to the positive integers was interchangeable with Church's lambda-definable notion within that specific context.

However, if one seeks to declare computational systems equivalent or nonequivalent in the context of implementational limitations, resource constraints, with regard to certain classes of problems, etc., then one must compare both systems within those contexts. The limitations argument seeks to apply implementational limitations to connectionist systems without either examining the effect of corresponding limitations on Turing-equivalent computation or significant argument to the effect that this limitation, and only this limitation, is relevant for a comparison between notions of computability within the context of computational explanations in cognitive science.

In order to show that the class of functions computable by each system is identical (or that the connectionist functions are a subset of the classical functions) these arguments appeal to the implementational limitations in brains. Once done they then revert to a theoretical notion of classical computation to "establish" that a classical machine can compute the connectionist functions. Once one crosses over to finite instantiations of theoretically defined machines, then one must establish (1) the relevance for the context at hand (in this case explanations in cognitive science) of the assumptions, and only those assumptions, upon which one bases the equivalence and (2) one must apply those assumptions when establishing the computational abilities of each of the systems being compared. The limitations argument clearly fails to satisfy (2) since it reverts to a theoretical definition of Turing-equivalent systems after imposing implementational limitations on the dynamic computationalist. The limitations argument does satisfy (1) to a small extent in that it argues against unlimited precision in real systems. However, advocates of the limitations argument fail to offer any argument to establish that finite precision is the sole assumption relevant to establishing computational equivalence of systems in the context of explanations in cognitive science. Indeed, such a claim would run strongly contrary with deeply entrenched explanatory presuppositions within cognitive science. For instance, speed of computation is a well established factor in assessing explanatory accuracy within cognitive science. One judges the adequacy of explanations of object recognition, to take a case, by whether the proposed mechanisms can preform object recognition within the human time frame of 300msec. Likewise, such proposed mechanisms must prove effective in cases of rotation and partial occlusion.

I would suggest that to adequately address issues in (1) one must argue that the computational complexity profile of the finite classical machine either better fits apparent human complexity profiles [or that it's fit is identical to the connectionist system] in real world contexts. Computational complexity is a measure of problem difficulty used widely in computer science and mathematics. Traditional notions of computational complexity measure task difficulty either by number of steps necessary to compute a function or amount of memory necessary. In general, they are system-relative resource-based assessments of problem difficulty.

One can think of a complexity profile as a overall evaluation of a system's computational prowess in terms of a range of complexity assessments. Arguments based upon complexity profiles would assesses system equivalence, whether human or computer, by sorting significant cognitive tasks along a spectrum ranging from classes of problems that prove difficult or impossible to classes of problems that prove relatively easy. In this way a complexity profile is analogous to the sorts of benchmark tests used for PCs, with the caveat that the "winner" is not necessarily the fastest, but the closest to human performance across a spectrum of tasks. In the context of computational explanations in cognitive science such assessments must match human data across a range of human tasks and include not only not only resource constraints, but also general constraints like performance under degraded stimulus conditions and task-specific constraints like rotation and occlusion in object recognition.

In my view, little work has been done to formulate such weak functions arguments. Dynamic computationalists have far excelled classicalists in sensitivity to the above issues in their early weak functions arguments in favor of connectionism. (Bechtel 1985, Chuchland 1988, Feldman and Ballard 1982, McClelland, Rumelhart and Hinton 1986, Rumelhart 1989, Shastri 1991, and Smolensky 1991 and 1991a). As a body of work, though still not perfect, these writings make a much stronger case in favor of dynamic computationalism than the above classicalist arguments.

Finally, limitations arguments falsely suppose that the computation cannot occur in presence of noise based largely upon a widespread misunderstanding of the proper application of information theory. Shannon's (1948) notion of information was based upon the transmission of binary encoded signals in static channels. One defines noise as a decreased probability of the state of affairs at the source given the signal at the receiver. The notion of noise operant in the limitations argument is one of error in storage and computation of representations within a computational system. For noise in this sense, Jack Copeland has noted Segalmann and Sontag's (1994) result, noting that

If we do assume that noise can limit precision, but do not follow Chalmers in making the "key point" assumption, then we can show that a continuous "noisy" system can exhibit uncomputable behavior. (Copeland, p. 709)

VII. The Nature Argument. So, all versions of the functions argument presuppose some computational context, be it a theoretical context or a practical/implementational context. The difference, therefore, between weak and a strong functions arguments is largely an illusion. Nevertheless, strong functions arguments are more rigorous in that they are based on explicit, widely agreed upon assumptions about the requirements of theoretical proofs in mathematics. Weak functions arguments can gain rigor by careful formulation on the basis of computational complexity profiles. Such a profile would require explicit formulation and wide acceptance. In the current literature, dynamic computationalists far excel classicists both in the formulation of relevant functions arguments and in the recognition of the structure of such arguments.

There remains, however, the second type of computational equivalence argument, the nature argument, mentioned in the introductory quote and section. Recall that formulations of the nature argument follow the assumption that "when you understand the nature of a thing, you know what it is capable of." Nature arguments look at the theoretical definitions of computational systems, arguing for equivalence or non-equivalence of systems based upon a mapping of possible computational operations and primitives available to the respective systems. In this arena, as in the functions argument, the dynamic computationalist has a clear advantage.

On the theoretical level, dynamic computers can compute using operations on real valued quantities. As a result, theoretical proofs have long existed that connectionist systems can compute functions not computable by Turing-equivalent systems.2 This is not surprising from the perspective of the nature argument, since these systems can, theoretically, represent using real numbers (have infinite representational capacities) and can perform computations using any mathematically definable function (call upon an infinite variety of primitive computational operations).

On the implementational level, dynamic computational systems plausibly have limitations in both their representational capacity (in that they will operate on some quite possibly finite subset of the reals) and their computational operations (in that they will likely only operate using some small subset of the definable mathematical functions). As noted in the last section, limitations in precision need not result in equivalence between classical and dynamic architectures. Indeed, because connectionist systems operate on activation values (quantities as opposed to structured symbols) using primitive operations best described as quantitative functions(as opposed to read/write or restructuring functions), the two types of systems tend only to approximate each other's functioning. Even these approximations are often bought at great computational expense.

Moreover, dynamic computationalists have long recognized that connectionist systems are quite imperfect models of neuronal functioning (Smolensky 1988). Nevertheless, connectionist models can make some pretense of neuron semblance. The general dynamic computational framework is consistent with more realistic systems.3 Indeed, the main difficulty in implementing more realistic dynamic models is the high computational complexity of simulation on classical machines.

In contrast, there is literally no evidence from neuroscience to support the supposition that the brain implements a Turing-equivalent system computing using read/write rules on symbol strings with a recursive, combinatorial syntax and semantics. This shocking gap in the classicist cannon is no longer either explicable or excusable by a limited understanding of the brain. Classicists, I would suggest, rely upon functions arguments because the only evidence suggestive of their architecture is gross behavioral evidence.

VIII. Conclusion. The debate as to the nature of cognitive state transitions in cognitive science remains a debate regarding the appropriate notion of computation in cognitive science. Neither classicists nor dynamic computationalists have an unblemished record of forwarding relevant, well-defined arguments regarding the equivalence or non-equivalence of these two approaches to computation. Similar comments can be made with regards to the arguments as to the superiority of one conception of computation over the other in the context of explanations in cognitive science.

Nevertheless, dynamic computationalists have a clear advantage with regard to rigor and insight (or meta-insight) in both their functions arguments and their nature arguments. Weaknesses on the theoretical level of classicalist arguments have resulted in their conflating implementational and theoretical considerations in their functions arguments. Lack of any supporting evidence from neuroscience as caused them to ignore nature arguments all together.

Dynamic computationalists have largely won the debate, despite its seeming continued fervor in the cognitive science literature.

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Notes

1.) For example, Eliasmith 2001, and 1996.

2.) See for example, Hecht-Nielsen, (1991) for and adaption of Kolomogrov's theorem to three layer neural networks.

3.) In the philosophical literature, Plunkett 2001 gives an overview of such research.