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Dialectical psychology (Riegel, 1973) postulates that one's mental processes move freely back and forth among all the Piagetian stages, meanwhile "transforming contradictory experience into momentary stable structures." In this paper it is shown that the symmetric difference operation of set-theoretic topology, together with its complement, can be used to represent the fundamental operations of both dialectical logic and dialectical psychology. Thesis and antithesis are expressed by the symmetric difference; synthesis and context, by its complement. Applications of this algorithm are made to Piagetian developmental psychology, memory and learning, intelligence, quantitative psychology, creativity, and social psychology.

"Dialectic" is a word of many meanings (Rychlak, 1976). Here the meaning of "dialectic" will be taken to be number (8) in the article "Dialectic" in the Encyclopedia of Philosophy: "...the logical development of thought or reality through thesis and antithesis to a synthesis of these opposites." The distinction between an object or concept and what it is not leads inevitably to a dialectic view of the nature of the world. Think of figure and ground: everything has its contrast. Each thought is a composite consisting of some element (or elements) that belongs to a universal class (the thesis) coupled with what it is not within that universal (the antithesis) plus a means (the synthesis) of resolving the contradiction arising out of that discrimination.

Dialectical thinking consists of an exploration of contradictory possibilities that results in cognitions which reduce cognitive dissonance. Doubt searches out every belief: "It could be that way, but is it really?" To any appearance, there is the underlying reality ("the thing in itself"). Kahneman and Miller (1986) state that all perceived events are compared to counterfactual alternatives, counterfactual in that they constitute alternative realities to that experienced. Johnson-Laird (1995) in his study of mental models appropriate to deductive thinking, also notes the importance of counterfactuals. Knight and Grabowecky (1995) assert that counterfactuals are omnipresent in normal human cognition.

Curiosity, interest, and belief - "intentionality" - are inherent in such a dialectical structure though not in such a closed system as logical deduction. According to Marcel and Bisiach (1988), "intention/al/ity" can mean any one of the following:

- A totality of objects or attributes comprehended as a set or concept. This is "intension."
- An explicit goal or purpose - an "intention" - with respect to some particular thing.
- Content, reference, or indication - what something is about, "intention" again.

The first of these - set-theoretic "intension" - is the meaning employed in this paper. Intentionality here consists of psychological sets of attributes, perepts, and/or concepts.

Riegel's postulates for dialectical psychology are the same as Hegel's for dialectic:

- I. The unity and struggle of opposites.
- II. The transformation of quantitative into qualitative change.
- III. The negation of the negation.

In the present study these three "laws" will be expressed, in the contexts of both dialectical logic and dialectical psychology, by means of the set-theoretic operation of symmetric difference: "one or the other but not both together." There exists a bewildering variety of notations for the symmetric difference, most of which have already been preempted in other contexts. Here I introduce a new one, $, which is like "S" for "symmetric" and close to the set subtraction symbol \ as well; besides it is already on the keyboard.

Before discussing the dialectical "laws" in symbolic terms, it seems advisable at this stage to list the notations to be employed in the sequel:

- v.......... set-theoretic union or "sum" (non-disjoint)
- &.......... set-theoretic intersection or "product"
- $.......... symmetric difference of sets
- \ ......... set-theoretic subtraction, or, depending upon context, an arrow pointing diagonally downward
- - ......... the complement of the set that follows -
- / ......... "factorization" of a set to yield a quotient set, or, depending upon context, an arrow pointing upward to the right
- < ......... set containment symbol
- 0 ......... the null set

The symmetric difference operation $ upon two cognitive sets C and C' is thus defined symbolically by the expression:

C v C' \ (C & C')

Here $ denotes the symmetric difference of C and C'; & is the set product or intersection; v is the set-theoretic sum or union; -C denotes the set-theoretic complement of C, and similarly for -C'; and \ is the (relative) set-theoretic difference. A Venn diagram for $ looks like that in Figure 1.

*Figure 1. : A Venn diagram for the symmetric difference of two
sets C and C'. The shaded portion comprises C $ C'. U is the
universe of discourse.*

Exemplified in Eq. (1) and Figure 1 are the first two of the Hegel-Riegel "laws:" the unity and struggle of opposites and the replacement of the quantitative by qualitative, set-theoretic operations. Also apparent is the "law of the excluded middle." The symmetric difference and its complement are equivalent in logic to the Sheffer stroke, which can of itself generate the operations of Boolean algebra (Sheffer, 1913), and so, ordinary first order logic. Formal "logical" thought is thus accessible via the symmetric difference even though intuitive thought is more fundamental (Hoffman, 1980a,b; Riegel, 1973).

Synthesis of C and C', the generation of their commonality, comes about by taking the "negative of the negative." Here, however, the innermost of these two "negatives" is not the complement as such but rather the symmetric difference, as suggested by Eq. (1). The symmetric difference in Eq. (1) acts to strip C and C' of their commonality, C & C'. If now one takes the complement of (1), not only is the commonality C & C' of C and C' restored but also their context, "everything else" in U, as in Figure 2. The derivation goes as follows:

-[(C & -C') v (-C & C')] =

-(C & -C') & -(-C & C') =

(-C v C') & (C v -C') =

(C & C') v (-C & -C') =

(C & C') v -(C v C')

*
Figure 2. The "negation of the negation": not(C $ C') is the
set-theoretic union of the intersection of C and C'
with the intersection of notC and notC': -C & -C'.
*

It is postulated that Eq. (2) properly constitutes the realization of the third of the Hegel-Riegel "laws," the "negation of the negation," even though the latter is traditionally taken to be simple complementation. The first term on the right in Eq. (2) represents the synthesis, the commonality of C and C'. This is Hegel's "unity of opposites." The second term provides their context within the universe of discourse U.

So, via -(C & C'), we have not only a means of synthesis in this second phase but also divergent as well as convergent thinking in the relation of C and C' to "everything else," expressed by -C & -C' = -(C v C'). It is Eq. (2) that corresponds to Hegel's principle that "negation is determination" (Stace, 1955, p. 94).

*
... mere being, without discriminable character, is no
identifiable object. It has no content and is altogether
empty,
identifiable in effect with nothing.
*

Thus the symmetric difference of an object with itself is the null set 0, "nothing":

*(Equation 3)*

A passage of "nothing" into "not-being" takes place via $ and complementation in the comparison

(C & --0) v (-C & -0) =

0 v -C =

-C, notC

Next the symmetric difference of an object with "what it is not" becomes - is - "everything:"

(C & C) v (-C & -C) =

C v -C =

U, the universe of discourse

By the same token, if C should happen to be contained within C', C < C', as in Figure 3, then C $ C' leads to the relative complement of C in C'. Here $ acts as differentia in Hegelian dialectic:

(C v C') \ (C & C') =

C' \ C,

the relative complement of C in C'

* Figure 3. If C is a subset of C', then the symmetric
difference
of C and C' is the relative complement of C in C'.
*

Before proceeding on to the usages of the symmetric difference in Dialectical Psychology we note parenthetically that Hegel himself would no doubt have found such an algorithmic description of dialectic an abomination. Styazhkin (1969, p. 112) has commented in connection with Hegel's denunciation of Ploucquet's "logical calculus" that "One can only imagine what epithets Hegel would have bestowed on contemporary mathematical logic!" Yet Ploucquet's calculus for generating a complete description of all logical relations that was based only on an identity function and an inconsistency function seems rather close to Hegel's own ideas on synthesis and thesis-antithesis.

In psychology the "similarity-difference" paradigm is of course an old story (Friedman, 1984, p. 113ff.). However, the paradigm should read "difference-similarity," for this order appears more appropriate not only in terms of the symmetric difference algorithm but also for greater realism. It is more important to a creature's survival in "the jungle out there" to be aware of novel stimuli in the environment and properly classify them than to reflectively seek their commonalities (compare Figure 2).

* ... the option to operate simultaneously or in short
succession at different levels ... implies contradiction
andis dialectic in character.
*

One of the most compelling features of Riegel's theory was his resolution of the contretemps surrounding the fourth Piagetian stage of development: Formal Operations. The disconfirming experimental results of Lovell (1961), the finding that 37% of college students and 50% of adults fail to demonstrate Formal Operations thinking, as well as the prevalence of such grammatical errors as double negatives in everyday speech, all cast doubt upon "logical thought" as such. Rather, according to cognitive research (Bruner and Olver, 1963, p. 434), the mind acts to reduce the number of possible alternatives and select among these on the basis of their relative degrees of interest.

Riegel (1973, p. 354), using an approach based on McLaughlin's "Psychologic" for the Piagetian stages, organized the latter in terms of the number of classificatins that a child is able to carry out simultaneously at any given development period. During the period of Concrete Operations, the child is able to do double classifications and can form such logical constructs as C and C', C but not C', C' but not C, and neither C nor C' (note 1). Thus both the Piagetian period of Formal Operations and Riegel's dialectical operations stem from the two-phase cognitive operation based on the symmetric difference. Given the foregoing logical constructs, both C $ C' and not(C $ C') are implicit.

At the time that Formal Operations should be developing most adolescents seem to exhibit protean ways of thought more often based on value judgments and peer pressure than logical reasoning. The latter bear little resemblance to the scientific method as envisioned by Piaget. To resolve contradictions, dialectic is required, not logic; logic comes afterward, if at all. This is not to say that one cannot do logic, write structured prose, compose music, solve puzzles, play chess, or proceed in logical thought processes when the occasion demands. But such highly formal thought processes invariably build upon the trains of thought previously generated intuitively in a sort of "chasing around the cognitive diagram" (note 2). It is worthy of note in this connection that any novel - indeed any literary work with a plot - arouses the reader's interest largely because of the conflicts and contradictions posed in that plot and which the reader feels impelled to pursue further in order to resolve.

I will thus identify Working Memory (WM, also denoted by W in the subsequent analysis) with a small set of concepts or percepts, the set having been called up by directed thought from the Long Term Memory Store (hereafter LTM or L), that are presently in the forefront of attention (Baddeley, 1993). The elements of W interact with the Perceived Field (denoted hereafter by P = S/A or Q = imagery/A), where S is the sensory field, e.g., a non-attention-selected ensemble of forms that make up the visual field in its entirety. Thus S is "factored" by attention A to provide a subset P = S/A of the full perceptual manifold.

Many features of this triad of P, WM, and LTM seem well described by the symmetric difference algorithm. If the contents P of the perceived field have already been noted as W in WM, then either P < W or else P = W. When P < W, P $ W = W \ P. When, however, P and W are the same, then P $ W = 0, and closure takes place. In either case identification occurs within WM and attention can be safely directed elsewhere. If, on the other hand, P is not contained within W, them P $ W leads to classification and discrimination based on the differences between WM and the perceived field P $ W.

* Figure 4. The relations among P, WM, and LTM imparted by
the
symmetric difference. (a) Percepts or images already in WM.
(b) Novel percepts or images. *

Further, in the case of assimilation when P < L, the commonality of LTM with the interaction between P and WM simply yields the latter back again:

(L & W) $ (L & P) =

W $ P

since, as is evident, WM must be embedded in LTM, consciously or subconsciously.

It is axiomatic that no one's memory contains all knowledge, present and future. The latter therefore constitutes the universe of discourse, denoted in the present context by K. The complement of LTM in K therefore represents "the great unknown." This more general situation occurs when P is not necessarily contained within LTM. Working memory WM is always contained within LTM, W < L, but if P is not contained within L, this may necessitate accomodation.

The symmetric difference among L, P, and W leads not only to the commonality of P and W, but also "leads memory outside of itself" in an exploration of LTM for concepts outside of P and W. The argument runs as follows:

by associativity and commutativity; but always W < L, so that

[P & -(L & -W)] v [-P & (L & -W)] =

[(P & -L) v (P & W)] v [(-P & -W) & L] =

[P & W] v [L & -(P v W)] v [(P & K) \ L]

The components of this triple union may be interpreted as follows:

- a combining of the perceived field and Working Memory, and/or
- those elements of memory outside of this combination - a reaching out to "everything else" within the Long Term Memory Store, and/or
- a combination of the perceived field and all knowledge K which lies outside of what the subject presently knows - an open-ended encounter with "the great unknown."

*
Reflective thinking is controlled doing, involving a pushing
and pulling of concepts, putting them together and
separating
them again.
*

And also (ibid, p. 20),

*
Learning the meaning of a new piece of knowledge requires
dialog, exchange, sharing, and sometimes compromise.
*

Such processes seem very close to the two-phase thought processes postulated in the symmetric difference model.

The left side of the human brain (hereafter LHC) is supposed to be devoted mainly to mental processes of a sequential, analytical, "logical," or linguistic kind. The right brain (hereafter RHC), on the other hand, is supposed to process mainly psychological phenomena of a Gestalt nature: holistic and relational thinking that involves imagery, imagination, and spatial relations. RHC processing may thus be subsumed under the broad heading of "intuition." Of course such functions are principal but not exclusive functions of each hemisphere.

The situation is depicted in broad outline in Figure 5. The LHC acts to

*
Figure 5. Postulated brain structure separating percept/imagery
into discrimination/classification ($) and
synthesis-cum-context.
W = Working Memory, P = perceived field, C = commissure.
*

impose constraints upon the RHC via subcortical and commissural interactions. Given the nature of mental processing, it would appear that a stimulus or other mental pattern is separated into distinctive classifications in the LHC and processed for synthesis-Gestalt aspects in the RHC by a two-phase mental operation that has the character of the symmetric difference algorithm. Chen (1981) and Sinatra (1984) have extensively analyzed gifted intellect on the basis of such a LHC-RHC dichotomy and made a convincing argument for the importance of such a specialization. Zernhausern et al. (1981) have tested such a hypothesis and validated it for widely differing learning styles.

The symmetric difference thus provides a scale and dimensional properties for such a family of sets. Such sets may correspond in the present context to any contents of P and/or WM, in particular the standard psychometric measures.

Another important measure on sets is probability measure. A probability measure p(C) leads immediately, as a consequence of Eq. (1), to the formula

p(C) + P(C') - 2 p(C & C')

while for the complementary second phase,

p(C & C') + p(-C & -C') =

1 + p(C & C') - p(C v C').

Formulas (10) and (11) appear relevant to the subjective statistical processes involved in behavioral decision making. The doubling of the product term in Eq. (10) provides a sharper basis for classification and discrimination than the standard probability formula for the non-disjoint union of events, and in Eq. (11) a trade-off between judgments of commonality and context is apparent. Eq. (11) admits an immediate and extensive connection with context and LTM via the term involving notC and notC'. These formulas also appear to embody such concepts as Kahneman and Tversky's (1982) "principle of complementarity" and "anchoring" as intuitive subjective probability rather than classical formal probability theory. This "representativeness' theme that intuitive probability is context and memory dependent rather than ordinary numerical runs through many of Kahneman and Tversky's conclusions. Such a view is in accord with some further basic properties of $:

In these two results it is as if alternatives to the event C simply did not occur to the subject, a phenomenon frequently noted by Kahneman and Tversky.

Consider Riegel's (1977) simple dialogue between two speakers, A and B, as in Figure 6.

A --> A' --> A" --> ... \ / \ / \ / B --> B' --> B" --> ...

* Figure 6. Riegel's simple dialogue between two speakers A
and B.*

The primes and double primes denote successive times A denotes the content of speaker A's initial statement, A', the content of his statement on the second occasion, etc., and similarly for speaker B. Each speaker assimilates the other's statements: (A $ B), etc. and accomodates his own production -(A $ A') and -(B $ B') so as to elaborate and extend the preceding dialogue. The contexts of these operations can extend back in a set of paths to the full past history of the dialogue/negotiation. In such a simple dialectical process as that in Figure 6 we encounter the essence of social interaction.

The symmetric difference applied to two cognitive entities generates the standard recognition paradigm of discrimination and classification - "similarities and differences" - but in the opposite, more realistic order. A novel but more realistic interpretation of "negation" in classical Hegelian dialectic is then possible. Negation not of negation itself but of the symmetric difference leads to the second phase of the Dialectical Psychology process, which generates both synthesis and memory-cum-imagination. In the negation,

the presence of -C & -C', i.e., notC and notC', makes possible a more general intuitionistic logic in which the law of the excluded middle expressed by Eq. (1) need no longer hold.

- Note 1. The last of these is the Sheffer stroke, from which
all other
logical operations are derivable. See (Sheffer, 1913). [
*Editor's note: what is called the "Sheffer stroke" was actually originally derived by Charles S. Peirce toward the end of the previous century; Sheffer rediscovered it independently.*] - Note 2. Riegel (1973, p. 363) put it this way,
*It has never been shown convincingly that the highest level of operation, i.e., Formal Operational intelligence, characterizes the thinking of mature adults. Only under the most exceptional conditions of logical argumentation and scholastic disputes is it conceivable that a person would engage in such a form of thinking. ... Such forms of thinking merely provide the last straw in the process of scientific inquiry which is applied after intuitive thought is exhausted.*

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