Anatol Rapoport (1996) Probabilistic Choice and Lefebvre's Model of Human Reflexion. Psycoloquy: 7(05) Human Choice (8)

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Psycoloquy 7(05): Probabilistic Choice and Lefebvre's Model of Human Reflexion

Commentary on Lefebvre on Human-Choice

Anatol Rapoport
University College
University of Toronto
Toronto, Canada


Some points raised by commentators on Lefebvre (1995) are examined. It is argued that Lefebvre's efforts are an important contribution to injecting rigorous deductive reasoning into intuitive approaches to human reflexive cognition.


choice; computation; decision theory; ethical cognition; mathematical psychology; model building; parameter estimation; probability; rationality.
1. In reading Adams-Webber's (1995) and Kampis's (1995) comments on Lefebvre's discussion of human reflexion and the anthropic principle, I tried to imagine a discussion between them on that subject and found it difficult. Kampis seemed to be interested mainly in the logical structure of Lefebvre's thought processes and found it deficient. Adams-Webber wrote about the content of Lefebvre's arguments and found it to be rich. Both showed an interest in the anthropic principle and evaluated Lefebvre's ideas about it, Kampis negatively, Adams-Webber positively. I found no common ground.

2. In re-reading my own remarks (Rapoport, 1995), I realized that they barely touched upon the anthropic principle, which was supposed to be central in the discussion. It became clear to me that I was only mildly interested in the question posed and answered by the principle and consequently I contributed next to nothing to the central theme of the discussion. What I am intensely interested in is the phenomenon of human cognition and its place in ethics and in the philosophy of science. Specifically, I am interested in a science of human cognition rooted in its most directly perceived aspect (introspective consciousness) but developed according to strict principles of rigor and operational criteria of meanings. It is primarily for these reasons that I joined the discussion. And it is in this spirit that I continue it.

3. In replying to Kampis's comments, Lefebvre (1996a) points out that he does not regard the irreducibly probabilistic model of physical reality underlying quantum mechanics as evidence supporting the existence of freedom of will. I interpret this reply to mean that "probability" is a multi-level concept.

4. To take a simple example, consider the probability p that a tossed coin shows heads. In a way, p is a parameter representing one of the physical properties of the coin. In attempting to "determine" p, we initially assign to it a "prior density distribution." For example, to express our initial total ignorance about p, this prior density may be one with maximal entropy, that is, f(p)=1(0<=p<=1). This density is a particular "beta function," namely B(1,1). To get a better estimate, we toss the coin several times. The recorded outcomes modify the parameters of the prior density, whereby the "posterior" density B(m,n) with m>1, n>1 exhibits a maximum on the interval [0,1] and its variance (or entropy) decreases. Our confidence in the "true" value of p increases but never reaches certainty. We can now speak of the "probability that p lies within certain limits", that is of a "probability of a probability." This concept can be clearly extended to higher levels, thus reducing indeterminacy on one level to determinacy on another: we say "the probability that the value of p lies within such and such limits 'is' so and so." And so on.

5. I believe it is in order to avoid infinite regression that Lefebvre essentially treats "freedom of the will" as a primitive (hence undefined) term in his model. In this context then, the postulate of the existence of free will cannot be falsified. In fact, I am inclined to express a mild criticism of Lefebvre's emphasis on the unfalsifiability of his initial assumption (which nevertheless leads to empirically corroborated predictions). Actually unfalsifiability of initial assumptions, as well as undefinability of primitive terms pervades the most rigorous "hard" sciences, and there is no need to emphasize them.

6. In the context of hard science methodology, it is also held axiomatic that only theorems devoid of empirical content can be "proved". Empirically testable hypotheses can be only corroborated or refuted. For this reason, Kampis's dismissal of Lefebvre's allegedly attempted "proof" of the free will hypothesis misses the mark. The corroborations of consequences derived from Lefebvre's introspection model constitute not a "proof" that it is true but only evidence that it is theoretically fruitful. Construction of fruitful models is, after all standard practice in all development of theories.

7. This brings me to Lefebvre's distinction between two meanings of "introspection" (Lefebvre, 1996b). In one sense, the term refers to the human ability to "turn mental ability back on itself." In this sense, Lefebvre agrees, introspection plays a central part in his researches. In another sense, introspection refers to studying the human psyche through self-observation, a path pursued by Wundt and his followers. This method, Lefebvre points out, led to a dead end and was subsequently abandoned.

8. It seems to me that the main reason for abandoning introspection as a method of scientific psychology was the appearance of a strongly seductive rival, namely behaviorism, which offered much broader opportunities for designing theoretically fertile mathematical models and well controlled experiments, particularly on animals (whose "inner world" is inaccessible to us). In turning away from self observation and from concepts in which mental phenomena play an important part, the behaviorists believed that they were putting psychology on a more solid "objective" basis. Yet their conceptual schemes also rested on undefined terms and unverifiable (and unfalsifiable) assumptions, although not explicitly recognized as such.

9. Consider a standard experiment in instrumental conditioning. The experimenter's notebook contains a number of check marks indicating instances when a rat, after a signal was given, "pressed a lever, in consequence of which a mechanism delivered a food pellet, which the rat ate." Apparently this statement is a model of clarity and objectivity. Yet consider the tacit assumption on which it rests, namely, that all the events mentioned can be subsumed under well defined classes. In the same way, in recording the results of coin tosses, the experimenter tacitly assumes that every toss is like every other. This is manifestly false: every toss is different. The angle at which the coin is thrown, the number of times it turns over while falling, how it is caught all differ from toss to toss. Nevertheless all these variations are ignored for a very good reason, namely, that those differences really don't make a difference.

10. Let us return to the experiment with the rat learning to get a pellet by pressing a lever. While the pellet-delivering mechanism may act in more or less exactly the same way from trial to trial, the same cannot be said of the rat. It may use entirely different muscles in different sequence pressing the lever and eating the pellet. Yet all these events are also lumped into one observed event marked by checking a column in the experimenter's notebook. The only thing that varies and matters is the number of these check marks under various experimental conditions.

11. Now what is the faculty of human cognition that makes possible the act of classification, that is, lumping together innumerable events into a single "recognized" event? The word "recognized" identifies the faculty -- recognition. Of course we share it with other living beings, and take it for granted that it is "natural" for a dog or a cat to distinguish between a dog and a cat. But aside from identification of a few key stimuli, we know very little about the mechanism of recognition in higher animals. Witness the persistent and thus far only vaguely fruitful efforts to build artificial intelligence devices that possess really impressive faculties of recognition, say of the sort that we exhibit when we recognize the bee alighting on a flower and a man sending flowers to a woman as steps in the same process, namely, sexual reproduction.

12. Possibly introspection in Lefebvre's sense, that is, "turning mental ability back on itself" is a faculty unique to humans, something that in view of the inaccessibility of the inner worlds of non-humans cannot be verified. However the fact that we possess it can be verified, namely, by introspection in Wundt's sense. If only for this reason, it seems not advisable to abandon introspection (in Wundt's sense) as a method of cognitive science. To be sure the problem of integrating it into the highly disciplined process of scientific cognition remains. It seems to me that Lefebvre undertook that very task, and it is for this reason that I regard his work in this area of prime importance.

13. Lefebvre's attack on the problem is presented in his monograph, A PSYCHOLOGICAL THEORY OF BIPOLARITY AND REFLEXIVITY (1992). He introduces the function X1=f(x1,x2,X3), where the arguments xi(0<=x<=1, i=1,2,3) represent respectively pressure on the subject toward a positive pole (of behavior), proximity of the of the subject's image of the world to the positive pole, and intention to perform a choice, while their complements 1-x, represent pressures toward the opposite pole. Assuming X1 to be a linear function of each of the three arguments separately, Lefebvre derives an expression for X1 as a function of two variables: X1=F[x1,F(x2,X3)], where F(x,y)=1-y+xy. Moreover, he has shown that this representation of X1=f(x1,x2,X3) is unique. In his discussion in PSYCOLOQUY, Lefebvre writes:

    "Thus we have succeeded in obtaining information about the
    subject's, mental domain from a function related to his behavior,
    without reliance on introspective observation" (Lefebvre, 1996b).

We have here an example of how Wundt's method can be "hardened" so as to satisfy the criteria usually associated with scientific cognition.

14. Finally a word is in order concerning the relevance of the anthropic principle to cognitive science. As it is usually stated, the anthropic principle asserts that any theory of cosmic evolution must contain an assumption that its supposed course should incorporate the coming into existence of an observer, actually of an observer capable of self-observation, because we (who are convinced that we exist) are such observers. If however, modern philosophy of science admits the possibility of irreducible uncertainty in the course of events (of the sort underlying, say, quantum mechanics), then the anthropic principle cannot be stated in a categorical mode. The most that can be attributed to it is that it should provide for a finite probability that a self observing observer appears in the course of cosmic evolution. This interpretation suggests that the course of cosmic evolution may have spawned several universes some with self-observing observers, some without. Of the latter, we can, of course know nothing. All we know is that we have been "lucky" to have appeared.


Adams-Webber, J. (1995). A Pragmatic Constructivist Gambit for Cognitive Scientists. PSYCOLOQUY 6(34) human-choice.2.adams-webber.

Kampis, G. (1995). The Anthropic Principle of What? PSYCOLOQUY 6(38) human-choice.4.kampis.

Lefebvre, V. A. (1992). A Psychological Theory of Bipolarity and Reflexivity. Lewiston: The Edwin Mellen Press.

Lefebvre, V. A. (1995). The Anthropic Principle in Psychology and Human Choice. PSYCOLOQUY 6(29) human-choice.1.lefebvre.

Lefebvre, V. A. (1996a). Extracting Information about Subjective States from a Function Describing Human Behavior. PSYCOLOQUY 7(08) human-choice.6.lefebvre.

Lefebvre, V. A. (1996b). The Inexplicable Effectiveness of Metaphysical Reasoning in the Construction of Mathematical Models. PSYCOLOQUY 7(09) human-choice.7.lefebvre.

Rapoport, A. (1995). Human Reflexion and the Anthropic Principle. PSYCOLOQUY 6(37) human-choice.3.rapoport.

psycoloquy.97.8.05.human-choice.8.bulitko Saturday 22 March 1997 ISSN 1055-0143 (24 paragraphs, 4 references, 296 lines) PSYCOLOQUY is sponsored by the American Psychological Association (APA)

                Copyright 1997 V.K. Bulitko

                Commentary on Lefebvre on Human-Choice

                Valeriy K. Bulitko 
                Institute of Mathematics
                Odessa State University
                Odessa, Ukraine 

    ABSTRACT: We discuss Lefebvre's well-known models of human choice
    (1995) mainly in terms of the methodology of modeling. We attempt
    to clarify why these models have above all a "data packing"
    character. We also critique Lefebvre's argument and propose a
    different approach to the problem.


1. We start with a discussion of Lefebvre's (1995) article in order to show that his arguments concerning the anthropic principle is not sufficient. His "principles", "postulates", and "axioms" do not consist of an explicitly formulated system and his interpretation of their role is quite controversial. To be constructive, we provide here a draft of another approach to the same area of psychological effects. The following is the notation used in this article: xi - a variable with index i. If an index i also has own index j we write xij; b\in B - if b is an element of B; M\subset N - if M is a subset of N; x\exp n - x to the nth power; * represents multiplication; a/b denotes the ratio of a to b; A\B is the difference of sets A and B; -> represents implication; a\not=b means a is not equal to b.


II.1. "Anthropic Principle"

2. Principles like the "Anthropic Principle" are means of using an informational property to specify free parameters of models. Thus before using such means, one must have a model with free parameters. In par. 4 the author chooses an initial model of the following form:

    X1 = f(x1, x2, x3) 
    f(x1, x2, x3) = a0 + a1 * x1 + a2 * x2 + a3 * x3 + a4 *
    x1 * x2 + a5 * x1 * x3 + a5 * x2 * x3 + a7 * x1 * x2 *

and specifies the interpretation of its variables. It is important to note that the variables X1, x1, x2, x3 are given essentially different interpretations. The author intends to specify values of the parameters a0 - a7 by means of some principles which are analogous to the cosmological "Anthropic Principle".

3. In par. 7 one such principle, the "Principle of Freedom" is formulated as a "point of departure": "We have free will and under certain circumstances also freedom of choice". It is clear that this statement can not be used immediately to specify the parameters. The author's specification follows from axioms 1, 2, and 3. Any proper subset of them does not permit that. However, is it possible to say that these three axioms correctly reflect the "Principle of Freedom"? Indeed Axioms 2 and 3 state just about the absence of any freedom since they postulate independence of choice X1 on will x3 (they determine six parameters a1, a2, ..., a7). Hence they describe "circumstances" where a subject has no freedom of choice. However it is quite clear that their content is essentially more specific than the Principle.

4. Axiom 3 is called "the axiom of free choice" and is responsible for a0 and a1. It also contains a lot of arbitrariness with respect to the Principle and does not follow from the Principle. To provide freedom of choice, it is sufficient to postulate a3 \not= 0. For example we can set a0 = 0.1 and a3 = 1.6. Then we will have "free choice" even under the "Realist's condition" at x1 = a0/a3 = 1/16, x2 = (1 + a0 - a3)/(a0 - a3) = 1/3.

5. We can therefore propose a statement that is completely opposite to the following statement by Lefebvre (par. 34): "Heretofore, we have found a specific function from the class of functions given by Equation (1.1) without experimental estimation of the parameters a0, a2, ..., a7 but using an abstract principle of freedom, instead." To be correct, the author should have accepted just the conjunction of Axioms 1, 2, and 3 as the Principle of Freedom with obvious consequences as to the possibilities of philosophical speculations.

II.2. How Much Simplicity Is There in the "Postulate of Simplicity"?

6. Axioms 1, 2, and 3 determine function f on part of the surface of unit cube K = [0,1] x[0,1] x[0,1]. The surface has volume 0. So values of f are set on almost the entire unit cube just by the "Postulate of Simplicity" (PS). However there are different notions of simplicity. For example, the function f'(x1, x2, x3) = min{1 + x1 - min{1 + x2 - x3, 1}, 1} is a piecewise linear function and it satisfies Axioms 1,2, and 3. Moreover it consists only of a finite set of plane pieces whereas Lefebvre's value for f is not linear in any small inner sphere of the cube. However it is impossible to obtain the results about the golden section by the author's reasonings where f is changed to f'. The author did not give any other ground for PS apart from his "simplicity" argument. We do not see this ground as convincing.

7. To figure out PS's meaning let us restrict ourselves to the vertex set of the cube. For this set, Axioms 1, 2, and 3 completely define f. So f(x1, x2, x3) coincides with the boolean function F(x1, x2, x3) = (x3 -> x2) -> x1. Now let us take the last function as the point of departure. Suppose we are given a point (k1, k2, k3)\in K and we want to calculate f(k1, k2, k3). Let us consider an ensemble of Lefebvre's subjects, that is such subjects where their variables X1, x1, x2, x3 have only boolean values and where their behavior is described by function F. Let us suppose that boolean values of x1, x2, x3 are independently distributed on the ensemble with the average value of xi equal to ki, i = 1, 2, 3. We refer to such an ensemble as Lefebvre's ensemble (with parameters k1, k2, k3). It is then not hard to see that f(k1, k2, k3) is the average value of F on Lefebvre's ensemble with parameters k1, k2, k3. A similar algorithm exists for an ensemble of Lefebvre's "Realists" and so on. One can reformulate Lefebvre's entire theory in those terms working only with boolean values of the general model variables.

8. Thus PS is equivalent to the conditions of Lefebvre's ensemble. However the statistical independence of free variables seems to be an excessive simplification taking into account their interpretation.

II.3. On the "Realist" Condition

9. We mean the condition where X1 = x3. It seems like here we have another hidden postulate since the reasonings of par. 19 force at most the condition X1 = g(x3) where g is a function independent of x1 and x2.

II.4. Consciousness and Experience in the General Model

10. The models do not have any feedback of the form "behaviour" -> "consciousness", "behaviour" -> "the last experience". Therefore x2 and x3 are external parameters. To our mind this mean that subjects in Lefebvre's models have neither consciousness nor memory even though these things are presupposed in the interpretation. Thus we can say that the models are essentially incomplete.


11. The following section proposes an alternative approach that tries to overcome the above-mentioned shortcomings. We follow (Bulitko, 1986) to describe "subject - environment" interaction with some simplifications.

12. Let a subject O interact with an environment S by activating its own output y in order to obtain an output x from a set U of desirable environment outputs. We can consider an operator P describing the environment reaction: x = P(y). Hence the subject's task is to inverse the operator P on x\in U. This means that a correctly acting subject has to implement an appropriate operator F such that F(P(x)) = x. Thus x is a fixed point of operator composition F * P. We regard x as a set of elementary events percepted by a subject and y as a set of elementary actions done by this subject.

13. It is sometimes possible to substantiate that both operators P and F are enumeration operators for a finite period of time (see the exact definition, e.g., in Rogers, 1967). This is because the enumeration operator is the general model of total effective transformation of natural number sets to natural number sets. The set W of all enumeration operators has the natural and important numbering n: N -> W. Every index n(E), E\in W, is a fixed number code of a program computing E. The marvellous Kleene's theorem ("recursion theorem") states (in simplified form) that there is an effectively computable function k(n) for any given index n and for any given enumeration operator En some program to enumerate a fixed point of En. It is important to remark that the function k does not depend on the operator.

14. So let us imagine that the environment and the subject of a given kind are described by a collection of enumeration operators Pni and Fmi correspondingly, i = 1,..., r, with respect to a given collection S1, ..., Sr of situations. Then to choose correctly (i.e., to survive) the subject has to determine indexes of Fmi * Pni and then compute a fixed point of this operator composition in any situation Si. The last part is done automatically by means of a subject's built-in (finite) program for k. The situation itself is perceived with some "hypothesis block" H for the subject.

15. It is important that this scheme does not assume any consciousness for subjects even though one can find situations like the ones described by x1, x2, x3, X1 in Lefebvre's model.

16. Moreover some evolution is possible for a population of such subjects. Indeed, let us assume that subject M has posterity M1, ..., Mu. M gave them the same k and H1, ..., Hu correspondingly. Then the genus of M will survive if among H1, ..., Hu there is at least one Ht such that it will reflect future situations correctly.

17. Every enumeration operator P is monotonic, that is if M is a subset of N then P(M) is also a subset of P(N). However there are many natural (and simple) non-monotonic operators. Moreover they are computable (computable with respect to some oracle) in some domains. The main point for us however is that here we do not have such a general theorem as Kleene's Recursion Theorem. Therefore the fixed point program and Hypothesis Block are not sufficient to find a correct behavior. One of the possible ways is as follows: Let us suppose that we are given a non-monotonic operator P: D -> R and the task of finding a fixed point for it. Let us also assume it is possible to select several subdomains D1, ..., Dq in G and subsets R1, ..., Rq correspondingly such that P(Di) = Ri, i = 1, ..., q, and the partial orders with respect to relation "to be subset" on sets Di, Ri satisfying the well known Tarski's theorem about fixed point existence. Now a subject may pass from operator P to the monotonic operator P' by simplifying the ordering on the set D. To achieve this the subject has to mark every element of Di with a label "Di". Then any two elements of different subdomains are incomparable. Given such classifications of possible systems of actions, the subject needs to choose one of several possible cases (i.e., one Di of i = 1, ..., q) and to apply the mechanism described above for enumeration operators in order to automatically find an acceptable action plan.

18. However to proceed in this manner the subject has to have elements of consciousness to classify domain D and some logic to analyze the cases correctly. Indeed, duplication of reality is the basis of classification and an essential property of consciousness. On the other hand, logic is the means to correctly analyze the system of all possible cases.

19. Such reasoning, a brief draft of which we have just outlined, allows us to understand where consciousness provides an advantage over unconscious behavior. We can also see the role of logic in behavior generation.

20. The outlined approach gives wider and more powerful means than Lefebvre's models for interpreting subject behavior. Indeed we can obtain Lefebvre's basic formula X1 = (x3 -> x2) -> x1 by limiting ourselves to operators of the form P = P1 \ (P2 \ P3) where every Pi is a constant operator. In other words, we just have to set P(M) equal to P1(M) \ (P2(M) \ P3(M)). Let us suppose that arguments and values of Pi, for every i, belong to sets of the form {0, N}. Here 0 is the empty set and N is the set of natural numbers. For every choice of such operators P1, P2, P3 the operator P obviously has a fixed point. If we write out the table representing this fixed point as a function on operator Pi, values (i = 1, 2, 3), then we will be able to see the table that is completely analogous to the table of the boolean function (x3 -> x2) -> x1.

21. We can interpret the golden section on the basis of some fixed point existence theorem. Indeed, let P: S -> S be the operator where S is the set of subsets of the natural numbers. The problem of finding a fixed point of P can be reduced to the problem of finding a minimum of a function f on the segment [0,1] by the following method. We represent an arbitrary subset X of N by a real number r(X)\in [0,1] that is the sum of all numbers 2exp(-a), a\in X. (Here we omit some further details for simplicity.) Then the problem of finding a minimum of r(|X - P(X)|) as the function of r(X) represents the fixed point problem.

22. There exists one remarkable algorithm to find the minimum for the class of all continuous functions with one minimum on the segment [0,1] if we only have access to values of functions for the requested points of [0,1]. We are referring to Kiefer's algorithm (Kiefer, 1953). This algorithm minimizes the number of calculations of given function values in the process of approximating the point where a given function has its minimal value. (We can show that Kiefer's algorithm is analogous in some sense to the recursion theorem.) Now, if the number of steps in Kiefer's algorithm is not limited, then the first step is a computation of the given function value in the point dividing [0,1] in the golden section ratio.

23. It is possible to come up with an interpretation for the categorization experiments.


24. Lefebvre's models are brilliant, a significant achievement in the area of mathematical modeling of psychological phenomena, and deserving of serious analysis. However, in our opinion, the model's shortcoming is its excessive conceptual simplicity and consequent oversimplicity of its mathematical technique. On the one hand this simplicity provokes (unsuccessfully as we saw above) attempts to specify model parameters by means of metaphysical principles. On the other hand the excessive simplicity prevents the further development of the models. Indeed, it is hard to agree that such concepts as the "index of belief" or quantum mechanical analogies like "screen" and "discrete spectrum" are derivatives from the initial concepts of the models (i.e., the concepts were used to formulate Axioms 1-3 and the Simplicity Postulate).


Lefebvre V.A. (1995) The Anthropic Principle in Psychology and Human Choice. PSYCOLOQUY 6(29) human-choice.1.lefebvre.

Bulitko V.K. (1986) Modeling of processes in economico-ecological systems, "Naukova Dumka", Kiev, (in Russian).

Rogers H., Jr. (1967) Theory of recursive functions and effective computability, McGraw-Hill.

Kiefer J. (1953) Sequential minimax search for a maximum, Proceedings of the American Mathematical Society, 4, 502.

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