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