Bulitko (1997) has suggested expanding my model by replacing Axiom 1 with a more general assumption. It is shown in this response that such an expansion is incompatible with the principle of freedom. It is also proven that Axiom 1 can be replaced by a formal analogue of the statement that the subject has free will.
2. I chose the following equation as a formal basis for the model of man:
where variable x1 corresponds to the influence of the external world and variable x2 to the subject's image of this influence. Variable x3 reflects the subject's intention: the value of x3 is the probability of his intention to choose the positive pole. And variable X1 reflects the subject's behavior: it is the probability of his choosing the positive pole in reality.
3. It is clear that the values of x3 and X1 belong in the interval [0,1]. I assume that the values of x1 and x2 also belong in this interval.
4. Further, equation (1) is specified in such a way that it reflects the principle of freedom. First, I introduce a formal analogue of the statement "we have free will." The meaning of this statement is that our intentions have no restrictions. In relation to the representation of a subject by equation (1), this means that variable x3 can take on any value in the interval [0,1], independently of the values taken on by x1 and x2.
5. This definition is not trivial. Suppose we choose a function X1=f(x1,x2,x3) such that for a certain pair of values x1 and x2 there exists x3=c such that X1<0. Since the probability is never negative, we conclude that the subject cannot have intention x3=c, that is, he does not have free will. Therefore, we will have to reject this particular function.
6. Second, I introduce the formal analogue of the statement that "under certain circumstances we have freedom of choice." In relation to equation (1) this means that there must be at least one pair of values x1=a and x2=b such that any subject's intention turns into an action, that is, for any x3 equation f(a,b,x3)=x3 holds. Therefore, in the framework of our approach the concepts of free will and freedom of choice are different.
7. The statement that we have freedom of choice only under certain circumstances implies the assumption that there might be circumstances under which we do not have freedom of choice. In the target article (Lefebvre, 1995) I introduced formal analogues for such circumstances and specified the principle of freedom in the framework of the model of man represented by equation (1). As a result three axioms were formulated:
Axiom 1: f(0,0,x3)=x3, Axiom 2: f(0,1,x3)=0, Axiom 3: f(1,x2,x3)=1,
where x2 and x3 are any numbers from [0,1].
8. These limitations are not enough to find a specific function X1=f(x1,x2,x3). So I assumed that this function becomes linear if any pair of variables are fixed. I justified this choice by the desire to find the simplest function. Bulitko (1997) criticizes me for choosing the criterion of simplicity in this way. I agree that other criteria might also exist, but I will hold to this one because it reflects my intuitive understanding of a function's simplicity. This criterion allows me to represent the right-hand part of equation (1) in the form
and then, with the help of axioms 1, 2, and 3, to find the value of ai. As a result we obtain the following function:
The subject represented by this function has freedom of choice when x1=x2=0. He also has free will because for any x3, independent of x1 and x2, the value of X1 belongs in the interval [0,1].
9. Axiom 1 determines the values of parameters a0=0 and a3=1. Bulitko (1997, par. 4) attempted to expand the model by lifting the limitations given by Axiom 1 and providing a subject with freedom of choice at values of a0 and a3 other than a0=0 and a3=1. I will show later that such an expansion is incompatible with the principle of freedom.
10. By applying Axioms 2 and 3 (without Axiom 1) to equation (2) we obtain the function
The subject represented by equation (4) has freedom of choice, if the values of a0,a3,x1, and x2 are such that for any x3 from [0,1] the condition X1=x3 holds.
11. It follows from (4) that the necessary and sufficient conditions for the identity X1=x3 to hold are the following:
x1+a0(1-x1)(1-x2)=0, (5) a3(1-x1)(1-x2)=1. (6)
It follows from these equations that a0<=0 and a3>=1.
12. To clarify the main idea of the further analysis, we will consider an example. Suppose we want the identity X1=x3 to hold for x1=x2=(1/2). Substituting these values into (5) and (6) we find that a0=-2 and a3=4. So for this case the equation (4) looks as follows:
If x1=x2=0 the subject is represented by equation
It is easy to see that this subject cannot have intention x3<(1/2), since it would make X1<0, and probability cannot be negative. Therefore, although the subject represented by equation (7) has freedom of choice at x1=x2=(1/2), he does not have free will, because at x1=x2=0 a set of "forbidden" intentions x3<(1/2) appears. Thus, equation (7) must be rejected.
13. Let us consider now a general case given by equation (4). (A) Let a0<0 and x1=x2=0. With x3=0 the value of X1=a0<0. Therefore, at a0<0 the subject does not have free will, and the only acceptable value of a0 is 0. (B) Let a0=0, a3>1 and x1=x2=0. We choose x3=1 and find that X1=a3>1, but probability cannot be greater than 1, therefore, at a3>1 the subject cannot have intention x3=1, that is, he does not have free will, and the only acceptable value for a3 is 1.
14. Thus we have proven that any expansion of the model by allowing the variables a0 and a3 to take on values different from a0=0 and a3=1 leads to the subject's losing free will. Moreover, we have demonstrated that our Axiom 1 is superfluous from the formal point of view. It can be replaced by the condition that the subject has free will. To include this condition, it is enough to assume that the function X1=f(x1,x2,x3) is such that for any three values of x1, x2, x3 in the interval [0,1] the value of X1 also belongs in the interval [0,1]. With such introduction to this model, Axiom 1 acquires the status of a theorem.
15. In conclusion I would like to emphasize that the principle of freedom, as well as the anthropic principle, is a means for the rejection of models rather than for their deduction.
Bulitko, V.K. (1997) Lefebvre's Principle of Freedom and One Alternative Approach. PSYCOLOQUY 8(05) human-choice.8.bulitko.
Lefebvre, V.A. (1995) The Anthropic Principle in Psychology and Human Choice. PSYCOLOQUY 6(29) human-choice.1.lefebvre.