Logical Self-Reference as a Model for Conscious Experience

Journal of Mathematical Psychology 45, 720731 (2001) doi:10.1006jmps.2000.1349, available online at http:www.idealibrary.com on

Logical Self-Reference as a Model for Conscious Experience Andrei G. Khromov Troitsk Institute of Innovation and Thermonuclear Research, Troitsk, Russia

The structure of autoreferential statements that describe (or serve as a logical model for) the human conscious experience is analyzed. Autoreferential statements are introduced by autoreferential definitions, such as a=F(a, x), where F is a Boolean function and x is some atomic statement describing the content of the experience, or by analogous systems of inter-related definitions, such as a=F(b, x, y) and b=G(a, x, z). It is argued that only ``noncreative'' (systems of) definitions introduce statements that describe conscious experience, the noncreativeness meaning that no statement with nontautological content can be derived from these definitions. The structure of such (systems of) definitions is comprehensively characterized in a series of theorems. A potential of the model in addressing empirical data is illustrated by applying it to the choice between two alternatives in the absence of a preference criterion.  2001 Academic Press

1. INTRODUCTION

A typical human experience (image, intention, thought, belief) can be thought of as consisting of its content and the awareness of this content. Let x, y, z, ... be statements describing contents of various experiences, such as ``this apple is green,'' ``I intend to choose this action.'' Call such statements x, y, z, ... (atomic) content statements. From a given list of atomic content statements one can form other content statements by means of Boolean functions. For example, ``z=not-x or y'' defines a content statement z. Note that definitions, z=F(x, y, ...), are metalinguistic propositions and are not therefore content statements (or other experience-describing statements) themselves. The next step is to introduce awareness statements a, b, c, .... According to the model proposed in this paper, these statements are self-referential (autoreferential) and introduced by means of autoreferential definitions. Specifically, n awareness I am grateful to A. A. Ezhov for helpful discussions related to the subject of this paper. I am also indebted to E. N. Dzhafarov for thoroughly editing earlier drafts of the paper. His invaluable suggestions have significantly influenced the paper's content and structure. Address correspondence and reprint requests to A. G. Khromov, Borovskoe Shosse, 373-159, 119634, Moscow, Russia. E-mail: andrei.khromovusa.net. 0022-249601 35.00 Copyright  2001 by Academic Press All rights of reproduction in any form reserved.

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statements a 1 , ..., a n are introduced by means of an autoreferential system of n definitions, a i =F i (a 1 , ..., a n , x 1 , ..., x m ),

i=1, ..., n,

where F i are some Boolean functions and x 1 , ..., x m are some atomic content statements. As examples, a=F(a, x) is a single definition of a single autoreferential statement a, whereas a=F(b, x, y) and b=G(a, x, z) are a system of two definitions introducing two autoreferential statements, a and b. Once content statements x, y, z, ... and awareness statements a, b, c, ... are introduced, all experience-describing statements, according to the model, can be formed by means of Boolean functions, z=F(a, b, c..., x, y, z...). The question of how to interpret the awareness statements in common language is rather subtle. 1 Consider first a single autoreferential definition, a=F(a, x), where x is interpreted as ``the apple is green.'' The common-language interpretation of a in this definition can be expressed as ``I am aware of this,'' where ``this'' refers to the entire statement F(a, x) rather than merely to x. Examples include ``I am aware of this''=``(the apple is green) and (I am aware of this),'' ``I am aware of this''=``If (the apple is green) then (I am aware of this).'' Note that the equation sign is not a part of the experience-describing statements. It merely shows that the two experience-describing statements on its left and on its right have the same meaning. Since the word ``this'' in the two examples above refers to two different conscious experiences, it is reasonable to introduce certain (meta-linguistic) labeling: ``I am aware of [the state of affairs labeled 1]'' =``(the apple is green) and (I am aware of [the state of affairs labeled 1])''; ``I am aware of [the state of affairs labeled 2]'' =``If (the apple is green) then (I am aware of [the state of affairs labeled 2])''. Such a labeling is indispensable when one deals with an autoreferential system of definitions. Consider, for example, the system a=a and x and not-b, b=b and not-x and not-a. In this case a and b are two different but interrelated conscious experiences of one and the same person, and the common-language interpretation of the definitions above can be given as follows: 1

The common-language interpretation described in the subsequent text was suggested to me by E. N. Dzhafarov, to whom I express my gratitude.

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``I am aware of [the state of affairs labeled 1]''=``(I am aware of [the state of affairs labeled 1]) and (the apple is green) and (I am not aware of [the state of affairs labeled 2])''; ``I am aware of [the state of affairs labeled 2]''=``(I am aware of [the state of affairs labeled 2]) and (the apple is not green) and (I am not aware of [the state of affairs labeled 1])''. That awareness statements describing conscious experiences are always defined by autoreferential definitions is the main assumption of the model proposed in this paper. However, not all syntactically possible autoreferential definitions introduce legitimate awareness statements. The second assumption of the model is that in order to define a legitimate awareness statement an autoreferential definition should be noncreative (see, e.g., Rantala, 1991). In the case of propositional calculus considered in this paper the noncreativeness of a definition means the following: It is not possible to derive from this definition any nontautological content statement. For example, the definition a=not-a or x is creative, because one can derive from it the content statement x. 2 Therefore no conscious experience, according to the model, can be described by this definition. Obviously, this also excludes from consideration all internally contradictory definitions, such as a=not-a because from a contradiction one can derive anything. The reason for adopting this noncreativeness constraint is rather obvious: one must not be able to gain true knowledge of the state of nature (``the apple is green'') from a definition of a subjective experience. Put differently, the validity of a definition of one's subjective experience should not depend on whether the factual statements it includes are true or false. Obviously, the non-creativeness constraint should apply not only to a single autoreferential definition, but also to any system of such definitions. In the two subsequent sections I establish necessary and sufficient conditions for the noncreativeness of an autoreferential definition and an autoreferential system of definitions. These sections also provide a complete enumeration of all non-creative autoreferential definitions and systems of definitions containing a given set of atomic content statements. As an example, in Section 2 I derive the list of all noncreative definitions containing a single atomic content statement. This list is utilized in the concluding section of the paper, in which I tentatively formulate empirically testable application of my model to the situation when a person chooses between 2 As explained in the next section, to derive a statement from a definition means to derive it, by means of propositional calculus, from a statement of the equivalence between the definition's left-hand and right-hand expressions. For example, x in the example just given is derived from at(not-a or x).

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two alternatives (say, x = ``this is good'' and not-x=``this is bad'') without having a criterion of preference.

2. NONCREATIVE AUTOREFERENTIAL DEFINITIONS

In this section I establish necessary and sufficient conditions for the noncreativeness of a single autoreferential definition, and I provide the enumeration of all noncreative definitions with a given list of atomic content statements. I use the standard notation a, ab or a 6 b, a+b, and atb to designate negation, conjunction, disjunction, and equivalence, respectively. Constants 1 and 0 denote, respectively, the truth values ``true'' and ``false.'' It is assumed that a 0 =a, a 1 =a. A definition a= f (a, x 1 , ..., x m ) is called noncreative if no nontautological statement in which a does not occur can be inferred by means of propositional calculus from the corresponding equivalence statement atf (a, x 1 , ..., x m ). Lemma 1. The definition a= f (a, x 1 , ..., x m ) is noncreative if and only if, for any combination t 1 , ..., t n of truth values, the equation a= f (a, t 1 , ..., t m ) has a solution for a. Proof. Suppose that for some truth values t 1 , ..., t m the equation a= f (a, t 1 , ..., t m ) has no solutions for a. Then the equivalence statement atf (a, t1 , ..., t m ) is false for any truth value of a. Consequently, the nontautological statement x t11 6 } } } 6 x tmm , which is false only at x 1 =t1 , ..., x m =t m , is a logical consequence of the equivalence statement atf (a, x 1 , ..., x m ). Conversely, suppose that some logical consequence h(x 1 , ..., x m ) of the equivalence statement atf(a, x 1 , ..., x m ) is not a tautology. Then a combination t 1 , ..., t m of truth values exists for which the statement h(t 1 , ..., t m ) is false. Consequently the equivalence statement atf (a, t 1 , ..., t m ) is false for any truth value of the variable a, and the equation a= f (a, t 1 , ..., t m ) has no solutions for a. K Theorem 1. The definition a= f (a, x 1 , ..., x m ) is noncreative if and only if the Boolean function f (a, x 1 , ..., x m ) can be represented in the form f (a, x 1 , ..., x m )=.(x 1 , ..., x m ) a+(x 1 , ..., x m ).

(1)

Moreover, the Boolean functions . and  can always be chosen to be ``orthogonal,'' .=0.

(2)

With this choice of . and , representation (1) of the function f (a, x 1 , ..., x m ) is unique. Proof.

It is easy to check that f (a, x 1 , ..., x m )=.(x 1 , ..., x m ) a+(x 1 , ..., x m ) a,

(3)

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where .(x 1 , ..., x m )=f (1, x 1 , ..., x m ), (x 1 , ..., x m )=f (0, x 1 , ..., x m ). This representation can be rewritten as f =.(+ ) a+(.+. ) a, and an elementary transformation of this expression yields f =(. ) a+(.) a +(.).

(4)

We first assume that the definition a= f (a, x 1 , ..., x m ) is noncreative and prove that representation (1) satisfying the orthogonality condition (2) exists. If, for some combination t 1 , ..., t m of truth values, .(t 1 , ..., t m ) (t 1 , ..., t m )=1, then .(t 1 , ..., t m ) =0 and (t 1 , ..., t m )=1. By virtue of (3), this implies that the equation a= f (a, t1 , ..., t m ) has the form a=a and consequently has no solutions. Due to Lemma 1, this contradicts the assumption. Therefore .=0 identically, and (4) is equivalent to f =(. ) a+(.).

(5)

(. )(.)=0,

(6)

Obviously,

and the existence of representation (1) satisfying (2) is obtained from (5) and (6) by renaming . as . and . as . Assume now that representation (1) of the Boolean function f exists, where the functions .,  are not necessarily orthogonal. Then f is noncreative, because the equation a=.a+ can always be solved for a: one of the solutions is a=. It remains to prove the uniqueness of representation (1) provided that (2) is satisfied. Let f =.a+=.$a+$ be two such representations. Substitution a=0 yields =$. Substitution a=1 yields .+=.$+$. Multiplying the latter equality by . we obtain, by virtue of =$ and the orthogonality condition, that .=.$.. Analogously, multiplying the same expression by .$, we get ..$=.$. Hence .=.$. K m

Theorem 2. Altogether there are 3 2 Boolean functions f (a, x 1 , ..., x m ) for which the definition a= f (a, x 1 , ..., x m ) is noncreative. Remark. It is well known and it is easy to calculate (see, for example, Hilbert m+1 6 Bernays, 1968) that all together there are 2 2 Boolean functions of m+1 variables. Proof. A Boolean function of the variables x 1 , ..., x m is defined by the set consisting of all combinations of truth values of these variables for which the function equals 1. Let A be this set for the function . in (1) and B be this set for the function  in (1). The orthogonality condition, (2), is equivalent to stating that A and B are disjoint: A & B=<. Denote by E the set consisting of all possible combinations of

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truth values for x 1 , ..., x m . By Theorem 1, the functions f are in a one-to-one correspondence with the ordered pairs (A, B) of disjoint subsets of the set E. The set E consists of 2 m elements, each of which belongs to A, belongs to B, or else does not m belong to either. Consequently there are all together 3 2 ordered pairs (A, B) and the same number of functions f. K 1

Consider the simplest case, m=1. All together there are 3 2 =9 Boolean functions f(a, x), for which definition a= f (a, x) is noncreative. According to Theorem 1, all these functions can be presented as f (a, x)=.(x) a+(x),

(7)

with .(x) (x)=0. Since a Boolean function of x can only be x, x, 0, or 1, it is easy to see that if f (a, x) depends on both its arguments essentially (i.e., neither of the two variables is a dummy), then f (a, x) can only be xa,

xa+x,

xa,

xa+x.

or

(8)

The full disjunctive normal forms of the statement at f (a, x) for these functions are atxa

=xa+xa +xa,

atxa+x =xa+xa +xa, atxa

(9)

=xa +xa+xa,

atxa+x=xa+xa+xa. The remaining five functions f (a, x) introduced by noncreative definitions are x,

x,

a,

1,

0,

(10)

with the full disjunctive normal forms of the statement at f (a, x), atx=xa+xa atx =xa +xa, a ta=xa+xa +xa+xa,

(11)

a t1=xa+xa, a t0=xa +xa. Remark. Observe that any function f (a, x 1 , ..., x m ) uniquely determines the statement at f (a, x 1 , ..., x m ), and vice versa. Moreover, any Boolean function g(a, x 1 , ..., x m ) can be presented in the form at f (a, x 1 , ..., x m ), as one can easily

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see by replacing f with atg(a, x 1 , ..., x m ). These are consequences of the following simple but important proposition, whose proof is obvious: for any triad of Boolean functions a, f, g, the equalities g=at f and f =atg are equivalent. 3. NONCREATIVE AUTOREFERENTIAL SYSTEMS OF DEFINITIONS

In this section I establish necessary and sufficient conditions for the noncreativeness of an autoreferential system of definitions, and I provide the enumeration of all noncreative systems with a given list of atomic content statements. These results generalize the corresponding results from Section 2. After that I analyze the important class of Boolean functions g(a 1 , ..., a n ) such that every single definition a i = g(a 1 , ..., a n ), i=1, ..., n, is noncreative (considering a 1 , ..., a i&1 , a i+1 , ..., a n as if they were content statements). It is convenient to formulate the results of this section in a vector notation. Let a=(a 1 , ..., a n ), x=(x 1 , ..., x m ) be vector variables and f=( f 1 , ..., f n ) be a vector function. All vector functions are tacitly assumed to be n-dimensional. Thus an autoreferential system of definitions a i = f i (a 1 , ..., a n , x 1 , ..., x m ),

i=1, ..., n

can be written as a single vector definition, a=f(a, x).

(12)

Logical operations on Boolean vectors are performed componentwise, yielding Boolean vectors. The conjunction of a Boolean vector h=(h 1 , ..., h n ) and a Boolean scalar . is the Boolean vector h.=(h 1 ., ..., h n .). The vector definition a=f(a, x) is called noncreative if no (scalar) nontautological statement in which a 1 , ..., a n do not occur can be inferred by means of propositional calculus from the conjunction of the equivalence statements 6 ni=1 (a i t f i (a 1 , ..., a n , x 1 , ..., x m )). Lemma 2. The definition a=f(a, x) is noncreative if and only if, for any truth value t of the variable x, the equation a=f(a, t) has a solution for a. Proof. Suppose that at some t=(t 1 , ..., t m ) the equation a=f(a, t) has no solutions for a. Then, for every combination of truth values for variables a 1 , ..., a n , at least one of the equivalence statements a i t f i (a 1 , ..., a n , t 1 , ..., t m ), 1in, is false. So, the conjunction 6 ni=1 (a i t f i (a 1 , ..., a n , t 1 , ..., t m )) is false for any truth values of the variables a 1 , ..., a n . Consequently, the nontautological statement x t11 6 } } } 6 x tmm , which is false only at x 1 =t 1 , ..., x m =t m , is a logical consequence of the conjunction 6 ni=1 (a i t f i (a 1 , ..., a n , x 1 , ..., x m )). Conversely, suppose that some logical consequence h(x 1 , ..., x m ) of the conjunction 6 ni=1 (a i t f i (a 1 , ..., a n , x 1 , ..., x m )) is not a tautology. Then h(t 1 , ..., t m ) is false for some combination t=(t 1 , ..., t m ) of truth values. Consequently the conjunction 6 ni=1 (a i t f i (a 1 , ..., a n , t 1 , ..., t m )) is false at all truth values of the variables a 1 , ..., a n . It follows that for any combination of the truth values of the variables

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a 1 , ..., a n at least one equivalence statement a i t f i (a 1 , ..., a n , t 1 , ..., t m ), 1in, is false. Thus the system of equations a=f(a, t) has no solutions for a. K Theorem 3. Let g i (a), i=1, ..., N, be all possible distinct nonzero vector functions for which the equation a=g i (a) has a solution. The definition a=f(a, x) is noncreative if and only if the function f can be represented in the form N

f(a, x)= : g i (a) . i (x),

(13)

i=1

where . i (x) are scalar pairwise-orthogonal Boolean functions . i . j =0

(14)

whenever i{ j. The functions . i are defined by the function f uniquely. Remark. If n=1 then there are just two nonzero Boolean functions, a and 1, for which the equation a= g(a) has a solution. Substitution of these functions for g i (a) in (13) yields the decomposition (1) of Theorem 1. Proof.

It is easy to check that the decomposition 1

1

f(a, x)= : } } } : h i 1 } } } i m (a) x i11 } } } x imm , i1 =0

(15)

im =0

where h i 1 } } } im (a)=f(a, i 1 , ..., i m ), holds for any Boolean function. Suppose that the definition a=f(a, x) is noncreative. Then, by virtue of Lemma 2, for any combination of truth values i 1 , ..., i m the equation a=f(a, i 1 , ..., i m ) has a solution for a. This means that each nonzero function h i i } } } i m (a) should be one of the functions g i (a). Grouping the terms with identical functions g i (a) in (15), one comes to the representation (13), with . i (x) of the form  (i 1 , ..., i m) # Gi x i11 } } } x imm , where the sets G 1 , ..., G N of index vectors are pairwise disjoint. The orthogonality condition (14) for this representation follows from the fact that (x t$11 } } } x t$mm ) t" m 1 (x t" 1 , ..., t" m ). 1 } } } x m )=0 if (t$1 , ..., t$m ){(t" The orthogonality condition ensures that at a given value of the variable x only one function . i , if any, assumes the value 1, while the others vanish. From this the uniqueness of the representation from (13)(14) follows trivially. The same observation, together with Lemma 2, implies that if the function f(a, x) can be represented in the form (13)(14), then the definition a=f(a, x) is noncreative. K n

n

m

Theorem 4. Altogether there are [2 n2 &(2 n &1) 2 ] 2 functions f(a, x) for which the definition a=f(a, x) is noncreative. Proof. Denote by E n and E m the sets of all possible truth values of variables a and x, respectively. These sets consist, respectively, of 2 n and 2 m elements. All possible representations (13) satisfying the orthogonality condition (14) are in oneto-one correspondence with the sequences A 1 , ..., A N of pairwise disjoint subsets of the set E m : A i & A j =< for i{ j, where A i , i=1, ..., N, is the set of all x # E m for

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m

which . i (x)=1. All together there are (N+1) 2 such sequences. In order to calculate the number N+1, observe that g i (a) for which the equation a=g i (a) has a solution is a map of the set E n into itself that has a fixed point. Therefore N+1 n is the number of such maps. All together there are (2 n ) 2 maps of the set E n into n itself, and (2 n &1) 2 ones among them have no fixed points (because a point a # E n n in such a map can transform into any value different from a). Hence N+1=2 n2 & n (2 n &1) 2 . K Remark. Observe that the concluding remark of Section 2 generalizes to vector functions. If n>1, however, then the conjunctions of equivalence statements 6 ni=1 (a i t f i (a 1 , ..., a n , x 1 , ..., x m )) and 6 ni=1 (a i t g i (a 1 , ..., a n , x 1 , ..., x m )) may be logically equivalent even when f(a, x){g(a, x). This follows from the fact that two conjunctions of different sets of Boolean functions may be equivalent. Due to Theorem 3, the investigation of noncreative definitions is reduced to the problem of characterizing all the functions g(a) for which the equation a=g(a) has a solution. In this paper I only present a sufficient condition, stated in the following simple proposition. Theorem 5. Let each Boolean function g i (a 1 , ..., a n ), i=1, ..., n, be written as a disjunction of conjunctions of its variables, without negations. Then the system of equations a i = g i (a 1 , ..., a n ),

i=1, ..., n,

(16)

has a solution. Remark. It is assumed, as usual, that the disjunction and the conjunction of the empty set of Boolean terms equal 0 and 1, respectively. Proof. If one of the functions, for example, g n , is equal to 0 identically, then the equation a n = g n(a 1 , ..., a n ) is satisfied at a n =0. Substitution of a n =0 in the other functions, g 1 , ..., g n&1 , yields a system of n&1 equations in n&1 unknowns that satisfies the conditions of the theorem. Continuing in the same fashion, eventually we arrive either at the solution a=0 for the system of equations (16) or to a system of equations which satisfies the conditions of the theorem and in which none of the functions is equal to 0 identically. In the latter case the assignment of the truth value 1 to all the remaining variables yields a solution of the obtained system of equations, and, when combined with the zero truth values for the previous variables, a solution of the original system of equations. K The Boolean functions g(a 1 , ..., a n ) that can be written in a form of disjunction of conjunctions of its variables, without negations, are called monotonic Boolean functions. These functions are characterized by the following property: for any two combinations of truth values t$1 , ..., t$n and t"1 , ..., t"n , if the inequalities t$i t "i , i=1, ..., n, hold, then g(t$1 , ..., t$n )g(t"1 , ..., t"n ). It is interesting to note that the set of all monotonic functions forms a closed class of Boolean algebra (Jablonski, Gawrilow, 6 Kudrjawzew, 1966): in particular, this set is closed with respect to

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substitutions. The following theorem provides another characterization of monotonic functions that has greater relevance in the context of the present analysis. Theorem 6. A Boolean function g(a 1 , ..., a n ) can be written as a disjunction of conjunctions of its variables, without negations, if and only if every single definition a i = g(a 1 , ..., a n ), i=1, ..., n, is noncreative (considering a 1 , ..., a i&1 , a i+1 , ..., a n as if they were content statements). Proof. The ``only if '' part immediately follows from Theorem 1. Suppose now that every definition a i = g(a 1 , ..., a n ), i=1, ..., n, is noncreative. The proof is conducted by induction with respect to n . If n=1 then the statement follows from Theorem 1. Let the statement be true for the functions depending on n&1 variables. The function g can be written in the form g(a 1 , ..., a n )=.(a 1 , ..., a n&1 ) a n +(a 1 , ..., a n&1 ) a n ,

(17)

where .(a 1 , ..., a n&1 )=g(a 1 , ..., a n&1 , 1) and (a 1 , ..., a n&1 )= g(a 1 , ..., a n&1 , 0). By the induction hypothesis, the functions . and  can be written as disjunctions of conjunctions of their variables, without negations. If .(t 1 , ..., t n&1 )=0 for some truth values t 1 , ..., t n&1 , then (t 1 , ..., t n&1 )=0, because otherwise the equation a n = g(t 1 , ..., t n&1 , a n ) for a n would have no solutions. Consequently .=.+ identically, and (17) can be written in the form g=(.+) a n +a n . This implies g=.a n +.

(18)

Substituting in (18) the representations of the functions . and  as disjunctions of conjunctions of its variables, without negations, we arrive at the same representation of the function g. K 4. POSSIBLE RELATION TO OBSERVABLE DATA

In this section I formulate tentative assumptions which could link the model of conscious experience proposed in this paper to observable data. Suppose that a person has to classify a set of items into two categories (say, ``good'' and ``bad''), without being given a criterion of categorization. Denote by x the content statement ``this item belongs to the first class.'' Assume, in accordance with my model, that the person's experience when an item is being classified is described by a statement F(a, x), satisfying the noncreative definition a=F(a, x). Suppose further that the Boolean function F(a, x) depends on its two arguments essentially (i.e., none of the two arguments is dummy). All functions of this kind are listed in (8), with the corresponding equivalence statements atF(a, x) given in (9). Because of the absence of an objective criterion, the person is free to assign to a and x any truth values that preserve the truth of the equivalence statement atF(a, x). Suppose, for example, that F(a, x) describing the person's experience is xa , the first function in (8). Then the pairs of truth values for a and x that preserve

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the truth of the equivalence statement atxa=xa+xa +xa, the first one in (9), are (a=1, x=1), (a=0, x=1), and (a=0, x=0). Assume now that while F(a, x)=xa describes the person's experience throughout the entire classification process, the pairs of truth values just mentioned vary randomly from one item to another. In the absence of a systematic bias, the three pairs of truth values will be chosen equiprobably, because of which the probability of choosing x=1 (i.e., the probability with which an item is classified into the first class) will be 23=0.666... . The same probability is obtained if the person's experience is described by the second function in (8), F(a, x)=xa+x, while for the remaining two functions, F(a, x)=xa and F(a, x)=xa+x, 23=0.666... is the probability of choosing x=0 (i.e., the probability with which an item is classified into the second class). That in the absence of a criterion of preference the probability with which an item is classified into one of two groups is significantly different from 12 has been demonstrated in many experiments (Adams-Webber 6 Rodney, 1983; Benjafield 6 Green, 1978; Lefebvre, Lefebvre, 6 Adams-Webber, 1986; Lefebvre, 1990b; Osgood 6 Richards, 1973). People may differ in which of the two categories (say, ``good'' or ``bad'') they choose with a greater probability, but a given person tends to systematically prefer one to another (usually, the ``positive'' one to the ``negative''). In the experiments reported by Lefebvre and his co-workers the probability of a preferred category is found to be close (- 5&1)2r0.618 (the ``golden section''). In other studies, for example, in Benjafield and Green (1978), who asked the subjects to repeatedly evaluate themselves, the probability of positive self-evaluations is reported to be close to 0.666. It might appear that my approach to the dichotomous classifications in the absence of a criterion is clearly inferior to Lefebvre's (1990a) theory, as the latter successfully predicts the ``golden section'' probability. One should take into account, however, that the ``mental computer'' postulated by Lefebvre has been selected from a large family of possible ``computers,'' many of which can be given equally persuasive a priori justifications. Lefebvre's specific choice is geared toward predicting a specific numerical probability rather than being derived from fundamental principles. In this respect, the situation with the model presented in this paper is neither better not worse, as this model, too, can be adjusted to predict a wide spectrum of numerical probabilities. The assumptions that throughout an experiment the subject's experience is described by one and the same statement (essentially depending on both its arguments) and that all pairs of the truth values preserving the truth of this statement are equiprobable can only be taken as a crude approximation. It is more realistic to assume that the substantive heterogeneity in the items being classified would induce the person to occasionally switch from one function from the list (8) to another. It is also realistic to assume that the person may be systematically biased toward a specific truth value for a . If such assumptions are allowed to be made freely, the model becomes too flexible to be empirically falsifiable. It is reasonable to expect, however, that when applied to specific situations the model can be complemented by a set of plausible constraining principles, whose discussion is beyond the scope of this paper. My only claim at present is that the model treating consciousness as noncreative logical self-reference has a potential for addressing

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