THE LOGICAL STRUCTURE OF EVOLUTION AND EMANATION *1)
GOTTHARD GUNTHER: When I decided to accept Dr. Roland Fischer's generous invitation to read a paper at this conference I felt somewhat like a forger who passes on his counterfeit money to an unsuspecting public. It has been a time-honored tradition to say that logic and its laws are timeless and of eternal validity. This viewpoint has, of course, sometimes been subjected to a sceptical scrutiny, but all attempts to analyze time with the means of a logical calculus have come to nothing. Consequently, since the traditional viewpoint seems to be the correct one, it follows that a logician at a Time Conference should be a
persona non grata, and the currency with which he pays for admission should not be acceptable.
However, I feel that my presence has some justification. The logic discussed in all previous confrontations between Logic and Time was invariably the classic two-valued logic; but it might be proper to raise the old issue again when a logician claims that our traditional theory of thinking is not the only one and that a trans-classic system of rationality might be able to tackle the problem of time if more powerful methods of investigation were available. Since the classic theory of rationality is indissolubly linked with the concept of value, first of all one has to show that the whole "value issue" covers the body of logic like a thin coat of paint. Scrape the paint off and you will discover an unsuspected system of structural forms and relations suggesting methods of thinking which surpass immeasurably all classic theories. This was the purpose of my paper "Time, Timeless Logic and Self-Referential Systems." The trans-classic order which we discover beyond the classic theory of logic was called "kenogrammatic structure."
However, there seemed to be some doubt as to how I arrived at that kenogrammatic concept, and limited time permitted no discussion of the transition from value to kenogram. Consequently, the quintessence of my procedure seems to require some detailed explanation. Such an explanation I have given in an earlier publication,
[21] but, alas, only in strict logical terminology which may make it again difficult for an interdisciplinary audience to follow. In this dilemma, I turned to my colleague Heinz Von Foerster, a veteran in interdisciplinary meetings, to help out. He suggested that I present the development of these concepts in a mathematical vocabulary. But since this vocabulary is not my vehicle of mental propulsion, I let Von Foerster tell his story in his own words.
HEINZ VON FOERSTER: Perhaps the easiest way to see the emergence of the concept
Kenogram is to see it through the concept of the "inverse" of a logical function. The inverse of a logical function is derived in precise analogy to the inverse of a mathematical function.
Let y = f(x) be a mathematical function in which the "dependent" variable y is expressed in terms of the "independent" variable x, say y = x
2
with f(x) = x
2. Inversion is accomplished when the independent variable y is explicitly expressed in terms of the dependent x:
maintaining, however, the original functional relationship between the two variables. In the above example y = x
2 , the inverse function is
Please note two points in connection with the inversion of functions. The first point refers to the conservation of the domain of x and the range of y before and after inversion. The second point refers to the possibility of a unique function becoming a multiple-valued function after inversion or vice versa.
Let me exercise these two points on the above example. If we wish to remain in the realm of real numbers, then in the expression y = x
2 the domain of the independent variable x is the set of all real numbers and the range of y, the dependent variable, is the set of all non-negative real numbers, while in the inverted form the domain of the independent variable, now y, is the set of all non-negative real numbers and the range of the dependent variable, now x, is the set of all real numbers. This is clearly seen if one wishes to use a negative real number as an argument in the inverted function. The result is a complex number, in contradiction to our premise to stay in the realm of real numbers.
The second point of the emergence of multiple valuedness after inversion is easily seen by the (+) and (-) sign in front of the square root. For y = 4, for instance, the inverse of y = x
2 produces indeed the two solutions x
1 = + 2 and x
2 = - 2 as suggested by the expression for (+ 2 )
2 and (-2 )
2 both equal to 4.
I turn now to the inversion of logical functions where, hopefully, it can be seen that Gunther's kenograms are nothing else but the original dependent variables becoming independent after inversion. Since the range of the dependent variable in logical functions is restricted to the number of values m in the logical system, e.g., m = 2 in a two-valued logical system, and since one deals here with logical systems that admit only a few values (i.e., m is a small integer), I believe it is quite legitimate to use simple geometrical forms, say triangles, squares, etc., for representing various variables, rather than the mathematician's x, y, z, etc. However, let me continue for a moment with the mathematical notation.
Let X
n = {x
0, x
1, x
2,
º, x
n-1} be the independent variable in a logical function
where X
n is represented by an n-tuple of independent elementary variables x
i (i = 0
Æ (n - 1) ). The domain of these elementary variables x
i depends upon the choice of the valuedness of m of the logical system under consideration. In the classical two-valued system one has, of course, m = 2. Consequently, since the domain of x
i is m, the domain of the independent variable X
n is the set of all natural numbers between 0 and (m
n - 1), i.e., comprises m
n values.
The
modus operandi of a logical function is to associate with each of the m
n values of the independent variable precisely one value of the dependent variable y, the range of which is identical with the domain of the elementary variables x
i. A particular logical function is defined if for each of the m
n values of X
n a particular value for y within the range m is specified. This restriction produces a variety of precisely
different functions in an m-valued logical system with n independent elementary variables.
If I am not mistaken, in the history of the development of logical functions there exists nowhere a discrepancy in the terminology of "values" and "variables." These terms are used exactly in the sense as I used them before. However, a considerable variety in the use of
symbols and in the
interpretation of these symbols representing the "values" of these variables is to be noted.
Let me stay, for the moment, within the classical case of a two-valued logic, i.e., m = 2. One will find the two available values being represented in a variety of ways, for instance W, F (for "wahr", "falsch"; Wittgenstein); or T, F (for "true", "false"; Russell); or 0, 1 (Boole, Hilbert); or 1, 2 (Günther), etc. This variety of symbolic representation of the variables leads, of course, to a variety of representations of one and the same logical function as I shall demonstrate on one particular logical function, namely, the logical "and" symbolized by "&," and also sometimes called the "logical sum" or the "logical product" (
Þ) of the elementary variables x
0 and x
i.
If (x
i) and (x
ß) are interpreted as propositions, then it is clear that the representations (i) (ii) give the " truth- values" for the proposition "x
i & x
0", for "x
i and x
0" is only true (T, W) if and only if both x
i and x
0 are true separately. Otherwise, "x
i and x
0"
is false (F). Representation (iii) makes use of the oddity that if "true" is represented by the integer "one" and "false" by "zero" then the truth-values for the logical "and" are obtained by algebraic multiplication y = x
i Þ x
0. In (iv) the representations for "true" and "false" by the integers 1, 0 is reversed and the values for y are obtained by a pseudo- arithmetic addition in which 1 + 1 = 1. This latter interpretation of the integers 0, 1 has, however, the advantage that the logical function "inclusive or" can be represented as a proper algebraic product
I apologize for this somewhat lengthy narrative on an otherwise well-known story. However, I wished to stress in this account the arbitrariness by which certain symbols may be associated with two-valued logical values "true," "false" or "position", "negation," etc. In the above examples, "true" was in one case associated with integer "one" and in the other case with integer "zero," and
mutatatis mutandis:
With this introduction, I believe it is now easy to understand Günther´s mysterious triangles in example (v). Let the upright empty triangle stand for the integer 0 and the downward full triangle stand for the integer 1, then the function represented in example (iv) is obtained. If, moreover, T
Æ 0 and F
Æ 1, example (v) represents the logical function "&", which symbol may now be inserted into the yet empty space on top of the column representing y.
However, there is no need to insist on the interpretation suggested above, and we may as well reversely identify the upright empty triangle with the integer 1 and the other one with 0. But since this reversal does not affect the values of the independent variable x
2-a particular triangle is associated uniquely with a
set of values Of X
2, namely
such a reversal will alter the logical function:
I hope that this simple example clarifies the meaning of those symbols that Günther called "Kenograms" and which are represented here as triangles of different shapes. Since kenograms may assume different values, but different kenograms not like values:
they may be considered to be "elementary variables" y
i of the dependent variable y, with the condition
their range being the number of values m of the logical system. The indices i, j may assume values of the integers 1
Æ r
£ m, where r is the number of different values admitted to occur in the dependent variable. For example, r = 1 suggests that the dependent variable y admits only one value. For logical functions that are confined to two variables only (n = 2), this situation (r = 1) is given by the following scheme
in which by the particular sequence of like kenograms the logical functions C and T are represented which, in turn, may stand respectively for "Contradiction" and "Tautology," but only if "0" is associated with "true" and "1" with "false."
A particular sequence of kenograms has been called by Günther a "morphogram", M. However, such a "sequence" is not invariant to permutations of the sequential arrangement of the values of the independent variable X
n, The invariance of the association of a kenogram with a particular value of X
n, and hence the invariance of a morphogram with respect to permutations in a particular sequential arrangement of X
n can be established by defining the set of values X
ni which are associated with a particular kenogram:
To return to the previous example where the morphogram Ærepresented the two logical functions "&" and "" we
have:
[ (00); (01, 10, 11)] =
F(Æ;)
which is, of course, nothing else but a representation of the set of the inverted logical functions "&" and "&":
[ (00); (01, 10, 11)] =
F1(0,1) = (&)
-1
[ (00); (01, 10, 11)] =
F2(1,0) = ()
-1
From these remarks the general,, invariant, definition of a morphogram M can be deduced:
