The paradox invaded science and art. Not any more an accident or marginal phenomenon, it is placed now at the centre of human thinking. Outside the paradox we are not able to understand the world. We have to learn how to identify the paradox in its diverse manifestations. If so far a paradox was considered as the symptom of a pathological state, in the last decades it appears more frequently to indicate normality. How happened this change of paradigm is the question answered by a first-rate mathematician with exceptional knowledge of contemporary linguistics.

In an excellent piece of work, eminently readable, the paradox is presented not only as a statement that seems to say something opposite to common sense or to truth, but which may contain the truth (e.g. more haste, less speed) as the result of the logic we use. The way of dealing with sorites-like arguments (the bald man, the heap) is held to be a good criterion for any conceptual framework intended for handling vague concepts.

Accepting that tearing out one hair makes a non-bald man not bald leads--via an induction chain of the length equal to the number of hairs--to the paradoxical conclusion that when all hairs are lost, our man still is not bald. The fuzzy set approach tries to circumvent this paradox by denying bivalence and assigning truth values smaller than I.

A fuzzy set is a mathematical model of a generalized set whose partial elements are characterized by partial existence or, in other words, by degrees of existence. The modern theory of topoi dealing with generalized sets does not need the principle of excluded middle, neither the theory of fuzzy sets, embedded in the first one.

Analysing the works of Stephane Lupasco, Solomon Marcus discovers that the idea of a system starts exactly from this point. And more than that, any ontological system has a cybernetics. In an entropic system that means a feedback measuring a heterogenous state. In a biologic system the feedback measures a homogenous state. In a nuclear or neuropsychic system the feedback measures the equilibrium of the parameters governing both homogenous and heterogenous states.

The paradox seems to be the soul of dialectical thinking. The development of knowledge means detection of paradoxes and development of another framework to bypass the paradox (not to solve it). This is exactly the mechanism of pull-back as described in Fuzzy Systems (Abacus Press. Tunbridge Wells, UK, 1981) where I discussed at large the conceptualization emphasising pull-back versus feedback.

Why we need a dialectical logic, is a question answered by a first-rate philosopher with exceptional knowledge in symbolic logic. In another brilliant monograph, Lorenzo Peña. now a professor of philosophy at the University of Madrid, presents an ontological perspective dominated by the following theses:

  1. The theory of sets is a good theoretical approach easily expressing the Real. To possess a property is to belong to a class.
  2. Being means existence, i.e. a degree of reality, and the theory of fuzzy sets is a good framework to explain what Mircea Eliade names coincidentia oppositorum.

Many conclusions about the process of synthesis, found in the cybernetics of human systems (Dependence and Inequality, Pergamon Press, 1982) are well explained by Lorenzo Peña in a book which definitely is a landmark.

Both Marcus and Peña observe that every important progress in logic means a projection onto reality of certain structures of human consciousness. Yet the most amazing fact about modern science is that the structures have turned out to correspond to something «out there». Mathematicians, physical scientists, and philosophers of science are still trying to understand just how this is possible. Some of them say that the development of these projections in the history of modern thought has its origins in very specific infrastructures without which this development is most unlikely ever to have taken place.

I found these books stimulating and provocative. To those cyberneticians who have not previously looked at purposiveness from a logician's point of view, they will be particularly valuable. One can hardly imagine future research on artificial intelligence that would ignore these thought-provoking contributions.

CONSTANTIN NEGOITA
City University of New York
695 Park Avenue, NY 10021

Kybernetes 14/1 (1985), pp. 61-62
ISSN 0368-492X