SORITES ISSN 1135-1349
Issue #17 -- October 2006. Pp. 7-16
About Properties of L-Inconsistent Theories
Copyright © by Vyacheslav Moiseyev and SORITES
About Properties of L-Inconsistent Theories
Vyacheslav Moiseyev


Apparently, there have been two traditions in the history of logic, these are Line of Parmenide and Line of Heraclitus. Former is originated from the ideas of Parmenide-Aristotle and is based on the Law of Identity. This line constitutes formal logic. Latter is originated from the ideas of Heraclitus-Plato and has been expressed itself in the ideas of dialectics, or dialectical logic. Contemporary mathematical logic is the worthy result of the development of the first line. Possibility of good precision and clear procedures of justification is the most strong side of this line. On the other hand, dialectics always have been trying to deny the meaning of Law of Identity. Dialectical ideal have been expressed itself in the idea of contradiction. But a very big problem have been subsisted here. This is the problem which we shall call Problem of Logical Demarcation (PLD). Breafly speaking, essence of the problem is in the following idea. Mistakes are contradictions too and if dialectics does not want to be simply mistaken reasoning, then it must show a criterion with the help of which we could to separate contradictions-mistakes from dialectical contradictions (antinomies). We shall call such criterion as Criterion of Logical Demarcation (CLD). Although dialectics has not been able to show CLD but there have been many interesting attempts to find the Criterion. One can refer here to Plato, Nicholas from Cusa, Russian Philosophy of All-Unity, etc.

It seems to us that one of the interesting ideas here is the idea of some connection between CLD and concept of limit. For example, Nicholas from Cusa tried to express idea of God in the image of a straight line which is limit for the infinite sequence of tangent circumferences. Our paper is an attempt to extend this trend and to formulate a version of CLD, where dialectical contradictions (antinomies) can be expressed as limits of ifinite sequences of formulas in a formal language. Main new idea is here in the technique of work with the limiting sequences of formulas, not terms. This idea is fully correlated with the method of extension of rational numbers by irrational ones in mathematical analysis. As is well known, every irrational number can be represented by a limiting sequence of rational numbers. Then we can represent rational numbers itselves as a particular case of limiting sequences, i.e., as stationary sequences. Thus we are passing to a new type of objects and we can define operations with these objects generalizing of operations on rational numbers. The same approach is demonstrated below but in the logical sphere.


In our opinion, by the similar way another philosophical and religious antinomies may be interpreted in suitable L-inconsistent theories. Taking into account the analogy between method of construction of mathematical continuum and method of L-inconsistent theories formation, one may conclude that numerous antinomies, constantly have been reproduced in the history of human thinking, are examples of «logical irrationalities». These are antinomies of all the limiting concepts of philosophy, for example, «World», «Being», «Consciousness», «Will», «Freedom», «Personality», etc. And just as there exists common method of mathematical irrationalities expression, there could be a common method of logical irrationalities representation. Author hopes that ideas of this paper could to help us to come nearer to this method.

Vyacheslav Moiseyev
Moscow Medical Stomatological University
<vimo [at]>

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