Paraconsistent logics as a Way of Expressing Objective Contradictions in Science

Iashin Boris Leonidovich Doctor of Philosophy Professor; Department of Philosophy, Institute of Social and Humanitarian education, Moscow Pedagogical State University 117571, Russia, g. Moscow, ul. Prospekt Vernadskogo, dom 88, к.1 jabor123@rambler.ru Other publications by this author

One of the serious problems of scientific cognition, which manifested itself already in the twentieth century, not only in various fields of natural science, but also in the socio-humanitarian sciences, turned out to be a problem related to the need for adequate expression in language in a consistent way of movement and development, various kinds of changes that occur in the objective world. Because of this, many philosophers and scientists have attempted to develop logics that would be tolerant of logical contradictions. In other words, they had to be logics within which these contradictions became quite acceptable. These logics could legitimately contain the formula A & ~A, which is deducible under certain conditions, but does not lead to the triviality of the system itself. And such logics, which, at the suggestion of the Peruvian philosopher F. Miro-Quisado, began to be called paranoherent or paraconsistent logics ^[1], were created.

The ancestor of this kind of logic is rightfully our compatriot N. A. Vasiliev. It was he who, by analogy with the "imaginary geometry" of N. Lobachevsky, developed the "imaginary logic" in which a logical contradiction was recognized as legitimate. In one of his works, he wrote that we can quite "think a contradiction ... to think a contradiction is to form a special judgment of contradiction or an indifferent judgment next to an affirmative and negative judgment" ^{[2, p. 69]}. The first formal system of paranoherent logic was presented in 1925 by A. N. Kolmogorov in his article "On the principle of tertiura non datur" ^[3]. Another Russian philosopher and logician who developed his axiomatic paraconsistent system of logic in 1929 was I. E. Orlov ^[4].

At about the same time, the Polish logician and philosopher J. Lukasiewicz drew attention to the possibility of the existence of logics without the law of contradiction. And about two decades later, his student S. Yaskovsky developed the first system of paranoherent (paraconsistent) logic, recognized by scientists and philosophers, which was designed to analyze contradictory opinions of participants in the discussion and called by him debatable logic (later it became known as discursive). He was the first to formulate the criteria of paraconcilability and gave a definition of paraconcilable logic as logic that can be used as the basis for contradictory, but not trivial theories ^[5]. According to S. Yaskovsky, this kind of logic had to satisfy "three conditions: 1) contradiction should not trivialize the system, the law of Duns Scotus should not be feasible in it; 2) it should be rich enough to draw conclusions in it, 3) have an intuitive explanation" ^[5].

Para-contradictory logic, where the contradiction was quite legitimate, had such capabilities that classical formal logic did not have: in many cases it successfully coped with problems that could not be solved within the framework of this latter. But "the most significant reason for paraconsistent logic is," according to V. O. Lobovikov, "prima facie, the fact that there are theories that are inconsistent but nontrivial" ^[6].

Since the seventies of the last century, the development of paraconsistent logic has taken on an international character. Work in this area is quite active in Argentina, Australia, Belgium, Brazil, Canada, the Czech Republic, England, Germany, India, Israel, Japan, Mexico, New Zealand, Poland, Scotland, Spain, the USA and other countries. In the period from 1977 to the present, five World Congresses and a number of major international conferences on paraconsistent logic have been held ^[7]. All this taken together testifies to the ongoing and currently ongoing interest in paranoherent logics.

In the process of their development, the paranoherent logics contributed to the solution of many paradoxes found in the foundations of classical propositional calculus and predicate logic. The apparatus of these logics has been successfully used in the research of contradictory statements, various kinds of antinomies arising in private sciences, in order to limit the possible undesirable consequences that arise in information retrieval systems when inconsistent or contradictory information is received in them. The effectiveness of the currently existing large number of diverse para-contradictory logics is also confirmed by such results as a deeper understanding of development theories, in particular, dialectics; the development of non-traditional ontological models; proof of the weakness of some critical statements against dialectics; and some others ^{[8, p. 60]}.

All these achievements in their entirety contributed to the fact that the idea arose in philosophy and science that the apparatus of paranoherent logics is quite possible to use not only for the purpose of penetrating into the essence of any contradiction, but also for the formalization of such a theory as dialectics. This idea was connected with the belief that, if implemented, dialectics would become more rigorous and evidence-based. All attempts to implement this idea, according to I. Zelena, can be divided into two main groups ^[9]. One of them (the works of N. da Costa, A. Arruda, M. Quisada, etc.) substantiates the point of view according to which paraconsistent logics are considered not only as the final stage of the development of logical calculus in general, but also as a link between traditional logic and dialectics. In addition, in some works belonging to this group, it is shown not only the usefulness of paranoherent logics in proving the thesis of the existence of contradictions of material existence, but also that they can be used as a means of "critical analysis of the initial dialectical position in experience, revealing the truth of this dialectical thesis" ^[10].

In my opinion, the other group includes, first of all, the works of G. Priest and R. Routley, who insist that it is quite realistic to use paraconsistent logics in order to reinterpret in its entirety the tradition of dialectical thinking, laid down by Heraclitus ^[11]. This assumption, which is directly related to the philosophical and scientific consequences of paraconsistance, related both to the relationship with dialectics and to the interpretation of rationality, is undoubtedly more radical and ambitious. 

The purpose of applying paraconsistent logic in the works of N. da Costa and R. Wolf is not to formalize dialectics as such, but to strive to "make some "rules of the dialectic of movement" stricter, more precise, in order to "shed new light on dialectical logic" ^{[12, p.191]}. Developing their logic, which, in their opinion, corresponds to the dialectical concept of the "unity of opposites", N. da Costa and R. Wolf believe that only two possible versions of its interpretation can contribute to solving this problem.

The first option includes all cases where the law of the excluded third does not "work" in some temporal or non-temporal continuum. These are cases for which, in the continuum space between incompatible classes A and non-A, some third class is possible, which does not intersect with any of them and whose characteristics differ from both the first and second classes. The second option combines cases where in the space of the continuum there is such a region in which both A and non-A are true. In other words, N. da Costa and R. Wolf consider dialectical contradiction only in its two extreme cases, which not only significantly limits the scope of this concept, but also gives reason to doubt whether their model of dialectics corresponds to its prototype. The reason for these doubts is that an adequate expression of the dialectical contradiction by negating "it is not true that A and not A" is impossible in classical logic. This is due to the fact that this formula, which contains the expressions "independently true that A" and "independently true that not-A", corresponds to the negation of a dialectical contradiction. To be more precise, the expression "it is not true that A and not-A" cannot be considered a formula of dialectical contradiction, since this latter, being one of the basic concepts of ontology, is actually external to the calculus, the language of which is used in describing the contradiction of the objective. Both foreign ^[9] and our domestic philosophers have drawn attention to this discrepancy in their works ^[13].

Unlike N. da Costa and R. Wolf, who propose a variant of "dialectical paraconsistent logic" based on limiting the scope of the concept of "dialectical contradiction" only to extreme cases, G. Priest in his "dialectical temporal logic" ^{[14, p. 63]} relies on the idea of creating such a modification of formalism, where the implementation of objective, that is, the "true" contradictions, and, consequently, the description of physical and social reality is quite real. One of the most important ideas of this logic, it seems to me, is G. Priest's understanding of the moment of change. He believes that any object or system S is in the state S 0 before some point in time t ₀, and immediately at time t 0 is in the state S ₁. In other words, at time t ₀, the transition of the object (system) occurs from the state S ₀ to the state S ₁, i.e., a change of states, a change of object. Based on this, G. Priest is trying to find an answer to the question of what is the state of the object (system) S exactly at the time of the change t ₀.

He believes that there are only three possible answers to this question. In the first variant, the change is assumed such that the system S is in one and only one of the extreme states (either S ₀ or S ₁).  In the second, it is stated that the system S is not in any of the extreme states. And the third option is associated with such a possibility, when this system is simultaneously in the state S ₀ and in the state S ₁. In his research, G. Priest showed that in reality there are changes that do not relate to either the first or the second answer options, and there is such a relationship between the changes in the second and third options that the presence of changes in one of them necessarily implies the presence of changes in the other. From this, according to G. Priest, it follows that in thinking there are also three possible options for evaluating the truth of a certain judgment p and its denial of non-R. Firstly, the judgment of p can be either true or untrue, and then the judgment of non-p will be true; secondly, the judgment of p is neither true nor untrue; finally, thirdly, the judgment of p is both true and untrue. The third type of change, according to G. Priest, should be considered the realization of some real contradiction in his system. At the same time, the statement "Something is neither true nor incorrect if and only if it is true and incorrect at the same time" can be considered appropriate to this situation.

It should be noted that, although G. Priest's "dialectical temporal logic" turned out to be a fairly successful means of describing contradictions existing in the real world, it is nevertheless not a complete analogue of dialectics. This, from my point of view, is due to the fact that, on the one hand, dialectics is not limited only to the doctrine of contradiction as a source of development. On the other hand, due to the fact that the possibilities of G. Priest's logic are also limited by the presence of problems related to the formalization of certain specific areas of scientific knowledge.

G. Priest's "dialectical temporal logic" turned out to be not the only logic where the factor of temporality was of paramount importance. The development of such logical systems was stimulated by the tendency manifested in science during this period to study nonequilibrium, dynamic systems, which required the formal logical means necessary for this, with the help of which it is possible to adequately describe the process of transition of an object (system) from one state to another, i.e., a description of a contradiction. The need for such logic, with the help of which it would be possible to avoid the paradoxes that arise when describing the processes of transition of a system from one state to another, was caused by the specific features of the language of classical two-valued logic used in such cases, in statements of which only static states of the beginning and completion of this process could be expressed. Revealing the essence of this situation, A. A. Zinoviev, for example, wrote that "the statement "A changing object has and at the same time does not have some sign" is logically false as a special case of contradiction... not because this is the world around us, but because this is the language we speak, i.e. by virtue of the logical operators included in it" ^{[15, pp.223-224]}. A. A. Zinoviev managed to solve the problem of describing a contradiction in a consistent way in his "logic of change", which it allowed " to reason without contradictions ... on changes in the subject" ^{[15, p. 225]}, by introducing instead the concept of contradiction of the traditional logic of the concepts of internal and external negation and the operator of uncertainty.

Another variant of the "logic of change" is the four-digit logic system developed by V. G. Kuznetsov. In it, along with the meanings of "true" and "false", such intermediate truth values as "sub-true" and "sub-false" also coexist. The difference between this logic and classical logic is also the presence in it of such concepts as "ontological possibility" (Bp - it is ontologically possible that p), "reality" (Tr - it is true that p), "existence" (Or - exists so that p), as well as two pairs of opposing concepts: "protension" (already exists so that p) and "retention" (still exists so that p), "emergence" (begins to be so that p) and "disappearance" (ceases to be so that p), where the symbol p corresponds to some state of the object ^[16].

Based on his definition of motion, understood either as a transition from possible being to real being, or as a denial of the reality of the negation of p (Or? ~T~p), V. G.Kuznetsov builds a logical system in which the formulas Tr ? Or, B~p ? Or, TVr ? (Or · O~p), Bp ? O~p, as well as some other formulas expressing the dependencies between reality, existence and ontological possibility, are identically true. Among the totality of these formulas, the most important is the formula TVr ? (Or · O~r), which corresponds to the real situation when, when implementing an ontological possibility, at the same time "there is such that p and there is such that there is not p", i.e. a situation that represents a contradiction. In other words, the statement about the existence of movement (since movement is the realization of the ontologically possible) implies the statement about the existence of a contradiction. However, this does not lead to the falsity of this consequence, within the framework of this V. G. Kuznetsov system, it remains identically true. At the same time, importantly, the system itself does not become trivial.

There are other features in V. G. Kuznetsov's logic that are interesting from the point of view of the possibility of expressing contradictions with its help. In it, for example, formulas are identically true, which can well be used as an interpretation of the description of the "entry" of an object into a certain place and, conversely, the "exit" of this object from this place… This logic can serve as a completely effective tool in describing the transition from the emergence of a particular state of a changing object to the realization of the possibility of its formation or its complete disappearance. However, despite the rather promising possibilities of using V. G. Kuznetsov's "logic of change" to express the inconsistency of movement in a consistent way using its apparatus, its author assessed his results very modestly. He believed that this logic should not be cited as an example of the formalization of dialectics, since it represents only the application of the apparatus of formal logic to dialectics, to the study of dialectical problems ^{[16, p. 56]}.

The works of the famous Finnish logician and philosopher G.H. Vrigt, who was one of the first to substantiate the need to develop such non-classical logics as modal and deontic logic, logic of norms, assessments, preferences and actions, are currently of undoubted interest in solving the problem of describing motion in a consistent way ^[17]. Rightly pointing out the close connection between time, change and contradiction, G.H. Vrigt notes that "if we assume that changes occur and are given to us in experience, we must describe them with a sequence of contradictory states, otherwise we will get a contradiction. Metaphorically speaking, time is man's deliverance from contradiction" ^{[17, p. 529]}.

According to G.H. Vrigt, time, being what separates opposites, makes it possible to express a genuine change. The transition from the state of p to the state of q can be represented as two elementary changes as the transition from p to non-p (p T non-p) and the transition from non-q to q (non-q T q), where T is the time operator. And since both of them are connected to each other by a similar transition, the whole process of change has the form of a sequence (r T not-r T not-q T q) ^{[17, pp. 530-536]}. After G.H. Vrigt established the fact that there is an inextricable link between the real contradiction that determines the nature of change and the continuity of time, he makes a very important philosophical conclusion, the essence of which is that the temporal logic he created indicates that between the "great tradition" going from Aristotle to Frege, Russell There is a deep connection to modern symbolic logic and the tradition, mainly associated with the name of G. Hegel. In other words, G.H. Vrigt, in fact, argues that the dividing gap between modern formal logic and dialectics, the existence of which representatives of these sciences have insisted on for many, many years, does not actually exist, but, on the contrary, in the future it is quite possible to "reconcile" these opposites ^{[17, pp. 537-538]}.

Further development of logic has shown that the description of motion is possible not only using the apparatus of a particular logical system, where temporality is present in some form, but also using logics in which the time factor does not play a significant role. One of these logics is the logic of directionality, in which the beginning or end of the process of transition of a system from one state to another is modeled, but this process itself, which has a well-defined direction ^[18]. The language of this logic is such that it does not require a temporal reference of statements, and the law of non-contradiction prohibits only categorically asserting and simultaneously denying the occurrence or disappearance of the same property. In other words, the logic of directionality does not allow simultaneous display of oppositely directed changes. "An example of directional logic," according to N. I. Steshenko, "shows that it is possible to build calculi that simultaneously satisfy the principle of non-contradiction and describe changes... beyond the time parameters" ^{[19, p. 245]}.

Among the many logics that are not related to time parameters, the so-called relevant logics are of particular interest from the point of view of the possibility of describing motion. One of the most important principles of these logics is the exclusion of the principle (A & ~A) ? C, according to which any statement is a consequence of a contradiction. With the help of these logics, some characteristics of a number of logical and methodologically important general scientific concepts have been clarified, such as, for example, "proof" and "refutation", "explanation" and "prediction", "law of science", "counterfactual statement", "explicit" and "implicit definiteness of terms" and some others.

The effectiveness of using relevant logics in the description of dynamical systems was demonstrated in the works of R. Routley and R. Meyer, who managed to create logical systems that, in their opinion, allow formalizing dialectics ^[20]. An example of such a system is their DL system, which contains as one of the axioms a real contradiction, which R. Routley and R. Meyer call "dialectical logic". However, it should be noted that, according to the authors themselves, with the help of this system, it is possible to formalize not the entire dialectic, but only its individual fragments. In order to eliminate this disadvantage, R. Routley and R. Meyer developed the so-called "weak" relevant logic DM, which is an extension of DL. From the point of view of the authors, this logic is not only quite possible to be called the "static maxim of dialectical logic", but also considered one of the most promising basic systems for the formalization of dialectics. In their opinion, this is due to the presence of conditions in it that are not only analogous to double negation (~ ~A ? A and A ? ~ ~A) and the principle of contraposition (A ? B) ? (~In ? ~A), but also with the fact that it is quite possible to consider the axiom of double negation as a formal analogue of the dialectical law of negation of negation ^{[20, p. 10]}.

Agreeing with R. Routley and R. Meyer that relevant logics can be used to describe certain areas of the changing objective world, it must be said that there are currently no sufficiently convincing grounds to consider these logical systems identical to dialectics or even dialectical logic. One of the facts confirming this conclusion is, for example, that the axiom of double negation used in these systems, which, as is known, requires a complete coincidence of the content obtained with the help of double negation with the original content, is not identical to the dialectical law of negation of negation. This is due to the fact that the meaning of this law is that the new content, while retaining the old content, does not completely coincide with it: it is both identical and non-identical to him. To what has been said, we can also add that the impossibility of complete formalization of dialectics is also associated with the peculiarities of cognition of the objective world by the subject. The infinity of this process is not adequate either to the infinity of explication of possible conclusions that a cognizing subject can obtain, or to the infinite number of consequences that can be obtained from a finite number of axioms and rules of any formal theory.

Some other varieties of non-classical logics could be cited as examples of logical systems that, with certain reservations, can be used to describe the contradictions of real existence. However, it seems to me that it is quite clear that these logics are quite effective in building models of certain fragments of dialectics, which not only indicates a high level of their development, but also contributes to the development of dialectics as logic and as ontology, and also indicates that the apparatus of non-classical logics is a very powerful tool studying and explaining many and many problems of theoretical knowledge. At the same time, it should also be noted here that the formalization of dialectics does not represent a kind of universal key to solving all the difficulties encountered in its development. Therefore, it seems to me that one cannot disagree with M. Tabakov that maybe only a small part of them will turn out to be meaningful and solved when building formal models of individual areas of dialectics, because in striving for rigor and positive results on the path of formalization, one cannot do without ordinary meaningful and intuitive thinking. And also with the fact that the formalization of any meaningful theory, including dialectics, is not an end in itself; that formalization is necessary in order to better understand the specifics of dialectics itself ^{[21, pp. 66-67]}.

From the fact that the apparatus of non-classical logics is widely and effectively used in the creation of formal models of dialectics, it is impossible to draw far-reaching conclusions. In addition to the difficulties and problems that arise along this path, which have already been noted, I think it is necessary to name another important circumstance, which, in my opinion, can serve as a reason to restrain scientists and philosophers to a certain extent. This circumstance is due to the fact that the assumption of the exact correspondence of some formal model of dialectics to certain characteristics of objective reality can become the foundation not only for the conclusion about the possession of the properties of this formal logical structure of objective reality itself, but also for the displacement of objective dialectics from material existence. Which, of course, would be wrong.

In the context of considering the problem of formalization of dialectics, in my opinion, we must not forget that any formal model of dialectics is a kind of abstract construction, which itself is a model of the model. In other words, it is a model of a "transformed form", and not a model of actual material existence. The use of such models to describe and explain some phenomenon or some fragment of the objective world forces a scientist or philosopher to rely not only on abstract ontology, the foundation of which is a set of such philosophical categories as "matter" and "consciousness", "space" and "time", "necessity" and "chance", "contradiction"and "identity", etc., but also on ideas borrowed from any field of scientific knowledge, and on the products of one's own mental activity. All this taken together can become, and sometimes becomes, the basis for refusal to accept this kind of philosophical reasoning as significant by scientists working in such fields of science where there are quite clear and convincing scientific explanations of the phenomena of reality. In the same cases, when a philosophical explanation refers to situations for which there is no satisfactory scientific explanation yet, it, combining facts with a universal ontological picture of the world and without revealing the specific essence of phenomena, turns out to be useful, since "it puts them in a series of known phenomena, removes the mystery of their existence and indicates the most likely path to their scientific explanation" ^[22].

The above also applies to theoretical models of the dialectic of being. Such models, write, for example, N. da Costa and S. French, "may be "internally" contradictory or empirically (precisely) inadequate, as in the case of Newton's mechanics, but they continue to be used either due to lack of knowledge about a more adequate model, or because it makes sense to sacrifice greater accuracy and adequacy for the sake of speed of application and saving time and effort" ^{[23, p. 67]}. In other words, theoretical models of real reality (classical or non-classical), being different "slices" of being, different pictures of its fragments, its areas, ultimately "work" to enrich the ontology itself.

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