Non-Classical Logics in Modern Science

Among the many reasons why one of the greatest modern philosophers G.H. von Vrigt called the twentieth century. The "golden Age of Logic" ^[1] is, according to A. G. Kislov, "the deuniversalization of classical logic and the emergence of many non-classical logical systems" ^[2]. And it is quite possible to agree with this, since non-classical logicians, who have become in their totality an alternative to classical logic, which is quite rightly still considered a model of reasoning not only in mathematics and other sciences, but also in everyday practice, have significantly expanded the traditional field of using logical methods. Currently, there are "not just infinitely many logics of this kind, there are a lot of them on a continuum" ^[3]. By the way, among logicians and philosophers dealing with philosophical problems of logic, the question of what to consider non-classical logic has not yet been finally settled. Therefore, it is not surprising that recently a more neutral term has been used for their designation, namely "non-standard logics" ^[4].

It is well known that the first multivalued (and in this sense – non-classical) logic was developed by Y. Lukasevich three-digit logic, followed by the three-digit logic of A. Bochvar, the "quantum logics" of G. Reichenbach and P. Detush-Fevrie, the n-digit logic of E. Post, infinite-valued, probabilistic and other systems of logic. If the first non-classical logics were mostly created on the basis of criticism of classical logic as systems in opposition to it, in which the unlimited applicability of its laws such as excluded third and double negation was questioned, then later their capabilities became widely used in various branches of scientific knowledge. Polysemantic and fuzzy, intuitionistic and modal, relevant and paranoherent, temporal and other varieties of non-classical logic are widely used today in physics, computational mathematics, computer science, linguistics, jurisprudence, ethics and other areas of natural science and socio-humanitarian knowledge ^[5].

Physics turned out to be one of the first sciences (not counting logic itself) where such a kind of non-classical logic as multivalued logic was successfully used. It is here that, in order to solve the problems of describing the state of uncertainty in the physics of the microcosm, G. Birkhoff and J. Von Neumann created a three-digit "quantum logic" ^[6]. Due to the fact that it did not observe such a law of classical logic as the law of distributivity, with the help of its apparatus it was possible to describe the situation of uncertainty of the diffraction of an electron passing through two slits of the screen in the experiment of T. Jung ^[7]. Somewhat later, other logical systems of multivalued logic appeared, the main purpose of which was to minimize the language problems that arise in the process of studying and describing the phenomena of the microcosm. And although each of them had its own characteristics, its own methods of justification and its pros and cons, they all began to be called "quantum logics" or "microcosm logics". The most successful of all such logics turned out to be the three-digit logic of G. Reichenbach ^[8], with the help of which it became possible to cope with the problems that arise when describing the states of uncertainty of its "causal anomalies" that manifest themselves in physics of quantum phenomena.

An important feature of G. Reichenbach's logic was his introduction of such new concepts for already existing non-classical logical systems (systems of Y. Lukasevich, E. Post, A. Tarsky) as "total negation", "quasi-implication", "alternative implication" and "alternative equivalence". The characteristic features of this logic include the fact that the obligatory condition for the fulfillment of the law of non-contradiction in it was the use of the operation of complete negation only, as well as the fact that the law of the excluded third does not apply in it, which made it possible with the help of its means to describe the state of uncertainty, which, in fact, corresponded to the third meaning of the logic of G. Reichenbach – "indefinite".

Speaking about the possibilities of using G. Reichenbach's logic in physics, one more important circumstance should be noted, which was connected with his introduction of the principle of complementarity for statements into its apparatus. The essence of this principle, which partly corresponded to the principle of complementarity of N. Bohr, was that if conditions were fulfilled for two statements when the uncertainty of the truth of the second followed from the truth (falsity) of either of them, then each of them had to be considered additional to the other. For such statements in the logic of G. Reichenbach, the complementarity condition was symmetric: if the first of them complemented the second, then the second complemented the first. From his point of view, this relation could be extended to any number of statements, in cases when the truth or falsity of one of them entails the uncertainty of all the others ^{[9, pp. 206-207]}.

The ability to link statements about unobservable objects with statements about observables, which became the advantage of the logical interpretation of quantum mechanics by G. Reichenbach over the interpretation of N. Bohr and V. Heisenberg, served as the basis for its author not only to consider it "as another interpretation of quantum mechanics", but also to assert that it represents "the final form of quantum mechanics physics" ^{[11, 144 p.]}.

It should be noted that some of his contemporary physicists reacted rather skeptically to the last statement of G. Reichenbach. In particular, N. Bohr believed that "attempts to resort to trivalent logic, sometimes proposed as a way of considering the paradoxical features of quantum mechanics, do not seem too suitable for a clear illumination of the situation, since all experimental data, even if they cannot be analyzed from the point of view of classical physics, must always be expressed in the usual language, using ordinary logic" ^{[12, pp. 397-398]}.

The idea of using three-digit logic not only in physics, but also in other sciences and technology was expressed by some foreign and domestic scientists already in the second half of the last century. For example, N. P. Brusentsov, the developer and one of the creators of the first Setun digital computer in our country, which worked on the basis of this logic, wrote about its advantages as follows: "In a symmetric ternary code, logic is radically simplified and the number of necessary variants of arithmetic operations is reduced, the possibility of operating with words of different lengths opens up, the structure of operating devices is significantly simplified, significant savings are obtained in the time spent on performing operations, and in some cases also equipment" ^[13]. And although not all the useful properties of ternary code and three-digit logic were "implemented in this first computer," it "clearly demonstrated" its profitability ^[13], and later "turned out to be unusually favorable for creating automated systems for various purposes" ^[14]. Nevertheless, despite the significant advantages of this logic and the rapid development of computer technology and the advent of computers, three-digit logic in this period did not find widespread support here. She started looking for applications in other areas. One of such applications of three-digit, "quantum logic" has already become "a dialogue between two researchers who hold opposite points of view on the issue under discussion, but use a common language of dialogue" ^{[15, p. 276]}. And such a feature of quantum logic as the presence in it of the logical operation "square root of negation" not only "allows the emergence of non-determinism from a strictly defined set of initial conditions, which is relevant ... to the main question of philosophy", but also quite possibly "is able to have a significant impact on the development of the theory of dialectical reasoning" ^[16].

In the XX1 century, the obvious advantages of three–digit logic over two-digit in real calculations (comparison of two numbers in three-digit logic is performed in one step, and in two-digit logic - in two) were in demand.  And the research and development of various algorithms based on three-digit, four-digit and other multi-valued logics that exist today have become relevant in such areas as medicine and bionics, telecommunications, artificial intelligence and neural network modeling, computer engineering and automata design, in solving problems of verification of production knowledge bases, etc.

The expansion of the scope of multivalued logics is largely due to the fact that they allow the use of linguistic, i.e. qualitative variables instead of quantitative ones. This makes it possible to combine quantitative and qualitative indicators in one model and thereby significantly simplify complex probabilistic schemes for assessing reliability. This, in turn, opens the way for the most complete study of the model, which turns out to be especially effective "in situations where it is not possible to quantify the influence of a factor on the process; the use of qualitative variables provides additional opportunities in assessing the factors under study" ^{[17, pp. 72-73]}.

It must be said that in the totality of multivalued logics, a special place belongs to infinite-valued logics. Their characteristic feature is that the number of truth values of statements that they are able to accept is infinite. These values can be set in various ways. For example, a numerical segment ^{[1, 0]}, where the extreme values correspond to the values "true" and "false", and the intermediate values correspond to one or another of the rational fractions from some of their series constructed in accordance with some law. So, in one of such logics developed by A. D. Getmanova, this series is the sum of two numerical series (1/2)k and (1/2)k x (2k -1), i.e. a series of numbers: 1, 1/2, 1/4, 3/4, 1/8, 7/8, 1/16 ..., 0 ^{[18, p. 266)]}.  

Infinite-valued logics often include probabilistic or, as they are also sometimes called, inductive logics, characterized by the fact that the premises and conclusions of these logics are hypothetical, i.e. probabilistic statements that play an important role in science. The theoretical foundation of probabilistic logic was laid by G. V. Leibniz, and later its development was continued by J. Buhl, W. Jevons, J. Venn, G. Reichenbach, R. Mises, R. Carnap, A. N. Kolmogorov and other scientists.

All systems of probabilistic logic, characterized by the fact that each of the statements in them can have one or another valid value from 1 (true) to 0 (false), have as their foundation the theory of mathematical probability, which allows an objective assessment of the possibility of the occurrence of a particular event in certain repeatedly repeated conditions. Quite often, this estimate is calculated using the formula P(A) = r/s, where P(A) is the event of A, r is the number of favorable options for A, and s is the total number of all equally possible options. When throwing two dice, for example, there are 36 of all possible options. Given that the number of variants, the fallout, for example, of the number 6 is 5 (1 and 5, 2 and 4, 3 and 3, 4 and 2, 5 and 1), we get that in this case P (A) = 5/36.

In addition to logic proper, the apparatus of probabilistic logics finds its application in game theory and argumentation theory, in statistics and psychology, in bioinformatics, in pattern recognition and in artificial intelligence, and in addition – in the study of mass random phenomena occurring in nature and society.  Noting the wide range of fields of science and practice in which the apparatus of probabilistic logic is successfully used, it is necessary to mention certain difficulties that arise in this case. Most of them are connected with the fact that these logics not only "tend to multiply the computational complexity of their probabilistic and logical components", but also with the fact that they can sometimes lead to contradictory results ^[19].

Among the logical theories with an obvious applied effect, the greatest interest is currently aroused by the "ideologically close" to multivalued logic "fuzzy logic", which made it possible to carry out "the transfer of operations with the probabilistic laws of the quantum world to the world of logical reasoning" ^[16]. This logic, which makes it possible to give approximate truth estimates, is based on the theory of fuzzy sets, originating from an article by L. Zadeh in 1965 ^[20]. Fuzzy-valued, as it is also called, logic is the basis of reasoning, which uses precisely or quantitatively indefinable concepts. It is intended primarily for the analysis of systems in which human reasoning and "blurred" concepts take place. If this logic is currently actively used as the basis for automating complex technological processes, then the so-called "dynamic logic", which grew "out of a historical and philosophical interest in the logic of time," is used "today to provide procedures for the synthesis and verification of programs." And such non-classical logics as the logic of norms or deontic logic, the logic of assessments or axiological logic, the logic of questions, which is also called erotetic, the logic of knowledge or epistemic logic, the logic of opinions or doxatic logic are successfully used in logical studies of natural language and humanitarian knowledge ^[3].

In particular, the above-mentioned deontic logic is successfully used in modeling the processes of decision-making by a subject in a particular situation, relying on his basic value attitudes, which are used as axioms added to his accepted model of the world. Understanding the logical characteristics of norms turns out to be necessary when solving questions "about the place and role of norms in scientific and other knowledge, about the mutual relations of norms and assessments, norms and descriptive statements, etc." ^[21].

No less wide range of applications in modern scientific cognition are also logics, which are called paranoherent. They are actively used not only in natural, socio-humanitarian and technical sciences, but also "in probabilistic and inductive reasoning, in the theory of fuzzy concepts, in deontic logic (moral dilemmas), in doxatic logic (systems of belief)" ^[22]. The high efficiency of these logics has been shown in the works on identifying the causes of linguistic contradictions, logical and semantic antinomies arising in scientific cognition, as well as in solving problems of minimizing the harm that can be caused by contradictory information entering information search engines.

According to N. Da Costa, the following results of their use are evidence confirming the wide possibilities of using paranoherent logics in scientific cognition: a deeper understanding of development theories, in particular, dialectics; proof of the possibility of contradictory, but not trivial theories; development of unconventional ontological models; proof of the weakness of some critical statements against dialectics; proof of excessive rigidity of standard methodological requirements imposed on scientific theories, etc. ^{[23, p.119]}.

One of the first who, in the late 1880s, expressed the idea of the possibility of the existence of a paran-contradictory ("non-Euclidean", as he called it) three-digit logic was Ch. Pier. Following him, at the very beginning of the twentieth century, P. Karus wrote about the possibility of "imagining a fairy-tale world" in which there is a "curved" logic, and N. A. Vasiliev, in his trial lecture at Kazan University, proposed "a system of non-classical logic not as an idea, but as a completely finished formal construction satisfying the discerning requirements of the scientific community" ^[24]. In his work on the "imaginary", as he called it, logic, N. A. Vasiliev used the ideas of N. Lobachevsky's nonclassical geometry. At the same time, he did not doubt that a person can think a contradiction. Since any actual thought finds its expression in a judgment, "to think a contradiction," he argued, "means to form a special judgment of contradiction or an indifferent judgment next to an affirmative and negative judgment" ^{[25, p. 69]}. However, N. A. Vasiliev's ideas about the possibility of "thinking contradiction" during his lifetime were not accepted by either Russian or foreign scientists.

Polish scientist J. Lukasevich is quite rightly considered to be another ancestor of the paranoherent logic. If N. A. Vasiliev proposed to expand the traditional (Aristotelian) logic by introducing into it the concept of a self-contradictory judgment "S is and is not P" ^[26], then Y. Lukasevich built his three-digit logic, where, in addition to the values "true" and "false", the meaning "possible" or "neutral" was used, which it allowed to "bypass" contradictions, since the laws of non-contradiction and the excluded third of classical logic were no longer laws in it ^[27].

Interest in paraconcilable logics and their applications in various branches of scientific knowledge increased significantly by the middle of the twentieth century. One of the first para-contradictory logics developed specifically in order to resolve logical difficulties that often arise in the process of this or that kind of scientific discussions, where the coexistence of different, even contradictory points of view is quite possible, was the "debatable logic" of S. Yaskovsky. Her apparatus not only prevented the derivation of any arbitrarily taken statement from contradiction, but also left no room for the rule of introducing the conjunction {A, B} ? A & B^[22].

If at the very beginning of its development, paraconcilable logics were created primarily as a means of resolving and eliminating paradoxes in set theory, in the logic of statements and predicate logic, then later they began to be effectively used in natural, technical, social and humanitarian sciences. So, for example, at present they successfully help to solve problems related to the need to express contradictory information or incompatible data coming into a local information network or computer in a consistent manner, i.e. in situations where the simultaneous truth (validity) of some statement together with its denial is allowed ^{[28, p. 37]}. Their apparatus works in probabilistic and inductive reasoning, in the theory of fuzzy concepts, in the creation of non-trivial ontological models, etc.

The so-called relevant logics, which are formal systems in which the principle (A~A) is excluded, are currently distinguished in the whole set of paraconcilable logics ? B, and one of the fundamental concepts is the concept of logical following. To overcome the difficulties that often arose in classical logic due to the ambiguity of the interpretation of this concept, relevant logics were created.Currently, their apparatus is successfully used when it is necessary to clarify the content of certain general scientific concepts, for example, "proof" and "refutation", "explanation" and "prediction", "law of science", etc., as well as certain methodologically significant concepts, such as "counterfactual statement", "definiteness terms in theory", etc. Relevant logics are also used in solving other problems related to the need to apply the logic of following.

The attempts to use these logics in order to formalize dialectics, made in the seventies of the last century by R. Routley and R. Meyer, are quite well known. However, the first logical system DL created by them, which contained in its axiomatics the contradiction p 0 & p 0 and which they called "dialectical logic", in fact could formalize only some fragments of dialectics.  But the "weak" DM system developed by them, containing all the axioms of the DL system, and called by them the "statistical maxim of dialectical logic", could become, according to R. Routley and R. Meyer, quite promising in solving the problem of formalization of dialectics ^[29].Nevertheless, it should be noted that the apparatus of the "dialectical logics" of R. Routley and R. Meyer turned out to be quite possible to use (without serious negative consequences for the formal system itself) as the foundation of theories that are models of individual areas in a constantly changing contradictory objective world. In particular, these logics are currently in demand due to "the development of weak solvable systems that are used in computer science, modeling argumentation and knowledge change processes" ^[30].

Once again, we note that the increased interest in non-classical logics in recent years is explained by the fact that practical interests are coming to the fore to replace various philosophical, syntactic, semantic and metalogical problems previously discussed in the scientific community. This, according to A. S. Karpenko, "explains the "obsession" with fuzzy logic, which occupy almost a leading position in information technology. But the main source of such interest is their wide application in computer science, artificial intelligence and programming" ^[31].

In particular, at present, for example, the logic of causality is used in discussions related to the interpretation of such concepts as "law of nature", "ontological necessity" and "determinism". The apparatus of probabilistic logic, in addition to the areas already mentioned above, finds its application in the study of mass random phenomena occurring in nature and society. The languages of "temporal modal logics are widely used for modeling, specification and verification (correctness analysis) of software systems of logical control and "reacting" systems" ^{[32, p.7]}. And in solving problems of dynamic verification of production knowledge bases and expert systems, logics with vector semantics combining the features of fuzzy and a pair of consistent logics ^[33].

In addition to the systems of non–classical logics presented above in a brief form, in modern sciences and practice, to one degree or another, the following are used: deontic logic, which studies the logical connections of normative statements - in the field of morality and law (in particular, legal expert systems work successfully in jurisprudence); logic of absolute assessments, which studies the logical structure and logical connections evaluative statements – in political economy, linguistics, studies of morality and law, analysis of values; probabilistic logic – in artificial intelligence and logical technology; epistemic logic, sometimes called the logic of knowledge and belief, the basic concepts of which are the concepts of "refutable", "insoluble", "provable", "convinced", "doubts" and T. P. - in the theory of cognition, in the methodology of science, in linguistics and psychology; preference logic dealing with statements containing the concepts of "better", "worse", "equivalent", etc. – in economic theories and in the field of ethics; the logic of change, which speaks of change and becoming, is in dialectics, theories and practical research related to the study and description of the processes of the emergence and change of the state of real and ideal; the logic of causality is in all branches of scientific knowledge that study cause–and–effect relationships ^[34].

At the same time, it seems to me, it is impossible not to agree that, despite the "rich in successes and prospects of the development of logic over the past century," just a declaration of its important socio-practical role is clearly not enough ^[2]. Currently, many graduates of higher education have a low level of logical culture, and humanities specialists who study at least traditional logic in one volume or another, quite clearly reveal methodological flaws and low efficiency of using the material being mastered. It is not necessary to talk about mastering them with modern logical means yet. "The development of logic rich in successes and prospects during the past century," writes A. G. Kislov, "seems to please only the logicians themselves" ^[2]. Therefore, in my opinion, the need for all university students to study at least a course of classical logic is becoming more and more obvious.