Advances in Social Science, Education and Humanities Research, volume 356 2nd International Conference on Contemporary Education, Social Sciences and Ecological Studies (CESSES 2019) Formal Models of the Evolution of Scientific Theory
Nikolay Arkhiereev Alexander Chernyaev National Research University National Research University Bauman Moscow State Technical University (BMSTU) Moscow Institute of Physics and Technology (MIPT) Moscow, Russia Dolgoprudny, Russia E-mail: [email protected] E-mail: [email protected]
Abstract—The article discusses various formal models of rules TC. The set TC proved to be partially ordered by the the evolution of scientific theory. The cumulative model based relation of formal derivability. on the standard interpretation of a scientific theory presented the evolution of science as a continuous linear process. The The formalization of the theory in F.O.P.L. — 1 = taken abstract character of this model has become the object of fierce to be called standard. criticism from some representatives of the post-positivistic The interpretation of the theory as a set of statements of philosophy of science. This led to the statement about the some formalized language (axioms and theorems) ordered by incommensurability of alternating theories, of the unrealistic nature of the statements of scientific theories and the the deductive derivability relation is usually called the impossibility of applying to the theories of the natural sciences, standard or "statement" view of the theory. and formally strict concepts of truth/falsity. The set-theoretic On the basis of the standard interpretation of scientific model of the development of science, based on the theory received in logical positivism, a very specific generalization of the concept of a model of theory in the sense (cumulative) model of the development of scientific of A. Tarski, allows avoiding the extremes of cumulative knowledge was proposed, which was subjected to fierce representations and at the same time allows demonstrating criticism in the works of a number of representatives of the partial continuity between the models of alternating theories. so-called historical school in the philosophy of science. As a Keywords—theory; model; formalization; axiomatization; result, the very possibility of a logical reconstruction of the truth; verisimilitude dynamics of scientific knowledge was called into question [3]. I. INTRODUCTION This article reviews the logical-positivist and model- One of the distinguishing features of the program of theoretic strategies for analyzing of the development of foundation of scientific knowledge put forward by logical scientific knowledge. As a result, the conclusion about the positivism was a deliberate and persistent reliance on formal fruitfulness of using formal methods in the reconstruction of research methods in solving a number of pivotal problems in the processes of development and transformation of the philosophy of science [1]. The possibility of the scientific theories [4] is drawn. successful solution of some of these problems (such as the problem of a strict distinction between analytical and II. LOGICAL POSITIVISTIC MODEL OF DYNAMICS OF synthetic statements of the theory, the formulation of criteria SCIENTIFIC KNOWLEDGE of the meaningfulness of statements etc.), was supposed to According to logical-positivist ideas, theories that have be conditioned by their preliminary explication in a certain successfully passed a series of empirical tests have a high artificial language [2]. In modern philosophical and logical degree of reliability. At the same time, further literature the language of classical first-order predicate logic transformations of "pre-confirmed" theories can be with equality (hereinafter referred to as F.O.P.L. — 1 =) is implemented according to one of the following scenarios: 1) generally accepted as such a formalized language. At the as the measuring and experimental equipment improves, at same time, the scientific theory subjected to analysis was to the theory' s predictions emerge inconsistencies and errors be formulated as a partially interpreted axiomatic system, the that lower the degree of its confirmation by experience; this axioms of which were the fundamental laws of the narrows the field of applicability of the theory and in some corresponding theory, expressed in some theoretical exceptional cases can lead to its abandonment; 2) the high language LT. The observable consequences of these laws degree of reliability of the original theory allows us to ought to be formulated in a separate observation language expand the scope of its application; 3) several disparate LO, and the relationship between concepts, belonging to LT theories, each of which has a high degree of reliability, are and LO, was to be expressed via so-called correspondence included in a single, "generalizing". rules C (or rules of reduction, as they were dubbed in later versions of this program). In this case, only concepts from Options 2) and 3) are considered as the most common. the LO language had literal semantic meanings. The In the second of the possible options for the development scientific theory was identified with a set of conjunctions of of the theory, the original set of theoretical postulates and theoretical postulates of the theory and the correspondence
Copyright © 2019, the Authors. Published by Atlantis Press. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/). 1280 Advances in Social Science, Education and Humanities Research, volume 356 correspondence rules of the TC is replaced by the set of TC, problems in a certain field of knowledge — became a in which the rules of correspondence C are designed to paradigmal itself [7]. "adapt" the conceptual apparatus of the original theory to a Kuhn's idea, in its own turn, aggravated the controversy wider area of observable phenomena. The theory TC should between scientific realism and anti-realism in interpretation successfully pass an empirical test and, in addition, the of the nature and functions of scientific theory. terminological apparatus of the original TC theory should not differ significantly from the terminological apparatus of Crucial ideas concerning those topics were presented in the TC theory. The latter requirement determines the the classical work of Larry Laudan "A Confutation of homogeneity of the dictionaries of both theories, which, in Convergent Realism" [8]. turn, makes it possible to derive deductively the statements The essence of scientific realism (or, in the terminology of the TC theory from the set of statements of the TC theory. of L. Laudan, convergent epistemic realism) can be Therefore, this model of the evolution of a theory is, in characterized as a sequence of the following theses: essence, a scheme for reducing the statements of one theory to statements of another. Mature scientific theories are (at least) approximately true in the sense of correspondence; at the same time, Other variations of this theory evolving suggest the the subsequent theories are "closer to the truth" than possibility of expanding the list of theoretical postulates of the preceding theories which relate to the same the original theory while maintaining the reduction rules subject area. (transition from TC to TC) or expansion of both lists of statements of the original theory (transition from TC to TC). Both observable and theoretical terms that are part of "mature" theories have real referents — there really The third of the possible options for the development of are objects in the world that correspond to the the theory suggests a slightly different type of reduction, in postulated elements of the ontology of scientific which (at least some) formulations of the laws of the original theory. theory/theories contain descriptive terms that are absent in the formulations of theoretical postulates and/or In mature science, the preceding theories turn out to correspondence rules of the "generalizing" theory. For the be "limiting cases" of subsequent ones (subsequent correct implementation of the reduction of this type, the theories in mature science retain the referents of following conditions must be satisfied: 1) the meanings of previous theories and the interconnections that bind the theoretical terms of both theories must be unambiguously them). determined; 2) for each theoretical term of the initial Each subsequent theory is capable of explaining why theory, which does not occur in the dictionary of the the previous theory was "successful" in predictive and generalizing theory, a hypothetical correspondence between explanatory terms (if, of course, the previous theory the meaning of the term and the theoretical term in the was indeed successful in the indicated sense). dictionary of the generalizing theory, (presumably) reflecting the properties of the value of is postulated; 3) on the basis The term "approximate truth" or "verisimilitude" is of hypothesis 2) all the laws of the original theory should be considered as some kind of "weakening" of the formally logically derived from theoretical postulates and the logical concept of the truth of the theory proposition [9]. correspondence rules of the generalizing theory; 4) all As a result, from the point of view of epistemic realism, additional hypotheses used in the conclusion should be the development of science is a "convergent" process, which empirically confirmed [5]. inevitably brings us closer to the "increasingly credible" As a rule, a reduction of this type is an explanation of a picture of reality. certain phenomenological theory or a set of experimental In particular, from the point of view of scientific realism, laws relating to one subject area, using a theory formulated the following statements should be correct: for another (albeit related) domain of exploration (for example, explanation of the laws of the motion of the planets If the theory is "plausible" / "approximately true," according to Kepler by means of Newton's mechanics). then it successfully fulfills its explanatory and predictive functions. As a result, science appears to be a linear cumulative process of continuous accumulation of knowledge. The If the theory successfully fulfills its explanatory and previous theories are not discarded, but are reduced to the predictive functions, then it is approximately true. new ones by the aforementioned ways [6]. With regard to the statement B), L. Laudan notes: one This technique, rather cumbersome and excessively can easily name the number of theories that had successfully schematized, has become a favorite object of criticism of implemented their explanatory and predictive functions for opponents of the logical-positivist program of foundation of quite a long period of time and, nevertheless, turned out to be scientific knowledge. virtually false eventually (Ptolemy and Aristotle's astronomy Thus, T. Kuhn's idea about the " incommensurability of is a classic example of such a theory). paradigms" , alternating fundamental scientific theories that Regarding the statement A) L. Laudan drew readers' determine the very way of positing and solving of scientific attention to the following fact: not a single attempt to define
1281 Advances in Social Science, Education and Humanities Research, volume 356 strictly the notion of "approximate truth" of the theory has concepts of truth — namely, it implicitly uses the first proved successful so far. concept instead of the second one. One of the most well-known variants of such a definition This fact is due to the insufficiently developed semantic is the following formulation. problematics in the logical-positivist program of formalization of scientific knowledge. Let T1 be a standardly formalized theory; Let CtT (T1) be the cardinality of the set of its true conclusions (true The concept of partial truth/verisimilitude of the theory propositions logically derivable from the postulates (axioms) has been successfully clarified in the set-theoretic approach of T1), CtF (T1) is the cardinality of the set of false to the foundation of scientific knowledge. The crucial notion conclusions of T1. underlying the set-theoretic approach is the generalization of the semantic concept of the model in the sense of A. Tarski. Then the T1 theory is verisimilar (approximately true) if and only if the cardinal number of the set of its true Generally speaking, in formal semantics a model of a corollaries is greater than the cardinal number of the set of its theory is usually understood as some possible realization of a false corollaries: theory that satisfies its axioms. A possible realization of the theory, in its own turn, is a set-theoretic object of the Ct T (T1) > Ct F (T1) corresponding logical type — for example, an ordered As L. Laudan notes, this definition is unsatisfactory: we sequence of elements
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n N. Da Kosta, S. French, I. Maikenberg, R. Chuaqui Then the pragmatic structure D, Ri , P admits the proposed a generalization of the concept of a model that existence of an A-normal structure, if and only if (iff) the set would successfully neutralize the described technical MP is consistent. difficulties and naturally explicate the concept of approximate truth of the theory, which demonstrates the The statement is quasi-true in the pragmatic structure n n validity of the main theses of convergent realism [11] [12]. D, Ri , P, i. e. there is an A-normal structure D, R i, in which is true (in the sense of Tarski). Otherwise, is The starting points for this approach are the concepts of a quasi-false in a pragmatic structure. partial relation (defined on some non-empty domain D), a partial structure, and a quasi-truth. The latter concept is a Finally, some statement is quasi-true (approximately formally precise explication of the concept of approximate true), if there is some pragmatic structure A and the truth. corresponding A-normal structure B such that is true in B (in the sense of Tarski). Otherwise, is quasi-false. Let D be a non-empty domain, some objects from which may eventually prove to be non-existent. IV. CONCLUSION n n The partial N-place relation on D is a triple R1 , R2 , The formal apparatus proposed in the framework of the n n n n R3 , where R1 , R2 , R3 are mutually disjoint sets, the set-theoretic strategy of analyzing of scientific knowledge n n n union of which coincides with D: R1 R2 R3 = D allows us to construct adequate models of the dynamics of n n science, reflecting the possibility of conceptual and structural R1 is such a set of n from D, which satisfies R, R 2 is a changes of theories and, at the same time, preserving (partial) set of n, which does not satisfy R, and R3 is the set of n, continuity between them. Thus, it can be said that the thesis relative to which it is not figured out yet, whether it satisfies of the "confutation" of convergent realism and, in particular, R or not. the complete incommensurability of changing fundamental n theories appears to be the result of an incorrect absolutization A partial structure is an ordered pair D, Ri (n1, i1), n of the standard interpretation of scientific theory proposed by where D is a non-empty set, and Ri is a family of partial relations on D. logical positivism. The next necessary notion in this approach is the concept of A-normal structure. REFERENCES n n Let A=D, Ri be a partial structure. Then B=
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the International Conference on Contemporary Education, Social Sciences and Ecological Studies (CESSES 2018). Series " Advances in Social Science, Education and Humanities Research" , vol. 283, pp. 801-805, 2018. DOI: 10.2991/cesses-18.2018.177 [10] V. Yu. Ivlev, M.L. Ivleva, "Peculiarities of Aristotelian scholastic logic," Proceedings of the 2nd International Conference on Contemporary Education, Social Sciences and Humanities (ICCESSH2017). Series " Advances in Social Science, Education and Humanities Research" , vol. 124, pp. 91-95, 2017. DOI: 10.2991/iccessh-17.2017.20 [11] N. Da Costa, S. Frensh, Science and Partial Truth. A Unitary Approach to Models and Scientific Reasoning. Oxford: Oxford University Press, 2003. [12] L Mikenberg, N. da Costa and R. Chuaqui, "Pragmatic Truth and Approximation to Truth," The Journal of Symbolic Logic, no. 51, pp. 201-221, 1986.
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