Сборник составили статьи, посвященные отдельным направлениям работы философа, воспоминания его друзей и коллег, а также поздние тексты самого Л. Н. Митрохина, в которых он рассказывает о своем жизненном и творческом пути.

Lev Nikolaevich Tolstoy’s departure from this earth took place in 1910, a year that has since become a common touchstone for Russians. The twentieth century came to a symbolic close, beginning a te...

First English-language translation of Lev Shestov's early writings on faith, ancient philosophy and biblical revelation.

Progressions of iterated reflection principles can be used as a tool for the ordinal analysis of formal systems. We discuss various notions of proof-theoretic ordinals and compare the information obtained by means of the reflection principles with the results obtained by the more usual proof-theoretic techniques. In some cases we obtain sharper results, e.g., we define proof-theoretic ordinals relevant to logical complexity Π1 0 and, similarly, for any class Π n 0 . We provide a more general version of the (...) fine structure relationships for iterated reflection principles (due to U. Schmerl [25]). This allows us, in a uniform manner, to analyze main fragments of arithmetic axiomatized by restricted forms of induction, including IΣ n , IΣ n − , IΠ n − and their combinations. We also obtain new conservation results relating the hierarchies of uniform and local reflection principles. In particular, we show that (for a sufficiently broad class of theories T) the uniform Σ1-reflection principle for T is Σ2-conservative over the corresponding local reflection principle. This bears some corollaries on the hierarchies of restricted induction schemata in arithmetic and provides a key tool for our generalization of Schmerl's theorem. (shrink)

We suggest an algebraic approach to proof-theoretic analysis based on the notion of graded provability algebra, that is, Lindenbaum boolean algebra of a theory enriched by additional operators which allow for the structure to capture proof-theoretic information. We use this method to analyze Peano arithmetic and show how an ordinal notation system up to 0 can be recovered from the corresponding algebra in a canonical way. This method also establishes links between proof-theoretic ordinal analysis and the work which has been (...) done in the last two decades on provability logic and reflection principles. Because of its abstract algebraic nature, we hope that it will also be of interest for non-prooftheorists. (shrink)

We deal with the fragment of modal logic consisting of implications of formulas built up from the variables and the constant ‘true’ by conjunction and diamonds only. The weaker language allows one to interpret the diamonds as the uniform reflection schemata in arithmetic, possibly of unrestricted logical complexity. We formulate an arithmetically complete calculus with modalities labeled by natural numbers and ω, where ω corresponds to the full uniform reflection schema, whereas n<ω corresponds to its restriction to arithmetical Πn+1-formulas. This (...) calculus is shown to be complete w.r.t. a suitable class of finite Kripke models and to be decidable in polynomial time. (shrink)

For “natural enough” systems of ordinal notation we show that α times iterated local reflection schema over a sufficiently strong arithmetic T proves the same Π 1 0 -sentences as ω α times iterated consistency. A corollary is that the two hierarchies catch up modulo relative interpretability exactly at ε-numbers. We also derive the following more general “mixed” formulas estimating the consistency strength of iterated local reflection: for all ordinals α ⩾ 1 and all β, β ≡ Π 1 0 (...) T ω α · , α ≡ Π 1 0 T β + ω α . Here T α stands for α times iterated local reflection over T , T β stands for β times iterated consistency, and ≡ Π 1 0 denotes mutual Π 1 0 -conservativity. In an appendix to this paper we develop our notion of “natural enough” system of ordinal notation and show that such systems do exist for every recursive ordinal. (shrink)

Prominent thinkers such as Jurgen Habermas and Michael Sandel are warning that biomedical enhancements will undermine fundamental political values. Yet whether biomedical enhancements will undermine such values depends on how biomedical enhancements will function, how they will be administered and to whom. Since only few enhancements are obtainable, it is difficult to tell whether these predictions are sound. Nevertheless, such warnings are extremely valuable. As a society we must, at the very least, be aware of developments that could have harmful (...) consequences. Indeed, if important values were to be jeopardized, we should take appropriate measures to protect them. This paper focuses on four central values: solidarity, personal responsibility, equality and autonomy. It delineates the conditions under which biomedical enhancements would undermine these values. It also details the circumstances under which these values would be unaffected by enhancements as well as those under which they would be promoted. Specifying these conditions is valuable; it would enable society to prepare appropriate ethical guidelines and policy responses in advance. (shrink)

This article is a study of the compatibility of the theory and practice of Marxism-Leninism and Orthodox Christianity. The great importance of this topic is due to its direct connection with the improvement of society and the ongoing controversy on this issue with attempts to integrate communism with Christianity. The work provides a historical and theological analysis based on a critical study of the works of the founders of Marxism-Leninism, their associates, historical and theological works, as well as historical documents (...) on this issue. The article examines the approaches of the teachings of Orthodox Christianity, as well as Marxism-Leninism to atheism, materialism, religion and the Church. For this purpose, the relevant texts of Holy Scripture, the opinions of the Church fathers and the views of the founders and followers of Marxism-Leninism are studied. The novelty of this study is that some documents are considered for the first time in the context of the problem under study. In addition, a more detailed and systematic historical and theological analysis of the comparison of the teachings of Marxism-Leninism and Orthodox Christianity on this topic is carried out in comparison with the available works. The purpose of the work is to study historical and theological works, as well as historical documents related to the problem under study. The article concludes that Marxism-Leninism in this matter is not compatible with Christianity and is directly opposed to it. Since the Second World War in the USSR, relations between the state and the Church have improved from time to time, but the official state course on Marxism-Leninism did not leave the possibility of their reconciliation. Currently, there are attempts to produce a symbiosis of these two teachings. Many Communists have a positive attitude towards the Church, so a dialogue between them is possible and necessary. But as long as the ideology of the Communists remains Marxism-Leninism, antagonistic to Christianity and any religion in general, in their ranks will always remain hostile to religion. Therefore, the synthesis of these teachings is impossible. (shrink)

Analyzes principal forces that determined the direction of Camus' thought on ethics and political values.

The main focus of the book is the presentation of the 'inertial' view of population growth. This view provides a rather simple model for complex population dynamics, and is achieved at the level of the single species without invoking species interactions. An important part of this account is the maternal effect. Investment of mothers in the quality of their daughters makes the rate of reproduction of the current generation depend not only on the current environment, but also on the environment (...) experienced by the previous generation. (shrink)

A well-known result states that, over basic Kalmar elementary arithmetic EA, the induction schema for ∑n formulas is equivalent to the uniform reflection principle for ∑n + 1 formulas . We show that fragments of arithmetic axiomatized by various forms of induction rules admit a precise axiomatization in terms of reflection principles as well. Thus, the closure of EA under the induction rule for ∑n formulas is equivalent to ω times iterated ∑n reflection principle. Moreover, for k < ω, k (...) times iterated ∑n reflection principle over EA precisely corresponds to the extension of EA by k nested applications of ∑n induction rule.The above relationship holds in greater generality than just stated. In fact, we give general formulas characterizing in terms of iterated reflection principles the extension of any given theory by k nested applications of ∑n or Πn induction rules. In particular, the closure of a theory T under just one application of ∑1 induction rule is equivalent to T together with ∑1 reflection principle for each finite Π2 axiomatized subtheory of T.These results have closely parallel ones in the theory of subrecursive function classes. The rules under study correspond, in a canonical way, to natural closure operators on the classes of provably recursive functions. Thus, ∑1 induction rule precisely corresponds to the primitive recursive closure operator, and ∑1 collection rule, introduced below, corresponds to the elementary closure operator. (shrink)

Provability logic GLP is well-known to be incomplete w.r.t. Kripke semantics. A natural topological semantics of GLP interprets modalities as derivative operators of a polytopological space. Such spaces are called GLP-spaces whenever they satisfy all the axioms of GLP. We develop some constructions to build nontrivial GLP-spaces and show that GLP is complete w.r.t. the class of all GLP-spaces.

This article compares the sections on the qualities desirable in a wife in Abū Ḥāmid al-Ghazālī’s Iḥyāʾ ʿulūm al-dīn and Bar Hebraeus’s Ethicon, which the West Syrian writer modeled on al-Ghazālī’s work. The article first establishes that al-Ghazālī based his profile of the ideal wife on a jurisprudential discussion of the topic by his teacher, Imām al-Ḥaramayn al-Juwaynī, expanding it, however, by adding anecdotes from Sufi literature and reasoned arguments on how “the good wife” will best facilitate her husband’s devotion (...) to God. The article then moves to consider how Bar Hebraeus appropriates, reconfigures, and reframes as Christian teaching al-Ghazālī’s discussion. Finally, the article argues that the similarity between the two texts demonstrates that both carve out a particular notion of male piety intelligible and resonant in their different religious traditions and in the broader monotheistic culture of the premodern Middle East. (shrink)

We characterize the bimodal provability logics for certain natural (classes of) pairs of recursively enumerable theories, mostly related to fragments of arithmetic. For example, we shall give axiomatizations, decision procedures, and introduce natural Kripke semantics for the provability logics of (IΔ 0 + EXP, PRA); (PRA, IΣ 1 ); (IΣ m , IΣ n ) for $1 \leq m etc. For the case of finitely axiomatized extensions of theories these results are extended to modal logics with propositional constants.

A well-known polymodal provability logic inlMMLBox due to Japaridze is complete w.r.t. the arithmetical semantics where modalities correspond to reflection principles of restricted logical complexity in arithmetic. This system plays an important role in some recent applications of provability algebras in proof theory. However, an obstacle in the study of inlMMLBox is that it is incomplete w.r.t. any class of Kripke frames. In this paper we provide a complete Kripke semantics for inlMMLBox . First, we isolate a certain subsystem inlMMLBox (...) of inlMMLBox that is sound and complete w.r.t. a nice class of finite frames. Second, appropriate models for inlMMLBox are defined as the limits of chains of finite expansions of models for inlMMLBox . The techniques involves unions of n -elementary chains and inverse limits of Kripke models. All the results are obtained by purely modal-logical methods formalizable in elementary arithmetic. (shrink)

We introduce the logics GLP Λ, a generalization of Japaridze’s polymodal provability logic GLP ω where Λ is any linearly ordered set representing a hierarchy of provability operators of increasing strength. We shall provide a reduction of these logics to GLP ω yielding among other things a finitary proof of the normal form theorem for the variable-free fragment of GLP Λ and the decidability of GLP Λ for recursive orderings Λ. Further, we give a restricted axiomatization of the variable-free fragment (...) of GLP Λ. (shrink)

By a result of Paris and Friedman, the collection axiom schema for $\Sigma_{n+1}$ formulas, $B\Sigma_{n+1}$ , is $\Pi_{n+2}$ conservative over $I\Sigma_n$ . We give a new proof-theoretic proof of this theorem, which is based on a reduction of $B\Sigma_n$ to a version of collection rule and a subsequent analysis of this rule via Herbrand's theorem. A generalization of this method allows us to improve known results on reflection principles for $B\Sigma_n$ and to answer some technical questions left open by Sieg (...) [23] and Hájek [9]. We also give a new proof of independence of $B\Sigma_{n+1}$ over $I\Sigma_n$ by a direct recursion-theoretic argument and answer an open problem formulated by Gaifman and Dimitracopoulos [8]. (shrink)

The Many-Worlds Interpretation (MWI) is an approach to quantum mechanics according to which, in addition to the world we are aware of directly, there are many other similar worlds which exist in parallel at the same space and time. The existence of the other worlds makes it possible to remove randomness and action at a distance from quantum theory and thus from all physics.

We study the fragment of Peano arithmetic formalizing the induction principle for the class of decidable predicates, $I\Delta_1$ . We show that $I\Delta_1$ is independent from the set of all true arithmetical $\Pi_2-sentences$ . Moreover, we establish the connections between this theory and some classes of oracle computable functions with restrictions on the allowed number of queries. We also obtain some conservation and independence results for parameter free and inference rule forms of $\Delta_1-induction$ . An open problem formulated by J. (...) Paris is whether $I\Delta_1$ proves the corresponding least element principle for decidable predicates, $L\Delta_1$ (or, equivalently. the $\Sigma_1-collection$ principle $B\Sigma_1$ ). We reduce this question to a purely computation-theoretic one. (shrink)

The purpose of the study is to reveal the main actions of the major central banks during the COVID - 19 pandemic and their main impact on the world stock markets. The scientific novelty consists in identifying the key results of the impact of the pandemic in general and the restrictive measures of national governments, in particular, on the dynamics of the state of the stock markets of the world, namely, the level of decline in the main stock indexes of (...) the S &P 500, Dow Jones and NASDAQ. The measures taken by the central banks of the United States, Japan, England and other developed and developing countries of the world to provide financial assistance to commercial banks and other financial organizations related to the purchase of government bonds at a "zero" rate, targeting the yield of government bonds, reducing key rates and other measures are summarized. It is indicated that the influx of new private investors provided a great help to the stock market, for which many Central Banks introduced special programs and preferences. As a result, the main measures taken by the Central Bank of the Russian Federation to stabilize the financial situation in the country and the domestic stock market, in particular, are summarized. (shrink)

The subject of the study is the epistemological specificity of art through the «prism» of the Paleolithic and modern eras. The focus of the research is aimed at analyzing the phenomenon of «eidetism», which is a link between modern and primitive art. The purpose of the article is to comprehend the epistemological specifics of art, which began with the «psychophysiology» of the primitive world and developed into forms of «practical philosophizing» of the modern era. The research methodology includes a review (...) of literary sources, a dialectical approach to historical and art historical reconstruction, analysis of paleopsychological data, comparative analysis, elements of systems psychology and sociology. An element of novelty is the author's position that already in the lower, early layers of its evolution, the status of art «as one of the most ancient forms of human activity» is enriched by the status of «a form of practical philosophizing». The generalization is made that the art of the XX-XXI centuries, on the one hand, abandoned both mimesis and humanism, on the other – it continued the path of «practical philosophizing» begun in ancient times, based on the perceptual-empirical method of cognition of reality. The evolution of art from its primitive origins through archaic, classical to its modern state is, at the same time, the transformation of human consciousness, and the comprehension through artistic and creative activity of one's being and place in the universe, etc. - the development of its «depth» (the content of art). This is the development of art «in breadth» (the development of artistic form): the proliferation of its types and genres, the growth of stylistic and technological polyphony, etc. As a result, «art has come a long way from its purely psychophysiological hypostasis to «practical philosophizing». Artistic and creative activity itself as a whole continues to be the most important component of the global «mechanism» of mastering reality. (shrink)

_Russian Philosophy in the Twenty-First Century: An Anthology_ presents a variety of contemporary philosophic problems found in the works of prominent Russian thinkers, ranging from social and political matters and pressing cultural issues to insights into modern science and mounting global challenges.

This is a philosophical paper in favor of the many-worlds interpretation of quantum theory. The necessity of introducing many worlds is explained by analyzing a neutron interference experiment. The concept of the “measure of existence of a world” is introduced and some difficulties with the issue of probability in the framework of the MWI are resolved.

What is the most important reason for using Computer Vision methods in humanities research? In this article, I argue that the use of numerical representation and data analysis methods offers a new language for describing cultural artifacts, experiences and dynamics. The human languages such as English or Russian that developed rather recently in human evolution are not good at capturing analog properties of human sensorial and cultural experiences. These limitations become particularly worrying if we want to compare thousands, millions or (...) billions of artifacts—i.e. to study contemporary media and cultures at their new twenty-first century scale. When we instead use numerical measurements of image properties standard in Computer Vision, we can better capture details of a single artifact as well as visual differences between a number of artifacts–even if they are very small. The examples of visual dimensions that numbers can capture better then languages include color, shape, texture, contours, composition, and visual characteristics of represented faces, bodies and objects. The methods of finding structures and relationships in large numerical datasets developed in statistics and machine learning allow us to extend this analysis to very big datasets of cultural objects. Equally importantly, numerical image features used in Computer Vision also give us a new language to represent gradual and continuous temporal changes—something which natural languages are also bad at. This applies to both single artworks such as a film or a dance piece and also to changes in visual characteristics in millions of artifacts over decades or centuries. (shrink)

We investigate the bimodal logics sound and complete under the interpretation of modal operators as the provability predicates in certain natural pairs of arithmetical theories . Carlson characterized the provability logic for essentially reflexive extensions of theories, i.e. for pairs similar to . Here we study pairs of theories such that the gap between and is not so wide. In view of some general results concerning the problem of classification of the bimodal provability logics we are particularly interested in such (...) pairs that is axiomatized over by ∏1-sentences only, and, for each n 1, proves the n-times iterated consistency of . A complete axiomatization, along with the appropriate Kripke semantics and decision procedures, is found for the two principal cases: finitely axiomatizable extensions of this sort, like e.g. ), ), etc., and reflexive extensions, like ¦n 1\s}), etc. We show that the first logic, ICP, is the minimal and the second one, RP, is the maximal within the class of the provability logics for such pairs of theories. We also show that there are some provability logics lying strictly between these two. As an application of the results of this paper, in the last section the polymodal provability logics for natural recursive progressions of theories based on iteration of consistency are characterized. We construct a system of ordinal notation , which gives exactly one notation to each constructive ordinal, such that the logic corresponding to any progression along coincides with that along natural Kalmar elementary well-orderings. (shrink)

We study the arithmetical schema asserting that every eventually decreasing elementary recursive function has a limit. Some other related principles are also formulated. We establish their relationship with restricted parameter-free induction schemata. We also prove that the same principle, formulated as an inference rule, provides an axiomatization of the Σ2-consequences of IΣ1.Using these results we show that ILM is the logic of Π1-conservativity of any reasonable extension of parameter-free Π1-induction schema. This result, however, cannot be much improved: by adapting a (...) theorem of D. Zambella and G. Mints we show that the logic of Π1-conservativity of primitive recursive arithmetic properly extends ILM.In the third part of the paper we give an ordinal classification of -consequences of the standard fragments of Peano arithmetic in terms of reflection principles. This is interesting in view of the general program of ordinal analysis of theories, which in the most standard cases classifies Π-classes of sentences. (shrink)

Denić ( 2018, 2019, To appear ) observes that the availability of distributive inferences—for sentences with disjunction embedded in the scope of a universal quantifier—depends on the size of the domain quantified over as it relates to the number of disjuncts. Based on her observations, she argues that probabilistic considerations play a role in the computation of implicatures. In this paper we explore a different possibility. We argue for a modification of Denić’s generalization, and provide an explanation that is based (...) on intricate logical computations but is blind to probabilities. The explanation is based on the observation that when the domain size is no larger than the number of disjuncts, universal and existential alternatives are equivalent if distributive inferences are obtained. We argue that under such conditions a general ban on ‘fatal competition’ (Magri 2009a, b, Spector 2014 ) is activated, thereby predicting distributive inferences to be unavailable. (shrink)