This article examines one of the most significant trajectories in the reception of the existential philosophy of Lev Shestov in the USSR. His principal successor in the Soviet context was Yakov Druskin, who developed a distinctive practice of phenomenological reduction that extended Shestov’s notion of the “apotheosis of groundlessness” through the poeticization of experience. This approach was realized in practice by the Chinari, a circle associated with OBERIU. During the period of the Khrushchev Thaw, the works of the Chinari experienced a revival and gained renewed attention. At the same time, in Minsk, a group of samizdat poets emerged who may be understood as a case of intellectual convergence: independently and without direct influence, they rediscovered and rearticulated the phenomenological reduction developed by Druskin and the Chinari, but within the specific context of Belarusian culture. These authors are now referred to as the Minsk School of Poetry—non-Soviet poets of the Belarusian samizdat. This development is particularly significant given that, during the same period, official philosophy in the Belarusian SSR largely took the form of a colonial discourse shaped by intellectual centers in Moscow and Leningrad. Alongside this, an unofficial but widespread Soviet New Age milieu also influenced a number of Belarusian authors, particularly those engaged in pseudoscientific theorizing. Against this backdrop, the Minsk School of Poetry stands out as the non-Soviet and noncolonial form of philosophical expression in Soviet Belarus, articulating a version of existentialism closely aligned with that of Shestov and Druskin. The study of this school is therefore of considerable importance for the broader project of decolonizing the philosophical and literary legacy of the BSSR.

This paper proposes a novel approach to Stoic and Aristotelian-Epicurean modal reasoning by replacing traditional Kripkean relational semantics with a topological framework. In the Stoic account, causality and modality are associated with the density of a space, whereas in the Epicurean tradition they are grounded in discreteness and atomic structure. Motivated by this distinction, we argue that causality and modality can be interpreted through the topology of neighborhoods: the Stoic framework corresponds to a density-based notion of logical determinism, while the Aristotelian-Epicurean framework corresponds to a discreteness-based notion of logical contingency. This perspective allows modal operators to be defined via limit-point and interior structures, rather than Kripke's accessibility relations. Also, to capture Stoic logical reasoning, we introduce the notion of logemes, which we model geometrically as simplicial complexes. This representation reflects the flexible, non-linear, and competitive character of ancient inference rules. The proposed topological interpretation not only clarifies the internal structure of Stoic logic but also enables systematic comparison with Aristotelian and Epicurean modal frameworks. In addition, the Stoic interpretation of modality admits concrete applications in data analysis: in particular, it leads to the construction of modal filters that implement a deterministic, topology-based mechanism for noise rejection.

This article investigates the computational foundations of Immanuel Kant’s transcendental logic and its relevance to artificial intelligence and machine consciousness. We demonstrate that while Kantian analytic judgments map to decidable computations on standard deterministic Turing machines, synthetic a priori judgments require extended models, specifically, Turing machines equipped with oracles, to capture their non-algorithmic, self-positing nature. Building on Johann Gottlieb Fichte’s formulation of consciousness as a self-negating loop (“I = not-I”), we formalize a consistent syllogistic system that accommodates non-trivial self-reference. By interpreting this system topologically via spanning simplicial complexes, we show that reflexive reasoning remains both logically consistent and polynomial-time decidable. Furthermore, we introduce quantitative evaluation frameworks, including a Turing Simulation Test and a Reflexion Loop Consciousness Test, to assess whether large language models exhibit minimal conscious-like behavior, such as escaping semantic stagnation and integrating contextual grounding. Thus, we argue that transcendental logic constitutes a viable, computationally tractable framework for modeling self-aware, synthetic reasoning in artificial agents.

This paper proposes a phenomenological approach to philosophical logic by reexamining the overlooked yet deeply insightful position of Zaremba in his historical dispute with leading Polish logicians of the early 20th century. Rather than treating this debate as a closed chapter, we argue that Zaremba’s critique remains highly relevant for understanding the limits of formal logic and the need for a more structurally sensitive model of reasoning. His emphasis on the relational and contextual nature of logic challenges the formalist vision of mathematics as a system reducible to axioms. To address this challenge, we introduce a formal mathematical language designed to represent and analyze logemes – the minimal, context-dependent units of logical reasoning. This framework employs simplicial complexes as a topological structure to model how local fragments of reasoning interconnect. Each vertex represents a basic proposition, while higher-dimensional simplices capture the logical relations among them, forming coherent yet flexible substructures. This approach makes it possible to represent interactions among diverse logical systems (such as classical and intuitionistic logic) in a unified, geometrically intuitive manner. By modeling logic not as a unified system but as a dynamic network of interrelated fragments, this work offers a new foundation for philosophical logic – one that is capable of expressing ambiguity, incompleteness, and contextual variation, and that aligns more closely with the actual structure of mathematical and human reasoning. This approach may have a wide and rich range of practical applications in artificial intelligence. It provides a universal representation of databases in the form of logemes, as well as universal models for non-monotonic reasoning. Together, these features make it possible to model key phenomena of consciousness, such as context sensitivity, revision of beliefs, and circularity. By offering a mathematically rigorous framework for non-monotonicity, the approach opens new possibilities for integrating logical, topological, and cognitive structures in AI systems, potentially contributing to more flexible, adaptive, and cognitively plausible models of artificial intelligence.

This paper proposes a geometric framework for representing non-monotonic relationships using simplicial complexes. We introduce the notion of a logeme, defined as a fragment of a logical system represented by a simplicial complex derived from a partially ordered structure of semantic relations. Unlike graphs, which encode only pairwise relations, simplicial complexes allow the representation of higher-order dependencies and hierarchical interactions among propositions. This makes them suitable for modeling complex logical fragments that arise in natural language reasoning, knowledge representation, and defeasible inference systems.

We model the fungal mycelium as a time-dependent simplicial complex. We prove that this representation is unique up to homeomorphism, establishing topology as a faithful invariant of mycelial architecture. Within this framework, logical inference emerges as cytoplasmic flow, when derivations correspond to directed paths, yielding monotonic reasoning in tree-like regions and non-monotonic, defeasible inference in loopy zones. According to this interpretation, any consistent system encoding arithmetic and exhibiting logical completeness cannot collapse to a discrete set of points but must contain non-bounding 1-cycles. Consequently, such systems inherently support undecidable propositions and non-monotonic dynamics.

In this paper, we introduce the concept of logemes, a novel framework for reasoning that bridges nontrivial fragments of formal logic and topological spaces. Logemes are defined as logic diagrams that capture logical relationships relevant to specific datasets or problems. Unlike standard symbolic logic, which emphasizes completeness and formal validation within an algebraic framework, logemes focus on pragmatic, nontrivial subsets of inference rules tailored to real-world reasoning tasks. This approach builds on historical tools like logic diagrams (e.g. the square of opposition) while leveraging the combinatorial and topological richness of simplicial complexes. The paper explores the interplay between syntax and semantics through logemes, using topology to compare logical fragments across different systems. By focusing on structural properties rather than syntactic or semantic details, logemes reveal deeper similarities and differences between reasoning patterns, even when derived from distinct logical frameworks. The narrative alternates between bottom-up and top-down approaches: starting with basic components to build complex systems and simplifying complete systems into manageable fragments. Finally, logemes are connected to homotopy types, positioning them as a bridge between reasoning and topology. This unified framework offers a modular, flexible tool for studying logical fragments. By grounding reasoning in the topology of simplicial complexes, logemes provide a novel paradigm for analysing datasets and logical systems.

This paper explores the relationship between legal argumentation and totalitarianism by examining the legal systems of Nazi Germany and the Soviet Union. It critically assesses Gustav Radbruch’s thesis which suggests that Nazi Germany adhered to a system of legal positivism, asserting that laws were governed by logical rules of inference. The paper challenges this view, arguing that legal theorists in Nazi Germany, including Carl Schmitt and others, rejected legal argumentation in favor of party loyalty and ideological conformity. Similarly, Soviet law under Marxism–Leninism substituted legal reasoning with political conviction. Both systems, though differing in ideological content, shared a disregard for legal argumentation, resulting in unjust laws that served the interests of the ruling party. The paper also distinguishes between totalitarian and authoritarian regimes, showing how, in totalitarian societies, legal norms are subordinated to political ideology, leading to a collapse of objective legal reasoning. In authoritarian systems, laws may imitate democratic principles but fail in practice due to a breakdown in the logical consistency of legal argumentation. This emphasizes that legal systems devoid of argumentation undermine justice and legal certainty, regardless of whether they exist under totalitarian or authoritarian regimes.

The paper substantiates the prospects of normativism in law, namely the possibility of using purely logical means to make a judgment as a logical conclusion. The main criticism of normativism is based on the possibility of conflicts of judicial decisions and especially on conflicts of the norms themselves, when a court under similar circumstances can make opposing decisions which are formally valid. Critics of normativism argue that logic is helpless in resolving the conflicts and that higher justice must be guided by the discretion of judges who share common values or a common ideology, especially in the case of totalitarian societies. However, we show that there are two types of logical argumentation in the elimination of these conflicts: (a) Aristotelian, when a more general basis is sought, from which only one of the contradictions follows; (b) non-Aristotelian, when a compromise is reached between two contrary statements. In the second case, this technique is used mainly for civil cases. Moreover, this technique is quite formal and does not require an appeal to paraconsistent or other non-classical logics for eliminating the conflict. Using two recent court cases, we demonstrate how the proposed two types of logical argumentation work.

This paper explores the origins and global significance of Judaic hermeneutics as a foundational logical culture, arguing that it constitutes one of the earliest and most sophisticated systems of reasoning in human history. Far beyond a method of religious interpretation, Rabbinic hermeneutics represents a logic in practice: a structured, culturally embedded framework of inference rules (middôt), such as qal wāḥōmer (a fortiori reasoning), that guided legal deliberation and textual exegesis. By comparing Judaic hermeneutic methods with Greco-Roman rhetoric, Indian logic, and Chinese philosophy, this study reveals that similar logemes—elementary reasoning units—appear only in these four ancient traditions. All emerged within a narrow geographic corridor (32–38° N latitude) historically linked by trade routes, particularly the Silk Road. Drawing on legal documents and logic history, this paper argues that logical cultures did not arise from isolated individuals, but from collective intellectual traditions among elites engaged in commerce, law, and education. Judaic hermeneutics, with its roots in Babylonian legal traditions and its codification in the Talmud, offers a clear example of logic as a communal, evolving practice. This study thus reframes the history of logic as a pluralistic, global phenomenon shaped by cultural, economic, and institutional contexts.

Mould or fungal propagation can serve as a universal model for a massive-parallel game of many players, moving concurrently. Based on this model, a game of politicians is introduced to occupy as many political keywords as possible in the media. If a politician begins to be mentioned every time with some word or group of words, then this means that he or she has occupied this word or group of words. In the election campaign, the winner is the politician who has occupied the most words. The study was carried out on the example of Polish politicians. In addition, a space for keywords related to mentions in the messages of 780 politicians of the Sejm and Senate of the last two terms of office was created. Thus, the reflection game of politicians for keywords was implemented. In this game, you can explain the role of politicians in their activities for their party or for themselves. The period of the game is for three years. This game shows the dynamics of political discourse in Poland during this period. In different media, this game is realized in different ways. For example, party-related keywords in the media correlate differently with party program texts. This means that journalists from various portals influence this game of politicians. Data modeling as a game of politicians for keywords makes it possible to identify hidden players and their degree of reflexivity.

The interview given by Gershon Trestman (born July 29, 1947, Minsk), a Russian-language Belarusian and Israeli poet, prose writer, publicist, and playwright. He is a member of the Union of Writers of Israel, the Commonwealth of Russian-Speaking Writers of Israel “Stolitsa,” and the International Federation of Russian Writers. His work has been recognized with the Yu. Stern and Yu. Nagibin awards, as well as a gold medal for “outstanding achievements in literature and the arts” from the California Academy of Sciences. Selected works: The One Who Crossed the River (Tel Aviv, 1996); Golem, or Faust’s Curse (Moscow, 2007); A Small Country with a Great History (Israel, 2008, foreword by Avigdor Lieberman); The Great History of a Small Country (Israel, 2011); The Scroll of Esther (Jerusalem, 2013); The Land of Olive Guardians (Jerusalem, 2013); The Israeli Knot: The History of the Country – The History of Confrontation (Book-Sefer, 2014); The Land of Olive Guardians (Jerusalem, KKL-JNF, 2014); Job (Minsk, New Wineskins, 2014); ... Where There Are No Coordinates. Poems and Epics (Jerusalem, 2017); The Book of Non-Being (Minsk, Logvinov, 2019); Alphabet for Elderly Children (Jerusalem, 2023).

We have developed a theory that allows us to accurately characterize information waves in social networks such as Twitter. This theory is an extrapolation of a mathematical theory that studies particle drifts in the acoustic field. We have noticed that each post on Twitter can be considered as a particle, and its radius is the number of followers of the author of the post. Then the number of likes and retweets of this entry can be considered as a drift of this particle. We have found that our hypothesis is correct, and in this interpretation, Twitter posts behave like particles in the acoustic field. In particular, if the author of the entry has no more than 15,000 followers, then this behavior is characterized as a radiative drift in the acoustic field, and if more than 15,000, then as a viscous drift. Based on this behavior of particles (posts on Twitter), we can mathematically reconstruct an appropriate information wave.

In the article introducing this special issue, we consider the prospects of cultural diffusionism. We show that diffusion is not a uniform phenomenon since it includes direct, partial, multi-layered, and reverse forms. The complex approach to diverse forms of diffusions is called by us the crossroads concept. It aligns with cultural relativism which examines cultural traits through diffusion and modification. In world-systems analysis, cultural diffusion is analyzed within the world-economy framework, rooted in the classical Marxism view of economic systems as foundational with culture as a superstructure. Neo-Marxism and dependency theory highlight a division between developed and developing countries, positing that cultural influences often flow from the center to the periphery. However, this view oversimplifies cultural diffusion’s complexities, as we demonstrate. Moreover, cultural diffusion often precedes trade route establishment, with religious diffusion frequently facilitating subsequent trade communications. This special issue, edited by us, explores cultural diffusion and its varied forms, challenging the notion that it fits neatly into a center-to-periphery movement.

In the article, we aim to understand the responses of living organisms, exemplified by mycelium, to external stimuli through the lens of a Turing machine with an oracle (oTM). To facilitate our exploration, we show that a variant of an oTM is a cellular automaton with an oracle, which aptly captures the intricate behaviours observed in organisms such as fungi, shedding light on their dynamic interactions with their environment. This interaction reveals forms of reflection that can be interpreted as a minimum volume of consciousness. Thus, in our study, we interpret consciousness as a mathematical phenomenon when an arithmetic function is arbitrarily modified. We call these modifications the hybridization of behaviour. oTMs are the mathematical language of this hybridization.

The first logical treatise in Chinese was written by Mozi and entitled the Dialectical Chapters (Mòbiàn). In this work we cannot detect a single logical system, but it contains many valuable logemes (logical fragments that are consistent). In particular, we find a non-Aristotelian square of opposition in which the existential quantifier is understood in the narrow sense as some and only some. We remember that Aristotle treats the existential quantifier in a broad sense as some, or perhaps all. Mozi was the first logician who proposed considering logic not as a single system, but as a set of different logemes, realized on some partially ordered sets (posets). This is a semantic approach to logic, which was then further developed in the Chinese school of names. In this paper, I logically formalize this approach, proposing to use simplicial complexes from topology as logical diagrams. The point is that these complexes uniquely represent finite posets up to homeomorphisms. Therefore, within the framework of simplicial complexes we can consider different logemes realized on different finite posets at once.

We uncover certain universal features of Turing machines (TM) as operating in a perpetually changing environment which can have sudden and highly random influence on TMs themselves. TM adapts and assimilates the changed environment and performs its computational functioning in new conditions. We show that this transcends essentially Turing computability relative to a ground model of ZFC. We distil the formal counterparts responsible for the adaptation and assimilation of TM and propose they may underlie the conscious behaviour of general systems including living creatures. However, in this last case more work leading to the layer’s structure of TMs is needed. We also make an attempt toward social TMs by finding the way how TMs can group and cooperate.

In the article, we studied the continuous problem for multi-agent pathfinding. We show continuity of the path-traversing time functional and the existence of the optimal path for a single agent. Also, we consider game theory interpretation for multi-agent pathfinding with continuous routes as a game with strategies in a Banach space. Finally, we briefly discuss near-optimal routes and connection of heuristics for pathfinding and integral geometry problems.