Abstract
This article systematically examines the ideological content and contemporary assessments of "imaginary logic," proposed by the Russian logician Nikolai Vasiliev (1880–1940) in 1910, and explores its potential value for consciousness studies and theories of the generation of subjective reality. The work first describes the key constructs of imaginary logic—the diagnosis of the "world-dependence of logical laws," the introduction of "indifferent" judgments, and the distinction between levels of "metalogic." Then, through the history of its reception by contemporary logicians, it reveals Vasiliev's status as a precursor of paraconsistent logic, many-valued logic, and "logical pluralism," while also presenting internal criticism of his ideas. On this basis, the article turns to the context of consciousness studies, analyzing how paraconsistent logic can provide a formal foundation for the model of an "inconsistent mind." Finally, it attempts a dialogue between imaginary logic and Ramil Garifullin's theory of the brain's "existence" (existom), arguing that if subjective reality, as the existom theory claims, is holographic and generated by "existential lenses" and "virtuons" in intentional flows, then imaginary logic may be able to provide a key logical instrument for the formal description of this generative process.
Keywords: Vasiliev; imaginary logic; paraconsistent logic; consciousness; subjective reality; existom; metalogic
1. Introduction: A Logician Delayed by a Century
In the chronicle of logic, there are thoughts that outpace their time so greatly that their authors do not live to witness their understanding and acceptance. Nikolai Alexandrovich Vasiliev is one such thinker. As a professor of logic, psychology, and literature at Kazan University, he first systematically expounded the concept of "imaginary logic" (воображаемая логика) in a lecture on May 18, 1910, and then in 1912 published the programmatic article "Imaginary (Non-Aristotelian) Logic." Vasiliev was clearly aware of the structural analogy between his work and Nikolai Lobachevsky's creation of non-Euclidean geometry—both bore the name "imaginary," both opened a new theoretical space by suspending a seemingly unshakeable axiom.
However, unlike Lobachevsky, Vasiliev's ideas were almost completely ignored during his lifetime. In the 1920s he withdrew from academic activity due to mental illness and died in 1940 in a psychiatric hospital. His works lay in oblivion for almost half a century, until they were rediscovered in the 1960s and systematically published in the 1980s–1990s by V.A. Bazhanov and others. Today Vasiliev is recognized as one of the most important precursors of modern non-classical logic—above all paraconsistent logic and many-valued logic. In 2024–2025, the second expanded Russian edition of his "Imaginary Logic: Selected Works," edited by Bazhanov and including previously unpublished manuscripts and lifetime reviews, marks the continuing attention of the international logical community to this Kazan thinker.
This article attempts to accomplish a dual task: first, to conduct a systematic literary review of the key constructs of imaginary logic and its contemporary reception; second, to introduce this logical heritage into the contemporary discussion of consciousness studies and theories of the generation of subjective reality, paying special attention to its potential resonance with Garifullin's theory of the brain's "existom." This work is necessary not only because Vasiliev's academic style, closely intertwining logic with psychology and philosophy, makes such an interdisciplinary dialogue possible, but also because contemporary consciousness studies face a fundamental logical dilemma: if consciousness by its very nature cannot be fully encompassed by any consistent formal system, then a logic that permits "local contradictions" without the collapse of the system may be precisely what we need.
2. The Key Constructs of Imaginary Logic
2.1 The "World-Dependence of Logical Laws"
The starting point of Vasiliev's imaginary logic is not a simple negation of Aristotelian logic, but a kind of "diagnosis": those laws that in Aristotelian logic are considered absolutely necessary—above all the law of contradiction and the law of excluded middle—are in fact empirical generalizations, rooted in the structure of "our world" and the properties of human sensory perception. In other words, the law of contradiction is not a formal condition of thinking itself, but an induction about the co-existence of objects in a particular world (that is, in the world in which we live and which we apprehend through limited sense organs).
Vasiliev here employs the Kantian framework of distinction, albeit in a critical manner. He holds that Aristotelian logic conflates the "empirical" (material) and "formal" levels. The formal part of logic—which he calls "metalogic" (металогика)—should contain only the minimal set of laws necessary for reasoning in any "imaginary world": the law of identity and the law of absolute non-self-contradiction (that is, the absolute distinction between truth and error). As noted in one work analyzing the epistemology of Vasiliev's logic: "Metalogic is the logic applicable to every world, however peculiar the structure of the objects of that world may be." And the law of contradiction and the law of excluded middle belong to the mutable, empirical level: in a world with a different "material" organization, an object may well simultaneously possess and not possess a certain property.
2.2 "Indifferent" Judgments and the "Law of Excluded Fourth"
The most original construct of imaginary logic is the expansion of the traditional classification of judgments. In the Aristotelian "square of opposition," judgments are divided into affirmative and negative. Vasiliev introduces a third type of judgment: indifferent judgments (индифферентные суждения), whose logical form is "S is and is not P." This is not merely the addition of a "third value" to judgment—it means that negation in imaginary logic is not atomic: a negative judgment is not equivalent to the assertion of the falsity of the corresponding affirmative judgment, but possesses an independent, primary status.
Accordingly, Vasiliev formulates the "law of excluded fourth" (закон исключённого четвёртого): in a logical space containing indifferent judgments, besides truth and falsehood there exists a third cognitive state—a state in which "contradiction is grasped as existing." However, Vasiliev here preserves a fundamental limitation: even in imaginary logic, the metalogical level of distinguishing truth and falsehood remains operative. The subject can grasp a contradiction, but cannot simultaneously assert its existence and non-existence—otherwise the entire system of reasoning would lose its operability. As Vasiliev himself emphasized: "Whoever ceases to distinguish truth from error ceases to think logically."
2.3 From "Imaginary" to "Paraconsistent": The Genealogy of a Concept
Vasiliev himself did not use the term "paraconsistent"—that term was introduced by the Peruvian logician Francisco Miró Quesada in 1976. But he constructed precisely such a system: in it, from a contradiction no arbitrary assertion is derived (that is, the "principle of explosion" does not hold), so a local contradiction does not destroy the entire system of knowledge. Vasiliev is therefore rightly considered the spiritual ancestor of paraconsistent logic. The Brazilian logician Ayda Arruda, in her 1979 article "On Vasiliev's Imaginary Logic," directly asserted: "We hold that any formalization of Vasiliev's imaginary logic will lead to a paraconsistent logic. Whether it is at the same time a many-valued logic is merely a matter of details or interpretation."
The work of Graham Priest, the leading contemporary representative of paraconsistent logic, also confirms Vasiliev's pioneering status. Priest's "dialetheism"—that is, the position that there exist propositions that are simultaneously true and false—intuitively resonates deeply with Vasiliev's "indifferent judgments." Priest himself participated in several collections devoted to Vasiliev's ideas, which testifies that the contemporary logical community regards imaginary logic as one of the indispensable sources of the paraconsistent tradition.
3. Contemporary Assessments: Reception, Criticism, and an Unfinished Project
3.1 The Establishment of Pioneering Status
In an article marking the centenary of imaginary logic, Bazhanov summarized Vasiliev's contribution as "the role of Lobachevsky in logic." This assessment has gained broad recognition in contemporary historiography of logic. Vasiliev's priority manifests in several dimensions: he was the first to explicitly formulate "logical pluralism"—that is, the idea that there exist multiple logically valid systems whose validity depends on the presupposed structure of the "world"; he was also the first to realize this idea through the actual construction of a semi-formal system.
Markin (V.I. Markin) and Zaitsev (D.V. Zaitsev) reformalized Vasiliev's "imaginary logic-2" as an intensional syllogistic theory, in which the terms of categorical judgments are associated with "sets of features" rather than "sets of individuals." In this reconstruction, imaginary logic appears in a guise substantially different from standard first-order logic: its semantics is feature-oriented rather than object-oriented. In 2025, Konkova (A.V. Konkova) and Markin proposed a "logic of existence judgments of imaginary logic" (logic ILΥ), embedding Vasiliev's indifferent judgments into a semantic framework with three extensional characteristics: not only the "volume" and "anti-volume" of a term, but also the "contradictory region" relative to that term. This formalization work transforms Vasiliev's system from a historical document into an object accessible to analysis by contemporary logical instruments.
3.2 Internal Criticism and Unresolved Problems
However, Vasiliev's project is not without difficulties. In one representative critical article it is pointed out that Vasiliev's use of the analogy with Lobachevsky contains a categorial error: in non-Euclidean geometry, the negation of the fifth postulate does not change the rules of deductive inference themselves, but only changes the set of axioms; but in imaginary logic Vasiliev attempts to "suspend" precisely the rules of inference (the application of the law of contradiction in reasoning). Critics hold that Vasiliev did not sufficiently justify whether the simultaneous rejection of the law of contradiction and the preservation at the metalogical level of the "law of absolute difference" is truly coherent.
A more fundamental difficulty consists in the fact that Vasiliev never completed the formalization stage of his project. Bazhanov notes that Vasiliev, although broadly acquainted with the development of mathematical logic, in fact did not apply its methods to construct imaginary logic. Therefore imaginary logic remained to a significant extent at the stage of a "semi-formalized" program—it gave an intuitive description of the system and part of the rules, but not a complete axiomatization and proofs of metatheorems. This incompleteness is simultaneously both the charm of Vasiliev's thought (it preserves a rich space of interpretations) and one of the reasons for its long neglect.
It is worth noting that in contemporary interpretations of the relationship between Vasiliev and Aristotle there also exist disagreements. Some scholars attempt to interpret Vasiliev's "judgments of contingency" as "possible belonging" in the Aristotelian modal doctrine, thereby arguing that many ideas of imaginary logic "are perhaps not so revolutionary." Such an interpretation, although it reduces Vasiliev's originality, on the other hand shows: imaginary logic did not arise out of nothing, but is deeply rooted in the soil of traditional logic.
4. The Paraconsistent Turn in Consciousness Studies
4.1 The "Inconsistent Mind" Model and the Holographic Paradigm
If one places the heritage of imaginary logic in the context of consciousness studies, the most direct connection is found in the so-called model of the "inconsistent mind" (inconsistent mind). The key idea of this model: the global inconsistency of the mind does not necessarily lead to cognitive collapse, since the mind consists of many local modules, each of which may be internally consistent, while the interfaces between modules may generate contradictions. As long as the system is in a paraconsistent logical regime, these local contradictions do not "explode" into globally arbitrary reasoning.
In recent years a more radical theory of consciousness has begun to emerge: it defines consciousness as the capacity of a system to detect, retain, and integrate internal paradoxes, rather than to eliminate them. This theory explicitly adopts paraconsistent logic as its formal foundation. It forms an interesting consonance with David Bohm's "implicate order" and Karl Pribram's holographic model of the brain. Pribram proposed that the act of consciousness is generated not by the transmission of nerve impulses along axonal-synaptic circuits, but by microprocesses in the dendritic tree—that is, interference patterns of electrical oscillations in a network of fine fibers. In such holographic information storage, each local region contains information about the whole. As noted in one analysis: "The holographic object operates on the principle of 'part is whole'… any small part of the hologram contains information about the real object, and therefore can restore it." If this holographic paradigm is correct, then the "contradiction" of the mind manifests not as the negation of one module by another, but as a phase confrontation between different interference patterns—and this confrontation can be processed by a strategy of "localization," analogous to paraconsistent logic, without the need for a global revaluation of truth values.
4.2 The Potential Impact on Penrose's Argument
The most radical significance of paraconsistent logic for consciousness studies is perhaps manifested in its challenge to Roger Penrose's argument about the "non-computability of consciousness." Penrose's key argument relies on Gödel's theorem: if consciousness is identical to some algorithmic system, then the formal consistency of this system will not allow it to "see" the truth of its own Gödel sentence. Consequently, consciousness must transcend any consistent formal system.
However, Penrose's argument is valid only for consistent formal systems. In paraconsistent systems (such as Priest's LP), the proof of Gödel's theorem does not go through, since the system does not satisfy the consistency conditions on which the proof is based. If the mind can be modeled as a paraconsistent system—that is, a system in which contradictions are locally tolerated rather than globally detonated—then Penrose's Gödelian argument loses its support. Consciousness does not necessarily "transcend" a formal system; it may simply function in a different logical regime. This line of thought forms an interesting consonance with Vasiliev's concept of "metalogic": the "law of absolute difference" at the metalogical level preserves the operability of distinguishing truth and falsehood, while tolerance of contradictions at the object level allows the mind to function continuously under conditions of conflicting information.
5. The Intersection of the Existom Theory and Imaginary Logic
5.1 The Foundations of Garifullin's Theory of the Brain's "Existom"
The theory of the brain's "existom" (экзистом), proposed in recent years by Ramil Garifullin (Kazan Federal University), offers an original ontological framework for the generation of consciousness and subjective reality. Its key concepts include:
· Existential lenses (existlens): multilayer biological diffraction structures-codings, which are the basis for the formation of standing wave interference patterns—"virtuons" (virtuons); virtuons correspond to innate meanings and concepts formed in ontogenesis.
· Existom: the totality of all existential lenses, subject to the holographic principle—each existential lens preserves all the information stored in the entire existom.
· Intentional flows (intentional flows): wave flows generated by the thalamus, hippocampus, and other structures, directionally acting on the brain's existom and exciting dynamic patterns of virtuons.
Garifullin explicitly contrasts this theory with neural network models. In his view, "the neural network (connectome) is not the basis and carrier of our subjective reality and mental content. The carrier is the existom." This means that the generation of subjective reality follows the logic of wave interference, not the logic of synaptic transmission. As he describes it: "The brain's existom can be regarded as a totality of certain complex and multilayer 'strings' (existential lenses), some of which begin to 'sound' only under the resonant influence of vibrations emitted by the brain's intentions."
5.2 Imaginary Logic as the "Metalogic" of Existom Generation
If Garifullin's theory is correct—subjective reality is generated by the interference patterns of existential lenses, and these patterns are dynamically activated and recombined by intentional flows—then a fundamental question arises: what is the logical structure of this generative process?
Here imaginary logic offers a possible answer. Consider the following correspondences:
First, multiple "worlds" and multiple existential lenses. Vasiliev's key insight: the applicability of logical laws depends on the structure of the "world." In the existom theory, each existential lens can be regarded as a microscopic "world": it possesses its own principles of organizing meaning, its own distribution of "innate concepts" and "concepts formed in experience." When an intentional flow simultaneously activates several existential lenses, the resulting virtuon patterns may carry mutually conflicting meanings—one virtuon may mark "danger" in lens A and "opportunity" in lens B. This is precisely the neuro-phenomenological correlate of Vasiliev's "indifferent judgment": meaning simultaneously exists and does not exist in the given category.
Second, metalogic as the formal expression of the holographic principle. Vasiliev's "metalogic" contains only the minimal laws necessary for reasoning in any world. In the context of the existom theory, this "metalogic" corresponds to the holographic principle itself: regardless of which existential lens is activated, regardless of how the virtuon patterns change, the accessibility of holistic information in each local part remains unchanged. The "law of identity" of metalogic corresponds to the self-identity of the existential lens—that is, each existential lens in any state of activation preserves its structural integrity as a "lens." The "law of absolute difference" of metalogic corresponds to the distinguishability of virtuon patterns—even if meanings are contradictory, different interference patterns can still be distinguished in the existom.
Third, paraconsistent stability and the integration of paradoxes. The most complex part of Garifullin's theory is probably the situation when an intentional flow simultaneously activates incompatible existential lenses. If the existom is a consistent system, such simultaneous activation will lead to a "holographic explosion"—all meaning patterns collapse into indistinguishable chaos. But if the existom is a paraconsistent system—that is, it is capable of localizing contradictions, so that conflicting interference patterns preserve coherence in different "regions" of the existom—then subjective reality can be continuously generated in the tension of contradictions without losing its structural character.
The concept of the "integration of paradoxes" in contemporary theories of consciousness receives a concrete explanation in the existom theory: when the interference between existential lenses generates a destructive phase confrontation, the intentional flow does not simply "resolve" the conflict, but reorganizes the activation pattern of the existom, so that the conflicting virtuons are redistributed into a new interference configuration, generating a new subjective state that is coherent at a higher level. This is precisely what Vasiliev presupposed in imaginary logic: contradiction does not destroy the logical space, but expands it—through the introduction of the "indifferent judgment," the logical space expands from two-dimensional (truth/falsehood) to three-dimensional (truth/falsehood/indifference), thereby accommodating initially inexpressible meanings.
5.3 A Tentative Integrative Thesis
On the basis of the foregoing analysis, the following tentative integrative thesis can be formulated:
The generation of subjective reality, at its formal level, can be described as a paraconsistent, many-worlds logical process; Vasiliev's imaginary logic—especially the distinction between "metalogic" and "indifferent judgment"—provides the earliest conceptual framework for this process; and Garifullin's existom theory provides a concrete neuro-holographic mechanism for the "material" level of this framework.
The testable consequence of this thesis is as follows: if the existom indeed functions in a paraconsistent manner, then under experimental conditions in which an intentional flow simultaneously activates conflicting existential lenses, one should observe the phenomenon of "coexistence of contradictions" in subjective reports (for example, the simultaneous sensation of something as "familiar and unfamiliar," "safe and dangerous"), and one should not observe cognitive collapse or loss of meaning. This prediction agrees in direction with the prediction of contemporary paraconsistent theories of consciousness about a "temporary increase in signal during the period of paradox integration."
6. Conclusion: A Logic Not Yet Fully Written
Vasiliev's imaginary logic was born in 1910—almost sixty years before the official birth of paraconsistent logic. Its key insights—the empirical character of logical laws, the local tolerance of contradictions, and the preservation of the "law of absolute difference" at the metalogical level—today appear not only significant for the history of logic, but also as providing a unique analytical instrument for consciousness studies. The publication in 2024–2025 of the second expanded Russian edition of Vasiliev's works, as well as the continuing attention to his ideas on the part of Brazilian, Russian, and world logicians, testifies that this "century-delayed" logic is acquiring its deserved academic status.
This article has attempted to show that between imaginary logic and Garifullin's existom theory there exists a structural kinship: both refuse to reduce consciousness/subjective reality to the operation of two-valued logic, both seek the driving force of the generation of consciousness in the coexistence of a "multiplicity of worlds," both regard "contradiction" as a phenomenon that must be integrated rather than eliminated. Of course, this kinship remains at the stage of tentative consideration. The incompleteness of imaginary logic—the absence of its full formalization—is simultaneously both a difficulty and an invitation: it invites contemporary logicians and consciousness researchers to jointly complete a project begun a century ago—to build a possible bridge between the rigor of formal science and the richness of subjective experience.
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Ramil Garifullin, Associate Professor of the Department of Cognitive Science and Artificial Intelligence at Kazan Federal University
