Vasiliev's Imaginary Logic

Origin. Nikolai Aleksandrovich Vasiliev (1880–1940) taught at Kazan University, where Lobachevsky had built non-Euclidean geometry, and took that construction as his explicit model: as geometry had been freed from the parallel postulate, logic was to be freed from the law of non-contradiction. Three papers carry the system — "On particular judgments, the triangle of opposition, and the law of the excluded fourth" (1910), "Imaginary (non-Aristotelian) logic" (1912), and "Logic and metalogic" (1913). They predate Łukasiewicz's three-valued tables (1920) and Post (1921) and were arrived at independently. Written in Russian and unnoticed abroad, the work was recovered by V. A. Smirnov in the 1960s and brought into the paraconsistent literature by Ayda Arruda (1977) and later by Puga and da Costa (1988). Bazhanov's studies supply the biography and the textual history.

Models. There is no semantics in the modern sense; the construction proceeds by varying the categorical judgment. Vasiliev separates the empirical content of logic, which depends on the world one is reasoning about, from its formal content, which does not. The law of non-contradiction falls on the empirical side: it holds of our world because perception never delivers both that S is P and that S is not-P about the same object, and an imaginary world where it did would be governed by a logic with three qualities of judgment rather than two. What is not negotiable is the metalogical law of absolute difference between truth and falsity, which holds in every world and keeps the object-level contradiction from spreading.

Formalism.

Three judgment types: Affirmative: S is P. Negative: S is not-P. Indifferent: S is and is-not P. Third type added.

Indifferent judgment: Not contradiction. Objects that both have and lack property. Possible in imaginary world. Non-Aristotelian.

Accidental vs essential: Accidental: might not hold. Essential: must hold. Law of excluded middle for essential. Not for accidental.

Metalogical laws: Law of absolute difference between truth and falsity. Holds at metalevel. Object-level: can have overlap. Two-level structure.

Imaginary logic: Logic of worlds unlike ours. In our world: excluded middle holds. In imaginary: indifferent propositions. Non-Euclidean analogy.

Law of excluded fourth: S is P, or S is not-P, or S is both. Exhaustive but not exclusive. Three values, one true. Extension of excluded middle.

Generalization to n qualities: n = 2: Aristotelian, law of the excluded third. n = 3: imaginary, law of the excluded fourth. General: law of the excluded (n+1)st. The classical case is one member of a series.

Triangle of opposition: Square of opposition replaced. Vertices: affirmative, negative, indifferent. Particular judgments treated as accidental, not formal. Contrariety among three, not two.

Symbols.

Symbol	Unicode	Name	Meaning
S	U+0053	Latin capital S	Subject term of a categorical judgment
P	U+0050	Latin capital P	Predicate term
¬	U+00AC	Not sign	Negation; in Vasiliev, predicate negation forming not-P
∧	U+2227	Logical and	Used in the indifferent form, S is P and not-P
∨	U+2228	Logical or	Used in the law of the excluded fourth
n	U+006E	Latin small n	Number of judgment qualities; the law of the excluded (n+1)st
утвердительное	—	Utverditel'noe	Affirmative judgment: S is P
отрицательное	—	Otritsatel'noe	Negative judgment: S is not-P
безразличное	—	Bezrazlichnoe	Indifferent judgment: S is both P and not-P
воображаемая логика	—	Voobrazhaemaya logika	Imaginary logic: the logic of a world unlike ours
металогика	—	Metalogika	Metalogic: the world-invariant part, including absolute difference

Metatheory. Vasiliev left no formal language, no axioms and no calculus, so there is nothing here to be proved sound or complete; the claims are structural. The central one is that Aristotelian and imaginary logic differ only in the number of judgment qualities, so the syllogistic transforms systematically as that number varies, with the law of the excluded (n+1)st replacing excluded middle and the classical case recovered at n = 2. The second is the two-level architecture: because the metalogical law of absolute difference between truth and falsity is world-invariant, a proposition and its negation cannot both be true even in a world where an object both has and lacks a property, so the object-level contradiction is contained rather than propagated. Formal reconstructions came sixty years later, from Arruda (1977) and from Puga and da Costa (1988). Arruda's analysis locates the contradiction in predicate negation, inside the term structure of the categorical form, while the propositional layer stays classical — which is the sense in which imaginary logic anticipates paraconsistency without being a paraconsistent propositional logic.

Applies to. History of logic. Paraconsistency. Russian logic. Many-valued logic.

Limitations. The three judgment types are qualities of judgment, not truth values: nothing assigns them to sentences, nothing computes a compound from its parts, and there is no matrix, so the system does not connect to the many-valued tables it is said to anticipate. The indifferent judgment is asserted rather than given satisfaction conditions, and no model of an imaginary world is exhibited, so the analogy with Lobachevsky is programmatic — non-Euclidean geometry came with consistency proofs by interpretation, and this does not. The metalogical law of absolute difference confines revision to the object level, so the propositional layer is untouched and explosion is not blocked where it is normally formulated. The scope is syllogistic throughout: there is no account of the connectives, no quantification beyond the four categorical forms, and no inference beyond the figures.