Vasiliev's Imaginary Logic

Origin. Nikolai Vasiliev (1910). Russian logician. Non-Aristotelian logic. Restriction of the law of non-contradiction (excluded middle is retained, extended to the law of the excluded fourth). Foundation of paraconsistent thought.

Models. Third type of judgment beyond affirmation/negation. Indifferent propositions. Metalogic vs object logic. Imaginary worlds. Pre-Łukasiewicz many-valued.

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.

Historical significance: 1910: before Łukasiewicz (1920). Russian precedent. Largely unknown in West. Paraconsistent precursor.

Symbols.

Metatheory. Three judgments. Metalogic distinction. Imaginary worlds. Non-Aristotelian.

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

Limitations. Historical. Limited development. Isolation from West. Incomplete system.

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