Imaginary Logic Research Articles

Intensional Semantics for Syllogistics: what Leibniz and Vasiliev Have in Common

This article deals with an alternative interpretation of syllogistics, different from the classical (extensional) one: an intensional one, in which subject and predicate are not associated with a set of individuals (the extension of the concept) but a set of attributes (the content of the concept). The authors of the paper draw attention to the fact that this approach was first proposed by Leibniz in works on logical calculus, which for a long time remained in the shadow of his other philosophical works. Currently, the intensional approach is gaining more and more popularity due to the development of non-classical logics, and the article will present several existing intensional formal syllogistic semantics. The paper will also consider another historical approach to syllogistics, associated with the name of the Russian logician Nikolai Vasiliev, who is not only one of the founders of non-classical (non-Aristotelian logic) but also of a different intensional interpretation of such logic. The authors, along with the already known formalizations of Vasiliev’s ideas, present two new systems. One of them is a reconstruction of one type of imaginary logic with statements of three qualities: affirmative and two types of negative statements (with absolute and ordinary negation). The second system is the one that is adequate to semantics, in which instead of the four classical ones, only three types of statements are presented (two particular statements are replaced by one - accidental), and their significance is determined through the relation of the classical logical entailment. Both of them are interpreted intensionally. The intensional approach in logic and, in particular, in syllogistics allows us to expand the class of accepted principles (which occurs due to the expansion of the class of correct moods of syllogisms).

Воображаемая логика – 2 Н.А. Васильева как силлогистическая теория.

Воображаемая (неаристотелева) логика Н.А. Васильева, одного из основоположников современной неклассической логики, является его самой известной логической системой. Дедуктивная система, преложенная им, представляет собой силлогистику особого типа, в языке которой вместе с формами утвердительных и отрицательных суждений содержатся формы противоречивых (индифферентных) суждений. Последние представляют собой суждения нового качества и содержат связку <<есть и не есть одновременно>>. Статья посвящена изложению результатов исследования одного из вариантов Воображаемой логики Николая Александровича Васильева. Этот вариант Воображаемой логики отличается не только от традиционной силлогистики, но и от основной версии. Здесь с каждым термином категорического суждения связывается не множество индивидов, а совокупность признаков, а силлогистические константы рассматриваются как обозначающие интенсиональные отношения между понятиями. Исследование предполагало рассмотрение этого варианта Воображаемой логики как силлогистической теории и производилось на базе системы, построенной для данной логики В.И. Маркиным и Д.В. Зайцевым. В статье кратко изложена система Воображаемой логики – 2, доказаны: законы тождества, законы противоположностей, законы подчинения, законы обращения и законы исключённого четвёртого, принимаемые в этой логике. Далее последовательно рассмотрены все четыре фигуры силлогизмов. В каждой фигуре рассмотрены все возможные комбинации посылок (36 комбинаций, для каждой фигуры) со всеми возможными заключениями (шесть возможный заключений, для каждой комбинации посылок). Произведено доказательство правильных силлогизмов каждой фигуры, а также приведены примеры возможных контрмоделей для опровержения всех неправильных модусов. Результатом работы является выделение законов данной силлогистической теории, а также намечено дальнейшее направление исследования, ставится вопрос о возможности формулирования общих правил силлогизма данного варианта Воображаемой логики.

N. A. Vasil’ev’s Logic and the Problem of Future Random Events

The solution of the problem of the future random events truth is considered in Vasil’ev’s logic. N. A. Vasil’ev graded the logic according to two levels—the level of facts, i.e. time fixed events, and the level of notions or rules, governing these facts. The mathematical construction previously suggested for imaginary Vasil’ev’s logic, extends to the early variant of his logic—a logic of notions. In the paper, we investigate the meaning of problematic and uncertain assertions introduced by Vasil’ev. As a result, we developed a model of Vasil’ev’s logic of facts that resolves also the truth problem of future random events. The imaginary logic has also been extended to the level of notions, and the law of the excluded eighth is gotten in it. The correspondence between Vasil’ev’s terms “some” and “all” and modern quantifiers is discussed.

Nicolai Vasiliev’s Imaginary Logic and Semantic Foundations for the Logic of Assent

Le philosophe russe Nicolai Vasiliev est connu en tant que précurseur des logiques essentiellement non-classiques, c'est-à-dire de logiques qui diffèrent de la logique classique par l'abandon de principes qui sont corrects en logique classique. La gamme de telles logiques couvre la logique intuitionniste, la logique plurivalente, la logique paraconsistante et les logiques de la pertinence. Dans la première partie de ce texte, j'analyse brièvement les vues de Vasiliev, à savoir sa « logique imaginaire », qu'il présente comme une nouvelle logique non-aristotélicienne. Dans les sections suivantes je discute l'impact des approches de Vasiliev sur la logique du consentement. Ici, les idées essentiellement non-classiques de Vasiliev seront reliées à des logiques non- essentiellement non-classiques, qui ne constituent pas une révision de la logique classique, mais étendent sa puissance expressive.

What trends in non-classical logic were anticipated by Nikolai Vasiliev?

In this paper we discuss a question about the trends in non-classical logic that were exactly anticipated by Niko- lai Vasiliev. We show the influence of Vasiliev’s Imaginary logic on paraconsistent logic. Metatheoretical relations between Vasiliev’s logical systems and many-valued predicate logics are established. We also make clear that Vasiliev has developed a sketch of original system of intensional logic and expressed certain ideas of modal and temporal logics.

The dawn of paraconsistency: Russia's logical thought in the turn of XX century

The paper deals with the factors which enabled N. A. Vasiliev to put forward in 1910 - 12 the idea of logics free of the laws of contradiction and excluded middle, the idea of metalogic and to construct his imaginary logic as novel non-classical system. It is shown that background of Vasiliev's ideas lies deeply in Russia's culture and particular approach to logical discourse. Several Russian scholars expressed ideas similar to Vasiliev's though not in such explicit form. This period might be called the prehistory of paraconsistency. Real history of paraconsistency starts with N.C.A. da Costa's works.

Non-Classical Stems from Classical: N. A. Vasiliev’s Approach to Logic and his Reassessment of the Square of Opposition

In the XIXth century there was a persistent opposition to Aristotelian logic. Nicolai A. Vasiliev (1880–1940) noted this opposition and stressed that the way for the novel – non-Aristotelian – logic was already paved. He made an attempt to construct non-Aristotelian logic (1910) within, so to speak, the form (but not in the spirit) of the Aristotelian paradigm (mode of reasoning). What reasons forced him to reassess the status of particular propositions and to replace the square of opposition by the triangle of opposition? What arguments did Vasiliev use for the introduction of new classes of propositions and statement of existence of various levels in logic? What was the meaning and role of the “method of Lobachevsky” which was implemented in construction of imaginary logic? Why did psychologism in the case of Vasiliev happen to be an important factor in the composition of the new ‘imaginary’ logic, as he called it?

VISIÓN NERUDIANA DEL DESIERTO NORTINO

The imaginary logic of Nerudian poetry, governed by water assumes as antilandscape, as infernal space, the figure of the northern desert: it is the place of pain, of silence, of death, of emptyness. A distance is then established between the lyric voice and the desert surpassed, in a secondary moment of the poetic meditation, by the precarious lives of the people and the workers' strife in the middle of the desert. At the end, that sentiment permits the reconciliation of stone and water, of the poet and the desert throught images constructed on oxymorons.

Vasil'Év and Imaginary Logic

This paper is about the ‘Imaginary Logic’ developed by the Russian logician Nicholas Vasil'év between about 1910 and 1913, a logic that is often claimed to be a forerunner of different sorts of modern nonclassical logics. The paper describes the content of that logic (not by trying to interpret it in modern logic, as some commentators have done, but by describing it in its own terms). It then looks at the philosophical underpinnings of the logic. Finally, in the light of the preceding, it discusses Vasil‘év's place in the history of logic.

Imaginary (non-aristotelian) logic

Imaginary (non-aristotelian) logic

The fate of one forgotten idea: N. A. Vasiliev and his imaginary logic

The fate of scientific ideas is whimsical. Very often time only gives us the opportunity to appraise the strength of forecasting of the scholar, the scale of his ideas, their orientation to the future, and the role and place of the scholar in the intellectual history of mankind. As one aphorism put it: "Time has a difficult delivery, but never a miscarriage" (F. R. Lamenne). Today, the significance of the logical works ? only a few articles ? of Kazan University philosophy professor, N. A. Vasiliev (1880? 1940), is becoming ever more apparent. He was the first to introduce considerations of inconsistency into formal logic, manifesting at that time an understanding of dialectical spirit (in Kant's or Hegel's sense). In his early years, Vasiliev was attracted to poetry. As if predicting the fate of his own logical ideas, he wrote: "We are the quickly dying flame. And again a burning fire". Actually, his ideas, expressed at the very beginning of the 20th century, entitle us to call Vasiliev a thinker who anticipated the development of many parts of contemporary non-classical logic, especially its most pioneering and novel parts. Already in 1910 Vasiliev abandoned the law of contradiction and constructed a logic without this law. That is why he is justifiably considered as a forerunner of para consistent logic, which incarnates the idea of non-Aristotelian logic. Through his critique (and rejection) of the law of excluded middle, Vasiliev anticipated the birth of another alternative to classical logic, viz., intuitionist logic. Moreover, due to introduction of new classes of judgements (and, respectively, new meanings of truth) he may be viewed as the predecessor of many ? valued logic, which expanded the capability of classical logic. All of this enables us to compare cum grano salis the role of Vasiliev in logic with that of N. I. Lobachevsky in geometry: Lobachevsky's ideas gave rise to non-Euclidian, non-classical geometry; Vasiliev's ideas gave rise to non-Aristotelian, non-classical logic. Lobachevskv called his geometry "imaginary"; Vasiliev, too,