Abstract
In this paper we analyze some of Vasiliev’s main theses on non-Aristotelian logics, in order to show that some of his ideas do indeed match those of present-day paraconsistent logic. Considered from a historical perspective, Vasiliev’s contribution to the history of logic has great intentional value, and his work contains many new ideas that could be extended by others in various ways.
Part of this paper corresponds to part of the results of the PhD Thesis of the second author, which was defended at the Institute of Philosophy and Human Sciences of the University of Campinas (Unicamp) in December 2013 under the advisory of the first author (see Gomes 2013).
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Notes
- 1.
Sometimes also translated as Vasiliev, Vassilieff, and even Wassilieff; see da Mata (2013).
- 2.
Nowadays universal logic can be seen as a general theory of logics, but also can be conceived as an actual field of theoretical inquiry. Its objective is understand the common substratum to all known particular logics through the analysis of the notion of logical consequence and the minimum requisites to the completeness of these systems. Beziau and Costa-Leite (2005, p. 5) argue: “In the same way that universal algebra is a general theory of algebraic structures, universal logic is a general theory of logical structures. During the twentieth century, numerous logics have been created: intuitionistic logic, modal logic, many-valued logic, relevant logic, paraconsistent logic, non-monotonic logic, etc. Universal logic is not a new logic, it is a way of unifying this multiplicity of logics by developing general tools and concepts that can be applied to all logics.”
- 3.
Yet in 1911, K. A. Smirnov published in Russian the paper N. A. Vasiliev and the law of Excluded Forth (see Smirnov 1911b). From 1960 there is a series of works on Vasiliev by V. A. Smirnov (see, for instance, Smirnov 1962, 1989b); Smirnov and Stiazhkin (1960) publish the entry Vasiliev, Nicolai Aleksandrovich in the Russian Philosophical Encyclopedia; in 1989, Smirnov (1989a) appears in the Proceedings of the Eighth International Congress of Logic, Methodology and Philosophy of Science; in the same year, Smirnov edits, in Russian, the book Imaginary Logic: Selected Works, with some of Vasiliev writings in the original version (see Vasilev 1989). Comey (1965) presents a review of Smirnov (1962). In 1988, V. A. Bazhanov publishes, in Russian, two papers, Nikolai Aleksandrovich Vasiliev. 1880–1940, and On the Attempts of a Formal Representation of the Imaginary Logic of N. A. Vasiliev, followed by a series of works on Vasiliev papers (see Bazhanov 1988a,b, 1998, 2011); in 2009, he publishes, also in Russian, the book N. A. Vasiliev and his imaginary logic: the rebirth of a forgotten idea (see Bazhanov 2009). More recently, Raspa and Di Raino publish a volume containing translations into Italian of Vasiliev’s logical texts (see Vasiliev 2012).
- 4.
It is not our objective in this work to present the technical details of the imaginary systems of logic proposed by Arruda.
- 5.
References inside square brackets correspond to the edition or translation of the work quoted. In this case, we quote the only translation into English of Vasiliev’s famous paper; see Vasiliev (2003).
- 6.
- 7.
- 8.
These distinctions take into account the scholastic notions of proprium and accidens, as the author himself explains.
- 9.
Adapted from the translation into Portuguese of the original Russian, translated by Edmundo da Silva Braga and Ione Mota Braga; see Arruda (1990).
- 10.
For Lobachevski seminal work, on the foundation of geometry, see Kagan (1946–1951), Vol. I
- 11.
Vasiliev (1912 [2003], p. 128) explains his metaphor thus: “It is not difficult to see that these designations are analogous to those of the ‘new geometry’ created by Lobachevski. He called it imaginary geometry; later on, the name ‘non-Euclidean geometry’ has been adopted. To the analogy of names, there corresponds an inner analogy between non-Aristotelian logic and non-Euclidean geometry, which consists in a logical identity of their methods”.
- 12.
- 13.
- 14.
We use square brackets […] to introduce information, and angle brackets <…> to indicate supplements to a text added by an editor, a translator, or ourselves.
- 15.
Vasiliev also brings forward similar pragmatical and anthropological premises in order to conclude that: “We are simply accustomed to believe in the uniqueness of logic. We believe in a single logic exactly in the same way as a people in the primitive stage of their cultural development believe that their language is the only one possible. When such a people is confronted with a neighboring one, the latter gives them the impression of being a people without language, a people of mutes. It would be an offense to our intellectual maturity if we too, encountering logical operations different from ours, would arbitrarily deny them the right to be called ‘logical’.” (Vasiliev 1912 [2003], p. 128)
- 16.
- 17.
A theory which has a negation symbol in its language, let us say ‘¬’, is consistent if, for any closed formula A of the language, A and ¬A are not demonstrable; in the contrary case, the theory is said to be inconsistent. If a theory Δ, in whose language the negation symbol occurs, is trivial, then Δ is also inconsistent; the reciprocal case, however, is not necessarily valid, as in the case of paraconsistent theories. However, the presence of contradiction in a theory Δ whose underlying logic is, for example, classical or intuitionist logic, has as an immediate consequence the trivialization of Δ. A logic is paraconsistent when it is the underlying logic of inconsistent but non-trivial theories.
- 18.
We here use bold capital Latin letters standing for metavariables for formulae.
- 19.
- 20.
See Footnote 18 on p. 66.
- 21.
- 22.
There are two general conceptions of paraconsistency. The first one is broad sense paraconsistency, which applies to paraconsistent theories in which the ex falso is merely restricted. In such theories only specific kinds of formulae are deducible from a contradiction – this is the case, for instance, with the minimal intuitionistic logic of Kolmogorov and Johansson (Kolmogorov 1925; Johansson 1936). The second one is strict sense paraconsistency, which applies to paraconsistent theories in which the ex falso does not hold in general; for instance, da Costa’s paraconsistent logics C n , 1 ≤ n ≤ ω (da Costa 1963, 1974).
- 23.
In two passages of the Metaphysics (Γ7, 1012a 24–28; Γ8, 1012b 11–13) Aristotle advances an argument similar to that of Vasiliev, arguing that the indistinction between the true and the false implies the dissolution of the logico-rational enterprise, an idea which is analogous to the theoretical phenomenon of trivialization. See Gomes and D’Ottaviano (2010).
- 24.
- 25.
- 26.
We quote V. L. Vasyukov translation into English of the Vasiliev’s paper in 1993; see Vasiliev (1993).
- 27.
- 28.
In fact, if one were to make a syllogistic with only one term, this syllogistic would be a theory of pure identity in the affirmative syllogistic modes derived from the valid mode Barbara, as well as a theory of absolute negation or confutation, as in the syllogistic modes dependent on the valid mode Celarent. See Łukasiewicz (1951, p. 53).
- 29.
According to Arruda (1990, p. 11), “Vasiliev did not make clear which of the laws of his metalogic, or of logic, he intended to construct, discussing only the laws of [non-] contradiction and of non-self-contradiction. However, given the context, it is clear that for him there did not exist a single imaginary logic, but as many as there are imaginable worlds.”
- 30.
See Vasiliev (2003).
- 31.
See, for instance, Priest and Routley (1989).
References
Arruda, A. I. (1977). On the imaginary logic of N. A. Vasilev. In A. I. Arruda, N. C. A. da Costa, & R. Chuaqui (Eds.), Non-classical logics, model theory and computability (Studies in Logic and the Foundations of Mathematics, Vol. 89, pp. 3–22). Amsterdam: North-Holland.
Arruda, A. I. (1980). A survey of paraconsistent logic. In A. I. Arruda, N. C. A. da Costa, & R. Chuaqui (Eds.), Latin American symposium on mathematical logic, IV – 1978, Santiago (Mathematical logic in Latin America: Proceedings, pp. 1–41). Amsterdam: North-Holland. (Synthese Historical Library, 9).
Arruda, A. I. (1984). N. A. Vasilev: a forerunner of paraconsistent logic. Philosophia Naturalis, 21, 427–491.
Arruda, A. I. (1989). Aspects of the historical development of paraconsistent logic. In G.Priest, R. Routley, & J. Norman (Eds.), Paraconsistent logic: essays on the inconsistent (pp. 99–130). München: Philosophia Verlag.
Arruda, A. I. (1990). N. A. Vasilev e a lógica paraconsistente. [N.A. Vasiliev and Paraconsistent logic, in Portuguese]. Campinas, SP: Centro de Lógica, Epistemologia e História da Ciência, Universidade Estadual de Campinas. (Coleção CLE, 7).
Bazhanov, V. A. (1988a). Nikolai Aleksandrovich Vasil’ev. 1880–1940. Moskva: Nauka.
Bazhanov, V. A. (1988b). O popytkakh formal’nogo predstavleniia voobrazhaemoi logiki N. A. Vasil’eva [On the attempts of a formal representation of the imaginary logic of N. A. Vasiliev in Russian]. In Metodologischeskii analiz osnovanii matematiki. [The methodological analysis of the foundations of mathematics, in Russian] (pp. 142–147). Moskva: Nauka.
Bazhanov, V. A. (1998). Towards the reconstruction of the early history of paraconsistent logic: the prerequisites of N. A. Vasiliev’s imaginary logic. Logique & Analyse. Brussels, 161–162–163, 17–20.
Bazhanov, V. A. (2009). N. A. Vasil’ev i ego voobrazhaemaia logika. Voskreshenie odnoi zabytoi idei. [N. A. Vasil’ev and his imaginary logic: the rebirth of a forgotten idea, in Russian]. Moscow: Kanon+.
Bazhanov, V. A. (2011). The dawn of paraconsistency: Russia’s logical thought in the turn of XX century. Manuscrito: Internacional Journal of Philosophy, 34(1), 89–98. Campinas, SP.
Béziau, J. -Y., & Costa-Leite, A. (2005). What is universal logic? In J.-Y. Béziau & A. Costa-Leite (Eds.), Handbook of the first world congress and school on universal logic. Montreux: UNILOG’05.
Brouwer, L. E. J. (1908). De onbetrouwbaarheid der logische principes. [The unrealibility of the logical principles, in Dustch] Tijdschrift voor Wijsbegeerte 2, 152–158.
Brouwer, L. E. J. (1913). Intuitionism and formalism. Bulletin of the American Mathematical Society, 20, 81–96.
Church, A. (1936). A bibliography of symbolic logic. The Journal of Symbolic Logic, 1(1), 121–216.
Church, A. (1956). Introduction to mathematical logic. Princeton: Princeton University Press.
Comey, D. D. (1965). Review of Smirnov (1962). The Journal of Symbolic Logic, 30(3), 368–370.
da Costa, N. C. A. (1963). Sistemas formais inconsistentes. [Inconsistent formal systems, in Portuguese]. Thesis (Mathematical analysis and superior analysis, full professorship) – Faculdade de Filosofia, Ciências e Letras, Universidade Federal do Paraná, Curitiba.
da Costa, N. C. A. (1974). On the theory of inconsistent formal systems. Notre Dame Journal of Formal Logic, 15(4), 497–510.
da Costa, N. C. A. (1980). Ensaio sobre os fundamentos da lógica. [Essays on the foundations of logic, in Portuguese], São Paulo: Edusp/Hucitec.
da Costa, N. C. A. (1997). Logiques classiques et non classiques: essais sur les fondements de la logique. Traduit du portugais et complété par Jean-Yves Béziau. Paris: Masson.
da Mata, J. V. T. (2013). Review of Bazhanov (2009). In Logic and logical philosophy (Vol. 22, pp. 131–135). Toruń: Nicolaus Copernicus University.
D’Ottaviano, I. M. L. (1990). On the development of paraconsistent logic and da Costa’s work. The Journal of Non-Classical Logic. Campinas, 7(1/2), 89–152.
Gomes, E. L. (2013). Sobre a história da paraconsistentcia e a olra de da Costa: a instavraçăo da lógica paraconsistente [On the history of paraconsistency and da Costa’s work: the establishment of paraconsistent logic, in Portuguese]. 535p + appendixes. Thesis (Ph.D. in Philosophy) – [Instituto de Filosofiae, ciências Humans, Universidade Estadoal de Campinas, SP, in Portuguese].
Gomes, E. L., & D’Ottaviano, I. M. L. (2010). Aristotle’s theory of deduction and paraconsistency. Principia: International Journal of Epistemology. Florianópolis, SC, 14(1), 71–97.
Gomes, E. L., & D’Ottaviano, I. M. L. (2017). Para além das Colunas de Hércules, uma história da paraconsistência: de Heráclito a Newton da Costa [Beyond the Columns of Hercules, a history of paraconsistency: from Heraclitus to Newton da Costa, in Portuguese]. Campinas: Editora da Unicamp; Centro de Lógica, Epistemologia e História da Ciência. (Série Unicamp Ano 50, 50; Coleção CLE, 80).
Hessen, S. I. (1910). Retsenziia na stat’iu N. A. Vasil’eva: “O chastnykh suzhdeniiakh, o treugol’nike protivopolozhnostei, o zakone iskliuchennogo chetvertogo” [Review of N. A. Vasiliev’s paper “On particular judgments, the triangle of oppositions, and the Law of the Excluded Fourth” in Russian]. Logos, 2, 287–288.
Jaśkowski, S. (1948). Rachunek zdah dla systemów dedukcyjnych sprezecznych [Un Calcul des Propositions pour les Systèmes Déductifs Contradictoires, in Polish]. Studia Societatis Scientiarum Torunesis, 1(5), sectio A. 55–77.
Jaśkowski, S. (1999). A propositional calculus for inconsistent deductive systems. Logic and Logical Philosophy. Toruń, 7, 35–56.
Johansson, I. (1936). Der Minimalkalküll, ein reduzierter intuitionistischer Formalismus. Compositio Mathematica, 4, 119–136.
Kagan, V. F. (1946–1951). N. I. Lobachevsky - Complete Colleted Works [in Russian]. Moscow, Leningrad, GITTL. 4 vols.
Kolmogorov, A. N. (1925). On the principle of excluded middle. In J. Heijenoort (Ed.), From Frege to Gödel: A source book in mathematical logic 1879–1931 (pp. 414–437). Lincoln: toExcel, 1999.
Lobachevsky, N. (1840). Geometrical researches on the theory of parallels. In R. Bonola (Ed.) (1955). Non-euclidean geometry. Translated into English by Carslaw, H. S. New York: Dover Publications.
Łukasiewicz, J. (1910a). Über den Satz des Widerspruchs bei Aristoteles. Bulletin Internationale de l’Académia Sciences de Cracovie, Classe d’Histoire de Philosophie (pp. 15–38).
Łukasiewicz, J. (1910b). O zasadzie sprezeczności u Aristotelesa: Studium Krytyczne, Krákow.
Łukasiewicz, J. (1930). Philosophische Bemerkungen zu mehrwertigen Systemen des Aussagenkalküls. Comptes Rendus des Séances de la Société des Sciences et des Lettres de Varsovie, Cl. iii, 23, pp. 51–77.
Łukasiewicz, J. (1951). Aristotle’s syllogistic from the standpoint of modern formal logic. enlarged (2nd ed.). New York: Oxford University Press.
Łukasiewicz, J. (1970). In L. Borkowski (Ed.), Selected works of Łukasiewicz. Amsterdam: North-Holland.
Łukasiewicz, J. (1971). On the principle of contradiction in Aristotle. The Review of Metaphysics, 24(3), 485–509. Translated by Vernon Wedin.
Łukasiewicz, J. (2003). Del Principio di Contraddizione in Aristotele. Traduzione dai polacco di Grazyna Maskowsia. Macerata: Quodlibet. (Quaderni Quodlibet, 14)
Peirce, C. S. (1897). The logic of relatives. The Monist, 7(2), 161–217.
Priest, G., & Routley, R. (1989). First historical introduction: A preliminary history of paraconsistent and dialethic aproaches. In G. Priest, R. Routley, & J. Norman (Eds.), Paraconsistent logic: Essays on the inconsistent (pp. 3–75). München: Philosophia Verlag.
Sautter, F. T. (2009). Silogísticas paraclássicas: um estudo de caso sobre a relação entre lógica clássica e lógicas não-clássicas [Paraclassical syllogistics: A case of study on the relation between classical logic and non-classical logics, in Portuguese]. Principia: Revista Internacional de Epistemologia. Florianópolis, SC, 13(2), 185–194.
Smirnov, K. A. (1911a). Retsenziia na sta’tiu N. A. Vasil’eva “O chastnykh suzhdeniiakh, o treugol’nike protivopolozhnostei, o zakone iskliuchennogo chetvertogo” [Review of N. A. Vasiliev’s paper “On particular judgments, the triangle of oppositions, and the Law of the Excluded Fourth” in Russian]. žurnal Ministérstva Narodnago Prosvěščéniá [The Journal of the Ministry of Educacion]. New series (Vol. XXXII, pp. 144–154). Sankt-Peterburg: Senatokaia tipografiia.
Smirnov, K. A. (1911b). Vasil’ev i ego zakon iskliuchennogo chetvertogo [N. A. Vasil’ev and the Law of Excluded Fourth in Russian]. Sankt-Peterburg: Senatokaia tipografiia.
Smirnov, V. A. (1962). Logičéskié vzglády N. A. Vasil’éva [The logical views of N. A. Vasil’iev in Russian]. In Očérki po istorii logiki v Rossii [Essays in the history of logic in Russian]. Izdatél’stvo Moskovskogo Univérsitéta, Moscow (pp. 242–257).
Smirnov, V. A. (1989a). The logical ideas of N. A. Vasil’ev and modern logic. In J. E. Fenstad, I. T. Frolov, & R. Hilpinen (Eds.), Logic, methodology and philosophy of science, VII (Proceedings of the eight international congress of logic, methodology and philosophy of science, Moscow, 1987). Amsterdam/New York/Tokyo: North-Holland.
Smirnov, V. A. (1989b). Logischeskie idei N. A. Vasil’eva i sovremennaia logika [The logical ideas of N. A. Vasil’ev and modern logic in Russian]. In Vasilev, N. A. (1989). V. A. Smirnov (Ed.), Voobrazhaemaia logika. Izbrannye trudy [Imaginary logic. Selected works in Russian] (pp. 229–259). Moskva: Nauka.
Smirnov, V. A., & Stiazhkin, N. I. (1960). Vasil’ev, Nikolai Aleksandrovich. In Filosofskaia Entsiklopediia [Philosophical encyclopedia in Russian] (Vol. 1, pp. 228). Moskva: Sov. Entsiklopediia.
Vasiliev, N. A. (1910). O cǎstnyh suždéniálh, o tréugol’niké protivopoložnostéj, o zakoné isklučēnnog čétvērtogo [On particular judgments, the triangle of oppositions and the law of excluded fourth, in Russian]. Učēnié zapiski Kanzan’skogo Universitéta (42p).
Vasiliev, N. A. (1911). Voobražaémaá logika: konspékt lektsii [Imaginary logic (conspectos of a lecture), in Russian] (6p).
Vasiliev, N. A. (1912). Voobražaémaá (néaristotéléva) logika. [Imaginary (non-Aristotellian) logic, in Russian] žurnal Ministérstva Narodnago Prosvěščéniá (Vol. 40, pp. 207–246).
Vasiliev, N. A. (1913). Logika i métalogika. [Logic and metalogic, in Russian] Logos, 2/3, 53–58.
Vasiliev, N. A. (1925). Imaginary (non-Aristotelian) logic. In Atti dei V Congresso Internazionale di Filosofia, Napoli (pp. 107–109).
Vasilev, N. A. (1989). In V. A. Smirnov (Ed.), Voobrazhaemaia logika. Izbrannye trudy [Imaginary logic. Selected works in Russian]. Moskva: Nauka.
Vasiliev, N. A. (1993). Logic and metalogic. Translated into English by Vladimir L. Vasyukov. Axiomathes, 4(3), 329–351.
Vasiliev, N. A. (2003). Imaginary (non-aristotelian) logic. Translated into English by R. Vergauwen and Evgeny A. Zaytsev. Logique & Analyse. Brussels, 46(182), 127–163.
Vasiliev, N. A. (2012). Logica immaginaria. A cura di Venanzio Raspa e Gabriella Di Raino. Roma: Carocci. [Italian translation of Vasiliev 1910, 1911, 1912 and 1913].
Acknowledgements
We would like to thank José Veríssimo Teixeira da Mata and an anonymous referee for their helpfulness in pointing out to us recent studies and translations related to Vasiliev’s work.
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D’Ottaviano, I.M.L., Gomes, E.L. (2017). Vasiliev’s Ideas for Non-Aristotelian Logics: Insight Towards Paraconsistency. In: Markin, V., Zaitsev, D. (eds) The Logical Legacy of Nikolai Vasiliev and Modern Logic. Synthese Library, vol 387. Springer, Cham. https://doi.org/10.1007/978-3-319-66162-9_5
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