https://orcid.org/0000-0003-3290-746X
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ABSTRACT
Ayda Ignez Arruda (1936–1983) was a key figure in the development of the Brazilian school of Paraconsistent logic and the first person to write a historical survey of the field. Despite her importance, the only paper entirely devoted to her works is Da Costa and De Alcântara's “The Scientific Work of Ayda I. Arruda”. In this paper, after offering motivation for an investigation of Arruda's work, on the basis of biographical and bibliographical research we present her intellectual development and the originality of her contributions in a new light. With this newly articulated survey of Arruda's thought, we hope to lay the groundwork for future research on her legacy and thus contribute to a new, more inclusive narrative of the history of the analytic tradition in Latin America.
Acknowledgements
The authors are grateful to Ítala M. L. D’Ottaviano for her very detailed answers to our questions as well as for all the attention and encouragement she gave to our project; to Diderik Batens for his interview; to Andrés Bobenrieth for his valuable contributions and illuminating conversations on the first version of this investigation; to Frank Thomas Sautter and João Marcos Almeida for their helpful comments on early drafts; to John Mumma for comments and corrections of our non-native speaker English; to Tamires Dal Magro for helping us to access part of the primary sources at Unicamp and to CLE Historical Archives for conceding access to parts of the documentation of the Ayda Ignez Arruda Fund.
Notes
1 Recent examples of specific research on Brazilian women philosophers are Paulo Margutti's book on Nísia Floresta (1810–1885) – the first feminist Brazilian writer and a philosopher of education – the chapter dedicated to her in the second volume of the História da Filosofia no Brasil, 412–524 and Secco and Pugliese, “Teaching Nísia Floresta” on the philosophical arguments found in her 1852 Opúsculo humanitário.
2 According to Pérez, in his Análisis filosófico, lenguaje y metafísica (from 1975), Rabossi characterizes analytic philosophy in terms of family resemblances such as:
A positive attitude toward scientific knowledge; a cautious attitude toward metaphysics; a conception of philosophy as a conceptual task, which takes conceptual analysis as a method; a close relationship between language and philosophy; a concern with seeking argumentative answers to philosophical problems; search for conceptual clarity.
3 It is noteworthy that other women philosophers are mentioned throughout the text. By stressing that the presence of women is not a proper topic in Pérez text, we are less criticizing it and more offering motivation for our own work. It is important to recognize that by its very theme and nature, Pérez's approach plays the vital role of mapping the history and main original developments of Latin American analytic philosophy in an otherwise pretty anglophone encyclopaedia.
4 Before Miró Quesada's baptism in 1976, the formal systems first proposed by Da Costa, whose main characteristics is to accommodate contradictions without trivialization, were called non-trivial inconsistent systems (see Bobenrieth, Inconsistencias, ¿por qué no?). In what follows, we shall use the term paraconsistent regardless.
5 Though it is true that there were earlier developments on paraconsistent logics like Jaśkowski's “A Propositional Calculus for Inconsistent Deductive Systems” and Halden's Logic of Nonsense (all well documented in the literature of the history of the movement – Arruda, “A Survey of Paraconsistent Logic”; Bobenrieth, Inconsistencias, ¿por qué no?; Gomes and D’Ottaviano, Para além das Colunas de Hércules), it is important to emphasise that da Costa was the first to lead a group of investigation focused mainly and primarily in the development and study of paraconsistent systems with the clear philosophical motivation of constraining the Law of Non-Contradiction. Therefore, we believe that the epithet of ‘leading figure in the creation of Paraconsistent logic’ suits him well.
6 The consolidation of classical (first-order) logic as the ‘standard logic’ is still a matter of historical inquiry. An interesting starting point on the matter is Eklund, “On How Logic Became First-order”.
7 “One way of understanding it is to say that from the assertion of two mutually contradictory statements any other statement can be deduced; hence, it would be better to refer to it as ex contradictione sequitur quodlibet (ECSQ) (see also Bobenrieth, 1996: p. 103) or, as Priest (1987) does, ex contradiction quodlibet.” (Bobenrieth, “The Origins of the Use of the Argument of Trivialization in the XX Century”).
8 For details, see Bobenrieth, Inconsistencias, ¿por qué no?.
9 Personal communication, 7th December 2017.
10 This information is given in an interview available online (in Portuguese) at http://www.filosofiajuridica.com.br/arquivo/arquivo_60.pdf
11 Asenjo is addressing the author of the paper, Evandro Gomes, a historian of Logic in Brazil who interviewed him.
12 D’Ottaviano and Gomes in “On the Development of Logic in Brazil I” report that the Symposium was specifically organized to host Alfred Tarski, who was then a visiting professor at the Pontifícia Universidad Católica de Chile as an invitee of Rolando Chuaqui. These events are detailed in Suguitani, Viana and D’Ottaviano, “Alfred Tarski: Lectures at Unicamp in 1975”. According to D’Ottaviano (in our forthcoming interview with her), the proceedings of this Symposium were prepared and typed by Arruda, on computers that she bought with her resources, as then the IMECC did not have staff available for this type of task. As another proof of her dedication to the cause of Logic in Latin America, it is worth mentioning also that Batens reports (in our interview with him) that Arruda typed the manuscripts of the proceeding of all the early Latin-American logic meetings published by North-Holland and Marcel Dekker.
13 These and other memories of Arruda will be published in a detailed interview with D’Ottaviano. Other relevant elements of the interview will be presented in section five.
14 Da Costa and De Alcântara consider two of her works as minor, merely expositive: Arruda, “Uma questão de lógica” and “A evolução do método axiomático”. The first one is her only work on the philosophy of logic. D’Ottaviano says that although Arruda was interested in philosophy, “she was always very careful, because hers was a mathematician's training” (D’Ottaviano, interview with the authors).
15 In a footnote, she exemplifies this claim by referring to two works on set theory: Fraenkel and Bar-Hillel's Foundations of Set Theory (1958) and Wang and McNaughton's Les systèmes axiomatiques de la théorie des ensembles (1953).
16 As Arruda's quote plainly shows, concerns underlying the discussion between monism and pluralism in logic antedates Beall and Restall's influential paper on the topic (“Logical Pluralism”), but we recognize that using this terminology could be considered somewhat anachronistic.
17 After paraphrasing this passage, Da Costa and Alcântara say that the explanation of this situation of double dependency (on conventional and unconventional parts of logic) “would be one of the principal tasks of the philosophy of logic” (“The Scientific Work of Ayda I. Arruda”, 7).
18 In those years, Arruda's professor defended a methodological view of philosophy inspired by logical positivism. In the beginning of her paper Arruda refers to da Costa, “conceptualización de la filosofia científica”, which distinguishes between ‘scientific’ and ‘speculative’ philosophy – the first being a form of philosophical inquiry that “follows the lines of logical positivism, and particularly Reichenbach's, together with some of Russell's ideas” (Bobenrieth, Inconsistencias, ¿por qué no?, 182).
19 It may be noteworthy to quote Da Costa and Alcântara on the matter:
Problems of various kinds gave rise to Paraconsistent logic. For instance, the paradoxes of set theory, the semantic antinomies, and some issues originated by dialectics (in particular, by the conceptions of Hegel, Marx, and modern Marxists), by Meinong's theory of objects, by some psychoanalytic theories (Lacan), and by the theory of fuzziness.
20 Da Costa's Portuguese neologism is dialetizar, for ‘turning it into something dialectic’. It is important not to identify this idea with Priest's dialetheism, the view that there are genuine contradictions. For brief clarification of this point, see the beginning of Priest, Tanaka, and Weber, “Paraconsistent Logic”. For a Brazilian version of the distinction between the Brazilian and the other schools of paraconsistency see Da Costa and Bueno, “Paraconsistent Logic”.
21 In naïve Set Theory it is relatively easy to prove that this set does and does not belong to itself. If the logic subjacent to this argument is classic, the Ex falso applies.
22 “As surprising as it may be at first sight, there is an analogy here with the way intuitionism is partially justified. As Heyting stressed (see Heyting 1956), intuitionism as a mathematical position can only subsist as far as professional mathematicians occupy themselves with its problems” (Arruda, Considerações sobre os Sistemas Formais , 47).
23 In less specialized jargon, this amounts to saying that one cannot use a truth table method for discerning if a formula of the system is valid or not. This result is the main one of Arruda's many results developed by João Marcos Almeida in his 1999 MA dissertation on the semantics of possible translations.
24 It is enough to prove something for to safely state that every other related system proves the same.
25 As shown by Almeida (“Semânticas de traduções possíveis”, 202), although Alves and Da Costa attribute this independency result do Arruda, she did not prove it – at least not in the way they present the calculations, which contain errors that the author spotted before proposing alternative calculi to finally prove the independence of the axioms. One hypothesis for explaining the reverberation of this misconception about Arruda's independence result could be that her fellows trusted her mathematical skills, not having taken the trouble to check the correctness of her calculations. On the other hand, Bobenrieth points out that Axiom 13 was not independent after all, a fact proven by Marcel Guillaume and acknowledged by Arruda in “Remarques sur les systemes ” (see also Bobenrieth, Inconsistencias, ¿por qué no?, 192). Future and more detailed historical research are needed to clarify this and other specific topics that we cannot deal with in the space of this paper.
26 The authors refer to Arruda, “A evolução do método axiomático” “Transformadas no calculo restrito de predicados”, “Remarques sur les systemes Cn”, and to a paper to the last period, Arruda and Da Costa, “Une sémantique pour le calcul ”. Their labelling in the references is: [2], [8], [22] and [26].
27 In set theory, the Separation Schema states the conditions under which, given a set and a property, one is allowed to distinguish a subset whose elements are all and only the elements for which the property holds; the Axiom of Choice, on the other hand, is a non-constructive principle that guarantees the existence of a set made from arbitrary elements taken from a certain collection of sets. Both principles are closely related to the paradoxes engendered in naïve set theory (as the Russell Set; see infra).
28 In our interview with Batens, he remembers that:
Ayda was making a trip through Europe, partly on a grant and partly on her own money. She came to Ghent from Poland (where, much to her amazement, some people had refused to give her preprints of unpublished papers). The seminar was about Ayda's work on paraconsistent set theories – the first two items in the bibliography of our joint paper – and was attended by the philosophy students and some colleagues and researchers. I remember that Ayda presented the materials in a pedagogically outstanding way; giving enough information to clarify the problem and results without overloading the audience.
29 Were these systems authentic formalizations of Vasiliev's ideas or Arruda's versions of it? Da Costa and De Alcântara claim that:
since Vasiliev's stance is vague and obscure, it is difficult to judge whether her systems constitute faithful formalizations of Vasiliev's points of view. Perhaps it would be more appropriate to say that her systems were inspired by the reading of Vasiliev's papers.
30 Since Arruda was not fluent in Russian, she worked with colleagues from the Institute of Physics (unfortunately, the sources could not remember their names) and by José Veríssimo da Matta, a colleague from the Philosophy Department who spoke Russian and had experience with translations (D'Ottaviano, interview with the authors).
31 Something like ‘usurpers’ in Brazilian Portuguese.
32 An approach in which the importance of women role-models in logic is considered is in Janssen-Laurent, “Making Room for Women in our Tools for Teaching Logic”.