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Articles

History of logic in Latin America: the case of Ayda Ignez Arruda

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Pages 384-408 | Received 13 Oct 2020, Accepted 10 Nov 2021, Published online: 11 Jan 2022

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.

1. Logic and analytic philosophy in Latin America: what about the women?

Systematic research recovering the neglected history of women philosophers in Latin America is relatively rare – a notable exception is Ana Miriam Wuensch's panorama, which shows through many examples that women have been doing philosophy in the continent for centuries (Wuensch, “Acerca da existência de pensadoras no Brasil e na América Latina”).Footnote1 Explanations for this state of affairs should start by considering contextual specificities such as Latin America's colonial history, its enduring social inequalities (together with its well-known negative impacts with respect to the education of women), and the volatile political settings in which the heterogeneous philosophical cultures of the southern hemisphere were born. Now, when one looks more specifically for research on women in the history of analytic philosophy and logic, scarcity reigns. In her recent presentation of the analytic tradition in Latin America for the Stanford Encyclopaedia of Philosophy (SEP), Diana Inés Pérez draws attention to its colonial context. We understand the reception of authors like Frege, Russell, Quine, Carnap and Wittgenstein only when we acknowledge the European philosophies already rooted in the continent since early colonial times. She also highlights the social and political contexts of the tradition by supplementing Eduardo Rabossi's characterization of the tradition with two specific Latin American traits:Footnote2

First, since analytic philosophy was introduced when other philosophical traditions were dominant, philosophical reflections in the analytic tradition frequently go hand in hand with metaphilosophical issues (e.g., the nature of philosophy, its role in society, its specific way of teaching, the relationships amongst various philosophical traditions, etc.). Second, since the introduction of analytic philosophy in Latin American countries was related to a quest to change conservative intellectual institutions, social and political structures, and their forms of management, the critical and constructive spirit of analytic philosophy led many of its practitioners in Latin America to engage politically in a variety of ways in their home countries.

(Pérez, “Analytic Philosophy in Latin America”, 1, 4)
However, with respect to how women have contributed to the analytic tradition, Pérez is for the most part silent. Although she discusses the philosophical works of some women, the only explicit mention of their participation in the development of analytic philosophy in Latin America occurs when the creation of Argentinian Society for Philosophical Analysis (SADAF in the Spanish acronym), in 1972, is described: “Many women philosophers also participated in this enterprise [the creation of SADAF], such as Cecilia Hidalgo, Cristina Gonzalez, Diana Maffia, Gladys Palau and Nora Stigol.”Footnote3 (Pérez, “Analytic Philosophy in Latin America”, 2.1, 8) Concerning the historical development of analytic philosophy in Brazil (discussed in section 2.3) and the original developments of the Latin American analytical tradition in the field of theoretical philosophy (section 3.1), the absence of two women intellectuals in Pérez's account is remarkable: Ayda Ignez Arruda and Ítala Maria Loffredo D’Ottaviano. The first was a key figure in the origin and first developments of Paraconsistent logic and its historiography and the latter is the main figure in the growth and the consolidation of the Paraconsistent tradition across the world – as attested by the fact that D’Ottaviano is the first Latin American woman ever elected to the Académie Internationale de Philosophie des Sciences. Even if most of their written output is in logic and not analytic philosophy, the close connection between both fields, especially in the case of Paraconsistent logic, would justify at least a mention of their work.

Thus, the broader aim of this text is to contribute to the creation of a new narrative in the history of analytic philosophy in Latin America that includes the works of women whose activities as philosophers and logicians are insufficiently recognized. In particular, we will present an original overview of the intellectual life of Ayda Ignez Arruda (1936–1983). In section two, after offering some biographical information, we give a brief assessment of the treatment of Arruda's intellectual trajectory in previous works, the main one being Da Costa and De Alcântara, “The Scientific Work of Ayda I. Arruda”. The originality of our approach lies in its methodological aspect. First, instead of a classification of Arruda's works according to the themes of her research, we use a chronological criterion to organize her oeuvre into three periods. Secondly, benefiting from a partial access to the Ayda Ignez Arruda Fund, a vast documentation of Arruda's life and work founded in 1990 at the Center for Logic, Epistemology and History of Science (CLE), at the University of Campinas (UNICAMP), as well as from interviews with two important associates (her colleague and friend Ítala Maria Loffredo D’Ottaviano and her last co-author Diderick Batens), our analysis incorporates one of the two specific aspects of analytical philosophy in Latin America highlighted by Pérez, namely, the typical political engagement of its actors. Thus, sections three, four, and five present the three phases of Arruda's intellectual development – from the early days of Paraconsistent logic, when her ideas were not substantially distinguished from those shared by the Curitiba Group (section three), through the intermediate period, marked by the mutual influence between her and da Costa (section four), to the last phase. The last period is where one sees her most original thinking, typically independent from Da Costa's ideas (section five). It is precisely during the last period of her rather abruptly abbreviated life that we will be able to illustrate the political aspect of Arruda as a philosopher in the Latin American analytic tradition. This is because Arruda was an impeccable example of the alliance between professional competence, engagement with the cause of logic and philosophy of science, and political courage during the dark years of the military dictatorship in Brazil. Finally, by articulating Sara Hutton's recent investigation on women's contribution and influence on the history of philosophy (“‘Context’ and ‘Fortuna’ in the History of Women Philosophers”) and Karin Beiküfner report on woman's place in the history of logic (Beiküfner and Reichenberger, “Women and Logic”), we conclude by stressing the importance of recovering Arruda's legacy with respect to the development of Paraconsistent logic and its historiography, emphasizing some methodological challenges faced in tracing the fortuna of women in the history of logic and philosophy in Latin America.

2. Introducing Ayda Arruda and paraconsistent logic

Ayda Ignez Arruda was born on 27th June 1936 in Lajes, a small town in Santa Catarina (a state in the south of Brazil). She studied mathematics at the Catholic University of another Southern state, Paraná, having received a bachelor's degree in 1958, and a license to teach mathematics in the following year. In 1960 she started her professional career as a professor of Mathematical Analysis at the Faculty of Philosophy, Sciences and Letters of the Federal University of Paraná. In 1964 she published her PhD dissertation in Mathematics, becoming a Full Professor at the same university. Her work was about some formal systems that, from 1976 on, would be designated as paraconsistent.Footnote4 Her supervisor was Newton C. A. da Costa, the leading figure in the creation of Paraconsistent logic.Footnote5 According to the text presenting the Ayda Ignez Arruda Fund at the CLE website:

During her scientific training, she had significant contacts with Brazilian and foreign logicians and mathematicians. Among them, we highlight the professors: Mário Tourasse Teixeira, from the Faculty of Philosophy of Rio Claro, São Paulo; Marcel Guillaume, from the Université de Clermont-Ferrand in France; Antonio Monteiro, from the University of Bahía Blanca, Argentina; and Andrés Raggio, from the University of Córdoba, Argentina.

Before presenting more biographical facts, it is worth considering the conceptual and institutional context of Arruda's PhD research. As for the conceptual background, let us start by reminding the reader that so-called ‘classical logic’ does not constitute a unique formalism but rather a broad family of formal apparatuses whose mathematical properties and philosophical motivations can be considered relatively ‘orthodox’. The conceptual core of this orthodoxy is partly inherited from the Aristotelian tradition, but it was ultimately consolidated by the time analytic philosophy was born.Footnote6 A logic is heterodox or ‘non-classical’ when it is based on any formalism or philosophical position that deviates from that orthodoxy in one way or another. A typical example is Intuitionistic Logic that, through the rejection of the Principle of Excluded Middle, challenges some preconceptions on the meaning of mathematical proofs and revisits the concept of validity within mathematics.

Given the possibility of proving that, when a contradiction can be derived in a system, everything whatsoever follows (i.e. every formula of the system is a theorem, the system is trivial), classical logic rejects contradictions. This refusal is already embodied somewhat in the ancient Principle of Non-Contradiction. This logical possibility, that in medieval times was called Ex falso quodlibet and nowadays is known as the Principle of ExplosionFootnote7 began to be the object of suspicion by some logicians such as Jan Łukasiewicz, Nikolaj Vasiliev, and Stanisław Jaśkowski during the first half of the twentieth century. However, the first conscious attempt to constrain its validity – to formalize logical systems that accept the presence of contradictions without trivialization – is due to da Costa in his 1963 doctoral dissertation, Sistemas formais inconsistentes [Inconsistent formal systems]. In this seminal work, several formal systems that are tolerant of contradictions are presented. The basic system is C1, an axiomatic calculus almost identical to the classical one, but the axiom for introducing negation, (AB)((A¬B)¬A)) – which may be read as: ‘if A allows deducing both B and not-B, then conclude not-A’ – is weakened under the condition that B itself should not be contradictory. This is expressed in the system by the following notation: B=¬(B¬B). Together with axioms ensuring the proper syntactic circulation of this clause of ‘good behaviour’, the resulting system can support inconsistent theories (theories where both a formula and its negation are proved) without being trivial (having no no-theorems). With C1 as his basis, Da Costa presents other results: the hierarchy of systems Cn(1n<ω) with its limit, the system Cω; the predicate calculi Cn and Cn=(1nω), the description theories Dn Dn=(1nω) and the axiomatic type theory NFn(1n<ω).Footnote8

As we shall see in detail in section three, Arruda's publications from 1964 to 1975 all concern da Costa's systems. According to the sources, Arruda met da Costa in Curitiba in 1957 or 1958 when she was finishing her undergraduate studies, and he was already a professor at the Federal University of Paraná. Da Costa recalls that “by the time I usually did speak about the group I was forming for the investigation in topics of Logic and Philosophy of Science, and she went into the group. Unfortunately, I don't remember the precise dates”.Footnote9

Both dissertations, Da Costa's and Arruda's, were the two main fruits of the group he gathered for studying logic and the philosophy of formal sciences, that over time began to be known as the Curitiba Group. Da Costa is known as the most preeminent of its members; Arruda was not a partner like the others, but his first studentFootnote10 and his closest collaborator. The sources are emphatic in stressing this point. In his conjoint paper with De Alcântara, Da Costa says that:

[S]he contributed greatly to the organization of the group of young logicians and mathematicians that in the sixties constituted the so-called Group of Curitiba (an expression coined by Professor L.W. Witta). To the group belonged A. I. Arruda, J. Cardoso, H.C.A da Costa, N.C.A da Costa, I. Inoue, Z.M. Pavão, and A. Santa Rosa, among others. Its members were primarily interested in logic, abstract algebra, foundations of mathematics, the methodology of science, and the teaching of mathematics. In a certain sense, the origin of research in logic in Brazil may be traced back to the activities of this group.

(“The Scientific Work of Ayda I. Arruda”, 2)
The Argentinian philosopher Florencio G. Asenjo, who worked on the same topics as da Costa, once said:

In Argentina, there has been nothing comparable with what Newton da Costa has done in Brazil. Not only his work, his papers, interesting and beautiful as they really are, but also the way in which him, together with Arruda, sponsored a group of disciples. He really created a movement that continues nowadays. In fact, youFootnote11 and so many others who are doing an excellent job, Carnielli, D’Ottaviano, Coniglio, Marcos and others. This is a beautiful job and there is nothing comparable with this development in Argentina.

(Asenjo in Gomes, Sobre a história da paraconsistência e a obra de Da Costa, 481 [our emphasis])
Marcel Guillaume remembers that during his visit to Curitiba in 1964, the group was “rarely more than three – Newton, the young Ayda Arruda and I”, and that nonetheless, “substantial progress was made in the study of Cn during that time” (Gomes and D’Ottaviano, Para além das Colunas de Hércules, 405).

The Curitiba Group disappeared in the mid-60s when political problems related to the military regime forced da Costa to migrate to the University of São Paulo. At the same time, Arruda was invited to join the Institute of Mathematics, Statistics and Computer Science (IMECC in the Portuguese acronym) of the recently founded UNICAMP.

The documentation about Arruda's career shows her vigorous engagement in the promotion of logic in the continent: besides having several experiences as visiting scholar or professor in international institutions – such as “the Universities of Bahía Blanca, Buenos Aires, Clermont-Ferrand, Lyon, Warsaw, Torun, Gent, the Catholic University of Chile, The Bulgarian Academy of Sciences and The Polish Academy of Sciences” (Da Costa and De Alcântara, “The Scientific Work of Ayda I. Arruda”, 2) – she was the editor of the Boletim da Sociedade Paranaense de Matemática [Bulletin of the Paranaense Mathematics Society] in 1960, having presided over this organization between 1964 and 65 and being vice-president in the next period; she coordinated teaching projects at the IMECC between 1972 and 1974, and in 1975 she organized the Simpósio de Lógica Matemática [Symposium on Mathematical Logic]Footnote12; she was the president of the III Simpósio Latino-Americano de Lógica Matemática (SLALM, the Latin-American Symposium of Mathematical Logic) in 1976, as well as of the first and second Encontro Brasileiro de Lógica [EBL, the Brazilian Logic Meeting] in 1977 and 1978; she was in the organizing committee of the fourth and fifth editions of SLALM (in 1978 and 1981), one of the founding members of the Brazilian Logic Society, its vice-president between 1979 and 1980, and the president of the Brazilian Mathematical Society from 1981 to 1982; she was a member of the Association for Symbolic Logic starting in 1979, and belonged to the Assessor's Council for Latin America between 1979 and 1981 within it. As D’Ottaviano recalls, Arruda always participated enthusiastically in the events she organized. The last conference she ever attended, between two surgeries due to a cancer diagnosed in 1982, was the VI SLALM (held in Caracas, Venezuela, in 1983). She passed away in Campinas, alongside her friends Ítala and Mercedes, on October 13th, 1983. Her wake and homages were held at UNICAMP.Footnote13

Now, although panoramic surveys of Paraconsistent logic invariably mention Arruda's contributions to the field (Bobenrieth, Inconsistencias, ¿por qué no?; D’Ottaviano and Gomes, “On the Development of Logic in Brazil” I and II; Gomes and D’Ottaviano, Para além das Colunas de Hércules; Da Costa and Bueno, “Paraconsistent Logic”; Priest, Tanaka and Weber's SEP entry on “Paraconsistent Logic”), the only paper dedicated to her intellectual development is “The Scientific Work of A. I. Arruda” by Da Costa and De Alcântara. In this paper, Arruda's writings are arranged through a thematic criterion, in two main areas:

  1. History of Logic:

    1. History of paraconsistent logic

    2. Imaginary Logic of A. N. Vasiliev

    3. The Axiomatic Method

  2. Non-Classical Logic:

    1. Paraconsistent first-order logic and related calculi

    2. Paraconsistent set theory

    3. Griss’ Logic

Although comprehensive, this exposition of Arruda's works is problematic. Firstly, because of the enormous disparity between the amount of work she produced on these topics: she wrote only one paper on the axiomatic method, and if one considers her writings integrally, it can be seen only as a minor work (Arruda, “A evolução do método axiomático”).Footnote14 Secondly, her writings on Paraconsistent first-order logic and Paraconsistent set theory represent more than half of her scientific production. Thus, Da Costa and De Alcântara's thematic exposition does not present Arruda's intellectual development adequately. Without ignoring their approach, in what follows, we will arrange Arruda's lifework by applying a chronological criterion. Within this bidimensional – thematic and historical – matrix that renders a tripartite classification, it will be possible to characterize her work better and better to understand the development and evolution of her thought.

3. The young Arruda and the youth of paraconsistent logic

It is undeniable that the interactions within the Curitiba Group shaped Arruda's intellectual interests, especially during her doctorate studies (from 1959 to 1966). In her first publication, an eminently philosophical (and short) paper, “Uma questão de lógica” [A question about logic], the young Arruda presents a summary of some fundamental conceptual problems raised by logic and discussed by the group. The main issue is, addressed right at the beginning of the paper: “Among the various possible logical systems what is, if it is unique, the true one?” (Arruda, “Uma questão de lógica”, 261). Although making clear that it was not her intention to offer a definitive answer, the strategy for responding to the question builds on a distinction between logic as a science – a sense in which logic is an ongoing knowledge, continually developing – and logic as a toolbox instrument with multiple uses. According to Arruda, even in the latter sense, logic cannot be identified with a unique logical system,

for this could lead us to a situation similar to that which occurred with classical logic in relation with the other sciences: instead of being a fertile instrument, it was nothing more than an obstacle to the evolution of the other sciences, of the deductive sciences in particular.

(Arruda, “Uma questão de lógica”, 262)
Given the fact that by that time, many different logical systems were being developed,Footnote15 Arruda proposes a discussion about the kind of arguments that would justify the claim that a single logical system is the true one. One difficulty for those defending the position currently called logical monism is that any criteria would depend on logic itself. As for those defending logical pluralism,Footnote16 the difficulty is the choice between a pragmatic criterion and the absence of any. Regarding the first, “since if a system is sufficient resolving problems in the face of which it was constructed, it is true” (Arruda, “Uma questão de lógica”, 262). She considers it too idiosyncratic and invariably related to a realist position “grounded in a superstition that the logical laws are once and for all univocally determined” (“Uma questão de lógica”, 263). As for the latter, the nonexistence of any standard could imply a problematic sort of conventionalism (something like ‘any logical system is in principle true’).

Building on an analogy between logic and physical geometry – both would have an essential unconventional core and a conventional periphery – Arruda advocates for a conciliatory position:

If logic were reduced to its nonconventional core, it would be practically finished, for nothing else could justify its current progress. Nevertheless, we still cannot abandon the nonconventional core, for, amongst the attempts to build logical systems in which the negation of the principle of non-contradiction is valid, almost nothing would be obtained. We cannot also totally deny the conventions, because one of the most defended positions nowadays, that has been bringing many problems to a solution, is the linguistic interpretation of logic; insofar that any language consists in large part of conventions, there is no doubt that logic will have a conventional part.

(Arruda, “Uma questão de lógica”, 264)Footnote17
In concluding the piece, Arruda shows caution when it comes to philosophy: based on the present-day logical developments and acknowledging that if one approaches the same problems within a more speculative perspective (‘opposed to the scientific, positive methods’) the results could be different, she highlights the open character of the middle term solution that she proposes. In the last paragraph, we see the subtlety of her philosophical sensibility in her biological metaphor for defining logic as a science. Right after championing the positivist methodology in philosophy,Footnote18 she writes:

However, one point is untroubled for us: either we make of logic a living science, treating it accordingly, in such a way that no result is absolute, or what we call logic does not deserve the name of science, for it would be something that does not evolve, where nothing fundamentally new will ever be achieved.

(Arruda, “Uma questão de lógica”, 264)
The conceptual context of Arruda's and Da Costa's dissertation is partially specified by the paper we have just analysed, but also by her second paper, “A evolução do método axiomático” [The evolution of the axiomatic method]. The philosophical relevance of this work lies in its methodological reflection. After approaching the axiomatic method from a historical perspective, Arruda suggests that the most fruitful, although still unexplored, aspect of axiomatics is the exploratory character of its applications. Her idea is that working within an axiomatic setting allows new logical and scientific discoveries. Referring to the reductio ad absurdum as a potent principle or form for obtaining logical consequences, Arruda writes:

Denying a principle to scrutinize its true essence may be one form of this principle [the reductio ad absurdum], but this does not prevent one from negating it in order to test its power […] Thus proceeding, the logician is not only making use of her right to complete freedom in constructing her systems but will also consider the internal structure and the consequences of certain principles, insofar as in many cases it is only by negating or suppressing a principle that one can have an idea of its consequences.

(“A evolução do método axiomático”, 220)
The paper finishes in a motivational tone, appealing to the promising aspect of this heuristic dimension of the axiomatic method, not only in the explanation of known systemsFootnote19 but “in the edification of new theories, thus opening an admirably vast field of research in all sciences” (“A evolução do método axiomático”, Arruda, 1964a, p. 220).

Despite the philosophical facet of Arruda's first publications, her PhD dissertation is a strictly logical work. Her general aim consists in advancing some aspects not dealt with by da Costa in his dissertation. In particular, the objectives stated in the short (two-pages long) preface were:

  1. to develop the particularities of da Costa's hierarchy of systems for n>1: “In many occasions, it will be shown that increasing the number of postulates, or adequately reforming postulates or definitions, as well as other similar artifices, make clearer the similarities between NF0 and NFi” (Considerações sobre os Sistemas Formais NFn, ii);

  2. “to obtain something similar to classical mathematics within those systems” in a broader way when compared with da Costa (1964) (Considerações sobre os Sistemas Formais NFn, ii);

  3. to verify the possibility of “applying, … the most known incompleteness and undecidability results (of Gödel, Church and Rosser) to the inconsistent formal systems” (Arruda, Considerações sobre os Sistemas Formais NFn, iii).

In order to grasp what was at stake in the dissertation, one should remember that, despite the conceptual fertility of Set Theory, within the context of the famous Grundlagenkrise that affected mathematics by the end of the nineteenth century its simpler form (today called ‘naïve’) contains some features that led to trivialization. Therefore, the twentieth century saw continuous efforts in logic and mathematics to construct set theories capable of repairing the unwanted features of the naïve version without sacrificing its expressive power. One of these efforts was the axiomatic system NF (New Foundations) proposed by W. O. Quine in 1937. Da Costa's hierarchy NFn(1n<ω) ‘dialethizes’Footnote20 Quine's original system, allowing certain contradictions while avoiding trivialization. In particular, the hierarchy allows contradictions concerning the so-called ‘Russell Set’, ‘the set of every set that does not belong to itself’.Footnote21 It happens that da Costa did not develop an inquiry on different mathematical features and other properties of this set in NFn; this precisely was aim (A) of Arruda's dissertation. One of the several merits of her work lies in the reformulation of da Costa's definitions and the complementation of his axioms (Bobenrieth, Inconsistencias, ¿por qué no?), which gives a broader homogeneity to his hierarchy, proving Quine's original system NF to be equivalent with NF0.

As for aim (B), “her analysis is broader in scope and perspective from da Costa's” (Gomes and D’Ottaviano, Para além das Colunas de Hércules, 469), while for (C), she proves that the inconsistent formal systems under consideration are affected by Gödel's incompleteness theorems. She also proves that if the systems NFn(0nω) are arithmetically consistent, they are undecidable.

Arruda concludes the dissertation with the following five points:

  1. that the developments she offers for da Costa's NFi systems show Paraconsistent systems to have a life on their own, and that within NFi Hilbert and Poincaré's maxim that in mathematics only what is consistent exists “almost no longer makes sense” (Arruda, Considerações sobre os Sistemas Formais NFn, 47);

  2. that the theory of Paraconsistent systems will establish itself as a mathematical discipline only insofar as logicians and mathematicians become interested in it, relating the systems with more traditional branches of the deductive sciencesFootnote22;

  3. that Paraconsistent systems seem to formalize in a more natural way ordinary mathematical thinking, for in the everyday mathematical talk, there is contradiction, although, from contradictions, no proposition is effectively deduced. “Amongst inconsistent formal systems it seems the NFω type (the infinitely trivializable ones) is the more adequate to formalize the aspect that we have just referred to” (Arruda, Considerações sobre os Sistemas Formais NFn, 48)

  4. Regarding the meaning or conceptual impact of these systems, it was still too soon to say something significant. It is only after developing them, exploring their formal particularities, knowing better the range of their consequences that new theories such as the Paraconsistent one could be properly evaluated;

  5. that the absolute character of logical principles does not seem safe anymore. The creation of the early paraconsistent systems, says Arruda,

    [C]onfirm our opinion that one can build set theories from other systems where traditional principles are not valid as naturally as one can construct set theories from classical logic. With this idea, we want only to raise another question that seems to immediately follow from the current state of the researches in mathematical logic: the freedom logicians have to construct systems as they wish, given their fulfillment of desirable and well-determined conditions.

    (Arruda, Considerações sobre os Sistemas Formais NFn, 49)

As we shall see, Arruda's intellectual development followed a path in which she progressively sought more freedom with respect to the first period of her career.

4. The middle period

Arruda's publications from 1964 to circa 1975 all concern da Costa's systems. They published several collaborative papers together, investigating the properties of Paraconsistent logics and the theories based on them. Da Costa and De Alcântara mention several results of this period:

  1. The calculi Cn(1nω) were found undecidable by finite matrixesFootnote23;

  2. It was proved that each calculus of the hierarchy Cn(1nω) is strictly stronger than the calculi following itFootnote24;

  3. It was proved that none of the primitive connectives (,,,¬) within the systems Cn(1nω) can be defined in terms of the others;

  4. It was proved that the postulates of each Cn(1nω) are independentFootnote25;

  5. The first-order systems based on the calculi Cn(1nω),with or without identity, were found to have a valuation semantics which generalizes the usual Tarskian semantics for first-order logic.Footnote26

She also developed systems resembling some of da Costa's and explored their properties. As for her contributions on Paraconsistent set theory, Arruda:

[A]nalysed carefully how the fundamental notions of set theory can be smoothly defined in NFω, indeed a very weak system; she showed that most concepts, which seem to require negation in order to be defined, can be characterized without this connective, by a positive procedure. Thus, it is possible to build a positive counterpart of several set-theoretic notions, and a great deal of elementary, naïve set theory may be reproduced in Cω [sic].

(Da Costa and De Alcântara, “The Scientific Work of Ayda I. Arruda”, 9)
In this field, her investigation with da Costa was primarily oriented towards the problem of finding safe versions of the Separation Schema and the Axiom of ChoiceFootnote27 within their systems. A significant part of their joint publications around the middle sixties and seventies (Arruda and Da Costa, “Sur le schéma de la separation”, “Le schéma de la separation et les calculs Jn”, 1977) revolve around this topic. These are the years where her intense collaboration with da Costa is most evident. As indicated above, the role of Da Costa's leading intellectual partner during this time is the most common picture of Arruda in the historical works about the development of Paraconsistent logic in Brazil. Since our interest is to revisit Arruda's own thought and legacy, we will now focus on the aftermath of Da Costa and their parting ways, when she forged her own intellectual path.

5. A work of her own: Arruda's late writings and a note on her legacy

The last of Arruda's joint publications with da Costa, Une sémantique pour le calcul C1= appeared in 1977. After this, her attention was directed to some relatively new topics, both for her and for the paraconsistent inquiry in general:

  • (LA1) Griss’ logic (Arruda, “Some Remarks on Griss’ Logic of Negationless Intuitionistic Mathematics”);

  • (LA2) The logic of vagueness (Alves and Arruda, “Some Remarks on the Logic of Vagueness”, “A Semantical Study of Some Systems of Vagueness Logic”);

  • (LA3) Paraconsistent Logic and Set Theory (Arruda, “The Russell paradox in the systems NFn”, Arruda and Batens, “Russell's Set versus the Universal Set in Paraconsistent Set Theory”);

  • (LA4) Vasiliev's logic (Arruda, “On the Imaginary Logic of N. A. Vasil’év”, “N. A. V asil’év: a forerunner of paraconsistent logic”, Vasiliev e a Lógica Paraconsistente);

  • (LA5) The history of paraconsistent logic (“A Survey of Paraconsistent Logic”, 1989).

Arruda actually first worked on LA1 during her middle period, the result being a paper on Griss' propositional logic (Arruda, “On Griss’ Propositional Calculus”). She followed up on it in her 1978 paper on Griss’ sui generis proposal of a logic for intuitionistic mathematics without negation. In this late period paper, Arruda explores a variety of themes related to this then-unknown work in Latin America. She made a meticulous metamathematical analysis of Griss’ logic, “proving new formulas, showing the equivalence of certain schemes apt to function as axiom schemes” (Da Costa and De Alcântara, “The Scientific Work of Ayda I. Arruda”, 11–12) and demonstrated that Griss’ formalization of the propositional calculus is not trivializable in at least two different senses. In his approach to the development of intuitionistic logic, Mark Van Atten mentions Arruda's paper as one of the two texts offering algebraic interpretations of the negationless logic, stressing that she also made a comparison between Griss’ and Vredenduin's systems (“The Development of Intuitionistic Logic”, 6.2.2, 30).

The more substantial logical work of Arruda in her last year's concerns (LA3). Together with Batens – whom she visited at the University of Ghent in the late seventies during a work trip to Europe – Footnote28 Arruda studied and discovered several properties of the Russell Set. In particular:

  • (RS1) She proved that the singleton containing the Russell Set belongs to the Russell Set; that is, the set that only contains the Russell Set is at the same time contained in the Russell Set;

  • (RS2) Subsequently, the singleton containing the Russell Set does not belong to itself;

  • (RS3) She proved that all the subsets of the Russell Set belong to the Russell Set. This is decidedly forbidden for consistent sets. As a matter of fact, Cantor proved that the power set (i. e. the set of all subsets) of a set is always greater than the set itself, no matter its cardinality. Arruda not only proved that this is false in the case of the Russell Set, but she also managed to state an even more scandalous result: that the iterated power sets of the Russell Set are ordered under strict containment. This is usually stated as: rrrr

  • (RS4) She proved that the union of all the sets within the Russell Set is equivalent to a Universe, that is, the set of all sets.

All these results are extremely relevant, since they relate to Russell's legendary discovery of the paradox – engendered by Axiom V of Frege's Grundgesetze der Arithmetic – produced by the existence of a set that belongs and does not belong to itself (see Van Heijenoort, From Frege to Gödel). As far as we know, there is no positive outcome about the Russell Set in the literature before Arruda and Batens’ results – a fact that highlights the importance of their conjoint work. Da Costa and De Alcântara dubbed (RS4) in particular as the “most interesting” of all the results of Arruda in this area given that, “Loosely speaking, one could say that in very many set theories the existence of Russell's set implies the existence of strongly inaccessible sets” (p. 9).

Regarding (LA4), Arruda was the first to link Vasiliev's system with paraconsistency (Da Costa and De Alcântara, “The Scientific Work of Ayda I. Arruda”; De Moraes and Teixeira, “Alguns Aspectos da História da Lógica Paraconsistente”), having argued satisfactorily for the claim that his sketch of a logic in which a particular version of the Principle of Non-Contradiction (that no object can have a predicate that contradicts it) can be derogated challenges the validity of the Principle of Non-Contradiction (Bobenrieth, Inconsistencias, ¿por qué no?). On June 13th 1976, during the third SLALM, Arruda presented the first version of her studies on (LA4) with the paper “On the Imaginary Logic of N. A. Vasilev”, later published in the proceedings of the event.

In this work, Arruda explains that Vasiliev did not believe in real contradictions, but that his logic was to Aristotelian logic as Lobachevsky's imaginary geometry was to Euclidean geometry. She also defends the hypothesis that because Vasiliev's extemporaneous ideas were presented in Russian (in three papers from 1910 to 1913), these works gained the attention of only some of the first non-classical logicians, mainly those working on logics with more than two truth-values. For this same reason, until Arruda's historical work, the mainstream interpretation of Vasiliev logic (by Kline, Rescher, Jammer, and Church) classified it as an antecedent of the many-valued logics, rather than as an early work on Paraconsistent logic. On this matter, she held that:

[I]t should be emphasized that, according to Vasiliev, a contradiction does not invalidate (or does not trivialize) his system. Therefore, his imaginary logic, even just sketched, satisfies the necessary condition for a logic to be called paraconsistent. Consequently, we hold that any formalization of Vasiliev's imaginary logic should lead to a paraconsistent logic. Were it also polyvalent [polivalente], this would be a matter of further detail or interpretation.

(Arruda, “On the imaginary logic of N. A. Vasil’év”, 7; Arruda, Vasiliev e a Lógica Paraconsistente, 13)
In the same paper, she contested the mainstream interpretations and also put forward a formalization of Vasiliev's logical conceptions that gave rise to three different paraconsistent systems.Footnote29

A last but no less relevant fact about Arruda's work on Vasiliev attests to her dedication to the cause of logic in Latin America. Between October and December of 1978, Arruda was a visiting professor at the Catholic University of Chile, where she not only worked within the organizing committee of the IV SLALM but also prepared a manuscript of great historical and pedagogical value: her organization of a series of extracts from Vasiliev's texts, that soon would be translated into Portuguese under her guidance.Footnote30 This material, first referred to by Da Costa and De Alcântara as an IMMEC internal research report (Arruda, “N. A. Vasiliev e a lógica paraconsistente”), was edited by Ítala D’Ottaviano and published as the seventh volume of the Coleção CLE [CLE Collection] (Vasiliev e a Lógica Paraconsistente.). Because of Arruda's interest and dedication, Vasiliev's logical ideas were available in Portuguese before they were translated into English.

As for (LA5), Arruda presented the communication “A Survey of Paraconsistent Logic” during the fourth version of SLALM (in 1978). The subsequent paper, which was also published in Spanish, is the first historiographical work on paraconsistent logic, both in Brazil and abroad. It is also noteworthy that after her death, her view on the historical development of Paraconsistent logic was published as the second chapter of Priest, Routley, and Norman's Paraconsistent Logic, Essays on the Inconsistent.

A last result that Arruda obtained around these years should not be disregarded: the one concerning da Costa's logics. Since 1963, both he and Arruda had stated that NF1 was “inconsistent, but apparently nontrivial” (Gomes and D’Ottaviano, Para além das Colunas de Hércules), but a rigorous proof of this fact was still to be found. In the fifth version of SLALM (in 1981) Arruda showed her proof that NF1 was, after all, trivial (unfortunately, this is a still unpublished paper). Her result forced da Costa to revise his original hierarchy of paraconsistent systems and propose new formal systems to mitigate the uncomfortable outcome (see also Bobenrieth, Inconsistencias, ¿por qué no?, 201).

A final note on Arruda's late period brings us back to the biographical elements presented in section two above: the last academic event in which she participated was the VI SLALM, in Caracas (Venezuela), right before her death. D’Ottaviano remembers:

Ayda was diagnosed with cancer in 1982, when she was only 45 years old. She got a delicate surgery and bravely faced chemotherapy. After an improvement, we went together with Sette to Caracas, to participate in the 1983 SLALM. Right after our return, she underwent a second surgery and died shortly afterward, in October 1983.

(D’Ottaviano, interview with the authors)

The VII edition of SLALM, which took place at UNICAMP in 1985, was dedicated to her dear memory.

5.1. A note on Arruda's legacy

Given the above, one would not be overstating things in saying that the Brazilian school of Paraconsistent logic – which until today is responsible for the training of many highly qualified logicians and philosophers of the formal sciences – owes a great deal to Ayda Ignez Arruda. The final words of Da Costa and De Alcântara paper aim to convey the extent of her impact:

Owing to her pioneering scientific work, her capacity for administration, and the role she played in the development of Logic in Brazil, A.I. Arruda will be considered as one of the leading Brazilian logicians.

(Da Costa and De Alcântara, “The Scientific Work of Ayda I. Arruda”, 12)

However, as we have shown through our different arrangement of Arruda's published works – the cumulative result of gaining partial access to CLE's historical archives, interviews with colleagues, D’Ottaviano's edition of her translation of Vasiliev, and also by the historical and generational distance that separates us from Da Costa and Alcântara's perspective – Da Costa and De Alcântara do not do justice to the richness and nuances of Arruda's intellectual development. An additional reason for this lies precisely in their decision not to consider the strict relation between her administrative capacities and the political courage she had shown during the last period of her life. As we noted at the beginning, following Pérez's approach, one of the peculiar aspects of the Latin American analytic tradition lies in the political engagement of philosophers and, as Arruda's case shows, also of logicians.

As her colleague and friend Ítala D’Ottaviano reports – and a good amount of documentation still to be carefully scrutinized as the Ayda Ignez Arruda Fund attests – Arruda was the first democratically elected dean of UNICAMPs Institute of Mathematics, Statistics and Computer Science (IMECC in the Brazilian acronym). After a period working as the head of the Mathematics Department (from July 1979 to April 1980), she was chosen to be the dean of IMECC by her colleagues, staff, and students (and not appointed by the president of the university, as it was the case in those days of the military dictatorship). Yet,

[B]etween mid-1981 and early 1982, UNICAMP went through a serious institutional crisis. The president's succession process had begun, with some Directors of Teaching and Research Units having presented themselves for consultation with the university community, as candidates for the rectory. Ayda was not, however, among the candidates. On October 16th, 1981, President Plínio Alves de Moraes published an Ordinance exonerating eight Directors from Units, including Ayda Arruda, and appointed eight new directors, not belonging to UNICAMP staff. They were “baptized” by the community as “Interventores”.Footnote31 The IMECC interventor (…) showed up at IMECC, settling in the management room. The reaction of teachers, staff and students was a bang: we paraded, one by one, in front of the invader, expressing, without words, our revolt. Our protest was vibrant and eloquent, supported by colleagues who arrived from other Institutes. The interventor literally ran away from the IMECC on foot and never returned.

(D’Ottaviano, interview with the authors)

After a more formal manifestation of her colleagues, repudiating the interventor, protesting the exonerations and praising Arruda's professional qualities, she filed a judicial order against her dismissal from her position as IMECC's dean. This situation led the usurper to formally resign, and on December 24th of 1981, Ayda Arruda and the other deans won the injunction against UNICAMP in court, recovering their rights. “Arruda's behaviour and attitudes throughout the crisis were firm, courageous, consistent and exemplary”, says her friend Ítala, for whom “she was always an ethical and independent person, not being intimidated by institutional powers and defending what she considered the best options for the IMECC”.

Besides this illustration of her ethical behaviour and political courage, other qualities of Arruda were marked in the memory of her last co-author, Diderik Batens: her pedagogical strength and her openness to talk about “politics and socio-economic matters in Brazil, and academic life. She was friendly and very broadly interested. Ten minutes after we first met, talking to her was like talking to an old friend”. In this vein, Batens also remembers that “Although we did not explicitly talk about it, it struck me, even in those days, that Brazil was a country where machismo was still much deeper rooted than in Western Europe” (Batens interview with the authors).

The fact that after so many years Batens decided to mention his impression about the machismo of the academic environment in which Arruda was living and working acts as a clue for explaining others’ impressions on her. There is at least one record of a student of hers reporting that “she was tough, formal and authoritarian” (see Gomes and D’Ottaviano, Para além das Colunas de Hércules, 668). On the other hand, we have her closest friend and colleague saying that:

She was serious, correct, and extremely rigorous in professional practice. She was dedicated to her students. If she looked ‘tough and formal’, it was because she was shy and reserved. If she seemed ‘authoritarian’, it was because she spoke little, always with discretion.

(D’Ottaviano, interview with the authors)

6. Concluding remarks

The observations of the last quote could be seen as irrelevant in the eyes of those who prefer to ‘stick to the logic’, ignoring the cultural context in which logic was and is historically practised. However, we think that Arruda's case is precisely the kind to which Sarah Hutton's recent methodological reflection about the history of women philosophers applies (Hutton, “‘Context’ and ‘Fortuna’ in the History of Women Philosophers”). For her, the omission and the marginalization of women in the history of philosophy can be reassessed by an analysis of what she calls the fortuna (the fortune and misfortune) of a philosopher. Tracing the fortuna of an author (be it a philosopher or, as in Arruda's case, a logician) requires investigating the reception of her ideas and arguments in a contextual way. This means that not only her writings and technical results should be considered, but also the “the patterns of fragmentation and distortion, remembering and forgetting that has been the lot of most philosophical women” (Hutton, “‘Context’ and ‘Fortuna’ in the History of Women Philosophers”, 38). In this sense, it is our contention that the current acknowledgment of Arruda's place and influence in the development of the Brazilian school of Paraconsistent logic can be significantly improved by means of more interviews with other colleagues and students of hers and primarily through more systematic researches on the documentation available at the Ayda Ignez Arruda Fund – that could possibly imply a reorganization of the material and eventually to the desirable long overdue publication of her selected works.

As Karin Beiküfner stress, when it comes to the history of women logicians, almost everything is still to be done: biographical and bibliographical research, documentation and evaluation of logical works of women logicians, evaluation of the reception of these works, the history of the institutions where they have worked, as well as more thorough methodological discussions of the integration of women logicians in the overall context of recovery projects (Beiküfner and Reichenberger, “Women and Logic”, 7–8). Taking this into consideration in Arruda's case, we can conclude by saying that a fair amount of work has already been done – but not to the extent her case deserves. There is more for philosophers and historians of logic and philosophy in Brazil to do.

The only way to do justice to Arruda's legacy in the history of logic in Latin America, is to evaluate it more thoroughly: exploring the possible interactions with the analytic tradition on the continent, better scrutinising the internal connections between her works as a researcher and a professor, analysing the ways these works interacted with those developed in the same institutional context, and above all, without ignoring (but also not overestimating) the fact that during many years she was practically the only woman in an extremely male-dominated field. Our hope is that, in the future, a new narrative on the history of the analytic tradition in Latin America will be told, one in which women philosophers and logicians will finally be integrated and thus serve as the much-needed role models that they can and must be.Footnote32 May Ayda Ignez Arruda have a part in it.

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.

Additional information

Funding

Gisele Secco's work was supported by CAPES via the projects 23038.006944/2014-72 and 88887.371155/2019-00. CNPQ.

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.

(“Analytic Philosophy in Latin America”, 3)

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.

(“The Scientific Work of A. I. Arruda”, 3)

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 NFn, 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 C1 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 Cn” (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 C1=”. 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.

(“The Scientific Work of Ayda I. Arruda”, p. 6)
One could conjecture that this is the kind of assessment that inspired Arruda to develop (LA2), but this is not the place to go deeper into issues that demand more detailed historical research. Future works on this issue could also illuminate how much of the results presented in Da Costa and Puga, “On the imaginary logic of NA Vasiliev” are original and how much of it is derived from Arruda's works—another reason to keep working on her own, late thought.

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”.

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