Remarks Concerning Semantic and Syntactic Contributions to the History of Logic
Universidade Estadual de Maringá
Abstract. What the historian seeks to identify and analyze in the sources of the history of logic is the notion of logical contribution. This notion was analyzed by Vega Reñon (1997, p. 40–45), whose conceptual framework we summarize here. This author characterizes the definition of logical contribution in an intuitively recursive way. Let 𝑇 be a text. In principle, 𝑇 is logically significant if 𝑇 has to do with the presuppositions, questions, or applications in the field of knowledge covered by logic in a certain historical landmark 𝑀. Consequently,
(i) if 𝑇𝐿 is a logically significant text, then 𝑇𝐿∗ is a logical contribution with respect to the notions, problems, methods, or results that have characterized the cultivation of logic as a discipline at some moment of its historical course;
(ii) 𝑇𝐿∗𝑝 is a potential logical contribution in a determined historical landmark 𝑀, if 𝑇𝐿∗ can be recognized by practitioners of logic in 𝑀 as a logical contribution;
(iii) 𝑇𝐿∗𝑒 is an effective logical contribution in a determined historical mark 𝑀, if 𝑇𝐿∗ can be recognized or assumed by practitioners of logic in 𝑀 as a logical contribution;
(iv) 𝑇𝐿∗𝑙𝑡 is an historical contribution in the broad sense (a memorable contribution), if there is some historical landmark 𝑀 from which 𝑇𝐿∗ comes to be seen as either a potential or effective logical contribution;
(v) 𝑇𝐿∗𝑠𝑡 is an historical contribution in the strict sense, if there is a historical mark 𝑀 in which 𝑇𝐿∗ was an effective logical contribution 𝑇𝐿∗𝑒.
We propose that these notions can be adapted to the historiography of every logic or family of logical systems. For instance, in the case of the paraconsistent logics, we can specify the notion of logical contribution, and introduce by analogy the notion of contribution to the history of paraconsistent logic 𝑇𝐿∗𝑃.
To the historiographical categories presented so far, which give prominence to historical contextual elements, may be added other conceptual categories that are applicable and quite appropriate to the comparative historical study of logical systems. Categories such as syntactic and semantic contribution to the history of logic may be also required by an historiographical point of view as internal criteria. These categories permit the determination of clear chains of theoretical formation and make evident the interdependencies and intercorrelations of authors among themselves and among their logical contributions and their communities. Of course, these categories can also be refined with the objective of better assessing the historical development of a logical theory, a logic, or an entire branch of logical systems.
We consider a contribution to the history of logic syntactic or semantic depending on the degree of conscious motivation demonstrated by an author at the moment of the proposal of his/her contribution to the field of logic. Intentionality is thus a decisive factor in determining if a contribution to the history of logic is purely syntactic or also semantic. These notions can be stated more precisely as follows:
1. A syntactic or accidental contribution to the history of logic (or a purely formal one) occurs when an author proposes a logical innovation (in either the narrow or the strict sense) in accordance with interpretations proper to the historical mark 𝑀 within which it appears, and he or she does not offer an explicit interpretation of it or has little or no consciousness of what he or she has just proposed.
2. A semantic or intentional contribution to the history of logic occurs when a logical innovation (in either the narrow or the strict sense), in accordance with interpretations proper to the historical mark 𝑀 within which it appears, is introduced with an explicit motivation and with full awareness on the part of its contributor (see Gomes (2013), p. 9–12 e Gomes & D’Ottaviano (2017), p. 33–36).
These new categories permit objective analysis, including analysis of possible priority disputes in the history of logic, and also establish clear criteria which historically enumerate different logical systems, whether or not they are mutually dependent in a branch of the development of logic. In addition, such intentional character must be readily recognized in the relevant historical landmark 𝑀 in the community of practitioners of logic at the time.
For example, the history of paraconsistent logic, founded on these categories, is supported by the historiographic premise according to which effective logical contributions 𝑇𝐿∗𝑃 must be conscious or recognized within their historical-theoretical landmark – that is, within the state-of-the-art in the community of practitioners at the time (Gomes & D’Ottaviano, 2017, p. 31-45). For this reason, in order to consider a paraconsistent logician in the strict sense to be a precursor or pioneer as a founder of the theoretical field of paraconsistent logic, it is necessary that his/her contributions be intentional or semantic and that they be chronologically appropriate.
In light of the reasons given above, purely chronological criteria appear to be simplistic and inefficient. If such criteria were enough to determine priority in the discovery of paraconsistency, its inauguration would be placed far back in the history of formal Western logic. And just as it would be improper to attribute to Aristotle the notable role of the founder of paraconsistency, it would likewise be improper to do so with regard to authors such as Peter of Spain, William of Ockham, and others. Although parts of their logical theories may at present be considered paraconsistent in the broad sense, these thinkers did not perceive the unusual and non-classical character that their theories implied. A similar situation is found in the cases of Kolmogorov, Johansson, Nelson, and in the cases of other thinkers of our own era. These authors cannot, under the view we have adopted here, be considered founders of paraconsistency, even though their logical theories can today be interpreted and considered as paraconsistent in the broad or even in the strict sense. In terms of the historiographical premises here assumed, the contributions of Stanisław Jaśkowski (1906–1965) and Newton da Costa (1929) are situated on another level. Motivated by problems arising from the presence of contradictions in specific rational contexts, they proposed and developed logical systems capable of dealing with contradictions or inconsistencies without the trivialization of the theories implied by these systems, completely fulfilling the requirements of historical postulates (1) and (2) above.
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