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Abstract Consequence and Logics Abstract Consequence and Logics Essays in honor of Edelcio G. de Souza edited by Alexandre Costa-Leite Contents Introduction Alexandre Costa-Leite On Edelcio G. de Souza PART 1 Abstraction, unity and logic 3 Jean-Yves Beziau Logical structures from a model-theoretical viewpoint 17 Gerhard Schurz Universal translatability: optimality-based justification of (not necessarily) classical logic 37 Roderick Batchelor Abstract logic with vocables 67 Juliano Maranh~ao An abstract definition of normative system 79 Newton C. A. da Costa and Decio Krause Suppes predicate for classes of structures and the notion of transportability 99 Patr´ıciaDel Nero Velasco On a reconstruction of the valuation concept PART 2 Categories, logics and arithmetic 115 Vladimir L. Vasyukov Internal logic of the H − B topos 135 Marcelo E. Coniglio On categorial combination of logics 173 Walter Carnielli and David Fuenmayor Godel's¨ incompleteness theorems from a paraconsistent perspective 199 Edgar L. B. Almeida and Rodrigo A. Freire On existence in arithmetic PART 3 Non-classical inferences 221 Arnon Avron A note on semi-implication with negation 227 Diana Costa and Manuel A. Martins A roadmap of paraconsistent hybrid logics 243 H´erculesde Araujo Feitosa, Angela Pereira Rodrigues Moreira and Marcelo Reicher Soares A relational model for the logic of deduction 251 Andrew Schumann From pragmatic truths to emotional truths 263 Hilan Bensusan and Gregory Carneiro Paraconsistentization through antimonotonicity: towards a logic of supplement PART 4 Philosophy and history of logic 277 Diogo H. B. Dias Hans Hahn and the foundations of mathematics 289 Cassiano Terra Rodrigues A first survey of Charles S. Peirce's contributions to logic: from relatives to quantification 301 Jonas R. B. Arenhart and Sanderson Molick On the very idea of choosing a logic: the role of the background logic 321 Lorenzzo Frade and Abilio Rodrigues Some remarks on logical realism and logical pluralism 341 DuˇskoPrelevi´c Modal rationalism, logical pluralism, and the metaphysical foundation of logic 359 Fabien Schang Quasi-concepts of logic On Edelcio G. de Souza Edelcio Gon¸calves de Souza was born in the village of Bauru (in the State of S˜ao Paulo, Brazil) in the beginning of the sixties, precisely at the year 1960, May 22. In 1963, his family moved to the city of S˜ao Paulo, the place where Edelcio grow up, studied, works and lives up to recent days. In 1979, he started undergraduate studies in Physics at the Universi- dade de S˜ao Paulo (USP), but he decided to change his path and moved towards Philosophy in the eighties. Edelcio got an undergraduate degree and a PhD in the area of Philosophy (USP) supervised by the Brazil- ian philosopher and logician Newton da Costa. Since then, Edelcio’s researches are in the intersection of philosophy of science, mathematics and logic. Particularly, his investigations, up to now, are in topics such as the structure of scientific theories, category theory, model theory and abstract logics. During his career, he had research stays at Stanford University and Miami University. He worked in many Brazilian universities such as Pontif´ıcia Universidade Cat´olica de S˜ao Paulo (where he worked for 21 years), Fundac˜ao Escola de Sociologia e Pol´ıtica do Estado de S˜ao Paulo and also at the first center of philosophical research in Brazil: Faculdade de S˜ao Bento. In 2013, Edelcio started his activities as professor of Logic at the Universidade de S˜ao Paulo. Edelcio is a enthusiastic supporter of the soccer team Sociedade Es- portiva Palmeiras and an Aikido Master who is an strong admirer of Brazilian cuisine, especially feijoada and churrasco.Heismarriedwith Vanessa Boarati and they have two children: Helena and Maria Luiza. Edelcio G. de Souza has a plurality of interests from general abstract logic to the notion of quasi-truth. Departing from Bueno, O; de Souza, E.G. (1996). The concept of quasi-truth. Logique et Analyse, 39(153/154), pp. 183-199; de Souza, E. G. (2000). Multideductive logic and the theoretic- formal unification of physical theories. Synthese, 125, pp.253-262. Edelcio still tries to model and represent formally the essential philo- sophical idea of quasi-truth (developed by his professor Newton da Costa) in the domain of the philosophy of science. This research attempts to x establish the formal limits of any empirical scientific knowledge about reality. Besides that, considering multideductive logic and its role in abstraction and unification of theories in general, he naturally goes to the domain of model theory. His steps in this area are substantial and they were basically conducted with the mathematician Alexandre Au- gusto Martins Rodrigues (in memoriam). Some representative articles of their collaboration are Rodrigues, A.A.M; Miranda Filho, R.C; de Souza, E.G. (2006). In- variance and set-theoretical operations in first-order structures. Reports in Mathematical Logic, 40, pp.207-213; de Souza, E.G; Rodrigues, A.A.M. (2017). On extensions of iso- morphisms of substructures. South American Journal of Logic, 3(1), pp.123-130. A next very natural level also towards generic abstraction is to con- sider Tarskian consequence operators as a source of logical and philo- sophical investigation. Connecting abstract logics, category theory and paraconsistency, we (Edelcio, myself and Diogo H.B. Dias) were able to study in detail a particular technique of paraconsistentization (i.e. dif- ferent ways which can be used to transform any explosive logic into a paraconsistent formalism). The results appeared in de Souza, E.G; Costa-Leite, A; Dias, D.H.B. (2016). On a paracon- sistentization functor in the category of consequence structures. Journal of Applied Non-Classical Logics, 26(3), pp.240-250; de Souza, E.G; Costa-Leite, A; Dias, D.H.B. (2019). Paradeduction in axiomatic formal systems. Logique et Analyse, 62(6), pp.161-176. The title of the book Abstract Consequence and Logics intends to capture and unify Edelcio’s main interests in logic, mathematics and philosophy. Thanks to Prof. Dov Gabbay and to Jane Spurr for all outstanding work they have done and, of course, they are still doing in College Publications by means of avant-garde logical, philosophical and scientific publications. Thanks also to all contributors of this volume for providing original and excellent articles to celebrate Edelcio’s 60th birth- day. The year 2020 has been very weird and complicated for all of us and xi it shows precisely the contradictory and paradoxical condition of human life in Earth: at the same time that soon we will be probably launching James Webb Space Telescope (NASA) and other special technologies, which looks like amazing from our current primary technological per- spective, we, as species, are desperately (and anxiously) searching for a (successful) vaccine against coronavirus. It was in this inconsistent situation, in world quarantine, that Edelcio’s 60 years old arrived and in this context it was celebrated: obrigado por ser esse querido amigo, Tio Ed, sa´ude e paz, um forte abra¸co! Bras´ılia, September 15, 2020 Alexandre Costa-Leite Departamento de Filosofia Universidade de Bras´ılia [email protected] xii Figure 1: Edelcio G. de Souza at the III Col´oquio UnB-USP de L´ogica e Filosofia da L´ogica, Universidade de Bras´ılia, Bras´ılia, 2018. Part 1 Abstraction, unity and logic Logical structures from a model-theoretical viewpoint Jean-Yves Beziau Federal University of Rio de Janeiro (UFRJ), Brazilian Research Council (CNPq), Brazil Dedicated to Edelcio Gon¸calves de Souza for his 60th birthday Abstract We first explain what it means to consider logics as structures. In a second part we discuss the relation between structures and ax- ioms, explaining in particular what axiomatization from a model- theoretical perspective is. We then go on by discussing the place of logical structures among other mathematical structures and by giving an outlook on the varied universe of logical structures. Af- ter that we deal with axioms for logical structures, in a first part in an abstract setting, in a second part dealing with negation. We end by saying a few words about Edelcio. 1 Logics as structures It is usual nowadays to consider a logic as a structure of type L = F; where • F is a set of objects, called formulas. Sets of formulas are called theories. •is a binary relation between theories and formulas, i.e. ⊆ P(F)×F, called consequence relation. The idea is to consider this kind of structure in the same way as other mathematical structures. A structure than can be seen as a model of some axioms, similarly for example to a structure of order O = O; < where • O is a set of objects. • < is a binary relation between objects i.e., <⊆ O × O, called order relation. The situation for logical structures is a bit ambiguous, tricky, mys- terious because there is an interplay between the method used and the 4 objects under study. These objects are logics and the method used is part of logic, namely model theory. There are four mammals of mathe- matical logic, in order of appearance: • Set Theory • Proof Theory • Recursion Theory • Model Theory. In modern times there has been a proliferation of logical systems,that can be simply called logics and that we consider here as logical structures. The study of logical systems can be called metalogic. It is performed using the four mammals of modern logic. Since 1993 the present author has promoted universal logic [3], not as one system among the jungle of logical systems, not even as a super system. Universal logic is a general theory of all these logical systems, in a way similar to universal algebra, which is a general theory of algebraic systems,orsimplyalgebras.
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