Universal Logic

The method of studying inter-relations between logical systems by the analysis of translations between them was originally introduced by Kolmogoroff, in 1925.
The first known ‘translations’ involving classical logic, intuitionistic logic and modal logic were presented by Kolmogoroff, Glivenko in 1929, Lewis and Langford in 1932, Gödel in two papers of 1933, and Gentzen in 1933. Kolmogoroff, Gentzen and one of Gödel’s papers were developed mainly in order to show relative consistency of the classical logic with respect to the intuitionistic one.
Meanwhile, in spite of these papers dealing with inter-relations between the studied systems, they are not interested in the meaning of the term translation between logics. Several terms are used by the authors, such as translations, interpretations, transformations among others. Since then, translations between logics have been used to different purposes.
Prawitz and Malmnäs (1968) is the first paper in which a general definition for the concept of translation between logical systems is introduced.
Wójcicki (1988) and Epstein (1990) are the first works with a general systematic study on translations between logics, both studying inter-relations between propositional calculi in terms of translations.
Da Silva, D’Ottaviano and Sette (1999) propose a more general definition for the concepts of logic and of translation between logics, in order to single out what seems to be in fact the essential feature of a logical translation, that is, the preservation of consequence relation. In this paper, logics are characterized, in a very general sense, as pairs constituted by a set and a consequence operator, and translations between logics as consequence relation preserving maps.
In this Tutorial, we will begin by a historical survey of the use of translations for the study of inter-relations between logical systems, and will discuss and compare the different approaches to the use of the term ‘translation’.
We will present an initial segment of a theory of translations and will also investigate some connections between translations involving logics and uniformly continuous functions between spaces of their theories.
We also intend to analyse the stronger notion of translation between logics introduced in Coniglio (2005).
We will study an important subclass of translations that preserve and conserve consequence relations, the conservative translations, introduced and investigated in Feitosa (1997) and in Feitosa and D’Ottaviano (2001). We will prove that the class constituted by logics and conservative translations determines a co-complete subcategory of the bi-complete category whose objects are the logics and whose morphisms are the translations between them.
We will present some conservative translations involving classical logic, intuitionistic logics, modal logics, the many-valued logics of Lukasiewicz and Post and several known paraconsistent logics (see D’Ottaviano and Feitosa 1999, D’Ottaviano and Feitosa 2000).
Based on Scheer and D’Ottaviano (2006), we will also initiate the study of a theory of conservative translations involving cumulative non-monotonic logics.
By dealing with the Lindenbaum-Tarski algebraic structures associated to the logics, we will study the problem, several times mentioned in the literature, of the existence of conservative translations from intuitionistic logic and from Lukasiewicz infinite-valued logic into classical logic (see Cignoli, D’Ottaviano and Mundici 2000).
Finally, based on the concept of conservative translation, we will investigate a possible general definition for the concept of duality between logics, and will discuss the concepts of combining and fibring logics.


	Translations between Logics Ítala M. Loffredo D'Ottaviano Centre for Logic, Epistemology and History of Science(CLE) State University of Campinas - Brazil
	The method of studying inter-relations between logical systems by the analysis of translations between them was originally introduced by Kolmogoroff, in 1925. The first known ‘translations’ involving classical logic, intuitionistic logic and modal logic were presented by Kolmogoroff, Glivenko in 1929, Lewis and Langford in 1932, Gödel in two papers of 1933, and Gentzen in 1933. Kolmogoroff, Gentzen and one of Gödel’s papers were developed mainly in order to show relative consistency of the classical logic with respect to the intuitionistic one. Meanwhile, in spite of these papers dealing with inter-relations between the studied systems, they are not interested in the meaning of the term translation between logics. Several terms are used by the authors, such as translations, interpretations, transformations among others. Since then, translations between logics have been used to different purposes. Prawitz and Malmnäs (1968) is the first paper in which a general definition for the concept of translation between logical systems is introduced. Wójcicki (1988) and Epstein (1990) are the first works with a general systematic study on translations between logics, both studying inter-relations between propositional calculi in terms of translations. Da Silva, D’Ottaviano and Sette (1999) propose a more general definition for the concepts of logic and of translation between logics, in order to single out what seems to be in fact the essential feature of a logical translation, that is, the preservation of consequence relation. In this paper, logics are characterized, in a very general sense, as pairs constituted by a set and a consequence operator, and translations between logics as consequence relation preserving maps. In this Tutorial, we will begin by a historical survey of the use of translations for the study of inter-relations between logical systems, and will discuss and compare the different approaches to the use of the term ‘translation’. We will present an initial segment of a theory of translations and will also investigate some connections between translations involving logics and uniformly continuous functions between spaces of their theories. We also intend to analyse the stronger notion of translation between logics introduced in Coniglio (2005). We will study an important subclass of translations that preserve and conserve consequence relations, the conservative translations, introduced and investigated in Feitosa (1997) and in Feitosa and D’Ottaviano (2001). We will prove that the class constituted by logics and conservative translations determines a co-complete subcategory of the bi-complete category whose objects are the logics and whose morphisms are the translations between them. We will present some conservative translations involving classical logic, intuitionistic logics, modal logics, the many-valued logics of Lukasiewicz and Post and several known paraconsistent logics (see D’Ottaviano and Feitosa 1999, D’Ottaviano and Feitosa 2000). Based on Scheer and D’Ottaviano (2006), we will also initiate the study of a theory of conservative translations involving cumulative non-monotonic logics. By dealing with the Lindenbaum-Tarski algebraic structures associated to the logics, we will study the problem, several times mentioned in the literature, of the existence of conservative translations from intuitionistic logic and from Lukasiewicz infinite-valued logic into classical logic (see Cignoli, D’Ottaviano and Mundici 2000). Finally, based on the concept of conservative translation, we will investigate a possible general definition for the concept of duality between logics, and will discuss the concepts of combining and fibring logics.
			References CIGNOLI, R.L.O., D’OTTAVIANO, I.M.L., MUNDICI, D. (2000) Algebraic foundations of many-valued reasoning. Trends in Logic, v. 2, 223 p. Dordrecht : Kluwer Acad. Publ. CONIGLIO , M.E. (2005) Towards a stronger notion of translation between logics. Manuscrito, v. 28, n. 2, p. 231-262. D’OTTAVIANO, I.M.L., FEITOSA, H.A. (1999) Many-valued logics and translations. In: CARNIELLI, W. (Ed.) Multi-valued logics. *Journal of Applied Non-Classical Logics, v. 9, n. 1, p. 121-140. D’OTTAVIANO, I.M.L., FEITOSA, H.A. (2000) Paraconsistent logics and translations. Synthese, Dordrecht , v. 125, n. 1-2, p. 77-95. DA SILVA, J.J., D’OTTAVIANO, I.M.L., SETTE, A.M., (1999) Translations between logics. In: CAICEDO, X. , MONTENEGRO , C.H. (Ed.) Models, algebras and proofs. New York : Marcel Dekker, p. 435-448. (Lectures Notes in Pure and Applied Mathematics, v. 203) EPSTEIN, R.L. (1990) The semantic foundations of logic. Volume 1: Propositional logics. Dordrecht : Kluwer Academic Publishers. FEITOSA, H.A. (1997) Traduções Conservativas* (Conservative tranlations). Doctorate Thesis. Instituto de Filosofia e Ciências Humanas, Universidade Estadual de Campinas, Campinas. FEITOSA, H.A., D’OTTAVIANO, I.M.L. (2001) Conservative translations. *Annals of Pure and Applied Logic, Amsterdam , v. 108, p. 205-227. PRAWITZ, D., MALMNÄS, P.E. (1968) A survey of some connections between classical, intuitionistic and minimal logic. In: SCHMIDT, H. et alii. (Ed.) Contributions to mathematical logic. Amsterdam : North-Holland, p. 215-229. SCHEER, M.C., D’OTTAVIANO, I.M.L. (2006) Operadores de conseqüência cumulativos e traduções entre lógicas cumulativas. Revista Informação e Cognição, São Paulo, v. 4, p. 47-60. WÓJCICKI, R. (1988) Theory of logical calculi*: basic theory of consequence operations. Dordrecht : Kluwer, 1988. (Synthese Library, v. 199)