Categories of Logics and Translations
Mathematical Institute
Mathematical Institute
University of Düsseldorf, Germany
University of Düsseldorf, Germany
email: peter.arndt@uni-duesseldorf.de
email: peter.arndt@uni-duesseldorf.de
In his opening lecture at Unilog 2010 Jean-Yves Beziau named the following as (at the time) main questions of Universal Logic:
In his opening lecture at Unilog 2010 Jean-Yves Beziau named the following as (at the time) main questions of Universal Logic:
1. What is a logic?
1. What is a logic?
2. What is a translation between logics?
2. What is a translation between logics?
3. When are two logics equivalent?
3. When are two logics equivalent?
4. How to combine logics?
4. How to combine logics?
As soon as one agrees on an answer to the questions \What is a logic" and \What is a translation between logics?" one obtains a category whose objects are the logics and whose morphisms are the translations.
As soon as one agrees on an answer to the questions \What is a logic" and \What is a translation between logics?" one obtains a category whose objects are the logics and whose morphisms are the translations.
Immediately some standard questions about categories come to mind: Which (co)limits exist? How can we construct them? Is there a small set of generators, i.e objects from which one can construct all the others? Are there meaningful functors to other categories? Is the category enriched in some other category?
Immediately some standard questions about categories come to mind: Which (co)limits exist? How can we construct them? Is there a small set of generators, i.e objects from which one can construct all the others? Are there meaningful functors to other categories? Is the category enriched in some other category?
In special relation to logic one can further ask: Is there a category theoretic account of known facts or constructions from logic? Can we obtain answers to the third and fourth questions by means of category theory?
In special relation to logic one can further ask: Is there a category theoretic account of known facts or constructions from logic? Can we obtain answers to the third and fourth questions by means of category theory?
In this tutorial we will discuss these questions. Here is an outline of the lectures:
In this tutorial we will discuss these questions. Here is an outline of the lectures:
1. For concreteness we will declare a logic to be a pair (Fm; `) given by a formal language, i.e. an absolutely free algebra over some signature , and a Tarskian consequence relation, but we wil address other notions of logic as well. We will consider several different options for the notion of translation. We will see how to infer properties of the category of logics from those of the category of signatures, in particular the lack or existence of (co)limits, and their construction. We will discuss the meaning of and use of (co)limits) for combination of logics. We will draw the connection to Jansana/Moraschini's recent notion of Leibniz class and their results on the poset of all logics.
1. For concreteness we will declare a logic to be a pair (Fm; `) given by a formal language, i.e. an absolutely free algebra over some signature , and a Tarskian consequence relation, but we wil address other notions of logic as well. We will consider several different options for the notion of translation. We will see how to infer properties of the category of logics from those of the category of signatures, in particular the lack or existence of (co)limits, and their construction. We will discuss the meaning of and use of (co)limits) for combination of logics. We will draw the connection to Jansana/Moraschini's recent notion of Leibniz class and their results on the poset of all logics.
2. Another question of Universal Logic is \How algebraic is Logic?" There are certain measures of algebraicity in category theory, and we will see where logic stands with respect to these [AFLM]. The several notions of translation make a big di
erence here. We then turn to the above questions 3. and 4. Every category comes equipped with a notion of isomorphism and we will see how this is not an adequate answer to question 3. A remedy is to consider equivalence of logics as an additional external notion - this gives rise to a so-called (1; 1)-category, but in the special case of logics they turn out to be easily treatable within usual category theory, following [MM].
2. Another question of Universal Logic is \How algebraic is Logic?" There are certain measures of algebraicity in category theory, and we will see where logic stands with respect to these [AFLM]. The several notions of translation make a big di
erence here. We then turn to the above questions 3. and 4. Every category comes equipped with a notion of isomorphism and we will see how this is not an adequate answer to question 3. A remedy is to consider equivalence of logics as an additional external notion - this gives rise to a so-called (1; 1)-category, but in the special case of logics they turn out to be easily treatable within usual category theory, following [MM].
3. Finally we address the fourth question: Combining logics can be seen as a colimit construction [SSC], and given a satisfactory categorical treatment, whose scope is considerably enhanced by adding a notion of equivalence of logics [A]. If time remains, we address how algebraic is the (1; 1)-category of logics in the sense of the second lecture, and sketch a descent theory for logics, which addresses whether a logic was obtained by bring one of its fragments with another logic.
3. Finally we address the fourth question: Combining logics can be seen as a colimit construction [SSC], and given a satisfactory categorical treatment, whose scope is considerably enhanced by adding a notion of equivalence of logics [A]. If time remains, we address how algebraic is the (1; 1)-category of logics in the sense of the second lecture, and sketch a descent theory for logics, which addresses whether a logic was obtained by bring one of its fragments with another logic.
References
References
[A] P. Arndt, Homotopical Categories of Logics, in "The road to Universal Logic", p.13-58, Festschrift for the 50th Birthday of Jean-Yves Beziau Volume I, Birkhäuser 2014
[A] P. Arndt, Homotopical Categories of Logics, in "The road to Universal Logic", p.13-58, Festschrift for the 50th Birthday of Jean-Yves Beziau Volume I, Birkhäuser 2014
[AFLM] P. Arndt, R. A. Freire, O. O. Luciano, H. L. Mariano, A global glance on categories in Logic, Logica Universalis 1 (2007), 3-39
[AFLM] P. Arndt, R. A. Freire, O. O. Luciano, H. L. Mariano, A global glance on categories in Logic, Logica Universalis 1 (2007), 3-39
[JM1] R. Jansana, T. Moraschini. The poset of all logics I: Interpretations and lattice structure. To appear in the Journal of symbolic Logic, 2021. (pdf)
[JM1] R. Jansana, T. Moraschini. The poset of all logics I: Interpretations and lattice structure. To appear in the Journal of symbolic Logic, 2021. (pdf)
[JM2] R. Jansana, T. Moraschini. The poset of all logics II: Leibniz classes and hierarchy. To appear in the Journal of symbolic Logic, 2021. (pdf)
[JM2] R. Jansana, T. Moraschini. The poset of all logics II: Leibniz classes and hierarchy. To appear in the Journal of symbolic Logic, 2021. (pdf)
[J] E. Jerabek, The ubiquity of conservative translations, Review of Symbolic Logic 5 (2012), no. 4, pp. 666{678.
[J] E. Jerabek, The ubiquity of conservative translations, Review of Symbolic Logic 5 (2012), no. 4, pp. 666{678.
[MM] H. L. Mariano, C. A. Mendes, Towards a good notion of categories of logics, arXiv preprint, http://arxiv.org/abs/1404.3780,2014
[MM] H. L. Mariano, C. A. Mendes, Towards a good notion of categories of logics, arXiv preprint, http://arxiv.org/abs/1404.3780,2014
[SSC] A. Sernadas, C. Sernadas, C. Caleiro, Fibring of logics as a categorial construction, Journal of logic and computation, 9(2)(1999), 149-179.
[SSC] A. Sernadas, C. Sernadas, C. Caleiro, Fibring of logics as a categorial construction, Journal of logic and computation, 9(2)(1999), 149-179.