Representation theory of logics: a categorial approach
Abstract
The major concern in the study of categories of logics is to describe condition for preservation, under the a method of combination of logics, of meta-logical properties. Our complementary approach to this field is study the "global" aspects of categories of logics in the vein of the categories $\Ss, \Ls, \cA_s$ studied in \cite{AFLM3}. All these categories have good properties however the category of logics $\cL$ does not allow a good treatment of the "identity problem" for logics (\cite{Bez}): for instance, the presentations of "classical logics" (e.g., in the signature $\{\neg, \vee\}$ and $\{\neg',\rightarrow'\}$) are not $\Ls$-isomorphic. In this work, we sketch a possible way to overcome this "defect" (and anothers) by a mathematical device: a representation theory of logics obtained from category theoretic aspects on (Blok-Pigozzi) algebraizable logics. In this setting we propose the study of (left and right) "Morita equivalence" of logics and variants. We introduce the concepts of logics (left/right)-(stably) -Morita-equivalent and show that the presentations of classical logics are stably Morita equivalent but classical logics and intuitionist logics are not stably-Morita-equivalent: they are only stably-Morita-adjointly related.
- Publication:
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arXiv e-prints
- Pub Date:
- May 2014
- DOI:
- 10.48550/arXiv.1405.2429
- arXiv:
- arXiv:1405.2429
- Bibcode:
- 2014arXiv1405.2429C
- Keywords:
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- Mathematics - Category Theory;
- 18C99;
- 3B99
- E-Print:
- 10 pages