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The Poset of All Logics III: Finitely Presentable Logics

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Abstract

A logic in a finite language is said to be finitely presentable if it is axiomatized by finitely many finite rules. It is proved that binary non-indexed products of logics that are both finitely presentable and finitely equivalential are essentially finitely presentable. This result does not extend to binary non-indexed products of arbitrary finitely presentable logics, as shown by a counterexample. Finitely presentable logics are then exploited to introduce finitely presentable Leibniz classes, and to draw a parallel between the Leibniz and the Maltsev hierarchies.

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Acknowledgements

Thanks are due to James G. Raftery for rising the question about whether the theory of the Maltsev and Leibniz hierarchy could be, to some extent, unified. In addition, we gratefully acknowledge the comments and suggestions of the anonymous referees, which helped to improve the presentation of the paper. This research was partially supported by the research Grant 2017 SGR 95 from the government of Catalonia and by the Research Project MTM2016-74892-P from the government of Spain, which include feder funds from the European Union. Furthermore, the second author was supported by the Grant CZ.02.2.69/0.0/0.0/17_050/0008361, OPVVV MŠMT, MSCA-IF Lidské zdroje v teoretické informatice, and by the Beatriz Galindo grant BEAGAL18/00040 of the Spanish Ministry of Science, Innovation and Universities.

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Correspondence to Tommaso Moraschini.

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Jansana, R., Moraschini, T. The Poset of All Logics III: Finitely Presentable Logics. Stud Logica 109, 539–580 (2021). https://doi.org/10.1007/s11225-020-09916-z

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