Abstract
In this paper, we study some classes of logical structures from the universal logic standpoint, viz., those of the Tarski- and the Lindenbaum-types. The characterization theorems for the Tarski- and two of the four different Lindenbaum-type logical structures have been proved as well. The separations between the five classes of logical structures, viz., the four Lindenbaum-types and the Tarski-type have been established via examples. Finally, we study the logical structures that are of both Tarski- and a Lindenbaum-type, show their separations, and end with characterization, adequacy, minimality, and representation theorems for one of the Tarski–Lindenbaum-type logical structures.
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Notes
These conditions are usually stated in a compact form as follows. A transitive set well-ordered by \(\in \) is a von Neumann ordinal. We avoid the terminology here as this is beside the main aim of the present article.
The presentations of these results here differ from those in [5], mainly due to differing terminology. One can, however, easily prove that our formulations are equivalent to the original ones.
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Acknowledgements
The authors wish to express their gratitude towards Prof. Peter Arndt, Lt. Prof. John Corcoran for their suggestions and encouragement, and two anonymous referees for their comments on the extended abstract of an earlier version of the paper submitted to the 9th Indian Conference on Logic and its Applications (ICLA), 2021.
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The first author wishes to dedicate this paper to his grandparents.
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Roy, S., Basu, S.S. & Chakraborty, M.K. Lindenbaum-Type Logical Structures. Log. Univers. 17, 69–102 (2023). https://doi.org/10.1007/s11787-023-00322-2
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DOI: https://doi.org/10.1007/s11787-023-00322-2