Abstract
We study the general problem of strengthening the logic of a given (partial) (non-deterministic) matrix with a set of axioms, using the idea of rexpansion. We obtain two characterization methods: a very general but not very effective one, and then an effective method which only applies under certain restrictions on the given semantics and the shape of the axioms. We show that this second method covers a myriad of examples in the literature. Finally, we illustrate how to obtain analytic multiple-conclusion calculi for the resulting logics.
This research was funded by FCT/MCTES through national funds and when applicable co-funded EU funds under the project UIDB/50008/2020. Work done under the scope of the CT4L initiative of SQIG at Instituto de Telecomunicações. The authors are indebted to the anonymous referees for their valuable feedback, which helped improve a previous version of this paper.
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Notes
- 1.
For simplicity, in this and other examples, we omit the usual brackets of set notation when describing the truth-tables.
- 2.
Since not all the variables \(q_1,\dots ,q_n,r_1,\dots ,r_m\) need to occur in A, it may well happen that the subformula \({\copyright }(p_1,\dots ,p_k)\) ends up not appearing in the \(\Sigma ^d\)-simple formula B based on \({\copyright }\). For this reason, such a \(\Sigma ^d\)-simple formula can also be based on any available \(k'\)-place connective distinct from \({\copyright }\), as long as \(k'\ge k\) (more precisely, \(k'\) needs to be at least as big as the number of distinct variables \(p_j\) occurring in B).
- 3.
Note that, in our definition, \(\Theta _\Gamma \) is not simply the union of the look-ahead sets of each formula in \(\Gamma \). We not only want \(\Theta _\Gamma \) to be closed for taking prefixes, but we want \(\varepsilon \in \Theta _\Gamma \) even if \(\Gamma =\emptyset \) (a rather pathological case).
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Caleiro, C., Marcelino, S. (2021). On Axioms and Rexpansions. In: Arieli, O., Zamansky, A. (eds) Arnon Avron on Semantics and Proof Theory of Non-Classical Logics. Outstanding Contributions to Logic, vol 21. Springer, Cham. https://doi.org/10.1007/978-3-030-71258-7_3
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