Abstract
We extend classical work by Janusz Czelakowski on the closure properties of the class of matrix models of entailment relations—nowadays more commonly called multiple-conclusion logics—to the setting of non-deterministic matrices (Nmatrices), characterizing the Nmatrix models of an arbitrary logic through a generalization of the standard class operators to the non-deterministic setting. We highlight the main differences that appear in this more general setting, in particular: the possibility to obtain Nmatrix quotients using any compatible equivalence relation (not necessarily a congruence); the problem of determining when strict homomorphisms preserve the logic of a given Nmatrix; the fact that the operations of taking images and preimages cannot be swapped, which determines the exact sequence of operators that generates, from any complete semantics, the class of all Nmatrix models of a logic. Many results, on the other hand, generalize smoothly to the non-deterministic setting: we show for instance that a logic is finitely based if and only if both the class of its Nmatrix models and its complement are closed under ultraproducts. We conclude by mentioning possible developments in adapting the Abstract Algebraic Logic approach to logics induced by Nmatrices and the associated equational reasoning over non-deterministic algebras.
Second reader: Hanamantagouda P. Sankappanavar
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Notes
- 1.
Czelakowski’s results are reviewed in Section 3 of J. M. Font and R. Jansana’s chapter included in the present book.
- 2.
For the sake of readability, we write \({\textbf{A}}_1=\langle {A_1,\cdot _1}\rangle \) instead of \({\textbf{A}}_1=\langle {A_1,\cdot _{{\textbf{A}}_1}}\rangle \), etc.
- 3.
Recall that an ultrafilter on I is a set \({\mathcal U}\subseteq \wp (I)\) such that \(\varnothing \not \in {\mathcal U}\), if \(X\subseteq Y\subseteq I\) and \(X\in {\mathcal U}\) then \(Y\in {\mathcal U}\), if \(X,Y\in {\mathcal U}\) then \(X\cap Y\in {\mathcal U}\), and for every \(X\subseteq I\) either \(X\in {\mathcal U}\) or \(I\setminus X\in {\mathcal U}\).
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Research funded by FCT/MCTES through national funds and when applicable co-funded by EU under the project UIDB/50008/2020 of Instituto de Telecomunicações.
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Caleiro, C., Marcelino, S., Rivieccio, U. (2024). Some More Theorems on Structural Entailment Relations and Non-deterministic Semantics. In: Malinowski, J., Palczewski, R. (eds) Janusz Czelakowski on Logical Consequence. Outstanding Contributions to Logic, vol 27. Springer, Cham. https://doi.org/10.1007/978-3-031-44490-6_12
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