Abstract
We discuss nonmonotonic reasoning in terms of consequence relations and corresponding operators. Based on the matrix consequence that gives the monotonic case, we define a restricted matrix consequence that illustrates the nonmonotonic case. The latter is a generalization of the relation of logical friendliness introduced by D. Makinson. We prove that any restricted single matrix consequence, although it may be nonmonotonic, is always weakly monotonic and, in the case of a finite matrix, the restricted matrix consequence is very strongly finitary. Further, by modifying the definition of logical friendliness relation formulated specifically in a proof-theoretic manner, we show a possibility of obtaining other reflexive nonmonotonic consequence relations, for which a limited result towards finitariness is proved. This leads to numerous questions about nonmonotonic consequence relations in the segment between the monotonic consequence relation based on intuitionistic propositional logic and logical friendliness.
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Notes
See, e.g., [6].
The property \((\text {j}^{\dag })\) appears only as a conclusion in (xi). Thus, if we challenge \((\text {b}^{\dag })\), we can possibly have \((\text {j}^{\dag })\) only in some exotic situations. Therefore, we will not be considering it.
Compare this with a remark by E. W. Beth, “The replacement of one axiom by another which is not equivalent to the first thus conversely may affect the import of the remaining axioms.” Cf. [2], chapter II.
The difference between parameters and constants is apparent only in semantics.
See, e.g., [3], theorem 1.3.22.
Makinson proved the strong finitariness of friendliness in a stronger form, however, weaker than the one exposed in this proposition. The condition imposed on \(\Lambda \) in Makinson’s version is \(\Delta \subseteq \Lambda \subseteq \Gamma \).
For the equality \(\mathbf{Cn} _{\mathbf{B }_2}=\mathbf{Cn} _{\textsf {Cl}}\) see [4], Sect. 4.3.5.
Compare with the notion of a sequent in [14].
See, e.g., [11], theorem IX.5.4.
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Acknowledgements
I would like to express my gratitude to one of the two referees whose constructive criticism led to a significant revision of the original version of this paper and, hopefully, to the clarity of the results presented.
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Muravitsky, A. On Nonmonotonic Consequence Relations. Log. Univers. 15, 227–249 (2021). https://doi.org/10.1007/s11787-021-00275-4
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DOI: https://doi.org/10.1007/s11787-021-00275-4