Abstract
Following a proposal by Kooi and Tamminga, we introduce a conservative translation manual for every four-valued truth-functional propositional logic into a modal logic. However, the application of this translation does not preserve the intuitive reading of the truth-values for every four-valued logic. In order to solve this problem, we modify the translation manual and prove its conservativity by exploiting the method of generalized truth-values.
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Notes
The idea of exploiting the method of generalized truth-values firstly appeared in Kubyshkina (2017), but no technical details were provided.
For more details on the distinction between designated truth values for the premises and for the conclusion see Cobreros et al. (2012).
In this sense, LRA is a logic of generalized truth values.
For instance, if one considers that the weakest system that is suitable for knowledge representation should be the system T, with \(K\phi \rightarrow \phi \) as the characteristic axiom, the following property can be imposed on the frames: if \(X \in N_{w}\), then \(w \in X\). It is not difficult to see that this property will not change the structure of the conservativity proof that follows. Moreover, having this property the definition of the translations T1 and F1 can be simplified: T1(p) can be defined as Kp, and F1(p) can be defined as \(K \lnot p\).
\(\mathfrak {X}(\varGamma ) \models _{M}^{w} \mathfrak {Y}(\phi )\) is an abbreviation for ‘if \(M, w \models _{M} \mathfrak {X}(\psi )\) for all \(\psi \in \varGamma \), then \(M, w \models _{M} \mathfrak {Y}(\phi )\)’.
One can keep \(\emptyset \) without altering the proofs that follow. To do so, it suffices to define all the logical operators in a way that they do not distinguish the \(\emptyset \) and \(\{F1\}\) values. For example, if a formula \(\phi \) takes value \(\{F1\}\) and \(\lnot _{4} \phi \) takes value \(\{T1\}\), and if a formula \(\psi \) takes value \(\emptyset \), then \(\lnot \psi \) takes value \(\{T1\}\).
It is not difficult to see, that an order \(x \le y \Leftrightarrow x = x \wedge y\) and a linear lattice of truth-values can be defined for LRA.
The functions \(v^{4}_{1}(\phi ),\ldots ,v^{4}_{n}(\phi )\) are possible \(v^{4}\) valuations of \(\phi \). Keeping in mind the Lemma 6, where each valuation was associated with a 4-w-equivalent world, the use of the subscripts 1, ..., n facilitates notationally the association of the valuations \(v^{4}_{1}(\phi ),\ldots ,v^{4}_{n}(\phi )\) with their 4-w-equivalent worlds \(w_{1}\), ..., \(w_{n}\) respectively.
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Acknowledgements
I am grateful for helpful comments and suggestions by two anonymous referees of Synthese. Thanks are also due to Paul Egré, Andreas Herzig, Mattia Petrolo, Nicholas J. J. Smith and Allard Tamminga for their feedbacks on previous drafts of some of this material.
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Kubyshkina, E. Conservative translations of four-valued logics in modal logic. Synthese 198 (Suppl 22), 5555–5571 (2021). https://doi.org/10.1007/s11229-019-02139-3
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DOI: https://doi.org/10.1007/s11229-019-02139-3