Abstract
We introduce a family of modal expansions of Łukasiewicz logic that are designed to accommodate modal translations of generalized basic logic (as formulated with exchange, weakening, and falsum). We further exhibit algebraic semantics for each logic in this family, in particular showing that all of them are algebraizable in the sense of Blok and Pigozzi. Using this algebraization result and an analysis of congruences in the pertinent varieties, we establish that each of the introduced modal Łukasiewicz logics has a local deduction-detachment theorem. By applying Jipsen and Montagna’s poset product construction, we give two translations of generalized basic logic with exchange, weakening, and falsum in the style of the celebrated Gödel-McKinsey-Tarski translation. The first of these interprets generalized basic logic in a modal Łukasiewicz logic in the spirit of the classical modal logic S4, whereas the second interprets generalized basic logic in a temporal variant of the latter.
This project received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 670624).
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Notes
- 1.
Recall that formulas are constructed recursively by stipulating that p is a formula for each \(p\in \mathsf{Var}\), and further that if \(\omega \) is an n-ary connective symbol and \(\varphi _1,\dots ,\varphi _n\) are formulas, then so is \(\omega (\varphi _1,\dots ,\varphi _n)\). As usual, we write binary connectives using infix notation.
- 2.
Most studies refer to these algebras as bounded commutative GBL-algebras or GBL\(_{ewf}\)-algebras. Because we always assume boundedness and commutativity, we call them GBL-algebras in order to simplify terminology.
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Fussner, W., Zuluaga Botero, W. (2021). Some Modal and Temporal Translations of Generalized Basic Logic. In: Fahrenberg, U., Gehrke, M., Santocanale, L., Winter, M. (eds) Relational and Algebraic Methods in Computer Science. RAMiCS 2021. Lecture Notes in Computer Science(), vol 13027. Springer, Cham. https://doi.org/10.1007/978-3-030-88701-8_11
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