Geodesic
Repetition-free and infinitary analytic calculi for first-order rational Pavelka logic
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 1869-1899

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We present an analytic hypersequent calculus $\textnormal{G}^3\textnormal{\L}\forall$ for first-order infinite-valued Łukasiewicz logic $\textnormal{\L}\forall$ and for an extension of it, first-order rational Pavelka logic $\textnormal{RPL}\forall$; the calculus is intended for bottom-up proof search. In $\textnormal{G}^3\textnormal{\L}\forall$, there are no structural rules, all the rules are invertible, and designations of multisets of formulas are not repeated in any premise of the rules. The calculus $\textnormal{G}^3\textnormal{\L}\forall$ proves any sentence that is provable in at least one of the previously known analytic calculi for $\textnormal{\L}\forall$ or $\textnormal{RPL}\forall$, including Baaz and Metcalfe's hypersequent calculus $\textnormal{G}\textnormal{\L}\forall$ for $\textnormal{\L}\forall$. We study proof-theoretic properties of $\textnormal{G}^3\textnormal{\L}\forall$ and thereby provide foundations for proof search algorithms. We also give the first correct proof of the completeness of the $\textnormal{G}\textnormal{\L}\forall$-based infinitary calculus for prenex $\textnormal{\L}\forall$-sentences, and establish the completeness of a $\textnormal{G}^3\textnormal{\L}\forall$-based infinitary calculus for prenex $\textnormal{RPL}\forall$-sentences.
Keywords: many-valued logic, mathematical fuzzy logic, first-order rational Pavelka logic, proof theory, hypersequent calculus, proof search, infinitary calculus.
Mots-clés : first-order infinite-valued Łukasiewicz logic 
@article{SEMR_2020_17_a40,
     author = {A. S. Gerasimov},
     title = {Repetition-free and infinitary analytic calculi for first-order rational {Pavelka} logic},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {1869--1899},
     publisher = {mathdoc},
     volume = {17},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2020_17_a40/}
}
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A. S. Gerasimov. Repetition-free and infinitary analytic calculi for first-order rational Pavelka logic. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 1869-1899. http://geodesic.mathdoc.fr/item/SEMR_2020_17_a40/