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On semilattice relevant logics
Author(s) -
Kashima Ryo
Publication year - 2003
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.200310043
Subject(s) - semilattice , mathematics , equivalence (formal languages) , sequent , discrete mathematics , commutative property , pure mathematics , algebra over a field , semigroup
The semilattice relevant logics ∪ R , ∪ T , ∪ RW , and ∪ TW (slightly different from the orthodox relevant logics R , T , RW , and TW ) are defined by semilattice models in which conjunction and disjunction are interpreted in a natural way. For each of them, there is a cut‐free labelled sequent calculus with plural succedents (like LK ). We prove that these systems are equivalent, with respect to provable formulas, to the restricted systems with single succedents (like LJ ). Moreover, using this equivalence, we give a new Hilbert‐style axiomatizations for ∪ R and ∪ T and prove equivalence between two semantics (commutative monoid and distributive semilattice) for the contractionless logics ∪ RW and ∪ TW .