7 found
  1.  14
    Admissible Rules and the Leibniz Hierarchy.James G. Raftery - 2016 - Notre Dame Journal of Formal Logic 57 (4):569-606.
    This paper provides a semantic analysis of admissible rules and associated completeness conditions for arbitrary deductive systems, using the framework of abstract algebraic logic. Algebraizability is not assumed, so the meaning and significance of the principal notions vary with the level of the Leibniz hierarchy at which they are presented. As a case study of the resulting theory, the nonalgebraizable fragments of relevance logic are considered.
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  2.  19
    Adding Involution to Residuated Structures.Nikolaos Galatos & James G. Raftery - 2004 - Studia Logica 77 (2):181 - 207.
    Two constructions for adding an involution operator to residuated ordered monoids are investigated. One preserves integrality and the mingle axiom x 2x but fails to preserve the contraction property xx 2. The other has the opposite preservation properties. Both constructions preserve commutativity as well as existent nonempty meets and joins and self-dual order properties. Used in conjunction with either construction, a result of R.T. Brady can be seen to show that the equational theory of commutative distributive residuated lattices (without involution) (...)
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  3.  42
    Order algebraizable logics.James G. Raftery - 2013 - Annals of Pure and Applied Logic 164 (3):251-283.
    This paper develops an order-theoretic generalization of Blok and Pigozziʼs notion of an algebraizable logic. Unavoidably, the ordered model class of a logic, when it exists, is not unique. For uniqueness, the definition must be relativized, either syntactically or semantically. In sentential systems, for instance, the order algebraization process may be required to respect a given but arbitrary polarity on the signature. With every deductive filter of an algebra of the pertinent type, the polarity associates a reflexive and transitive relation (...)
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  4.  6
    Conserving Involution in Residuated Structures.Ai-ni Hsieh & James G. Raftery - 2007 - Mathematical Logic Quarterly 53 (6):583-609.
    This paper establishes several algebraic embedding theorems, each of which asserts that a certain kind of residuated structure can be embedded into a richer one. In almost all cases, the original structure has a compatible involution, which must be preserved by the embedding. The results, in conjunction with previous findings, yield separative axiomatizations of the deducibility relations of various substructural formal systems having double negation and contraposition axioms. The separation theorems go somewhat further than earlier ones in the literature, which (...)
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  5.  9
    Inconsistency Lemmas in Algebraic Logic.James G. Raftery - 2013 - Mathematical Logic Quarterly 59 (6):393-406.
  6.  7
    A Finite Model Property for RMImin.Ai-ni Hsieh & James G. Raftery - 2006 - Mathematical Logic Quarterly 52 (6):602-612.
    It is proved that the variety of relevant disjunction lattices has the finite embeddability property. It follows that Avron's relevance logic RMImin has a strong form of the finite model property, so it has a solvable deducibility problem. This strengthens Avron's result that RMImin is decidable.
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  7.  23
    Willem Blok's Work in Algebraic Logic.James G. Raftery - 2004 - Studia Logica 76 (2):155 - 160.