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  1. Singly Generated Quasivarieties and Residuated Structures.Tommaso Moraschini, James G. Raftery & Johann J. Wannenburg - 2020 - Mathematical Logic Quarterly 66 (2):150-172.
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  • On Rules.Rosalie Iemhoff - 2015 - Journal of Philosophical Logic 44 (6):697-711.
    This paper contains a brief overview of the area of admissible rules with an emphasis on results about intermediate and modal propositional logics. No proofs are given but many references to the literature are provided.
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  • Hereditarily Structurally Complete Superintuitionistic Deductive Systems.Alex Citkin - 2018 - Studia Logica 106 (4):827-856.
    Propositional logic is understood as a set of theorems defined by a deductive system: a set of axioms and a set of rules. Superintuitionistic logic is a logic extending intuitionistic propositional logic \. A rule is admissible for a logic if any substitution that makes each premise a theorem, makes the conclusion a theorem too. A deductive system \ is structurally complete if any rule admissible for the logic defined by \ is derivable in \. It is known that any (...)
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  • Algebraic Logic Perspective on Prucnal’s Substitution.Alex Citkin - 2016 - Notre Dame Journal of Formal Logic 57 (4):503-521.
    A term td is called a ternary deductive term for a variety of algebras V if the identity td≈r holds in V and ∈θ yields td≈td for any A∈V and any principal congruence θ on A. A connective f is called td-distributive if td)≈ f,…,td). If L is a propositional logic and V is a corresponding variety that has a TD term td, then any admissible in L rule, the premises of which contain only td-distributive operations, is derivable, and the (...)
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  • 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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  • An Abelian Rule for BCI—and Variations.Tomasz Kowalski & Lloyd Humberstone - 2016 - Notre Dame Journal of Formal Logic 57 (4):551-568.
    We show the admissibility for BCI of a rule form of the characteristic implicational axiom of abelian logic, this rule taking us from →β to α. This is done in Section 8, with surrounding sections exploring the admissibility and derivability of various related rules in several extensions of BCI.
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  • BCK is Not Structurally Complete.Tomasz Kowalski - 2014 - Notre Dame Journal of Formal Logic 55 (2):197-204.
    We exhibit a simple inference rule, which is admissible but not derivable in BCK, proving that BCK is not structurally complete. The argument is proof-theoretical.
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  • A Syntactic Approach to Unification in Transitive Reflexive Modal Logics.Rosalie Iemhoff - 2016 - Notre Dame Journal of Formal Logic 57 (2):233-247.
    This paper contains a proof-theoretic account of unification in transitive reflexive modal logics, which means that the reasoning is syntactic and uses as little semantics as possible. New proofs of theorems on unification types are presented and these results are extended to negationless fragments. In particular, a syntactic proof of Ghilardi’s result that $\mathsf {S4}$ has finitary unification is provided. In this approach the relation between classical valuations, projective unifiers, and admissible rules is clarified.
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  • Finitary Extensions of the Nilpotent Minimum Logic and (Almost) Structural Completeness.Joan Gispert - 2018 - Studia Logica 106 (4):789-808.
    In this paper we study finitary extensions of the nilpotent minimum logic or equivalently quasivarieties of NM-algebras. We first study structural completeness of NML, we prove that NML is hereditarily almost structurally complete and moreover NM\, the axiomatic extension of NML given by the axiom \^{2}\leftrightarrow ^{2})^{2}\), is hereditarily structurally complete. We use those results to obtain the full description of the lattice of all quasivarieties of NM-algebras which allow us to characterize and axiomatize all finitary extensions of NML.
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  • Almost Structural Completeness; an Algebraic Approach.Wojciech Dzik & Michał M. Stronkowski - 2016 - Annals of Pure and Applied Logic 167 (7):525-556.
  • Structural Completeness in Relevance Logics.J. G. Raftery & K. Świrydowicz - 2016 - Studia Logica 104 (3):381-387.
    It is proved that the relevance logic \ has no structurally complete consistent axiomatic extension, except for classical propositional logic. In fact, no other such extension is even passively structurally complete.
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  • Unification in Intermediate Logics.Rosalie Iemhoff & Paul Rozière - 2015 - Journal of Symbolic Logic 80 (3):713-729.
  • Admissible Rules in the Implication–Negation Fragment of Intuitionistic Logic.Petr Cintula & George Metcalfe - 2010 - Annals of Pure and Applied Logic 162 (2):162-171.
    Uniform infinite bases are defined for the single-conclusion and multiple-conclusion admissible rules of the implication–negation fragments of intuitionistic logic and its consistent axiomatic extensions . A Kripke semantics characterization is given for the structurally complete implication–negation fragments of intermediate logics, and it is shown that the admissible rules of this fragment of form a PSPACE-complete set and have no finite basis.
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  • Fuzzy Logic.Petr Hajek - 2008 - Stanford Encyclopedia of Philosophy.
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