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On rules. (English) Zbl 1336.03037
Summary: 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.

##### MSC:
 03B55 Intermediate logics 03B45 Modal logic (including the logic of norms)
Full Text:
##### References:
 [1] Baader, F., & Snyder, W. (2001). Unification theory, Handbook of automated reasoning: Elsevier. · Zbl 1011.68126 [2] Babenyshev, S; Rybakov, VV, Linear temporal logic LTL: basis for admissible rules, Journal of Logic and Computation, 21, 157-177, (2011) · Zbl 1233.03026 [3] Chagrov, A, A decidable modal logic with undecidable admissibility problem, Algebra and Logic, 31, 53-61, (1992) · Zbl 0782.03005 [4] Cintula, P; Metcalfe, G, Structural completeness in fuzzy logics, Notre Dame Journal of Formal Logic, 50, 153-183, (2009) · Zbl 1190.03027 [5] Cintula, P., & Metcalfe, G. Admissible Rules in the Implication-Negation Fragment of Intuitionistic Logic, Submitted. · Zbl 1225.03011 [6] Citkin, AI, On structurally complete superintuitionistic logics, Soviet Mathematics Doklady, 19, 816-819, (1978) · Zbl 0412.03009 [7] Dzik, W, On structural completeness of some nonclassical predicate calculi, Reports on Mathematical Logic, 5, 19-26, (1995) · Zbl 0343.02038 [8] Dzik, W, Chains of structurally complete predicate logics with the application of prucnal’s substitution, Reports on Mathematical Logic, 38, 37-48, (2004) · Zbl 1058.03013 [9] Ghilardi, S, Unification in intuitionistic logic, Journal of Symbolic Logic, 64, 859-880, (1999) · Zbl 0930.03009 [10] Ghilardi, S, Best solving modal equations, Annals of Pure and Applied Logic, 102, 183-198, (2000) · Zbl 0949.03010 [11] Ghilardi, S, A resolution/tableaux algorithm for projective approximations in IPC, Logic Journal of the IGPL, 10, 227-241, (2002) · Zbl 1005.03504 [12] Ghilardi, S, Unification, finite duality and projectivity in varieties of Heyting algebras, Annals of Pure and Applied Logic, 127, 99-115, (2004) · Zbl 1058.03020 [13] Goudsmit, J.G., & Iemhoff, R. On unification and admissible rules in Gabbay-de Jongh logics. Annals of Pure and Applied Logic. to appear. · Zbl 1316.03016 [14] Goudsmit, JG, The admissible rules of$$B$$$$D$$_{2} and gsc. logic group preprint series, Logic Group Preprint Series, 313, 1-23, (2013) [15] Goudsmit, JG, Admissibility and refutation: some characterisations of intermediate logics, Logic Group Preprint Series, 315, 1-23, (2013) [16] Iemhoff, R, On the admissible rules of intuitionistic propositional logic, Journal of Symbolic Logic, 66, 281-294, (2001) · Zbl 0986.03013 [17] Iemhoff, R, Intermediate logics and visser’s rules, Notre Dame Journal of Formal Logic, 46, 65-81, (2005) · Zbl 1102.03032 [18] Iemhoff, R. A syntactic approach to unification in transitive reflexive modal logics. Notre Dame Journal of Formal Logic. to appear. · Zbl 1436.03133 [19] Iemhoff, R; Metcalfe, G, Proof theory for admissible rules, Annals of Pure and Applied Logic, 159, 171-186, (2009) · Zbl 1174.03024 [20] Iemhoff, R., & Metcalfe, G. (2009). Hypersequent systems for the admissible rules of modal and intermediate logics In Artemov, S., & Nerode, A. (Eds.), Lecture Notes in Computer Science 5407 - Proceedings of LFCS ’09, (pp. 230-245): Springer. · Zbl 1211.03037 [21] Iemhoff, R., & Rozière, P. Unification in fragments of intermediate logics. Journal of Symbolic Logic. to appear. [22] Jeřábek, E, Admissible rules of modal logics, Journal of Logic and Computation, 15, 411-431, (2005) · Zbl 1077.03011 [23] Jeřábek, E, Complexity of admissible rules, Archive for Mathematical Logic, 46, 73-92, (2007) · Zbl 1115.03010 [24] Jeřábek, E, Independent bases of admissible rules, Logic Journal of the IGPL, 16, 249-267, (2008) · Zbl 1146.03008 [25] Jeřábek, E, Canonical rules, Journal of Symbolic Logic, 74, 1171-1205, (2009) · Zbl 1186.03045 [26] Jeřábek, E, Admissible rules of łukasiewicz logic, Journal of Logic and Computation, 20, 425-447, (2010) · Zbl 1216.03042 [27] Jeřábek, E, Bases of admissible rules of łukasiewicz logic, Journal of Logic and Computation, 20, 1149-1163, (2010) · Zbl 1216.03043 [28] Jeřábek, E. (2011). Blending margins, preprint, pp.1-9. · Zbl 0709.03017 [29] Johansson, I, Der minimalkalkül, ein reduzierter intuitionistischer formalismus, Compositio Mathematica, 4, 119-136, (1937) · JFM 62.1045.08 [30] Lorenzen, P. (1955). Einführung in die operative Logic und Mathematik, Volume 78 of Grundlehren der mathematischen Wissenschaften: Springer. · Zbl 0782.03005 [31] Minari, P; Wronski, A, The property (HD) in intermediate logics: a partial solution of a problem of H. ono, Reports on Mathematical Logic, 22, 21-25, (1988) · Zbl 0696.03009 [32] Mints, G, Derivability of admissible rules, in studies in constructive mathematics and mathematical logic, Zap. Nauchn. Sem. LOMI, V, 85-89, (1972) [33] Odintsov, S; Rybakov, VV, Unification and admissible rules for paraconsistent minimal johanssons logic J and positive intuitionistic logic IPC+, Annals of Pure and Applied Logic, 164, 771-784, (2013) · Zbl 1323.03029 [34] Olson, JS; Raftery, JG; Alten, CJV, Structural completeness in substructural logics, Logic Journal of the IGPL, 16, 453-495, (2008) · Zbl 1168.03012 [35] Pogorzelski, WA, Structural completeness of the propositional calculus, Bulletin de l’Academie Polonaise des Sciences, Série de sciences mathématiques, astronomiques et physiques, 19, 349-351, (1971) · Zbl 0214.00704 [36] Pogorzelski, WA; Prucnal, T, Structural completeness of the first-order predicate calculus, Mathematical Logic Quarterly, 21, 315-320, (1975) · Zbl 0312.02011 [37] Prucnal, T, On the structural completeness of some pure implicational propositional calculi, Studia Logica, 32, 45-50, (1973) · Zbl 0268.02013 [38] Prucnal, T, On two problems of harvey Friedman, Studia Logica, 38, 247-262, (1979) · Zbl 0436.03018 [39] Prucnal, T, Structural completeness of medvedev’s propositional calculus, Reports on Mathematical Logic, 6, 103-105, (1976) · Zbl 0358.02024 [40] Rozière, P. (1992). Regles admissibles en calcul propositionnel intuitionniste, PhD thesis, Université Paris VII. · Zbl 1058.03013 [41] Rybakov, V, Even tabular modal logics sometimes do not have independent base for admissible rules, Bulletin of the Section of Logic, 24, 37-40, (1995) · Zbl 0847.03014 [42] Rybakov, V. (1997). Admissibility of Logical Inference Rules: Elsevier. · Zbl 0872.03002 [43] Rybakov, V, Rules admissible in transitive temporal logic TS4, sufficient condition, Theoretical Computer Science, 411, 4323-4332, (2010) · Zbl 1209.03011 [44] Rybakov, V, Writing out unifiers in linear temporal logic, Journal of Logic and Computation, 22, 1199-1206, (2012) · Zbl 1259.03029 [45] Skura, R, A complete syntactical characterization of the intuitionistic logic, Reports on Mathematical Logic, 23, 75-80, (1989) · Zbl 0809.03007 [46] Slaney, J; Meyer, R, A structurally complete fragment of relevant logic, Notre Dame Journal of Formal Logic, 33, 561-566, (1992) · Zbl 0798.03019 [47] Słomczyńska, K, Algebraic semantics for the $$(↔ ,¬ ¬ )$$-fragment of IPC, Mathematical Logic Quarterly, 58, 29-37, (2012) · Zbl 1245.03016 [48] Troelstra, A., & Schwichtenberg, H. (2000). Basic Proof Theory: Cambridge University Press. · Zbl 0957.03053 [49] Visser, A, Rules and arithmetics, Notre Dame Journal of Formal Logic, 40, 116-140, (1999) · Zbl 0968.03071 [50] Visser, A, Substitutions of $$Σ _{1}^{0}$$-sentences: explorations between intuitionistic propositional logic and intuitionistic arithmetic, Annals of Pure and Applied Logic, 114, 227-271, (2002) · Zbl 1009.03029 [51] Williamson, T, An alternative rule of disjunction in modal logic, Notre Dame Journal of Formal Logic, 33, 89-100, (1992) · Zbl 0765.03011 [52] Williamson, T, Some admissible rules in nonnormal modal systems, Notre Dame Journal of Formal Logic, 34, 378-400, (1993) · Zbl 0803.03008 [53] Wojtylak, P, On a problem of H. Friedman and its solution by T. prucnal, Reports on Mathematical Logic, 38, 69-86, (2004) · Zbl 1053.03017 [54] Wolter, F; Zakharyaschev, M, Undecidability of the unification and admissibility problems for modal and description logics, ACM Transactions on Computational Logic, 9, article 25, (2008) · Zbl 1367.03026 [55] Wroński, A, Transparent unification problem, Reports on Mathematical Logic, 29, 105-107, (1995) · Zbl 0865.08002 [56] Wójcicki, R. (1988). Theory of Logical Calculi: Basic Theory of Consequence Operations, Synthese Library / Vol. 199: Kluwer Academic Publishers. · Zbl 0774.03005 [57] Zakharyaschev, M, Modal companions of superintuitionistic logics: syntax, semantics, and preservation theorems, Mathematics of the USSR, Sbornik, 68, 277-289, (1991) · Zbl 0709.03017 [58] Zakharyaschev, M, Canonical formulas for K4. part I: basic results, Journal of Symbolic Logic, 57, 1377-1402, (1992) · Zbl 0774.03005 [59] Zakharyaschev, M, Canonical formulas for K4. part II: cofinal subframe logics, Journal of Symbolic Logic, 61, 421-449, (1996) · Zbl 0884.03014 [60] Zakharyaschev, M, Canonical formulas for K4. part III: the finite model property, Journal of Symbolic Logic, 62, 950-975, (1997) · Zbl 0893.03006
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