Canonical rules
Emil Jeřábek
Source: J. Symbolic Logic Volume 74, Issue 4 (2009), 1171-1205.
Abstract
We develop canonical rules capable of axiomatizing all systems of multiple-conclusion rules over K4 or IPC, by extension of the method of canonical formulas by Zakharyaschev [37]. We use the framework to give an alternative proof of the known analysis of admissible rules in basic transitive logics, which additionally yields the following dichotomy: any canonical rule is either admissible in the logic, or it is equivalent to an assumption-free rule. Other applications of canonical rules include a generalization of the Blok—Esakia theorem and the theory of modal companions to systems of multiple-conclusion rules or (finitary structural global) consequence relations, and a characterization of splittings in the lattices of consequence relations over monomodal or superintuitionistic logics with the finite model property.
Primary Subjects: 03B45, 03B55
Keywords: Inference rule; modal logic; intermediate logic; admissible rule
Full-text: Access denied (no subscription detected)
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.jsl/1254748686
Digital Object Identifier: doi:10.2178/jsl/1254748686
References
Franz Baader and Paliath Narendran, Unification of concept terms in description logics, Journal of Symbolic Computation, vol. 31 (2001), pp. 277--305.
Mathematical Reviews (MathSciNet):
MR1814334
Zentralblatt MATH:
0970.68166
Digital Object Identifier: doi:10.1006/jsco.2000.0426
Patrick Blackburn, Johan van Benthem, and Frank Wolter (editors), Handbook of modal logic, Studies in Logic and Practical Reasoning, vol. 3, Elsevier, Amsterdam, 2007.
Zentralblatt MATH:
1114.03001
Willem J. Blok, Varieties of interior algebras, Ph.D. thesis, University of Amsterdam, 1976.
--------, On the degree of incompleteness in modal logics and the covering relation in the lattice of modal logics, Technical Report 78--07, Department of Mathematics, University of Amsterdam, 1978.
Alexander V. Chagrov, A decidable modal logic with the undecidable admissibility problem for inference rules, Algebra and Logic, vol. 31 (1992), no. 1, pp. 53--55.
Mathematical Reviews (MathSciNet):
MR1215086
Alexander V. Chagrov and Michael Zakharyaschev, Modal logic, Oxford Logic Guides, vol. 35, Oxford University Press, 1997.
Mathematical Reviews (MathSciNet):
MR1464942
W. Wistar Comfort and Stylianos Negrepontis, The theory of ultrafilters, Die Grundlehren der mathematischen Wissenschaften, vol. 211, Springer, Berlin, 1974.
Mathematical Reviews (MathSciNet):
MR396267
Zentralblatt MATH:
0298.02004
Leo L. Esakia, To the theory of modal and superintuitionistic systems, Logical inference. Proceedings of the USSR symposium on the theory of logical inference (V. A. Smirnov, editor), Nauka, Moscow, 1979, pp. 147--172 (Russian).
Mathematical Reviews (MathSciNet):
MR720228
Kit Fine, An ascending chain of $S4$ logics, Theoria, vol. 40 (1974), no. 2, pp. 110--116.
Mathematical Reviews (MathSciNet):
MR536316
--------, Logics containing K4, Part II, Journal of Symbolic Logic, vol. 50 (1985), no. 3, pp. 619--651.
Mathematical Reviews (MathSciNet):
MR805673
Digital Object Identifier: doi:10.2307/2274318
JSTOR: links.jstor.org
Harvey M. Friedman, One hundred and two problems in mathematical logic, Journal of Symbolic Logic, vol. 40 (1975), no. 2, pp. 113--129.
Mathematical Reviews (MathSciNet):
MR369018
Zentralblatt MATH:
0318.02002
Digital Object Identifier: doi:10.2307/2271891
JSTOR: links.jstor.org
Silvio Ghilardi, Unification in intuitionistic logic, Journal of Symbolic Logic, vol. 64 (1999), no. 2, pp. 859--880.
Mathematical Reviews (MathSciNet):
MR1777792
Zentralblatt MATH:
0930.03009
Digital Object Identifier: doi:10.2307/2586506
JSTOR: links.jstor.org
--------, Best solving modal equations, Annals of Pure and Applied Logic, vol. 102 (2000), no. 3, pp. 183--198.
Mathematical Reviews (MathSciNet):
MR1740482
Zentralblatt MATH:
0949.03010
Digital Object Identifier: doi:10.1016/S0168-0072(99)00032-9
Rosalie Iemhoff, On the admissible rules of intuitionistic propositional logic, Journal of Symbolic Logic, vol. 66 (2001), no. 1, pp. 281--294.
Mathematical Reviews (MathSciNet):
MR1825185
Zentralblatt MATH:
0986.03013
Digital Object Identifier: doi:10.2307/2694922
JSTOR: links.jstor.org
--------, Intermediate logics and Visser's rules, Notre Dame Journal of Formal Logic, vol. 46 (2005), no. 1, pp. 65--81.
Mathematical Reviews (MathSciNet):
MR2131547
Zentralblatt MATH:
1102.03032
Digital Object Identifier: doi:10.1305/ndjfl/1107220674
Project Euclid: euclid.ndjfl/1107220674
--------, On the rules of intermediate logics, Archive for Mathematical Logic, vol. 45 (2006), no. 5, pp. 581--599.
Mathematical Reviews (MathSciNet):
MR2231792
Zentralblatt MATH:
1096.03025
Digital Object Identifier: doi:10.1007/s00153-006-0320-8
V. A. Jankov, The relationship between deducibility in the intuitionistic propositional calculus and finite implicational structures, Mathematics of the USSR, Doklady, vol. 4 (1963), no. 4, pp. 1203--1204.
Emil Jeřábek, Admissible rules of modal logics, Journal of Logic and Computation, vol. 15 (2005), no. 4, pp. 411--431.
Mathematical Reviews (MathSciNet):
MR2157725
Zentralblatt MATH:
1077.03011
Digital Object Identifier: doi:10.1093/logcom/exi029
--------, Frege systems for extensible modal logics, Annals of Pure and Applied Logic, vol. 142 (2006), pp. 366--379.
Mathematical Reviews (MathSciNet):
MR2250549
Zentralblatt MATH:
1101.03038
Digital Object Identifier: doi:10.1016/j.apal.2006.04.001
--------, Complexity of admissible rules, Archive for Mathematical Logic, vol. 46 (2007), no. 2, pp. 73--92.
Mathematical Reviews (MathSciNet):
MR2298605
Zentralblatt MATH:
1115.03010
Digital Object Identifier: doi:10.1007/s00153-006-0028-9
--------, Independent bases of admissible rules, Logic Journal of the IGPL, vol. 16 (2008), no. 3, pp. 249--267.
Mathematical Reviews (MathSciNet):
MR2426605
Digital Object Identifier: doi:10.1093/jigpal/jzn004
Miroslav Kat\v etov, A theorem on mappings, Commentationes Mathematicae Universitatis Carolinae, vol. 8 (1967), no. 3, pp. 432--433.
Mathematical Reviews (MathSciNet):
MR229228
Zentralblatt MATH:
0206.51901
Marcus Kracht, An almost general splitting theorem for modal logic, Studia Logica, vol. 49 (1990), no. 4, pp. 455--470.
Mathematical Reviews (MathSciNet):
MR1114726
Digital Object Identifier: doi:10.1007/BF00370158
--------, Review of [ryb:bk?], Notre Dame Journal of Formal Logic, vol. 40 (1999), no. 4, pp. 578--587.
Project Euclid: euclid.ndjfl/1012429722
--------, Modal consequence relations, In Blackburn et al. [h-mod-log?], pp. 491--545.
Paul Lorenzen, Einführung in die operative Logik und Mathematik, Grundlehren der mathematischen Wissenschaften, vol. 78, Springer, 1955 (German).
Mathematical Reviews (MathSciNet):
MR72065
Larisa L. Maksimova and Vladimir V. Rybakov, Lattices of modal logics, Algebra and Logic, vol. 13 (1974), pp. 105--122.
Mathematical Reviews (MathSciNet):
MR363810
Wolfgang Rautenberg, Splitting lattices of logics, Archiv für mathematische Logik und Grundlagenforschung, vol. 20 (1980), no. 3--4, pp. 155--159.
Mathematical Reviews (MathSciNet):
MR603335
Vladimir V. Rybakov, Admissibility of logical inference rules, Studies in Logic and the Foundations of Mathematics, vol. 136, Elsevier, 1997.
Mathematical Reviews (MathSciNet):
MR1454360
--------, Logical consecutions in discrete linear temporal logic, Journal of Symbolic Logic, vol. 70 (2005), no. 4, pp. 1137--1149.
Mathematical Reviews (MathSciNet):
MR2194241
Zentralblatt MATH:
1110.03010
Digital Object Identifier: doi:10.2178/jsl/1129642119
Project Euclid: euclid.jsl/1129642119
--------, Logical consecutions in intransitive temporal linear logic of finite intervals, Journal of Logic and Computation, vol. 15 (2005), no. 5, pp. 663--678.
Mathematical Reviews (MathSciNet):
MR2172414
Zentralblatt MATH:
1091.03002
Digital Object Identifier: doi:10.1093/logcom/exi025
--------, Linear temporal logic with Until and Before on integer numbers, deciding algorithms, Computer science---theory and applications (Dima Grigoriev, John Harrison, and Edward A. Hirsch, editors), Lecture Notes in Computer Science, vol. 3967, Springer, 2006, pp. 322--333.
Frank Wolter, Tense logic without tense operators, Mathematical Logic Quarterly, vol. 42 (1996), no. 1, pp. 145--171.
Mathematical Reviews (MathSciNet):
MR1388183
Zentralblatt MATH:
0858.03019
Digital Object Identifier: doi:10.1002/malq.19960420113
Frank Wolter and Michael Zakharyaschev, Modal decision problems, In Blackburn et al. [h-mod-log?], pp. 427--489.
--------, Undecidability of the unification and admissibility problems for modal and description logics, ACM Transactions on Computational Logic, vol. 9 (2008), no. 4.
Mathematical Reviews (MathSciNet):
MR2444463
Digital Object Identifier: doi:10.1145/1380572.1380574
Michael Zakharyaschev, Modal companions of superintuitionistic logics: syntax, semantics, and preservation theorems, Mathematics of the USSR, Sbornik, vol. 68 (1991), no. 1, pp. 277--289.
Mathematical Reviews (MathSciNet):
MR1025689
--------, Canonical formulas for $K4$. Part I: Basic results, Journal of Symbolic Logic, vol. 57 (1992), no. 4, pp. 1377--1402.
Mathematical Reviews (MathSciNet):
MR1195276
Digital Object Identifier: doi:10.2307/2275372
JSTOR: links.jstor.org
--------, Canonical formulas for $K4$. Part II: Cofinal subframe logics, Journal of Symbolic Logic, vol. 61 (1996), no. 2, pp. 421--449.
Mathematical Reviews (MathSciNet):
MR1394608
Digital Object Identifier: doi:10.2307/2275669
JSTOR: links.jstor.org
--------, Canonical formulas for $K4$. Part III: the finite model property, Journal of Symbolic Logic, vol. 62 (1997), no. 3, pp. 950--975.
Mathematical Reviews (MathSciNet):
MR1472132
Digital Object Identifier: doi:10.2307/2275581
JSTOR: links.jstor.org
