Unified correspondence and proof theory for strict implication

The unified correspondence theory for distributive lattice expansion logics (DLE-logics) is specialized to strict implication logics. As a consequence of a general semantic consevativity result, a wide range of strict implication logics can be conservatively extended to Lambek Calculi over the bounded distributive full non-associative Lambek calculus (BDFNL). Many strict implication sequents can be transformed into analytic rules employing one of the main tools of unified correspondence theory, namely (a suitably modified version of) the Ackermann lemma based algorithm $\msf{ALBA}$. Gentzen-style cut-free sequent calculi for BDFNL and its extensions with analytic rules which are transformed from strict implication sequents, are developed.

[1]  Kazushige Terui,et al.  Expanding the Realm of Systematic Proof Theory , 2009, CSL.

[2]  Willem Conradie,et al.  Chapter 1 Unified Correspondence , 2014 .

[3]  Magdalena Fabisiak,et al.  THE LOGIC K OF STRICT IMPLICATION AND ITS RELATIVES , 2004 .

[4]  Valentin Goranko,et al.  Algorithmic correspondence and completeness in modal logic. I. The core algorithm SQEMA , 2006, Log. Methods Comput. Sci..

[5]  Naoki Kobayashi,et al.  Resource Usage Analysis for the Pi-Calculus , 2006, VMCAI.

[6]  Yde Venema,et al.  A Sahlqvist theorem for distributive modal logic , 2005, Ann. Pure Appl. Log..

[7]  Willem Conradie,et al.  Algebraic modal correspondence: Sahlqvist and beyond , 2016, J. Log. Algebraic Methods Program..

[8]  Félix Bou Moliner.  Strict-weak languages: an analysis of strict implication , 2004 .

[9]  Alessandra Palmigiano,et al.  Dual characterizations for finite lattices via correspondence theory for monotone modal logic , 2014, J. Log. Comput..

[10]  Dirk Pattinson,et al.  Correspondence between Modal Hilbert Axioms and Sequent Rules with an Application to S5 , 2013, TABLEAUX.

[11]  Nicola Olivetti,et al.  Nested Sequent Calculi for Conditional Logics , 2012, JELIA.

[12]  Alessandra Palmigiano,et al.  Jónsson-style canonicity for ALBA-inequalities , 2017, J. Log. Comput..

[13]  Michael Zakharyaschev,et al.  Modal Logic , 1997, Oxford logic guides.

[14]  Alessandra Palmigiano,et al.  Unified Correspondence as a Proof-Theoretic Tool , 2016, J. Log. Comput..

[15]  Kosta Došen,et al.  Modal Translations in K and D , 1993 .

[16]  Mohammad Ardeshir,et al.  Basic Propositional Calculus I , 1998, Math. Log. Q..

[17]  Paolo Maffezioli,et al.  Hypersequent and Labelled Calculi for Intermediate Logics , 2013, TABLEAUX.

[18]  Alessandra Palmigiano,et al.  Canonical extensions and relational completeness of some substructural logics* , 2005, Journal of Symbolic Logic.

[19]  Alessandra Palmigiano,et al.  Sahlqvist theory for impossible worlds , 2016, J. Log. Comput..

[20]  Willem Conradie,et al.  Canonicity results for mu-calculi: an algorithmic approach , 2017, J. Log. Comput..

[21]  Brian A. Davey,et al.  An Introduction to Lattices and Order , 1989 .

[22]  Kazushige Terui,et al.  Algebraic proof theory for substructural logics: Cut-elimination and completions , 2012, Ann. Pure Appl. Log..

[23]  Albert Visser,et al.  A propositional logic with explicit fixed points , 1981 .

[24]  Willem Conradie,et al.  Algorithmic correspondence and canonicity for non-distributive logics , 2016, Ann. Pure Appl. Log..

[25]  Willem Conradie,et al.  On Sahlqvist theory for hybrid logics , 2015, J. Log. Comput..

[26]  Lutz Straßburger,et al.  Label-free Modular Systems for Classical and Intuitionistic Modal Logics , 2014, Advances in Modal Logic.

[27]  Katsumasa Ishii,et al.  Sequent Calculi for Visser's Propositional Logics , 2001, Notre Dame J. Formal Log..

[28]  Johan van Benthem,et al.  Sahlqvist Correspondence for Modal mu-calculus , 2012, Studia Logica.

[29]  Katsuhiko Sano,et al.  Constructive Embedding from Extensions of Logics of Strict Implication into Modal Logics , 2016 .

[30]  M. Gehrke,et al.  Bounded Lattice Expansions , 2001 .

[31]  Ramon Jansana,et al.  A Closer Look at Some Subintuitionistic Logics , 2001, Notre Dame J. Formal Log..

[32]  Ian Hacking,et al.  What is strict implication? , 1963, Journal of Symbolic Logic.

[33]  Ramon Jansana,et al.  Bounded distributive lattices with strict implication , 2005, Math. Log. Q..

[34]  Timothy A. D. Fowler,et al.  Categorial Grammar: Logical Syntax, Semantics, and Processing Glyn V. Morrill (Universitat Politècnica de Catalunya)Oxford: Oxford University Press, 2011, xv+236 pp; hardbound, ISBN 978-0-19-958985-2, £ 80.00, $150.00; paperbound, ISBN 978-0-19-958986-9, £ 29.99, $55.00 , 2011, Computational Linguistics.

[35]  Silvio Ghilardi,et al.  The bounded proof property via step algebras and step frames , 2013, Ann. Pure Appl. Log..

[36]  M. de Rijke,et al.  Modal Logic , 2001, Cambridge Tracts in Theoretical Computer Science.

[37]  Kazushige Terui,et al.  From Axioms to Analytic Rules in Nonclassical Logics , 2008, 2008 23rd Annual IEEE Symposium on Logic in Computer Science.

[38]  M. de Rijke,et al.  Logic, Language and Computation , 1997 .

[39]  Ori Lahav,et al.  From Frame Properties to Hypersequent Rules in Modal Logics , 2013, 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science.

[40]  Willem Conradie,et al.  Algorithmic correspondence and canonicity for distributive modal logic , 2012, Ann. Pure Appl. Log..

[41]  Rajeev Goré,et al.  On the Correspondence between Display Postulates and Deep Inference in Nested Sequent Calculi for Tense Logics , 2011, Log. Methods Comput. Sci..

[42]  J. Lambek.  The Mathematics of Sentence Structure , 1958 .

[43]  J. Michael Dunn,et al.  Positive modal logic , 1995, Stud Logica.

[44]  Peter Jipsen,et al.  Residuated lattices: An algebraic glimpse at sub-structural logics , 2007 .

[45]  Mai Gehrke,et al.  Canonical Extensions, Esakia Spaces, and Universal Models , 2014 .

[46]  Johan van Benthem,et al.  Minimal predicates, fixed-points, and definability , 2005, Journal of Symbolic Logic.

[47]  Wojciech Buszkowski,et al.  Nonassociative Lambek Calculus with Additives and Context-Free Languages , 2009, Languages: From Formal to Natural.

[48]  Joachim Lambek,et al.  On the Calculus of Syntactic Types , 1961 .

[49]  Nick Bezhanishvili,et al.  Finitely generated free Heyting algebras via Birkhoff duality and coalgebra , 2011, Log. Methods Comput. Sci..

[50]  N. Kurtonina,et al.  Frames and Labels. A modal analysis of categorial inference , 1995 .

[51]  Greg Restall,et al.  On logics without contraction , 1994 .

[52]  Roy Dyckhoff,et al.  Proof analysis in intermediate logics , 2012, Arch. Math. Log..

[53]  Mohammad Ardeshir Behrostaghi.  Aspects of basic logic , 1995 .

[54]  Björn Lellmann,et al.  Axioms vs Hypersequent Rules with Context Restrictions: Theory and Applications , 2014, IJCAR.

[55]  Willem Conradie,et al.  Canonicity and Relativized Canonicity via Pseudo-Correspondence: an Application of ALBA , 2015, ArXiv.

[56]  Sara Negri,et al.  Proof Analysis in Modal Logic , 2005, J. Philos. Log..

[57]  Willem Conradie,et al.  Algorithmic correspondence for intuitionistic modal mu-calculus , 2015, Theor. Comput. Sci..

[58]  Giovanna Corsi,et al.  Weak Logics with Strict Implication , 1987, Math. Log. Q..

[59]  Heinrich Wansing,et al.  Displaying as Temporalizing , 1997 .

[60]  Michael Zakharyaschev,et al.  Modal companions of intermediate propositional logics , 1992, Stud Logica.

[61]  R. Ishigaki,et al.  Sequent Calculi for Some Strict Implication Logics , 2008, Log. J. IGPL.

[62]  Chris Fox,et al.  Expressiveness and Complexity in Underspecified Semantics , 2010 .

[63]  M. de Rijke,et al.  Diamonds and Defaults , 1993 .

[64]  Nuel Belnap,et al.  The pure calculus of entailment , 1962, Journal of Symbolic Logic.

[65]  M. Kracht.  Power and Weakness of the Modal Display Calculus , 1996 .

[66]  Kentaro Kikuchi.  RELATIONSHIPS BETWEEN BASIC PROPOSITIONAL CALCULUS AND SUBSTRUCTURAL LOGICS , 2001 .

[67]  Majid Alizadeh,et al.  On the linear Lindenbaum algebra of Basic Propositional Logic , 2004, Math. Log. Q..