Algorithmic correspondence and completeness in modal logic. IV. Semantic extensions of SQEMA. In [{it W. E. Conradie, V. F. Goranko} and {it D. Vakarelov}, Log. Methods Comput. Sci. 2, No. 1, Paper 5 (2006; Zbl 1126.03018)] we introduced the algorithm SQEMA for computing first-order equivalents and proving canonicity of modal formulae, and thus established a very general correspondence and canonical completeness result. SQEMA is based on transformation rules, the most important of which employs a modal version of a result by Ackermann that enables elimination of an existentially quantified predicate variable in a formula, provided a certain negative polarity condition on that variable is satisfied. In this paper we develop several extensions of SQEMA where that syntactic condition is replaced by a semantic one, viz. downward monotonicity. For the first, and most general, extension SemSQEMA we prove correctness for a large class of modal formulae containing an extension of the Sahlqvist formulae, defined by replacing polarity with monotonicity. By employing a special modal version of Lyndon’s monotonicity theorem and imposing additional requirements on the Ackermann rule we obtain restricted versions of SemSQEMA which guarantee canonicity, too.

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  1. Conradie, Willem; Robinson, Claudette: On Sahlqvist theory for hybrid logics (2017)
  2. Frittella, Sabine; Palmigiano, Alessandra; Santocanale, Luigi: Dual characterizations for finite lattices via correspondence theory for monotone modal logic (2017)
  3. Ma, Minghui; Zhao, Zhiguang: Unified correspondence and proof theory for strict implication (2017)
  4. Palmigiano, Alessandra; Sourabh, Sumit; Zhao, Zhiguang: Jónsson-style canonicity for ALBA-inequalities (2017)
  5. Palmigiano, Alessandra; Sourabh, Sumit; Zhao, Zhiguang: Sahlqvist theory for impossible worlds (2017)
  6. Galeazzi, Paolo; Lorini, Emiliano: Epistemic logic meets epistemic game theory: a comparison between multi-agent Kripke models and type spaces (2016)
  7. Lorini, Emiliano; Sartor, Giovanni: A STIT logic for reasoning about social influence (2016)
  8. Conradie, Willem; Fomatati, Yves; Palmigiano, Alessandra; Sourabh, Sumit: Algorithmic correspondence for intuitionistic modal mu-calculus (2015)
  9. Bezhanishvili, Nick; Ghilardi, Silvio: The bounded proof property via step algebras and step frames (2014)
  10. Lorini, Emiliano; Troquard, Nicolas; Herzig, Andreas; Broersen, Jan: Grounding power on actions and mental attitudes (2013)
  11. Ciabattoni, Agata; Galatos, Nikolaos; Terui, Kazushige: Algebraic proof theory for substructural logics: cut-elimination and completions (2012)
  12. Conradie, Willem; Palmigiano, Alessandra: Algorithmic correspondence and canonicity for distributive modal logic (2012)
  13. Schmidt, Renate A.: The Ackermann approach for modal logic, correspondence theory and second-order reduction (2012)
  14. Broersen, Jan: Deontic epistemic stit logic distinguishing modes of mens rea (2011)
  15. Dunin-Kȩplicz, Barbara; Nguyen, Linh Anh; Szałas, Andrzej: Converse-PDL with regular inclusion axioms: a framework for MAS logics (2011)
  16. Conradie, Willem; Goranko, Valentin; Vakarelov, Dimitar: Algorithmic correspondence and completeness in modal logic. V. Recursive extensions of SQEMA (2010)
  17. Broersen, Jan: A complete STIT logic for knowledge and action, and some of its applications (2009)
  18. Conradie, Willem; Goranko, Valentin; Vakarelov, Dimiter: Algorithmic correspondence and completeness in modal logic. III. Extensions of the algorithm SQEMA with substitutions (2009)
  19. Kikot, Stanislav: An extension of Kracht’s theorem to generalized Sahlqvist formulas (2009)
  20. Conradie, Willem; Goranko, Valentin: Algorithmic correspondence and completeness in modal logic. IV. Semantic extensions of SQEMA. (2008)

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