Simulating polyadic modal logics by monadic ones

We define an interpretation of modal languages with polyadic operators in modal languages that use monadic operators (diamonds) only. We also define a simulation operator which associates a logic Λ sim in the diamond language with each logic Λ in the language with polyadic modal connectives. We prove that this simulation operator transfers several useful properties of modal logics, such as finite/recursive axiomatizability, frame completeness and the finite model property, canonicity and first-order definability.

[1]  Michael R. Hansen,et al.  Duration calculus: Logical foundations , 1997, Formal Aspects of Computing.

[2]  Robert Goldblatt,et al.  Varieties of Complex Algebras , 1989, Ann. Pure Appl. Log..

[3]  Dirk Roorda,et al.  Dyadic Modalities and Lambek Calculus , 1993 .

[4]  Steven K. Thomason.  Reduction of Tense Logic to Modal Logic. I , 1974, J. Symb. Log..

[5]  A. Tarski,et al.  Boolean Algebras with Operators. Part I , 1951 .

[6]  Johan van Benthem,et al.  Modal Deduction in Second-Order Logic and Set Theory - II , 1998, Stud Logica.

[7]  Johan van Benthem,et al.  Modal Deduction in Second-Order Logic and Set Theory - I , 1997, J. Log. Comput..

[8]  maarten marx,et al.  Arrow logic and multi-modal logic , 1997 .

[9]  Steven K. Thomason,et al.  Reduction of second-order logic to modal logic , 1975, Math. Log. Q..

[10]  Marcus Kracht,et al.  Normal monomodal logics can simulate all others , 1999, Journal of Symbolic Logic.

[11]  Yde Venema,et al.  A Modal Logic for Chopping Intervals , 1991, J. Log. Comput..

[12]  A. Tarski,et al.  Boolean Algebras with Operators , 1952 .

[13]  I. Hodkinson,et al.  Relation Algebras by Games , 2002 .

[14]  Natasha Kurtonina.  Categorial Inference and Modal Logic , 1998, J. Log. Lang. Inf..

[15]  Marcus Kracht,et al.  About this book , 1999 .

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

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

[18]  Marcus Kracht,et al.  Simulation and Transfer Results in Modal Logic – A Survey , 1997, Stud Logica.

[19]  M. Kracht.  Tools and Techniques in Modal Logic , 1999 .

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