Galois Connections in Categorial Type Logic

The introduction of unary connectives has proved to be an important addition to the categorial vocabulary. The connectives considered so far are order-preserving; in this paper instead, we consider the addition of order-reversing, Galois connected operators. In §2 we do the basic model-theoretic and proof-theoretic groundwork. In §3 we use the expressive power of the Galois connected operators to restrict the scopal possibilities of generalized quantifier expressions, and to describe a typology of polarity items.

[1]  Rajeev Goré,et al.  Substructural Logics on Display , 1998, Log. J. IGPL.

[2]  M. Moortgat.  Generalized quantifiers and discontinuous type constructors , 1996 .

[3]  R. May.  The grammar of quantification , 1978 .

[4]  Johan van Benthem,et al.  Handbook of Logic and Language , 1996 .

[5]  Joachim Lambek,et al.  Type Grammars as Pregroups , 2001, Grammars.

[6]  Kosta Dosen,et al.  A Brief Survey of Frames for the Lambek Calculus , 1992, Math. Log. Q..

[7]  T. Stowell,et al.  Distributivity and Negation: The Syntax of Each and Every , 1997 .

[8]  Horst Herrlich,et al.  Galois Connections , 1985, Mathematical Foundations of Programming Semantics.

[9]  Michael Moortgat,et al.  Multimodal linguistic inference , 1995, J. Log. Lang. Inf..

[10]  M. Moortgat.  In Situ Binding: A Modal Analysis , 1996 .

[11]  V. Michele Abrusci.  Phase Semantics and Sequent Calculus for Pure Noncommutative Classical Linear Propositional Logic , 1991, J. Symb. Log..

[12]  Michael Moortgat,et al.  Structural control , 1997 .

[13]  Anna Szabolcsi.  Ways of Scope Taking , 1997 .

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

[15]  Raffaella Bernardi,et al.  Generalized Quantifiers in Declarative and Interrogative Sentences , 2003, Log. J. IGPL.

[16]  Michael Moortgat,et al.  Categorial Type Logics , 1997, Handbook of Logic and Language.

[17]  Joachim Lambek,et al.  Type Grammar Revisited , 1997, LACL.

[18]  Greg Restall,et al.  An Introduction to Substructural Logics , 2000 .

[19]  A. Giannakidou.  The Landscape of Polarity Items , 1997 .

[20]  Czeslaw Bylinski,et al.  Galois Connections , 1985, Mathematical Foundations of Programming Semantics.