Dependence logic with generalized quantifiers: Axiomatizations

We prove two completeness results, one for the extension of dependence logic by a monotone generalized quantifier Q with weak interpretation, weak in the meaning that the interpretation of Q varies with the structures. The second result considers the extension of dependence logic where Q is interpreted as “there exists uncountable many.” Both of the axiomatizations are shown to be sound and complete for FO(Q) consequences.

[1]  Juha Kontinen,et al.  Characterizing quantifier extensions of dependence logic , 2012, The Journal of Symbolic Logic.

[2]  Jon Barwise,et al.  Some applications of Henkin quantifiers , 1976 .

[3]  Wilfrid Hodges,et al.  Compositional Semantics for a Language of Imperfect Information , 1997, Log. J. IGPL.

[4]  Lauri Hella,et al.  Definability of Polyadic Lifts of Generalized Quantifiers , 1997, J. Log. Lang. Inf..

[5]  Perlindström.  First Order Predicate Logic with Generalized Quantifiers , 1966 .

[6]  H. Keisler.  Logic with the quantifier “there exist uncountably many” , 1970 .

[7]  M. Kaufmann.  Chapter IV: The Quantifier "There Exist Uncountably Many" and Some of Its Relatives , 1985 .

[8]  Johann A. Makowsky,et al.  Some model theory for monotone quantifiers , 1977, Arch. Math. Log..

[9]  Elias Dahlhaus,et al.  Skolem Normal Forms Concerning the Least Fixpoint , 1987, Computation Theory and Logic.

[10]  A. Mostowski.  On a generalization of quantifiers , 1957 .

[11]  Jouko A. Väänänen,et al.  Dependence Logic - A New Approach to Independence Friendly Logic , 2007, London Mathematical Society student texts.

[12]  Fredrik Engström,et al.  Generalized Quantifiers in Dependence Logic , 2012, J. Log. Lang. Inf..

[13]  Neil Immerman,et al.  Languages that Capture Complexity Classes , 1987, SIAM J. Comput..

[14]  Phokion G. Kolaitis,et al.  Generalized Quantifiers and Pebble Games on Finite Structures , 1995, Ann. Pure Appl. Log..

[15]  Edward L. Keenan,et al.  Erratum to: Stanley Peters and Dag Westerståhl: Quantifiers in language and logic , 2011 .

[16]  Juha Kontinen,et al.  Axiomatizing first order consequences in dependence logic , 2012, Ann. Pure Appl. Log..

[17]  K. Jon Barwise,et al.  On branching quantifiers in English , 1979, J. Philos. Log..

[18]  Juha Kontinen,et al.  On Definability in Dependence Logic , 2009, J. Log. Lang. Inf..

[19]  Stanley Peters,et al.  Quantifiers in language and logic , 2006 .