Expressing Second-order Sentences in Intuitionistic Dependence Logic

Intuitionistic dependence logic was introduced by Abramsky and Väänänen [1] as a variant of dependence logic under a general construction of Hodges’ (trump) team semantics. It was proven that there is a translation from intuitionistic dependence logic sentences into second order logic sentences. In this paper, we prove that the other direction is also true, therefore intuitionistic dependence logic is equivalent to second order logic on the level of sentences.

[1]  H. Enderton.  Finite Partially-Ordered Quantifiers , 1970 .

[2]  Wilfried Hodges,et al.  Some Strange Quantifiers , 1997, Structures in Logic and Computer Science.

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

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

[5]  Juha Kontinen,et al.  Team Logic and Second-Order Logic , 2011, Fundam. Informaticae.

[6]  Peter W. O'Hearn,et al.  The Logic of Bunched Implications , 1999, Bulletin of Symbolic Logic.

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

[8]  L. Goble.  The Blackwell guide to philosophical logic , 2001 .

[9]  Grzegorz Rozenberg,et al.  Structures in Logic and Computer Science, A Selection of Essays in Honor of Andrzej Ehrenfeucht , 1997 .

[10]  Patrick Suppes,et al.  Logic, Methodology and Philosophy of Science , 1963 .

[11]  J. Hintikka,et al.  Informational Independence as a Semantical Phenomenon , 1989 .

[12]  Ville Nurmi,et al.  Dependence Logic: Investigations into Higher-Order Semantics Defined on Teams , 2009 .

[13]  Jouko Väänänen,et al.  From IF to BI , 2009, Synthese.

[14]  David J. Pym,et al.  The semantics and proof theory of the logic of bunched implications , 2002, Applied logic series.

[15]  Wilbur John Walkoe,et al.  Finite Partially-Ordered Quantification , 1970, J. Symb. Log..