Review of : Sheaves, Games, and Model Completions

Abstract

Sheaves, Games, and Model Completionsby Silvio Ghilardi and Marek ZawadowskiPublished July 2002, by Kluwer, ISBN 1402006608.One of the most striking and useful results in the proof theory of intuitionisticlogic in the 1990’s is undoubtedly Andy Pitts’ Theorem, which gives aninterpretation of the second-order intuitionistic propositional calculus intoordinary intuitionistic propositional logic IPC: for every IPC-formula φ andfor every propositional variable p there are formulas ∃pφ and ∀pφ, involvingonly variables occurring in φ except p, such that for any other IPC-formulaψ which does not contain p, one has the following equivalences:IPC φ → ψ iff IPC ∃pφ → ψIPC ψ → φ iff IPC ψ → ∀pφThe book under review is an investigation into algebraic and model-theoreticaspects of this fact. The authors have written a number of joint papers, ofwhich this book can be seen as the synthesis; however, there is also materialthat was not published before.Their starting point is a re-interpretation of Pitts’ Theorem.