Sheaves, Games, and Model Completions

1. Front Matter

   Pages i-ix

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2. Introduction

   • Silvio Ghilardi, Marek Zawadowski

   Pages 1-14

3. Preliminary Notions

   • Silvio Ghilardi, Marek Zawadowski

   Pages 15-47

4. Model Completions

   • Silvio Ghilardi, Marek Zawadowski

   Pages 49-71

5. Heyting Algebras

   • Silvio Ghilardi, Marek Zawadowski

   Pages 73-113

6. Duality for Modal Algebras

   • Silvio Ghilardi, Marek Zawadowski

   Pages 115-162

7. Model Completions in Modal Logic

   • Silvio Ghilardi, Marek Zawadowski

   Pages 163-180

8. Algebraically Closed Models

   • Silvio Ghilardi, Marek Zawadowski

   Pages 181-201

9. Open Problems

   • Silvio Ghilardi, Marek Zawadowski

   Pages 203-205

10. Appendix

    • Silvio Ghilardi, Marek Zawadowski

    Pages 207-231

11. Back Matter

    Pages 233-245

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This book is an example of fruitful interaction between (non-classical) propo­ sitionallogics and (classical) model theory which was made possible due to categorical logic. Its main aim consists in investigating the existence of model­ completions for equational theories arising from propositional logics (such as the theory of Heyting algebras and various kinds of theories related to proposi­ tional modal logic ). The existence of model-completions turns out to be related to proof-theoretic facts concerning interpretability of second order propositional logic into ordinary propositional logic through the so-called 'Pitts' quantifiers' or 'bisimulation quantifiers'. On the other hand, the book develops a large number of topics concerning the categorical structure of finitely presented al­ gebras, with related applications to propositional logics, both standard (like Beth's theorems) and new (like effectiveness of internal equivalence relations, projectivity and definability of dual connectives such as difference). A special emphasis is put on sheaf representation, showing that much of the nice categor­ ical structure of finitely presented algebras is in fact only a restriction of natural structure in sheaves. Applications to the theory of classifying toposes are also covered, yielding new examples. The book has to be considered mainly as a research book, reporting recent and often completely new results in the field; we believe it can also be fruitfully used as a complementary book for graduate courses in categorical and algebraic logic, universal algebra, model theory, and non-classical logics. 1.

• Interpolation
• logic
• modal logic
• model theory
• proof
• simulation

• Università degli Studi di Milano, Milan, Italy

  Silvio Ghilardi

• Uniwersytet Warszawski, Warsaw, Poland

  Marek Zawadowski

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