Linking Game-Theoretical Approaches with Constructive Type Theory: Dialogical Strategies, CTT demonstrations and the Axiom of Choice (SpringerBriefs in Philosophy)

"This title links two of the most dominant research streams in philosophy of logic, namely game theory and proof theory. As the work’s subtitle expresses, the authors will build this link by means of the dialogical approach to logic. One important aspect of the present study is that the authors restrict themselves to the logically valid fragment of Constructive Type Theory (CTT). The reason is that, once that fragment is achieved the result can be extended to cover the whole CTT system. The first chapters in the brief offer overviews on the two frameworks discussed in the book with an emphasis on the dialogical framework. The third chapter demonstrates the left-to-right direction of the equivalence result. This is followed by a chapter that demonstrates the use of the algorithm in showing how to transform a specific winning strategy into a CCT-demonstration of the axiom of choice. The fifth chapter develops the algorithm from CTT-demonstrations to dialogical strategies. This brief concludes by introducing elements of discussion which are to be developed in subsequent work."--Publisher's description Preface Acknowledgments Contents List of Figures List of Tables 1 Brief Reminder of Constructive Type Theory 1.1 Fundamentals of the CTT Approach 1.2 The Basic CTT Framework for Intuitionistic Predicate Logic 1.2.1 About the Rules 1.2.2 Intuitionistic Logic in CTT References 2 Dialogues with Play-Objects 2.1 Standard Dialogical Games 2.2 The Formation of Propositions 2.3 Play-Objects 2.4 The Development of a Play 2.5 Example References 3 From Dialogical Strategies to CTT Demonstrations 3.1 Towards the Core 3.1.1 Getting Rid of Infinite Ramifications 3.1.2 Disregarding Formation Rules 3.1.3 Disregarding Irrelevant Variations in the Order of O-moves 3.2 From the Core to a CTT Demonstration 3.2.1 Generalities 3.2.2 The Algorithm 3.3 Adequacy of the Translation Algorithm References 4 The Dialogical Take on the Axiom of Choice, and Its Translation into CTT 4.1 The Dialogical Take on the Axiom of Choice 4.1.1 Two Plays on the Axiom of Choice 4.1.2 The Core of a Winning P-strategy 4.2 From the Dialogical Strategy to the CTT Demonstration of the Axiom of Choice 4.3 The Extensional Version of the Axiom of Choice 4.3.1 Martin-Löf on the Extensional Version with Uniqueness 4.3.2 The Dialogical Way References 5 Building a Winning P-strategy Out of a CTT Demonstration 5.1 Transforming CTT Demonstrations 5.1.1 Guidelines 5.1.2 The Procedure 5.2 Adequacy of the Algorithm References 6 Conclusions and Work in Progress References 7 Errata to: Linking Game-Theoretical Approaches with Constructive Type Theory Errata to: N. Clerbout and S. Rahman, Linking Game-Theoretical Approaches with Constructive Type Theory, SpringerBriefs in Philosophy, DOI 10.1007/978-3-319-19063-1 This℗ltitle links two of the most dominant research streams in philosophy of logic, namely game theory and proof theory. As the workĺls subtitle expresses, the authors will build this link by means of the dialogical approach to logic. One important aspect of the present study is that the authors restrict themselves to the logically valid fragment of Constructive Type Theory (CTT). The reason is that, once that fragment is achieved the result can be extended to cover the whole CTT system.℗lThe first chapters in the℗lbrief offer overviews on the two frameworks discussed in the book with an emphasis on the dialogical framework. The third chapter demonstrates the left-to-right direction of the equivalence result. This is followed by a chapter that demonstrates the use of the algorithm in showing how to transform a specific winning strategy into a CCT-demonstration of the axiom of choice. The fifth chapter develops the algorithm from CTT-demonstrations to dialogical strategies. This brief concludes by introducing elements of discussion which are to be developed in subsequent work