Treatise on Intuitionistic Type Theory

211 Pages·2011·1.51 MB·English

by  Johan Georg Granström

Treatise on Intuitionistic Type Theory
LOGIC,EPISTEMOLOGY,ANDTHEUNITYOFSCIENCE
VOLUME22
Editors
ShahidRahman,UniversityofLilleIII,France
JohnSymons,UniversityofTexasatElPaso,U.S.A.
EditorialBoard
JeanPaulvanBendegem,FreeUniversityofBrussels,Belgium
JohanvanBenthem,UniversityofAmsterdam,TheNetherlands
JacquesDubucs,UniversityofParisI-Sorbonne,France
AnneFagot-Largeault,CollègedeFrance,France
BasvanFraassen,PrincetonUniversity,U.S.A.
DovGabbay,King’sCollegeLondon,U.K.
JaakkoHintikka,BostonUniversity,U.S.A.
KarelLambert,UniversityofCalifornia,Irvine,U.S.A.
GrahamPriest,UniversityofMelbourne,Australia
GabrielSandu,UniversityofHelsinki,Finland
GöranSundholm,UniversiteitLeiden,TheNetherlands
HeinrichWansing,TechnicalUniversityDresden,Germany
TimothyWilliamson,OxfordUniversity,U.K.
Logic,Epistemology, andtheUnityofScienceaimstoreconsiderthequestionoftheunityofscience
in light ofrecent developments in logic. Atpresent, no single logical, semantical or methodological
frameworkdominatesthephilosophyofscience.However,theeditorsofthisseriesbelievethatformal
techniques like, forexample, independence friendly logic, dialogical logics, multimodal logics, game
theoreticsemanticsandlinearlogics,havethepotentialtocastnewlightonbasicissuesinthediscussion
oftheunityofscience.
Thisseriesprovidesavenuewherephilosophersandlogicianscanapplyspecifictechnicalinsightsto
fundamentalphilosophicalproblems.Whiletheseriesisopentoawidevarietyofperspectives,including
thestudyandanalysisofargumentationandthecriticaldiscussionoftherelationshipbetweenlogicand
thephilosophyofscience,theaimistoprovideanintegratedpictureofthescientificenterpriseinallits
diversity.
Forfurthervolumes:
http://www.springer.com/series/6936
Johan Georg Granström
Treatise on Intuitionistic
Type Theory
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JohanGeorgGranström
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ISBN978-94-007-1735-0 e-ISBN978-94-007-1736-7
DOI10.1007/978-94-007-1736-7
SpringerDordrechtHeidelbergLondonNewYork
LibraryofCongressControlNumber:2011929986
(cid:2)c SpringerScience+BusinessMediaB.V.2011
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Contents
Contents v
List of Figures vii
List of Tables ix
Introduction xi
Chapter I. Prolegomena 1
§ 1. A threefold correspondence 1
§ 2. The acts of the mind 4
§ 3. The principle of compositionality 6
§ 4. Lingua characteristica 9
Chapter II. Truth and Knowledge 13
§ 1. The meaning of meaning 13
§ 2. A division of being 16
§ 3. Mathematical entities 18
§ 4. Judgement and assertion 22
§ 5. Reasoning and demonstration 25
§ 6. The proposition 26
§ 7. The laws of logic 36
§ 8. Variables and generality 44
§ 9. Division of definitions 49
Chapter III. The Notion of Set 53
§ 1. A history of set-like notions 53
§ 2. Set-theoretical notation 59
§ 3. Making universal concepts into objects of thought 59
§ 4. Canonical sets and elements 63
§ 5. How to define a canonical set 69
§ 6. More canonical sets 74
Chapter IV. Reference and Computation 77
§ 1. Functions, algorithms, and programs 78
§ 2. The concept of function 81
§ 3. A formalization of computation 86
§ 4. Noncanonical sets and elements 91
§ 5. Nominal definitions 97
vi CONTENTS
§ 6. Functions as objects 98
§ 7. Families of sets 102
Chapter V. Assumption and Substitution 107
§ 1. The concept of function revisited 107
§ 2. Hypothetical assertions 111
§ 3. The calculus of substitutions 120
§ 4. Sets and elements in hypothetical assertions 131
§ 5. Closures and the λ-calculus 135
§ 6. The disjoint union of a family of sets 142
§ 7. Elimination rules 144
§ 8. Propositions as sets 151
Chapter VI. Intuitionism 155
§ 1. The intuitionistic interpretation of apagoge 155
§ 2. The law of excluded middle 163
§ 3. The philosophy of mathematics 170
Bibliography 175
Index of Proper Names 187
Index of Subjects 191
List of Figures
1 The relation between object, concept, and expression 2
2 Meaning, referent, and term 14
3 Mediate vs. immediate reference 15
4 Division of modes of being into real and ideal 16
5 The threefold correspondence for universal concepts 18
6 The threefold correspondence for beings of reason 21
7 The disjoint union of a family of sets 104
8 Example of a dependently typed function 105
9 A context, telescopic in the sense of de Bruijn 113

List of Tables
1 Division of logic according to the acts of the mind 5
2 The interpretation of the propositional connectives 29
3 Classification of mathematical categorems 47
4 Four different notions of set 54
5 Classification of assertions and inference rules 70
6 Overview of seven different notions of function 108
7 The Curry-Howard correspondence 152