Contents
Foreword
xi
Part I The basics
1
1
Existence vs. essence
The opposition existence/essence
1.2 Essentialist and existentialist projects
1.3
and after
Essentialism vs.
Perfect vs. imperfect
2
Incompleteness
2.1 Technical statement
2.2 Hilbert in the face of incompleteness
2.3 Incompleteness is not a deficiency
2.4 Metaphorical readings
2. A More on the classification of predicates
2.B Formal arithmetic
2.C Techniques of incompleteness
2.D Incompleteness and truth
2.E Undecidability
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36
38
3
Classical sequents: LK
3.1 Generalities
3.2 The classical calculus
3.3 The cut-free system
3.4
the Hauptsatz
3.A Around sequent calculus
3.B Semantic aspects
3.C Infinitary proof-theory
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4
Intuitionistic logic: LJ, NJ
The intuitionistic sequent calculus LJ
4.2 The Hauptsatz in LJ
4.3 The natural deduction NJ
4.4 The signature in natural deduction
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Girard, Jean-Yves
The blind spot
cop. 2011
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digitalisiert durch
IDS Basel/Bern, im Auftrag der
Schweizerischen Nationalbibliothek
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Contents
4.B
4.C
4.D
Existence and disjunction in LJ
Natural deduction vs. sequent calculus
Around contraction
Logic programming
Kripke models
Part II Around Curry-Howard
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86
87
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95
5
Functional interpretations
5.1 Proofs as functions
5.2 Pure
5.3 The Curry-Howard isomorphism
Kreisel and functional interpretation
5.B Combinatory logic
5.C Other connectives
5.D
type theory
6
System F
6.1 System F
6.2 The normalisation theorem
Type theories
6.B Heyting's arithmetic
6.C
T
6.D Expressive power
Subtyping
6.F Essence, existence and typing
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126
129
129
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135
136
The category-theoretic interpretation
7.1 The three layers
7.2 Closed cartesian categories
7.3 Examples CCC
7.4 Logic
CCC
Classical logic
7.B Various interpretations
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140
147
150
152
154
157
7
Part III
8
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103
107
109
Linear logic
Coherent spaces
8.1 Grandeur and misery of Scott domains
8.2 Coherent spaces
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164
Contents
8.3
9
Interpretation of system F
Asymmetric interpretations
vii
171
175
Linear logic
9.1 Linearity in coherent spaces
9.2 Perfect linear connectives
9.3 Imperfect connectives
9.4 The logical system
Monoidal categories
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182
10
Perfection vs. imperfection
10.1 Phase semantics
10.2 A perfect world?
10.3 The world is imperfect
10.A Focalisation
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197
202
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210
11
Proof-nets
11.1 ILL
Multiplicative nets
11.3 The correctness criterion
More on multiplicatives
1
Syllogistic
General nets
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233
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240
Part IV
12
13
Polarised interpretations
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249
A hypothesis: polarisation
12.1 Faithfulness of coherent spaces
A prototype
12.3 Objections to polarisation
Logic and games
12.5 Proofs and tests
12.6 Hypersequentialised logic
Classical polarity
Intuitionistic logic
12.C Hypercoherences
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Designs and behaviours
13.1 Designs-dessins
13.2 Designs-desseins
13.3 Partial designs
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Contents
13.4
13.5
13.6
13.7
13.8
13.9
Nets and normalisation: dessins
Nets and normalisation: desseins
Analytical theorems
Introspective vs.
Behaviours
An example: the shift
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288
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298
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304
14
Ludics: the reconstruction
14.1 Additives
14.2 Multiplicatives
14.3 Quantifiers
14.A Faithfulness
14.B Bihaviours
14.C Parsimony
14.D
(?)
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15
Orthodox exponentials
15.1 The perennial perenniality
15.2 Exponential nets: normalisation
15.3 Categories and classical logic
15.4 The
LC
Exponentials and analytic functions
Exponential ludics
15.C Polarised linear logic
The
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334
337
340
342
343
350
Part V
Iconoclasm
355
16
Heterodox exponentials
16.1 The quarrel of images
16.2 Exponentials
16.3 Russell's antinomy
16.4
16.5 Expressive power
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366
17
Quantum coherent spaces
17.1 Logic vs. quantum
17.2 Probabilistic coherent spaces
17.3 Quantum coherent spaces
17.4 Additives
17.5 Multiplicatives
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Contents
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17.6 Discussion
17.A Initiation to
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Nets and duality
18.1 Duality and correctness
18.2 The original criterion
Trips and coherent spaces
18.B Non-commutative logic
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403
406
409
410
Part VI
Geometry of interaction
19
The feedback equation
19.1 Basic examples
19.2 Cut-systems
19.3 Solving the equation
19.4 The normal form
19.5 The first
Complements on operators
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417
419
421
425
432
439
20
Babel Tower vs. Great Wall
20.1 Idioms
20.2 The Babel Tower
20.3
20.A Von Neumann algebras
20.B Finite algebras
20.C Hyperfinite algebras
20.D The determinant
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446
448
450
453
456
459
21
Finite
21.1 Projects
21.2 Conducts
21.3 The social life of conducts
21.4 Polarised conducts
21.5 Exponentials
21.6 Lateralised logic
21.7 The social life of behaviours
Second-order quantification
21.B Truth
Truth and intersubjectivity
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462
469
473
475
477
480
482
485
488
491
x
Contents
Envoi. The phantom of transparency
The transparent world
Logics of transparency
Semantics
From semantics to the cognitive onion
Negation
497
497
498
499
501
504
Bibliography
509
Index
517