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El. knyga: Branching Space-Times: Theory and Applications

(Professor of Philosophy, University of Konstanz), (Professor of Philosophy, Jagiellonian University in Kraków), (Professor Emeritus of Philosophy, University of Pittsburgh)
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"This book develops a rigorous theory of indeterminism as a local and modal concept. Its crucial insight is that our world contains events or processes with alternative, really possible outcomes. The theory aims at clarifying what this assumption involves, and it does it in two ways. First, it provides a mathematically rigorous framework for local and modal indeterminism. Second, we support that theory by spelling out the philosophically relevant consequences of this formulation and by showing its fruitful applications in metaphysics. To this end, we offer a formal analysis of modal correlations and of causation, which is applicable in indeterministic and non-local contexts as well. We also propose a rigorous theory of objective single-case probabilities, intended to represent degrees of possibility. In a third step, we link our theory to current physics, investigating how local and modal indeterminism relates to issues in the foundations of physics, in particular, quantum non-locality and spatio-temporal relativity. The book also ventures into the philosophy of time, showing how the theory's resources can be used to explicate the dynamic concept of the past, present, and future based on local indeterminism"--

This monograph presents the first detailed exposition of the formal theory of Branching Space-Times. The theory presented here by Nuel Belnap, Thomas Muller, and Tomasz Placek describes how real possibilities can play out in our spatio-temporal world. In our world, some things that are really
possible in Cleveland are not really possible in San Francisco; other things were really possible in 1988 but are not really possible in 2021. The authors develop a rigorous, relativity-friendly theory of indeterminism as a local and modal concept, demonstrating that our world contains events with
alternative possible outcomes.

The book is divided into two parts. The first contains the exposition of the theory, including detailed proofs. The second contains three applications of Branching Space-Times in metaphysics and philosophy of science, focusing on the use of Branching Space-Times to represent pertinent forms of
indeterminism in each area. Some specific applications include a formal analysis of modal correlations and of causation and a rigorous theory of objective single-case probabilities, intended to represent degrees of possibility. The authors link their theory to current physics, investigating how
local and modal indeterminism relates to issues in the foundations of physics, particularly in quantum non-locality and spatio-temporal relativity. They also relate the theory to philosophy of time, showing how it may be used to explicate the dynamic concept of the past, present, and future based on
local indeterminism.

The Branching Space-Times theory has been in development over the past 25 years. This volume provides a much needed first systematic and comprehensive book-length exposition of both the theory and its applications.

This is an open access title available under the terms of a CC BY-NC-ND 4.0 International license. It is free to read at Oxford Scholarship Online and offered as a free PDF download from OUP and selected open access locations.

Recenzijos

Belnap, Müller, and Placek defend a novel theory of branching space-times in this mathematically rigorous, engaging monograph at the intersection of metaphysics, philosophy of time, and modern physics. * L. B. McHenry, CHOICE * This book develops a rigorous theory of indeterminism asa local and modal concept. Its crucial insight is that our world contains events or processes with alternative, really possible outcomes. * MathSciNet *

Preface xi
I. The Core Theory
1 Introduction
3(21)
1.1 Real possibilities
4(2)
1.2 Representing possibilities via branching vs. possible worlds
6(2)
1.3 Some thoughts on our modally rich world
8(5)
1.4 Branching in the landscape of possible-worlds theories: Some comments on modal metaphysics
13(9)
1.5 Outline of the book
22(1)
1.6 Exercises to
Chapter 1
22(2)
2 The Foundations of Branching Space-Times
24(19)
2.1 The underlying ideas of BST
24(3)
2.2 Histories
27(6)
2.3 Historical connection
33(3)
2.4 Density and continuity
36(1)
2.5 Weiner's postulate and spatio-temporal locations
37(3)
2.6 Axioms of the common core of BST
40(1)
2.7 Exercises to
Chapter 2
40(3)
3 Two Options for the Branching of Histories
43(34)
3.1 Indeterminism as the branching of histories
43(2)
3.2 On chains in common BST structures
45(3)
3.3 Extending common BST: two options
48(1)
3.4 BST92
49(11)
3.4.1 BST92 in formal detail
49(2)
3.4.2 Local possibilities
51(3)
3.4.3 The pattern of branching of BST92
54(4)
3.4.4 Transitions
58(2)
3.5 Introducing BSTNF
60(10)
3.5.1 The new Prior Choice Principle and BSTNF structures defined
60(3)
3.5.2 Local possibilities and the pattern of branching in BSTNF
63(2)
3.5.3 Facts about choice sets
65(5)
3.6 BST92 or BSTNF: Does it matter?
70(5)
3.6.1 Topological issues: An overview
70(2)
3.6.2 Translatability results: An overview
72(3)
3.7 Exercises to
Chapter 3
75(2)
4 Building upon the Foundations of Branching Space-Times
77(26)
4.1 A variety of events and their occurrence propositions
77(6)
4.2 Basic transitions
83(7)
4.2.1 Basic transitions in BST92
83(6)
4.2.2 A note on basic transitions in BSTNF
89(1)
4.3 Sets of basic transitions
90(2)
4.4 Topological aspects of BST
92(6)
4.4.1 General idea of the diamond topology
92(1)
4.4.2 Properties of the diamond topology in BST92
93(3)
4.4.3 The diamond topology in BSTNF
96(2)
4.5 A note on branching-style semantics
98(4)
4.6 Exercises to
Chapter 4
102(1)
5 Modal Funny Business
103(25)
5.1 Motivation for being interested in modal correlations
105(1)
5.2 Modal funny business
106(13)
5.2.1 Expected inconsistencies in sets of basic transitions
108(2)
5.2.2 Combinatorial funny business
110(2)
5.2.3 Explanatory funny business
112(3)
5.2.4 On the interrelation of combinatorial and explanatory funny business
115(4)
5.3 Some consequences of modal funny business
119(4)
5.4 On MFB in BSTNF
123(3)
5.5 Exercises to
Chapter 5
126(2)
6 Causation in Terms of causae causantes
128(30)
6.1 Causation: Causes and effects as BST transitions
128(6)
6.2 At least an inus condition
134(4)
6.3 Causae causantes in BST92 in formal detail
138(9)
6.3.1 Defining causae causantes in BST92
138(4)
6.3.2 Causae causantes are at least inus conditions
142(16)
6.3.2.1 Transitions to outcome chains or scattered outcomes
143(2)
6.3.2.2 Transitions to disjunctive outcomes
145(2)
6.4 Causation in the presence of modal funny business
147(6)
6.5 Causae causantes in BSTNF structures
153(2)
6.6 Conclusions
155(1)
6.7 Exercises to
Chapter 6
156(2)
7 Probabilities
158(65)
7.1 Two conditions of adequacy and two crucial questions
158(9)
7.1.1 Two conditions of adequacy
159(1)
7.1.2 Two crucial questions
160(2)
7.1.3 Propensities µ and probability measures p
162(5)
7.2 Causal probability spaces in BST
167(33)
7.2.1 Probabilities for transitions: The simplest case
167(2)
7.2.2 Two BST transitions, one basic transition
169(2)
7.2.3 Two or more transitions and some complications
171(14)
7.2.4 General probability spaces in BST
185(8)
7.2.5 Representing transitions in different causal probability spaces
193(2)
7.2.6 Probabilistic funny business
195(5)
7.3 Fending off objections to propensities
200(18)
7.3.1 Some remarks on propensities
201(1)
7.3.2 Humphreys's paradox
201(4)
7.3.3 Our diagnosis of Humphreys's paradox
205(8)
7.3.4 Salmon's corkscrew story: More on conditional propensities and inversion
213(5)
7.4 Conclusions
218(1)
7.5 Exercises to
Chapter 7
219(4)
II. Applications
8 Quantum Correlations
223(70)
8.1 Introducing quantum correlation experiments
225(2)
8.2 On the BST analysis of quantum correlations
227(4)
8.3 Explaining modal correlations via instruction sets
231(32)
8.3.1 Extensions of a surface structure by generic instruction sets
235(12)
8.3.1.1 The possibility of superdeterministic extensions
241(2)
8.3.1.2 Splitting in extended structures: The general case
243(4)
8.3.2 Non-contextual and contextual instruction sets
247(6)
8.3.2.1 Non-contextual instruction sets
248(2)
8.3.2.2 Contextual instruction sets
250(2)
8.3.2.3 On the interrelation of different types of instruction sets
252(1)
8.3.3 Instruction sets for GHZ
253(10)
8.3.3.1 The superdeterministic extension
256(1)
8.3.3.2 C/E independence
256(1)
8.3.3.3 Non-contextual instruction sets for GHZ
257(3)
8.3.3.4 Contextual instruction sets for GHZ
260(3)
8.3.4 Summary of the BST approach to modal structure extensions
263(1)
8.4 Probabilistic correlations
263(28)
8.4.1 Probabilistic hidden variables
264(2)
8.4.2 Extension of a probabilistic surface structure
266(10)
8.4.3 Single and multiple cases of PFB, and super-independence
276(4)
8.4.3.1 A structure with a single case of PFB
277(1)
8.4.3.2 A structure with multiple cases of PFB - super-independence
278(2)
8.4.4 The Bell-Aspect experiment
280(14)
8.4.4.1 The set-up of the Bell-Aspect experiment
281(1)
8.4.4.2 The surface structure for the Bell-Aspect experiment
282(1)
8.4.4.3 Probabilistic funny business
283(1)
8.4.4.4 Derivation of the Bell-CH inequality
284(3)
8.4.4.5 Analysis of the derivation
287(3)
8.4.4.6 Consequences from our analysis
290(1)
8.5 Exercises to
Chapter 8
291(2)
9 Branching in Relativistic Space-Times
293(49)
9.1 Minkowskian Branching Structures
294(14)
9.1.1 Basic notions
294(4)
9.1.2 Defining MBSs
298(8)
9.1.3 Taking stock
306(2)
9.2 Differential manifolds and BSTNF
308(8)
9.2.1 Differential manifolds
308(1)
9.2.2 Differential manifolds and MBSs
309(3)
9.2.3 Differential manifolds and BSTNF, generally
312(3)
9.2.4 Differential manifolds in GR
315(1)
9.3 GR space-times
316(23)
9.3.1 The initial value problem in GR
318(2)
9.3.2 An example of the failure of the IVP: Non-isometric extensions of Taub space-time
320(3)
9.3.3 Can non-Hausdorff manifolds in GR be interpreted modally?
323(1)
9.3.4 On bifurcating curves in GR
324(5)
9.3.5 Global and local determinism and indeterminism
329(5)
9.3.6 A note on closed causal curves and BST
334(4)
9.3.7 Summary on General Relativity
338(1)
9.4 Conclusions
339(1)
9.5 Exercises to
Chapter 9
340(2)
10 A Branching Space-Times Perspective on Presentism
342(47)
10.1 Introduction
342(1)
10.2 The problem of defining the present in special relativity
343(6)
10.3 Making room for an extended dynamic present
349(3)
10.4 The dynamic present, past, and future: Two approaches
352(3)
10.5 Dynamic time via causae causantes
355(9)
10.6 Dynamic time via the semantics of the open future
364(7)
10.7 The way to guarantee satisfactory dynamic time in BST: Sticky modal funny business
371(8)
10.8 What does dynamic time look like in MBSs?
379(3)
10.9 Conclusions
382(2)
10.10 Exercises to
Chapter 10
384(5)
III. Appendix
A Selected Proofs and Additional Material
389(22)
A.1 Dedekind continuity
389(1)
A.2 Formal details of the interrelation of BST92 and BSTNF
389(14)
A.2.1 Characterizing the transition structure of a BST92 structure
390(3)
A.2.2 BST92 transition structures are BSTNF structures
393(2)
A.2.3 From new foundations BSTNF to BST92
395(3)
A.2.4 Going full circle
398(4)
A.2.4.1 From BST92 to BSTNF to BST92
399(1)
A.2.4.2 From BSTNF to BST92 to BSTNF
400(2)
A.2.5 The translatability of some notions pertaining to MFB
402(1)
A.3 Proof of Theorem 5.1
403(2)
A.4 Additional material for
Chapter 8
405(6)
A.4.1 Extensions by one point or by multiple points?
405(2)
A.4.2 Proofs for
Chapter 8
407(4)
B Answers to Selected Exercises
411(10)
B.1 Answers to selected exercises from
Chapter 1
411(1)
B.2 Answers to selected exercises from
Chapter 2
412(1)
B.3 Answers to selected exercises from
Chapter 3
412(1)
B.4 Answers to selected exercises from
Chapter 4
413(1)
B.5 Answers to selected exercises from
Chapter 5
413(2)
B.6 Answers to selected exercises from
Chapter 6
415(3)
B.7 Answers to selected exercises from
Chapter 7
418(1)
B.8 Answers to selected exercises from
Chapter 8
419(1)
B.9 Answers to selected exercises from
Chapter 9
419(1)
B.10 Answers to selected exercises from
Chapter 10
419(2)
Bibliography 421(10)
Name Index 431(3)
Subject Index 434
Nuel Belnap is Professor Emeritus of Philosophy at the University of Pittsburgh and member of the American Academy of Arts and Sciences, specializing in logic, metaphysics, philosophy of science, and theory of action.

Thomas Muller is Professor of Philosophy at the University of Konstanz, Germany and member of the German National Academy of Sciences Leopoldina, specializing in logic, metaphysics, philosophy of science, and theory of action.

Tomasz Placek is Professor of Philosophy at the Jagiellonian University in Kraków, Poland, specializing in metaphysics and philosophy of physics.
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