From Zeno to Arbitrage

263 Pages·2016·1.29 MB·Portuguese

by  Brian Skyrms

From Zeno to Arbitrage
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From Zeno
to Arbitrage
Essays on Quantity,
Coherence, and Induction
Brian Skyrms
1
3
GreatClarendonStreet,Oxford,OX26DP,
UnitedKingdom
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Contents
Preface vii
Acknowledgements viii
Part I. Zeno and the Metaphysics of Quantity
Introduction 3
1. Zeno’sParadoxofMeasure 5
2. TractarianNominalism 38
3. LogicalAtomsandCombinatorialPossibility 46
4. StrictCoherence,SigmaCoherence,andthe
MetaphysicsofQuantity 62
Part II. Coherent Degrees of Belief
Introduction 81
5. Higher-OrderDegreesofBelief 84
6. AMistakeinDynamicCoherenceArguments? 107
7. DynamicCoherenceandProbabilityKinematics 117
8. Updating,Supposing,andMAXENT 139
9. TheStructureofRadicalProbabilism 160
10. DiachronicCoherenceandRadicalProbabilism 174
Part III. Induction
Introduction 187
11. CarnapianInductiveLogicforMarkovChains 189
12. CarnapianInductiveLogicandBayesianStatistics 211
13. BayesianProjectibility 229
Index 253
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Preface
Thechaptersinthebookcenterontheconceptofprobability.Whatisthe
framework within which probability comfortably lives? What are the
coherence principles that must be satisfied for degrees of belief to be
probabilities, and how do these principles generalize to probability
change? What is the relation between coherent degrees of belief, beliefs
aboutchances,andinductiveinference?Whatconstraintsdoescoherence
putoninductiveskepticism?
Probability,inallitsmanifestations,isaquantity.Itisnotthereforeoutof
the way that Part I deals with fundamental issues concerning the nature
ofquantity.ThesegobacktoZenoofEleaandextendtomoderntheoriesof
infinitesimals. I do not have in mind so much the well-known paradoxes
of motion,butinstead abasicparadoxinthefoundationsofthetheoryof
quantity—Zeno’sparadoxofmeasure.Theveryshortversionofalongstoryis
thatthekernelofZeno’sparadoxsurvivesinproofsoftheexistenceofnon-
measurable sets. These raise interesting, if underappreciated, philosophical
issueswhenthequantityinquestionisprobabilityratherthanthegeometrical
quantitiesthatZenohadinmind.
We have degrees of belief about frequencies; knowledge of frequencies
informsourdegreesofbelief.Theseconceptsdancetogether.Themathemat-
ics of frequency is clear, and hardly open to dispute. The mathematics of
coherentdegreesofbeliefisstillcontroversial,evenifwell-trodden,ground.It
isevenmorecontroversialtoholdthatbeliefchangeissomehowconstrained
by considerations of coherence. Part II is about how this is so. It can be
summedupbythetheoryofarbitrage,andisdonesoinChapter10ofPartII.
TheinteractionbetweenfrequencyandcoherenceisdiscussedinPartIII.
Coherentbeliefandbeliefchange,frequency,andindeedchancearealltiedtogether
ininductiveinference.DeFinetti’stheoremisthetouchstone.Ifyourbeliefs
aresuchthatonlyfrequencymatters(exchangeability),thenyoubelievein
induction.Youbelievethatchanceequalslimitingrelativefrequency,close
enoughasmakesnoodds.Youbelievethatinductionwillleadyoutolearn
the limiting relative frequency. There are generalizations of this theorem.
Thekeyconceptisthatofsymmetryofdegreesofbelief.Inthisframework,
questionsofinductiveskepticismcanbeaddressedinapreciseway.Itturns
outtoberemarkablydifficulttobeacompleteinductiveskeptic.
Acknowledgements
The chapters collected here appeared over a period of 26 years, and are
reprintedwiththekindpermissionoftheoriginalpublishers:
“Zeno’sParadoxofMeasure”(1983)inR.Cohenetal.(eds.),Philosophy,
PhysicsandPsychoanalysis.Dordrecht:D.Reidel,pp.233–54.
“TractarianNominalism”(1981)PhilosophicalStudies40:199–206.
“LogicalAtomsandCombinatorialPossibility”(1993)JournalofPhilosophy
90:219–32.
“StrictCoherence,Sigma Coherence, andthe Metaphysicsof Quantity”
(1995)PhilosophicalStudies77:39–55.
“Higher-OrderDegreesofBelief”(1980)inD.H.Mellor(ed.),Prospects
forPragmatism:EssaysinHonorofF.P.Ramsey.Cambridge:Cambridge
UniversityPress,pp.109–37.
“A Mistake in Dynamic Coherence Arguments?” (1993) Philosophy of
Science60:320–8.
“Dynamic Coherence and Probability Kinematics” (1987) Philosophy of
Science54:1–20.
“Updating, Supposing, and MAXENT” (1987) Theory and Decision 22:
225–46.
“TheStructureofRadicalProbabilism”(1996)Erkenntnis45:285–97.
“Diachronic Coherence and Radical Probabilism” (2006) Philosophy of
Science(SpecialIssueforPSA2004)73:959–68.
“Carnapian Inductive Logic for Markov Chains” (1991) Erkenntnis 35:
439–60.
“CarnapianInductiveLogicandBayesianStatistics”(1996)Statistics,Prob-
abilityandGameTheory30:321–36.
“BayesianProjectibility”(1994)inD.Stalker(ed.),Grue:EssaysontheNew
RiddleofInduction.LaSalle,IL:OpenCourt,pp.241–62.
They are unaltered except for some typographical corrections, and the
omissionofsomedigressionsfrom“Higher-OrderDegreesofBelief.”
ThankstoPeterMomtchiloff,myeditoratOxford,forsuggestingthis
projectandseeingitintoprint.
PART I
Zeno and the Metaphysics
of Quantity