Table Of ContentVarieties of Logic
Varieties of Logic
Stewart Shapiro
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Acknowledgments
Thisprojecthasbeenintheworksforalongtime—perhapstoolong—
and my debts are many. First and foremost, I thank my friend and
colleague, Kevin Scharp, for reading several versions of each of the
chapters and providing much insightful and useful feedback, especially
on how my approach relates to others who have defended or resisted a
pluralismconcerninglogic,andonhowthepresentmattersplaythem-
selves out with respect to general matters of contextualism and relativ-
ism. Thanks also to Graham Priest, Edward Mares, and Alexander
Paseau who also read entire drafts of this work, and provided helpful
feedback, helping me to avoid potential confusions, organize things
better, and strengthen the arguments (even when they disagreed with
theconclusions).Ihavebenefittedfromcommentsondraftsandgeneral
discussionwithmanyfriendsandcolleagues,includingStevenAwodey,
Herman Cappelen, Steven Hales, Geoffrey Hellman, Ole Hjortland,
CatarinaDutilhNovaes,TorfinnHuvenes,ChristopherKennedy,Teresa
Kouri, Robert Kraut,(cid:1)ystein Linnebo,David McCarty,Michael Miller,
Stephen Read, Craige Roberts, Eric Snyder, Neil Tennant, Gabriel Uz-
quiano,CrispinWright,andseveralanonymousreferees.Iapologizefor
any omissions from this list. I have also benefitted immensely from
feedback from various audiences, where I gave papers based on aspects
of this work. First and foremost, thanks to the Foundations of Logical
Consequence project at the Arché Research Centre, University of St
Andrews, for devoting four different sessions at various conferences,
workshops, and seminars to this work. Thanks to the University of
Paris I, the University of Connecticut, the Association for Symbolic
Logic,CambridgeUniversity,theInterUniversityCentreinDubrovnik,
UniversityofUppsala,EdinburghUniversity,andaconferenceonlogic
inMathematicsinNancy,France.ThanksalsotoPeterMomtchiloffand
CarolineHawleyfromOxfordUniversityPress,fortheirencouragement
andprofessionalexpertise inguiding this bookthroughthepublication
process.
Finally, I thank my wife, Beverly Roseman-Shapiro, for putting up
withallofmytravelandeverythingelsethatgoeswithbeingthespouse
ofaworkaholicacademic.
Contents
1. Relativism,Pluralism,Tolerance 1
2. VarietiesofPluralismandRelativismforLogic 17
3. Structure:AnEclecticPerspective 63
4. WeMeanWhatWeSay:ButWhatDoWeMean? 88
5. MeaningandContext 126
6. TheoryandMeta-theory;LogicandMeta-logicI:
PhilosophicalandFoundationalStudies 163
7. TheoryandMeta-theory;LogicandMeta-logicII:
Meta-theoreticPerspective 182
RecapitulationandConclusion 205
References 210
Index 221
1
Relativism, Pluralism,
Tolerance
1 Overview: Where We are Going
The aim of this book is to articulate various kinds of pluralism and
relativism concerning logic, and to defend some of them. There is
controversy over what logic is. Indeed, a certain flexibility concerning
the nature of logic drives at least some aspects of the project. Although
wewillbroachmoregeneralmattersfromtimetotime,themainfocusis
onrigorousdeduction,withinvariousmathematicaltheories.Ourques-
tions include these: Is there more than one logic that is employed—or
can be employed, or should be employed—in legitimate mathematical
theories? Is there more than one extension for the relation of logical
consequence, or validity, depending on the background theory (or
depending on something else)? If so, is the extension of logical conse-
quencesomehowrelativetosomething?
In the philosophical literature, there is no consensus on terms like
“pluralism,”“relativism,”and“contextualism.”SoImustbeginbybriefly
showing how I use the indicated terms. That is the business of the
succeeding sections of this chapter, which will also help motivate the
materialtofollow.Fornow,wewillkeeptheterminologicalmatterstacit,
hopingthattheresultingconfusionisminimal.
Chapter2developsanumberofdifferentsensesinwhichonecanbea
relativistand/orapluralistconcerninglogicalconsequenceandvalidity.
I argue, first, that the terms “logical consequence” and “validity” are
polysemousor,perhapsbetter,theyexpressclusterconcepts.Anumber
of different, closely related notions go by those names. They invoke
matters of modality, meaning, effectiveness, justification, rationality,
andform.Theupshotisakindofpluralism:therearedifferent,mutually
