Saving Truth From Paradox

by Hartry Field

Overview

Saving Truth From Paradox is an ambitious investigation into paradoxes of truth and related issues, with occasional forays into issues such as vagueness, the nature of validity, and the Godel incompleteness theorems. Hartry Field presents a new approach to the paradoxes and provides a systematic and detailed account of the main competing approaches.

Part One examines Tarski's, Kripke's, and Lukasiewicz's theories of truth, and discusses validity and soundness, and vagueness. Part Two considers a wide range of attempts to resolve the paradoxes within classical logic. In Part Three Field turns to non-classical theories of truth that restrict excluded middle. He shows that there are theories of this sort in which the conditionals obey many of the classical laws and in which all the semantic paradoxes (not just the simplest ones) can be handled consistently with the naive theory of truth. In Part Four, these theories are extended to the property-theoretic paradoxes and to various other paradoxes, and some issues about the understanding of the notion of validity are addressed. Extended paradoxes, involving the notion of determinate truth, are treated very thoroughly, and a number of different arguments that the theories lead to "revenge problems" are addressed. Finally, Part Five deals with dialetheic approaches to the paradoxes: approaches which, instead of restricting excluded middle, accept certain contradictions but alter classical logic so as to keep them confined to a relatively remote part of the language. Advocates of dialetheic theories have argued them to be better than theories that restrict excluded middle, for instance over issues related to the incompleteness theorems andin avoiding revenge problems. Field argues that dialetheists' claims on behalf of their theories are quite unfounded, and indeed that on some of these issues all current versions of dialetheism do substantially worse than the best theories that restrict excluded middle.

About the Author:
Hartry Field is University Professor and Silver Professor of Philosophy at New York University. He is the author of Truth and the Absence of Fact OUP, 2001

Product Details

ISBN-13:	9780199230747
Publisher:	Oxford University Press, USA
Publication date:	05/11/2008
Edition description:	New Edition
Pages:	432
Product dimensions:	9.10(w) x 6.10(h) x 0.90(d)

Table of Contents

Introduction     1
Grelling's paradox     1
Russell's paradox for properties     1
... v. Russell's paradox for sets     2
Solution routes to Russell's paradox for properties     4
Grelling again     11
Change of logic and change of meaning     14
Some connections to other issues     18
A Selective Background
Self-Reference and Tarski's Theorem     23
Self-reference and Godel-Tarski diagonalization     24
Tarski's "Undefinability Theorem"     27
Tarski's "Undefinability Theorem" states more than undefinability     30
Another form of Tarski's theorem     32
Can set-theoretic truth be defined?     33
Inductive characterizations and restricted truth definitions     36
Further remarks on explicit definition     39
Validity and the Unprovability of Soundness     42
Validity and the necessary preservation of truth     42
Truth in a model     43
The Kreisel squeezing argument     46
The unprovability of soundness     48
Kripke's Theory of Truth (Strong Kleene Version)     56
The Kripke construction for restricted languages     58
The Kripke construction forunrestricted languages     62
Conservativeness     65
Does truth coincide with semantic value 1 even for restricted languages? (KFS v. FM)     68
Gaps and gluts     70
The weaknesses of Kripke's construction     72
Acceptance and rejection in KFS     73
Defectiveness again     76
Kleene logic and other deMorgan logics     79
Adding a Conditional? Curry and Lukasiewicz     83
The Curry paradox     83
Continuum-valued semantics     86
What do the semantic values mean?     88
Determinate truth in continuum-valued semantics     89
Ultimate failure: quantifiers     92
Indeterminacy and rejection     94
The Conservativeness Theorem     97
Interlude on Vagueness, and the Paradoxes of Konig and Berry     100
Must vague predicates have sharp boundaries?     100
Penumbral connections and higher order vagueness     102
Must higher order vagueness collapse?     104
Linear order?     105
The Konig and Berry paradoxes     106
The role of a classical meta-theory for a non-classical language     108
Broadly Classical Approaches
Introduction to the Broadly Classical Options      117
Truth-Value Gaps in Classical Theories     121
Gaps and (T-OUT)     121
Kleene-style gaps versus supervaluation-style gaps     124
Declaring one's axioms untrue     130
Propositions to the rescue?     132
Truth-of, heterologicality, and properties     134
Restricted (T-OUT)     135
Does declaring one's axioms untrue destroy the purpose of truth?     138
Truth-Value Gluts in Classical Theories     142
Gluts and (T-IN)     142
(T-IN) theories     143
What do glut theories say about themselves?     146
Evaluation of gluts v. gaps     147
A Second Interlude on Vagueness     150
Indeterminacy in classical theories     150
Supervaluationism     153
Introduction to Supervaluational Approaches to Paradox     156
The simplest supervaluational fixed points     156
Indeterminacy, weak validity, and reasoning by cases     160
The status of the truth rules     162
Indeterminacy again     164
Boolean-valued semantics     166
Strong validity, and weak validity revisited     169
Yogi Berra's advice and ignorance interpretations      172
A Survey of Supervaluational and Revision-Rule Theories     176
Simple supervaluationism is very weak     176
General supervaluational fixed point theories     177
Avoiding the problems of simple supervaluationism     181
Rule-of-revision theories     186
Soundness proofs revisited     190
Are Supervaluational and Revision Theories Self-Undermining?     192
What do strong supervaluational and strong revision theories say about themselves?     192
What do medium supervaluational and medium revision theories say about themselves?     196
Are even the strong theories really "self-undermining"?     199
Godel's Second Incompleteness Theorem     200
Conclusion     204
Intersubstitutivity and the Purpose of Truth     205
Harmless gaps     206
Acceptance and rejection in weakly classical theories     208
The rest of the iceberg     209
Stratified and Contextual Theories     211
Contextual theories and "Strengthened Liar reasoning"     211
Stratified gap and glut theories     214
"The ghost of the Tarski hierarchy": stratified internal fixed point theories     222
Stratified determinacy predicates for weakly classical theories      225
Paracompleteness
What Is To Be Done?     231
A framework for generalizing continuum-valued semantics     231
Determinateness and the Liar hierarchy     235
More on the never-collapsing hierarchy of determinately operators     239
Fixed Points and Revision Rules for Conditionals     242
Yablo fixed points     244
Revisionism     249
The transfinite Liar hierarchy and other examples     253
The Fundamental Theorem     257
More on Revision-Theoretic Conditionals     259
Algebraic semantics     259
Conservativeness and schemas     262
Modal semantics     264
Laws and non-laws     266
Variations     271
What Has Been Done     275
More on Paracomplete Solutions
Validity, Truth-Preservation, and the Second Incompleteness Theorem     281
Another form of Curry's paradox     281
The validity argument     284
The Second Incompleteness Theorem     286
Other Paradoxes     291
Paradoxes of denotation     291
The naive theory of properties, relations, and propositions     294
Set theory     296
Paradoxes of truth and validity     298
"Non-bivalent" validity?     303
Do Paracomplete Solutions Depend on Expressive Limitations?     309
Boolean negation and "exclusion negation"     309
Intuitionist negation and the intuitionist conditional     312
Wright's argument     314
Restall's argument     316
Determinateness, Hyper-Determinateness, and Super-Determinateness     325
Transfinite iteration made rigorous     326
Hyper-determinateness: the problem     331
Hyper-determinateness: the solution     333
Expanding the language?     338
Higher-order resources     340
Super-determinateness     343
Determinateness, Stratification, and Revenge     347
Stratified truth v. iterated determinacy     347
Genuine costs     350
Trying to get revenge     353
Paraconsistent Dialetheism
An Introduction to Paraconsistent Dialetheism     361
Dialetheism, the truth schema, and intersubstitutivity     361
Acceptance, rejection, and degree of belief     363
Gluts, gaps, and intersubstitutivity again     364
Some Dialetheic Theories     368
Priest's LP     368
Dualizing paracomplete theories     369
Priest's conditionals     371
Paraconsistent Dialetheism and Soundness     376
The first incompleteness theorem, Curry's paradox, and truth-preservation     376
Can we get even restricted truth-preservation?     380
Hyper-Determinacy and Revenge     384
Model theory, designated values and truth     384
Sole truth and sole falsehood     386
Extended paradox?     390
References     393
Index     399