Saving Truth From Paradox

Contents......Page 12
2 Russell’s paradox for properties......Page 18
3 … v. Russell’s paradox for sets......Page 19
4 Solution routes to Russell’s paradox for properties......Page 21
5 Grelling again......Page 28
6 Change of logic and change of meaning......Page 31
7 Some connections to other issues......Page 35
PART I. A SELECTIVE BACKGROUND......Page 38
1 Self-Reference and Tarski’s Theorem......Page 40
1.1 Self-reference and Gödel–Tarski diagonalization......Page 41
1.2 Tarski’s "Undefinability Theorem"......Page 44
1.3 Tarski’s "Undefinability Theorem" states more than undefinability......Page 47
1.4 Another form of Tarski’s theorem......Page 49
1.5 Can set-theoretic truth be defined?......Page 50
1.6 Inductive characterizations and restricted truth definitions......Page 53
1.7 Further remarks on explicit definition......Page 56
2.1 Validity and the necessary preservation of truth......Page 59
2.2 Truth in a model......Page 60
2.3 The Kreisel squeezing argument......Page 63
2.4 The unprovability of soundness......Page 65
3 Kripke’s Theory of Truth (Strong Kleene Version)......Page 73
3.1 The Kripke construction for restricted languages......Page 75
3.2 The Kripke construction for unrestricted languages......Page 79
3.3 Conservativeness......Page 82
3.4 Does truth coincide with semantic value 1 even for restricted languages? (KFS v. FM)......Page 85
3.5 Gaps and gluts......Page 87
3.6 The weaknesses of Kripke’s construction......Page 89
3.7 Acceptance and rejection in KFS......Page 90
3.8 Defectiveness again......Page 93
Appendix: Kleene logic and other deMorgan logics......Page 96
4.1 The Curry paradox......Page 100
4.2 Continuum-valued semantics......Page 103
4.3 What do the semantic values mean?......Page 105
4.4 Determinate truth in continuum-valued semantics......Page 106
4.5 Ultimate failure: quantifiers......Page 109
4.6 Indeterminacy and rejection......Page 111
Appendix: The Conservativeness Theorem......Page 114
5.1 Must vague predicates have sharp boundaries?......Page 117
5.2 Penumbral connections and higher order vagueness......Page 119
5.3 Must higher order vagueness collapse?......Page 121
5.4 Linear order?......Page 122
5.5 The König and Berry paradoxes......Page 123
5.6 The role of a classical meta-theory for a non-classical language......Page 125
PART II. BROADLY CLASSICAL APPROACHES......Page 132
6 Introduction to the Broadly Classical Options......Page 134
7.1 Gaps and (T-OUT)......Page 138
7.2 Kleene-style gaps versus supervaluation-style gaps......Page 141
7.3 Declaring one’s axioms untrue......Page 147
7.4 Propositions to the rescue?......Page 149
7.5 Truth-of, heterologicality, and properties......Page 151
7.6 Restricted (T-OUT)......Page 152
7.7 Does declaring one’s axioms untrue destroy the purpose of truth?......Page 155
8.1 Gluts and (T-IN)......Page 159
8.2 (T-IN) theories......Page 160
8.3 What do glut theories say about themselves?......Page 163
8.4 Evaluation of gluts v. gaps......Page 164
9.1 Indeterminacy in classical theories......Page 167
9.2 Supervaluationism......Page 170
10.1 The simplest supervaluational fixed points......Page 173
10.2 Indeterminacy, weak validity, and reasoning by cases......Page 177
10.3 The status of the truth rules......Page 179
10.4 Indeterminacy again......Page 181
10.5 Boolean-valued semantics......Page 183
10.6 Strong validity, and weak validity revisited......Page 186
10.7 Yogi Berra’s advice and ignorance interpretations......Page 189
11.1 Simple supervaluationism is very weak......Page 193
11.2 General supervaluational fixed point theories......Page 194
11.3 Avoiding the problems of simple supervaluationism......Page 198
11.4 Rule-of-revision theories......Page 203
11.5 Soundness proofs revisited......Page 207
12.1 What do strong supervaluational and strong revision theories say about themselves?......Page 209
12.2 What do medium supervaluational and medium revision theories say about themselves?......Page 213
12.3 Are even the strong theories really "self-undermining"?......Page 216
12.4 Gödel’s Second Incompleteness Theorem......Page 217
12.5 Conclusion......Page 221
13 Intersubstitutivity and the Purpose of Truth......Page 222
13.1 Harmless gaps......Page 223
13.2 Acceptance and rejection in weakly classical theories......Page 225
13.3 The rest of the iceberg......Page 226
14.1 Contextual theories and "Strengthened Liar reasoning"......Page 228
14.2 Stratified gap and glut theories......Page 231
14.3 "The ghost of the Tarski hierarchy": stratified internal fixed point theories......Page 239
14.4 Stratified determinacy predicates for weakly classical theories......Page 242
PART III. PARACOMPLETENESS......Page 246
15.1 A framework for generalizing continuum-valued semantics......Page 248
15.2 Determinateness and the Liar hierarchy......Page 252
15.3 More on the never-collapsing hierarchy of determinately operators......Page 256
16 Fixed Points and Revision Rules for Conditionals......Page 259
16.1 Yablo fixed points......Page 261
16.2 Revisionism......Page 266
16.3 The transfinite Liar hierarchy and other examples......Page 270
Appendix: The Fundamental Theorem......Page 274
17.1 Algebraic semantics......Page 276
17.2 Conservativeness and schemas......Page 279
17.3 Modal semantics......Page 281
17.4 Laws and non-laws......Page 283
17.5 Variations......Page 288
18 What Has Been Done......Page 292
PART IV. MORE ON PARACOMPLETE SOLUTIONS......Page 296
19.1 Another form of Curry’s paradox......Page 298
19.2 The validity argument......Page 301
19.3 The Second Incompleteness Theorem......Page 303
20.1 Paradoxes of denotation......Page 308
20.2 The naive theory of properties, relations, and propositions......Page 311
20.3 Set theory......Page 313
20.4 Paradoxes of truth and validity......Page 315
20.5 "Non-bivalent" validity?......Page 320
21.1 Boolean negation and "exclusion negation"......Page 326
21.2 Intuitionist negation and the intuitionist conditional......Page 329
21.3 Wright’s argument......Page 331
21.4 Restall’s argument......Page 333
22 Determinateness, Hyper-Determinateness, and Super-Determinateness......Page 342
22.1 Transfinite iteration made rigorous......Page 343
22.2 Hyper-determinateness: the problem......Page 348
22.3 Hyper-determinateness: the solution......Page 350
22.4 Expanding the language?......Page 355
22.5 Higher-order resources......Page 357
22.6 Super-determinateness......Page 360
23.1 Stratified truth v. iterated determinacy......Page 364
23.2 Genuine costs......Page 367
23.3 Trying to get revenge......Page 370
PART V. PARACONSISTENT DIALETHEISM......Page 376
24.1 Dialetheism, the truth schema, and intersubstitutivity......Page 378
24.2 Acceptance, rejection, and degree of belief......Page 380
24.3 Gluts, gaps, and intersubstitutivity again......Page 381
25.1 Priest’s LP......Page 385
25.2 Dualizing paracomplete theories......Page 386
25.3 Priest’s conditionals......Page 388
26.1 The first incompleteness theorem, Curry’s paradox, and truth-preservation......Page 393
26.2 Can we get even restricted truth-preservation?......Page 397
27.1 Model theory, designated values and truth......Page 401
27.2 Sole truth and sole falsehood......Page 403
27.3 Extended paradox?......Page 407
References......Page 410
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Z......Page 423