Introduction to Metalogic: With an Appendix on Type-Theoretical Extensional and Intensional Logic

Cover......Page 1
Title......Page 2
Publication Data......Page 3
Dedication......Page 4
Acknowledgements......Page 5
Table of Contents......Page 6
1.1  The Subject Matter of Metalogic......Page 8
1.2  Basic postulates on languages......Page 9
1.3  Speaking about languages......Page 11
1.4  Syntax and semantics......Page 12
2.1  Grammatical Means......Page 14
2.2  Variables and Quantifiers......Page 17
2.3  Logical means......Page 22
2.4  Definitions......Page 26
2.5  Class Notation......Page 27
3.1  Definition and Postulates......Page 32
3.2  The Simplest Alphabets......Page 35
4.1  Inductive Definitions......Page 38
4.2  Canonical Calculi......Page 43
4.3  Some Logical Languages......Page 45
4.4  Hypercalculi......Page 49
4.5  Enumerability and Decidability......Page 54
5.1  What is an Algorithm?......Page 58
5.2  Definition of Normal Algorithms......Page 61
5.3  Deciding Algorithms......Page 65
5.4  Definite Classes......Page 68
6.1  What is a Logical Calculus?......Page 73
6.2  First-Order Languages......Page 74
6.3  The Calculus QC......Page 77
6.4  Metatheorems on QC......Page 79
6.5  Consistency. First-Order Theories......Page 81
7.1  Approaching Intuitively......Page 83
7.2. The Canonical Calculus Σ*......Page 85
7.3  Truth Assignment......Page 88
7.4  Undecidability: Church’s Theorem......Page 90
8.1  The Formal Theory CC......Page 92
8.2  Diagonalization......Page 94
8.3  Extensions and Discussions......Page 97
9.1  Preparatory Work......Page 100
9.2  The Proof of the Unprovability of Cons......Page 101
10.1  Sets and Classes......Page 105
10.2  Relations and Functions......Page 110
10.3  Ordinal, Natural, and Cardinal Numbers......Page 113
10.4  Applications......Page 117
C-F-G-H-K-M......Page 121
P-R-S-T......Page 122
A-B-C......Page 123
D-E-F......Page 124
G-H-I-J-K-L......Page 125
M-N-O-P......Page 126
Q-R-S......Page 127
T-U-V-W-Z......Page 128
List of Symbols......Page 129
Appendix (Lecture Notes): Type Theoretical Extensional and Intensional Logic......Page 130
Contents......Page 131
Technical Introduction......Page 132
1.1.2  The Grammar of the EL Languages......Page 134
1.1.3  Semantics for EL Languages......Page 136
1.1.4  Some Semantical Metatheorems......Page 137
1.1.5  Logical Symbols Introduced Via Definitions......Page 139
1.1.6  The Generalized Semantics......Page 140
1.2.1  Definition of EC......Page 141
1.2.2  Some proofs in EC......Page 142
1.2.3  EC-consistent and EC-complete sets......Page 149
1.2.4  The completeness of EC......Page 151
2.1.1  Montague’s Type Theory......Page 155
2.1.2  The Grammar of IL and IL*......Page 156
2.1.3  The Semantics of IL and IL*......Page 157
2.1.4  The Generalized Semantics of IL......Page 162
2.2.2  The Modal Laws of IC......Page 163
2.2.3  IC-Consistency and IC-Complete Sets......Page 166
2.2.4  Modal Alternatives......Page 167
2.2.5  The Completeness of IC......Page 169
2.3.1  A Fragment of English: ℒE......Page 173
2.3.2  Translation Rules from ℒE into ℒ(i)......Page 178
2.3.3  Reduction of Intensionality: Meaning Postulates......Page 183
2.3.4  Some Critical Remarks......Page 186
References......Page 189