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Tables of Contents for Predicate Logic
Chapter/Section Title
Page #
Page Count
Propositions and Propositional Logic
Logic
2
1
Propositions
Propositions and agreements
2
2
Other views of propositions
4
1
Words and Propositions as Types
5
2
Propositions in English
7
2
Exercises for Sections A-D
8
1
The Basic Connectives of Propositional Logic
9
2
A Formal Language for Propositional Logic
Defining the formal language
11
1
Realizations: semi-formal English
12
2
Exercises for Sections E and F
14
1
Classical Propositional Logic
The Classical Abstraction and truth-functions
15
5
Models
20
1
Validity and semantic consequence
20
2
Determining whether a wff is a tautology
22
1
Examples of Formalization
23
5
Exercises for Sections G and H
27
1
Relatedness Logic
28
7
The subject matter of a proposition
Relatedness relations
29
2
Subject matter as the content of a proposition
31
1
Models
32
3
An Overview of Semantics for Propositional Logics
35
4
Exercises for Sections J and K
38
1
The Internal Structure of Propositions
Things, the World, and Propositions
39
3
Names and Predicates
42
2
Propositional Connectives
44
1
Variables and Quantifiers
45
2
Compound Predicates and Quantifiers
47
1
The Grammar of Predicate Logic
48
2
Exercises
49
1
A Formal Language for Predicate Logic
A Formal Language
50
3
The Unique Readability of Wffs
53
2
The Complexity of Wffs
55
1
Free and Bound Variables
56
2
The Formal Language and Propositions
58
5
Exercises
59
4
Semantics
Syntax vs. Semantics as a Basis for Logic
63
1
Atomic Propositions
64
1
Names
A name picks out at most one thing
65
1
A name picks out at least one thing
65
3
Predicates
A predicate applies to an object
68
3
Predications involving relations
71
3
Other conceptions of predicates and predications
74
2
How many predicates are there?
76
1
Naming, Pointing, and What There Is
Agreements
77
2
Naming, pointing, and descriptions
79
2
Avoiding names completely?
81
1
Forms of pointing: what there is
81
3
Exercises for Sections A-E
84
3
The Universe of a Realization
87
3
The Self-Reference Exclusion Principle
90
2
Models
The assumptions of the realization: Form and Meaningfulness
92
2
Interpretations: assignments of references and valuations
94
4
The Fregean Assumption and The Division of Form and Content
98
1
The truth-value of a complex proposition
99
5
Truth in a model
104
4
Logics, Validity, Semantic Consequence
108
9
Exercises for Sections F-J
114
2
Summary Chapters II-IV.J
116
1
Tarski's Definition of Truth
117
6
Eliminating semantic terms: Convention T
118
4
Other logics, other views of truth
122
1
Extensionality
Intensional predicates
123
2
The Extensionality Restriction
125
2
Quantification and intensional predicates
Languages without names
127
1
Models in which every object is named
128
1
Inconsistent predications and quantification
128
1
Other Interpretations of the Quantifiers and the Use of Variables
A current variation on Tarski's definition
129
1
The substitutional interpretation
129
2
Naming all elements of the universe at once
131
1
Surveying all interpretations of the name symbols
132
1
Exercises for Sections K-M
133
4
The Logical Form of a Proposition
Rewriting English Sentences
137
2
Common Nouns as Subject and Object
Relative quantification: A
139
3
Relative quantification: E
142
2
Nouns into Predicates
144
1
Adjectives
145
3
Indexicals
148
1
Adverbs
149
2
Tenses
151
3
Collections and Qualities
154
3
Mass Terms
157
2
Aristotelian Logic
159
3
Formalizations Relative to Formal Assumptions
Analysis vs. formalization
162
1
Extending the scope of predicate logic
163
1
Formalizing a notion
164
1
Criteria of Formalization
165
7
Examples of Formalization
172
36
Exercises
201
7
Identity
Identity
208
2
The Equality Predicate
210
1
The Interpretation of `=' in a Model
211
2
The Identity of Indiscernibles
The Predicate Logic Criterion of Identity (p.l.c.i.)
213
2
The p.l.c.i. vs. the implicit identity of the universe
215
1
The p.l.c.i. and names
216
1
Validity
217
4
Is the Equality Predicate Syncategorematic?
221
4
Exercises
223
2
Quantifiers
The Order of Quantifiers
∀x∃y and ∃y∀x
225
1
∀x∃y and ∃x∀y
226
1
Superfluous quantifiers
226
1
The Scope of Quantifiers: Substituting One Variable for Another
227
3
Names, Quantifiers, and Existence
230
2
Is `--exists' a Predicate?
232
1
Quantifying Over a Finite Universe: ∀ as Conjunction, ∃ as Disjunction
233
1
Modeling Other Quantifiers
Positive quantifiers: `there are at least n'
234
2
Negative quantifiers: `there are at most n', `no', `nothing'
236
1
Exact quantifiers: `there are exactly n'
237
1
Quantifications we can't model
238
1
Relative Quantification
Nouns into Predicates revisited
239
1
Formalizations involving the same quantifier
240
2
Formalizations involving mixtures of quantifiers
242
1
Examples of Formalization
243
20
Exercises
257
6
Descriptive Names
Descriptive Names: A Problem in Formalization
263
2
Descriptive Names Relative to Formal Assumptions
265
1
Russell's Method of Eliminating Descriptive Names from Atomic Propositions
266
3
Eliminating All Names?
269
3
Examples of Formalization
272
10
Exercises
279
3
Functions
Name-Makers
282
3
Functions
A definition
285
1
Terms
286
1
The value of a function
286
2
Functions compared to predicates
288
1
A Formal Language with Function Symbols and Equality
289
2
Realizations and Truth in a Model
291
2
Partial Name-Makers
Russell's abstraction operator
293
4
The ε-operator
297
1
Examples of Formalization
298
6
Exercises
300
4
Quantifying Over Predicates: Second-Order Logic
Quantifying over Predicates?
304
1
Predicates and Things
305
1
Predicate Variables and their Interpretation: Avoiding Self-Reference
Predicate variables
306
2
The interpretation of predicate variables
308
3
Note: Higher-order logics
311
1
A Formal Language for Second-Order Logic: L2
312
1
Realizations
313
4
Identifying Predicates with Collections of n-tuples of the Universe
317
2
Exercises for Sections A-F
318
1
Models
319
3
Examples of Formalization
322
13
Exercises for Sections G and H
334
1
Predicates as Things: Reducing General Second-Order Logic to First-Order Logic
One universe for predicates and individuals
335
3
The translation
338
1
Proof that the mapping preserves consequences
338
6
Does the reduction preserve meaning?
344
1
Quantifying over Functions
Why quantify over functions?
345
1
A formal language: L2F
346
1
Realizations and models
347
2
The difficulty of reducing quantification over functions to first-order logic
349
1
Many-Sorted Languages
350
3
Exercises for Sections J-L
352
1
Language, the World, and Predicate Logic
The World
353
1
The Template Analogy
353
1
Eliminating Natural Languages?
354
1
Predicate Logic as a Model of or Guide to Reasoning
355
2
Appendices
A The Notion of Thing in Predicate Logic
357
5
B What There Is: Restrictions on the Universe of a Realization
362
1
C Primitives and Assumptions of Predicate Logic
363
6
D Formalization: Criteria and Agreements
369
7
Bibliography
376
5
Index of Examples
381
7
Index of Notation
388
2
Index
390