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Tables of Contents for Propositional Logics
Chapter/Section Title
Page #
Page Count
Preface and Acknowledgements
The Basic Assumptions of Propositional Logic
What is Logic?
1
1
Propositions
Sentences, Propositions, and truth
2
1
Other views of propositions
3
2
Words and Propositions as Types
5
2
Propositions in English
7
2
Exercises for Sections A-D
7
2
Form and Content
9
1
Propositional Logic and the Basic Connectives
10
3
Exercises for Sections E and F
12
1
A Formal Language for Propositional Logic
Defining the formal language
13
1
Realizations: semi-formal English
14
2
Exercises for Section G
16
4
Classical Propositional Logic - PC -
The Classical Abstraction and the Fregean Assumption
20
1
Truth-Functions and the Division of Form and Content
21
3
Models
24
2
Exercises for Sections A-C
25
1
Validity and Semantic Consequence
Tautologies
26
2
Semantic Consequence
28
3
Exercises for Section D
31
1
The Logical Form of a Proposition
On logical form
31
1
Criteria of formalization
32
1
Other propositional connectives
33
1
Example of Formalization
Ralph is a dog or Dusty is a horse and Howie is a cat
Therefore: Howie is a cat
34
1
Ralph is a dog and George is a duck and Howie is a cat
34
1
Ralph is a dog or he's a puppet
35
1
Ralph is a dog if he's not a puppet
35
1
Ralph is a dog although he's a puppet
35
1
Ralph is not a dog because he's a puppet
36
1
Three faces of a die are even numbered
Three faces of a die are not even numbered
Therefore Ralph is a dog
36
1
Ken took off his clothes and went to bed
36
1
The quotation marks are signals for you to understand what I mean; they are not part of the realization
37
1
Every natural number is even or odd
37
1
If Ralph is a dog, then Ralph barks
Ralph barks
Therefore Ralph is a dog
37
1
Suppose {sn} is monotonic. Then {sn} convereges if and only if it is bounded
38
1
Dedekind's Theorem
38
1
Exercises for Sections E and F
39
2
Further Abstractions: The Role of Mathematics in Logic
41
3
Induction
44
3
Exercises for Sections G and H
46
1
A Mathematical Presentation of PC
The formal language
47
3
Exercise for Section J.1
50
1
Models and the semantic consequence relation
51
1
Exercises for Section J.2
52
1
The truth-functional completeness of the connectives
53
2
The choice of language for PC
55
1
Normal forms
55
1
The principle of duality for PC
56
1
Exercises for Sections J.3-J.6
57
1
The decidability of tautologies
58
1
Exercises for Section J.7
59
1
Some PC-tautologies
60
2
Exercises for Sections J.8
62
2
Formalizing the Notion of Proof
Reasons for formalizing
64
1
Proof, syntactic consequence, and theories
64
3
What is a logic?
67
2
Exercises for Section K
69
1
An Axiomatization of PC
The axiom system
69
2
Exercises for Section L.1
71
1
A completeness proof
72
2
The Strong Completeness Theorem for PC
74
1
Exercises for Section L.2 and L.3
75
1
Derived rules: substitution
76
1
Exercises for Section L.4
77
1
Other Axiomatizations and Proofs of Completeness of PC
History and Post's proof
77
1
A constructive proof of the completeness of PC
78
1
Exercises for Section M.1 and M.2
79
1
Schema vs. the rule of substitution
80
1
Independent axiom systems
81
1
Proofs using rules only
81
1
Exercises for Sections M.3-M.5
82
1
An axiomatization of PC in L(⌝, →, ∧, ∨)
82
1
An axiomatization of PC in L(⌝, ∧)
83
1
Exercises for Sections M.6 and M.7
84
1
Classical logic without negation: the positive fragment of PC
85
3
Exercises for Section M.8
88
1
The Reasonableness of PC
Why classical logic is classical
88
1
The paradoxes of PC
89
4
Relatedness Logic: The Subject Matter of a Proposition - S and R -
An Aspect of Propositions: Subject Matter
93
3
The Formal Language
96
1
Properties of the Primitive: Relatedness Relations
96
3
Subject Matter as Set-Assignments
99
3
Exercises for Sections A-D
101
1
Truth and Relatedness-Tables
102
2
The Formal Semantics for S
Models based on relatedness relations
104
2
Models based on subject matter assignments
106
1
Non-symmetric relatedness logic, R
106
1
Exercises for Sections E and F
107
1
Examples
107
4
If the moon is made of green cheese, then 2+2=4
108
1
2+2=4
Therefore: If the moon is made of green cheese, then 2+2=4
108
1
If Ralph is a dog and if 1=1 then 1=1, then 2+2=4 or 2+2≠4
108
1
If John loves Mary, then Mary has 2 apples
If Mary has 2 apples, then 2+2=4
Therefore: If John loves Mary, then 2+2=4
109
1
If Don squashed a duck and Don drives a car, then a duck is dead
Therefore: If Don squashed a duck, then if Don drives a car, then a duck is dead
110
1
Exercises for Section G
111
1
Relatedness Logic Compared to Classical Logic
The decidability of relatedness tautologies
111
1
Every relatedness tautology is a classical tautology
112
1
Classical tautologies that aren't relatedness tautologies
112
1
Exercises for Section H
113
1
Functional Completeness of the Connectives and the Normal Form Theorem for S
113
4
Exercises for Section J
116
1
Axiomatizations
S in L(⌝, →)
117
2
Exercises for Section K.1
119
1
S in L(⌝, →, ∧)
120
1
R in L(⌝, →, ∧)
120
1
Substitution
121
1
The Deduction Theorem
122
2
Exercises for Section K.2-K.5
124
1
Historical Remarks
125
2
A General Framework for Semantics for Propositional Logics
Aspects of Sentences
Propositions
127
2
The logical connectives
129
1
Two approaches to semantics
129
1
Exercises for Section A
130
1
Set-Assignment Semantics
Models
130
4
Exercises for Section B.1
134
1
Abstract models
135
1
Semantics and logics
136
1
Semantic and syntactic consequence relations
137
2
Exercises for Section B.2-B.4
139
1
Relation-Based Semantics
140
1
Exercises for Section C
141
1
Semantics Having a Simple Presentation
141
2
Some Questions
Simply presented semantics
143
1
The Deduction Theorem
143
1
Functional completeness of the connectives
144
1
Representing the relations within the formal language
145
1
Characterizing the class of relations in terms of schema
145
1
Translating other semantics into the general framework
145
1
Decidability
146
1
Extensionally equivalent propositions and the rule of substitution
146
1
On the Unity and Division of Logics
Quine on deviant logical connectives
147
1
Classical vs. nonclassical logics
148
2
A Mathematical Presentation of the General Framework with the assistance of Walter Carnielli
Languages
150
1
Formal set-assignment semantics
150
1
Formal relation-based semantics
151
1
Exercises for Sections G.1-G.3
152
1
Specifying semantic structures
152
1
Set-assignments and relations for SA
153
1
Relations for RB
154
1
Wholly intensional connectives
154
1
Truth-default semantic structures
155
1
Exercises for Sections G.4-G.6
156
1
Tautologies of the general framework
156
1
Valid deductions of the general framework
Examples
157
1
The subformula property
157
1
Axiomatizing deductions in L(⌝, →, ∧)
158
2
Deductions in lanaguages containing disjunction
160
1
Exercises for Sections G.7-G.8
161
2
Dependence Logics -- D, Dual D, Eq, DPC --
Dependence Logic
The consequent is contained in the antecedent
163
3
The structure of referential content
166
1
Set-assignment semantics
167
3
Relation-based semantics
170
1
Exercises for Sections A.1-A.4
171
1
The decidability of dependence logic tautologies
171
1
Examples of formalization
Ari doesn't drink
Therefore: If Ari drinks, then everyone drinks
172
1
Not both Ralph is a dog and cats aren't nasty
Therefore: If Ralph is a dog, then cats are nasty
173
1
If Ralph is a bachelor, then Ralph is a man
173
1
If dogs barks and Juney is a dog, then Juney barks
Therefore: If dogs bark, then if Juney is a dog, then Juney barks
174
1
If dogs bark, then Juney barks
If Juney barks, then a dog has scared a thief
Therefore: If dogs bark, then a dog has scared a thief
174
1
If Ralph is a dog, then Ralph barks
Therefore: If Ralph doesn't bark, then Ralph is not a dog
175
1
Ralph is not a dog because he's a puppet
175
1
Dependence logic tautologies compared to classical tautologies
175
2
The functional completeness of the connectives
177
1
Exercises for Section A.5-A.8
177
1
An axioms system for D
178
3
Exercises for Section A.9
181
1
History
182
1
Dependence-Style Semantics
183
2
Exercises for Section B
184
1
Dual Dependence Logic, Dual D
185
3
Exercises for Section C
187
1
A Logic of Equality of Contents, Eq
Motivation
188
1
Set-assignment semantics
188
1
Characterizing Eq-relations
189
3
An axiom system for Eq
192
1
Exercises for Section D
193
1
A Syntactic Comparison of D, Dual D, Eq, and S
194
1
Content as Logical Consequences
195
4
Exercises for Section F
196
3
Modal Logics - S4, S5, S4Grz, T, B, K, QT, MSI, ML, G, G* -
Implication, Possibility, and Necessity
Strict implication vs. material implication
199
1
Possible worlds
200
2
Necessity
202
1
Different notions of necessity: accessibility relations
202
2
Exercises for Section A
204
1
The General Form of Possible-World Semantics for Modal Logics
The formal framework
205
3
Possibility and necessity in the formal language
208
3
Exercises for Section B
211
2
Semanatic Presentations
Logical necessity: S5
Semantics
213
1
Semantic consequence
214
3
Iterated modalities
217
1
Syntactic characterization of the class of universal frames
217
2
Some rules valid in S5
219
1
Exercises for Section C.1
220
1
K---all accessibility relations
220
2
Exercises for Section C.2
222
1
T, B, and S4
222
1
Decidability and the Finite Model Property
223
2
Exercises for Section C.4
225
1
Examples of Formalization
225
8
If roses are red, then sugar is sweet
226
1
If Ralph is a bachelor, then Ralph is a man
226
1
If the moon is made of green cheese, then 2+2=4
226
1
If Juney was a dog, then surely it's possible that Jueny was a dog
226
1
If it's possible that Juney was a dog, then Jueny was a dog
227
1
It's not possible that Juney was a dog and a cat
228
1
If it is necessary that Juney is a dog, then it is necessary that it is necessary that Juney is a dog
228
1
If this paper is white, it must necessarily by white
228
1
If Hoover was elected president, then he must have received the most votes
Hoover was elected president
Therefore: Hoover must have received the most votes
228
1
A sea fight must take place tomorrow or not. But it is not necessary that it should take place tomorrow; neither is it necessary that it should not take place. Yet it is necessary that it either should or should not take place tomorrow
229
1
It is contingent that US $1 bills are green
229
1
It is possible for Richard L. Epstein to print his own bank notes
230
1
If there wereno dogs, then everyone would like cats
230
1
It is permissible but not obligatory to kill cats
230
1
A dog that likes cats is possible
231
1
Example 2 of Chapter II.F is possible
231
1
Ralph knows that Howie is a cat
232
1
Example 18 is not possible
232
1
Exercises for Section D
232
1
Syntactic Characterizations of Modal Logics
The general format
Defined connectives
233
1
PC in the language of modal logic
233
1
Normal modal logics
234
1
Axiomatizations and completeteness theorems in L(⌝, ∧, □)
235
3
Axiomatizations and completeness theorems in L(⌝, →, ∧)
238
1
Exercises for Section E.1-E.3
239
1
Consequence relations
Without necessitation, ⊢ L
240
2
With necessitation, ⊢ L□
242
1
Exercises for Section E.4
242
1
Quasi-normal modal logics
243
1
Exercises for Section F
244
1
Set-Assignment Semantics for Modal Logics
244
16
Semantics in L(⌝, →, ∧)
Modal semantics of implication
245
2
Weak modal semantics of implication
247
1
Semantics in L(⌝, ∧, □)
248
1
Exercises for Section G.1 and G.2
249
1
Connections of meanings in modal logics: the aptness of set-assignment semantics
249
2
S5
251
2
Exercises for Section G.4
253
1
S4 in collaboration with Roger Maddux
254
1
Exercises for Section G.5
255
2
T
257
1
B
258
1
Exercises for Sections G.6 and G.7
259
1
The Smallest Logics Characterized by Various Semantics
K
260
1
QT and quasi-normal logics
261
1
The logic characterized by modal semantics of implication
261
2
Exercises for Section H
263
1
Modal Logics Modeling Notions of Provability
`□' read as `it is provable that'
263
2
S4Grz
265
1
G
266
2
G*
268
4
Intuitionism - Int and J -
Intuitionism and Logic
272
4
Exercises for Section A
275
1
Heyting's Formalization of Intuitionism
Heyting's axiom system Int
276
1
Kripke semantics for Int
277
3
Exercises for Section B
280
1
Completeness of Kripke Semantics for Int
280
8
Some syntactic derivations and the Deduction Theorem
281
2
Completeness theorems for Int
283
4
Exercises for Sections C.1 and C.2
287
1
On completeness proofs for Int, and an alternate axiomatization
287
1
Exercises for Section C.3
288
1
Translations and Comparisons with Classical Logic
Translations of Int into modal logic and classical arithmetic
288
2
Translations of classical logic into Int
290
4
Axiomatizations of classical logic relative to Int
294
1
Exercises for Section D
294
1
Set-assignment Semantics for Int
295
9
The semantics
296
2
Observations and refinements of the set-assignment semantics
298
3
Exercises for Section E.1 and E.2
301
1
Bivalence in intuitionism: the aptness of set-assignment semantics
302
2
The Minimal Calculus J
The minimal calculus
304
1
Kripke-style semantics
305
2
An alternate axiomatization
307
1
Kolmogorov's axiomatization of intuitionistic reasoning in L(⌝, →)
308
1
Exercises for Sections F.1-F.4
309
1
Set-assignment semantics
310
2
Exercises for Section F.5
312
2
Many-Valued Logics - L3, Ln, Lx, K3, G3, Gn, Gx, S5 --
How Many Truth-Values?
History
314
1
Hypothetical reasoning and aspects of propositions
315
1
A General Definition of Many-Valued Semantics
316
2
The Lukasiewicz Logics
318
18
The 3-valued logic L3
The truth-tables and their interpretation
319
4
Exercises for Section C.1.a
323
1
A finite axiomatization of L3
324
4
Exercises for Section C.1.b
328
1
Wajsberg's axiomatization of L3
328
1
Set-assignment semantics for L3
329
2
Exercises for Section C.1.d
331
1
The logics L3 and Lx
Generalizing the 3-valued tables
332
1
An axiom system for Lx
333
1
Set-assignment semantics for Lx
334
1
Exercises for Section C.2
335
1
Kleene's 3-Valued Logic
The truth-tables
336
2
Set-assignment semantics
338
1
Exercises for Section D
339
1
Logics Having No Finite-Valued Semantics
General criteria
339
1
Infinite-valued semantics for the modal logic S5
340
1
The Systems Gn and Gx
341
3
Exercises for Section F
344
1
A Method for Proving Axiom Systems Independent
344
5
Exercises for Section G
346
3
Paraconsistent Logic: J3 in collaboration with Itala M.L. D'Ottaviano
Paraconsistent Logics
349
1
The Semantics fo J3
Motivation
350
1
The truth-tables
351
4
Exercises for Section B.2
355
1
Definability of the connectives
356
1
The Relation Between J3 and Classical Logic
357
2
Exercises for Section C
359
1
Consistency vs. Paraconsistency
Definitions of completeness and consistency for J3 theories
359
2
The status of negation in J3
361
1
Axiomatizations of J3
As a modal logic
362
3
As an extension of classical logic
365
2
Exercises for Section E
367
1
Set-Assignment Semantics for J3
368
2
Truth-Default Semantics
370
5
Exercises for Sections F and G
373
2
Translations Between Logics
Syntactic translations in collaboration with Stanislaw Krajewski
A formal notion of translation
375
2
Exercises for Section A.1
377
1
Examples
377
4
Exercises for Section A.2
381
1
Logics that cannot be translated grammatically into classical logic
381
3
Exercises for Section A.3
384
1
Translations where there are no grammatical translations R ↪ PC and S ↪ PC
384
4
Exercises for Section A.4
388
1
Semantically faithful translations
388
13
A formal notion of semantically faithful translation
389
3
Exercises for Section B.1
392
1
Examples of semantically faithful translations
393
1
The archetype of a semantically faithful translation: Int↠S4
394
1
The translations of PC into Int
395
1
Exercises for Section B.4
396
1
The translation of S into PC
397
1
Different presentations of the same logic and strong definability of connectives
398
1
Exercises for Section B.6
399
1
Do semantically faithful tranlations preserve meaning?
399
2
The Semantic Foundations of Logic Concluding Philosophical Remarks
401
37
Summary of Logics
Classical Logic, PC
407
3
Relatedness and Dependence Logics S, R, D, Dual D, Eq, DPC
410
7
Classical Modal Logics S4, S5, S4Grz, T, B, K, QT, MSI, ML, G, G*
417
7
Intuitionistic Logics, Int and J
424
4
Many-Valued Logics L3, Ln, Lx, k3, G3, Gn, Gx, Paraconsistent J3
428
10
Bibliography
438
13
Glossary of Notation
451
4
Index of Examples
455
3
Index
458