Theories of probability: an examination of logical and qualitative foundations
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New Jersey [u.a.]
World Scientific
2007
|
| Series: | Advanced series on mathematical psychology
2 |
| Subjects: | |
| Online Access: | Table of contents only Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | IX, 219 S. |
| ISBN: | 9812708014 9789812708014 |
Staff View
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Record in the Search Index
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| adam_text | Contents
1
Elementary Concepts
1
1.1
Introduction
1
1.2
Preliminary Conventions and Definitions
3
2
Kolmogorov Probability Theory
7
3
Infinitesimals
ц
3.1
Introduction
11
3.2
Ultrafilters
15
3.3 Ultrapowers
of the Reals
18
3.4
Totally Ordered Field Extensions of the Reals
25
4
Qualitative Probability
29
4.1
Extensive Measurement of Probability
29
4.2
Probability Representations
36
4.2.1
Scott s Theorem
39
4.3
*R-probability Representations
43
4.4
Weak Probability Representations
50
4.5
Independence
53
4.5.1
Independent Experiments
54
5
Qualitative Utility
61
5.1
Brief Overview
61
5.2
Subjective Expected Utility
61
5.3
Dual Bilinear and Rank Dependent Utility
66
6
Axioms for Choice Proportions
75
6.1
Brief Overview
75
6.2
Luce s Choice Property
76
6.3
BTL Model of Choice
78
vii
viii CONTENTS
6.4
Basic
Choice Axioms
78
7
Conditional Belief
85
7.1
Introduction
85
7.2
Belief Axioms
89
7.3
Choice Axioms
96
7.4
Belief Support Functions
97
7.4.1
Fair Bets
99
7.4.2
Ellsberg s Paradox
100
7.5
Application to Support Theory
103
7.5.1
Empirical Phenomena
103
7.5.2
Tversky s and Koehler s Theory
104
7.5.3
Rottenstreich s and Tversky s Theory 111
7.6
Conclusions
112
8
Lattices
115
8.1
Introduction
115
8.2
Boolean Lattices
116
8.3
Representation Theorems of Stone
118
8.4
П
-Complemented Lattices
124
8.4.1
Lattices of Open Sets
125
8.4.2
Properties of lattices of open sets
128
8.5
Intuitionistic Propositional Logic
130
8.6
Non-Boolean Event Spaces
131
9
Belief for Scientific Events and Refutations
133
9.1
Introduction
133
9.2
Lattices of Scientific Events
134
9.3
Boolean Lattice of Refutations
137
9.4
Probability Theory for Scientific Events
140
9.5
An Example Involving American Law
145
10
The Psychology of Human Probability Judgments
149
10.1
Introduction
149
10.2
Support Theory
150
10.2.1
Judgmental Heuristics
150
10.2.2
Empirical Basis
151
10.2.3
Summary of the Theory
153
10.2.4
Focal Descriptions and Contexts in Support Theory
155
10.3
Concepts Used in the New Formulation
155
CONTENTS ix
10.3.1
Focal Descriptions
and Contexts in the New
Formulation
155
10.3.2
Semantical Representations
156
10.3.3
Cognitive Representations
157
10.3.4
Cognitive Model
159
10.3.5
Clear Instances
160
10.3.6
Single Probability Judgments
162
10.3.7
Multiple Probability Judgments
164
10.4
Frequency Based Judgments
164
10.5
Judgments Based on Representativeness
168
10.6
Probability Judgments for Binary Partitions
172
10.7
Topological Considerations
176
11
Metamathematical Considerations
179
11.1
Brief Summary
179
11.2
First Order Languages
180
11.3
Models
182
11.4
Los s Theorem
186
11.5
Compactness Theorem of Logic
190
11.6 Löwenheim-Skolem
Theorem
191
11.7
Axiomatizability and Uniqueness of Probability
Representations
196
11.8
Axiom of Choice and Probability Theory
202
11.8.1
Classical Results of
Vitali,
Hausdorff, Banach,
and
Ulam
202
11.8.2
Implications for
σ
-additivity
205
References
207
Index
213
|
| adam_txt |
Contents
1
Elementary Concepts
1
1.1
Introduction
1
1.2
Preliminary Conventions and Definitions
3
2
Kolmogorov Probability Theory
7
3
Infinitesimals
ц
3.1
Introduction
11
3.2
Ultrafilters
15
3.3 Ultrapowers
of the Reals
18
3.4
Totally Ordered Field Extensions of the Reals
25
4
Qualitative Probability
29
4.1
Extensive Measurement of Probability
29
4.2
Probability Representations
36
4.2.1
Scott's Theorem
39
4.3
*R-probability Representations
43
4.4
Weak Probability Representations
50
4.5
Independence
53
4.5.1
Independent Experiments
54
5
Qualitative Utility
61
5.1
Brief Overview
61
5.2
Subjective Expected Utility
61
5.3
Dual Bilinear and Rank Dependent Utility
66
6
Axioms for Choice Proportions
75
6.1
Brief Overview
75
6.2
Luce's Choice Property
76
6.3
BTL Model of Choice
78
vii
viii CONTENTS
6.4
Basic
Choice Axioms
78
7
Conditional Belief
85
7.1
Introduction
85
7.2
Belief Axioms
89
7.3
Choice Axioms
96
7.4
Belief Support Functions
97
7.4.1
Fair Bets
99
7.4.2
Ellsberg's Paradox
100
7.5
Application to Support Theory
103
7.5.1
Empirical Phenomena
103
7.5.2
Tversky's and Koehler's Theory
104
7.5.3
Rottenstreich's and Tversky's Theory 111
7.6
Conclusions
112
8
Lattices
115
8.1
Introduction
115
8.2
Boolean Lattices
116
8.3
Representation Theorems of Stone
118
8.4
П
-Complemented Lattices
124
8.4.1
Lattices of Open Sets
125
8.4.2
Properties of lattices of open sets
128
8.5
Intuitionistic Propositional Logic
130
8.6
Non-Boolean Event Spaces
131
9
Belief for Scientific Events and Refutations
133
9.1
Introduction
133
9.2
Lattices of Scientific Events
134
9.3
Boolean Lattice of Refutations
137
9.4
Probability Theory for Scientific Events
140
9.5
An Example Involving American Law
145
10
The Psychology of Human Probability Judgments
149
10.1
Introduction
149
10.2
Support Theory
150
10.2.1
Judgmental Heuristics
150
10.2.2
Empirical Basis
151
10.2.3
Summary of the Theory
153
10.2.4
Focal Descriptions and Contexts in Support Theory
155
10.3
Concepts Used in the New Formulation
155
CONTENTS ix
10.3.1
Focal Descriptions
and Contexts in the New
Formulation
155
10.3.2
Semantical Representations
156
10.3.3
Cognitive Representations
157
10.3.4
Cognitive Model
159
10.3.5
Clear Instances
160
10.3.6
Single Probability Judgments
162
10.3.7
Multiple Probability Judgments
164
10.4
Frequency Based Judgments
164
10.5
Judgments Based on Representativeness
168
10.6
Probability Judgments for Binary Partitions
172
10.7
Topological Considerations
176
11
Metamathematical Considerations
179
11.1
Brief Summary
179
11.2
First Order Languages
180
11.3
Models
182
11.4
Los's Theorem
186
11.5
Compactness Theorem of Logic
190
11.6 Löwenheim-Skolem
Theorem
191
11.7
Axiomatizability and Uniqueness of Probability
Representations
196
11.8
Axiom of Choice and Probability Theory
202
11.8.1
Classical Results of
Vitali,
Hausdorff, Banach,
and
Ulam
202
11.8.2
Implications for
σ
-additivity
205
References
207
Index
213 |
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| id | DE-604.BV022960843 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T19:04:40Z |
| indexdate | 2024-07-09T21:08:38Z |
| institution | BVB |
| isbn | 9812708014 9789812708014 |
| language | English |
| lccn | 2007018216 |
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| physical | IX, 219 S. |
| publishDate | 2007 |
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| series | Advanced series on mathematical psychology |
| series2 | Advanced series on mathematical psychology |
| spelling | Narens, Louis Verfasser (DE-588)1084236524 aut Theories of probability an examination of logical and qualitative foundations Louis Narens New Jersey [u.a.] World Scientific 2007 IX, 219 S. txt rdacontent n rdamedia nc rdacarrier Advanced series on mathematical psychology 2 Includes bibliographical references and index Probabilités ram Probabilities Wahrscheinlichkeit (DE-588)4137007-7 gnd rswk-swf Mathematische Psychologie (DE-588)4126558-0 gnd rswk-swf Mathematische Psychologie (DE-588)4126558-0 s Wahrscheinlichkeit (DE-588)4137007-7 s b DE-604 Advanced series on mathematical psychology 2 (DE-604)BV023117842 2 http://www.loc.gov/catdir/toc/ecip0717/2007018216.html Table of contents only Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016165219&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Narens, Louis Theories of probability an examination of logical and qualitative foundations Advanced series on mathematical psychology Probabilités ram Probabilities Wahrscheinlichkeit (DE-588)4137007-7 gnd Mathematische Psychologie (DE-588)4126558-0 gnd |
| subject_GND | (DE-588)4137007-7 (DE-588)4126558-0 |
| title | Theories of probability an examination of logical and qualitative foundations |
| title_auth | Theories of probability an examination of logical and qualitative foundations |
| title_exact_search | Theories of probability an examination of logical and qualitative foundations |
| title_exact_search_txtP | Theories of probability an examination of logical and qualitative foundations |
| title_full | Theories of probability an examination of logical and qualitative foundations Louis Narens |
| title_fullStr | Theories of probability an examination of logical and qualitative foundations Louis Narens |
| title_full_unstemmed | Theories of probability an examination of logical and qualitative foundations Louis Narens |
| title_short | Theories of probability |
| title_sort | theories of probability an examination of logical and qualitative foundations |
| title_sub | an examination of logical and qualitative foundations |
| topic | Probabilités ram Probabilities Wahrscheinlichkeit (DE-588)4137007-7 gnd Mathematische Psychologie (DE-588)4126558-0 gnd |
| topic_facet | Probabilités Probabilities Wahrscheinlichkeit Mathematische Psychologie |
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