The Theory of Probability

Front Cover
University of California Press, 1971 - 492 pages
 

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A comprehensive treatment, unique in covering probability theory independently of modern theory. New edition features additional problems, examples that show scope and limitations of various results, and enlarged chapters on laws of large numbers, extensions, and generalizations.

Contents

The probability concept of the language of everyday life
3
The historical development of the scientific concept of probability
5
Remarks about the plan of the book
10
INTRODUCTION TO SYMBOLIC LOGIC
13
The calculus of propositions
15
The method of derivation
23
The calculus of functions
26
The calculus of classes
33
CONTINUOUS EXTENSIONS OF THE CONCEPT OF PROBABILITY SEQUENCE
201
The geometrical interpretation of the axiom system
203
Definition of probability sequences with continuous attribute
205
Empirical determination of a probability function
209
The onedimensional attribute space
219
Manydimensional attribute spaces
223
Relative probability functions
228
Continuous probability sequences
237

Axiomatic systems
38
ELEMENTARY CALCULUS OF PROBABILITY
43
The probability implication
45
The abbreviated notation
49
The rule of existence
52
The axioms of univocality and of normalization
53
The theorem of addition
57
The theorem of multiplication
61
Reduction of the multiplication theorem to a weaker axiom
65
The frequency interpretation
67
The origin of probability statements
70
Derivation of the axioms from the frequency interpretation
72
The rule of elimination
76
The general theorem of addition
82
The rule of the product
90
The rule of reduction
96
The relation of independence
102
Complete probability systems
106
The mathematical formalization of the probability calculus
116
Appendix to chapter 3 Exercises and solutions
123
THEORY OF THE ORDER OF PROBABILITY SEQUENCES
129
The task of the theory of order
131
Phase probabilities
132
PAGE
133
Axioms concerning the theory of order
136
Sequences without aftereffect
141
Normal sequences
143
Some numerical problems referring to normal sequences
151
Mutually dependent normal sequences
154
Probability transfer
159
The probability lattice
167
PROBABILITY SEQUENCES WITH CO÷RDINATED AMOUNTS
175
The average of a sequence of quantities
177
Formation of an average when the summation is extended to infi nitely many terms
183
The dispersion
188
Average and dispersion for a combination of events
192
Average and dispersion in the lattice
195
Competition of chances
250
THE FREQUENCY PROPERTIES OF PROBABILITY SEQUENCES
259
The frequency sequences
261
The theorem of Bernoulli
262
The significance of Bernoullis theorem
274
The amplified Bernoulli theorem
281
The frequency dispersion
283
The dispersion of nonnormal sequences
290
A simple interpretation of the dispersion
292
A simple derivation of Bernoullis theorem
293
Bernoulli sequences
303
Probabilities of a higher level
311
Operations with probabilities of the second level
318
The dispersion in a horizontally inhomogeneous lattice
324
The problem
337
The necessity of the concept of the limit for the frequency inter
344
The assertability of probability statements in the frequency inter
350
The three forms of an a posteriori determination of degrees
359
The frequency interpretation of the probability of the single case
372
The logical interpretation of probability
378
A quantitative logic of an individual verifiability
389
Probability as a property of propositional sequences
395
Truth tables of the logic of modalities
403
The logic of weight
409
The quantitative negation
420
INDUCTION PAGE
427
The various forms of induction in empirical science
429
The probability of hypotheses
434
Induction by enumeration in advanced knowledge
442
The rule of induction
472
Anticipative posits in advanced knowledge 89 The method of correction 90 The hierarchy of posits 91 The justification of induction
475
Index
483
442
485
444
487
469
488
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