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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 Copyright