The Theory of ProbabilityUniversity of California Press, 1971 - 492 pages |
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 |
36 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 |
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 |
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 |
A simple derivation of Bernoullis theorem | 55 |
The theorem of addition | 57 |
The theorem of multiplication | 61 |
The inductive inference | 62 |
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 |
31 Some numerical problems referring to normal sequences | 151 |
Mutually dependent normal sequences | 154 |
Probability transfer | 159 |
The probability lattice | 167 |
PROBABILITY SEQUENCES WITH COÖRDINATED AMOUNTS | 175 |
The average of a sequence of quantities | 177 |
The onedimensional attribute space | 219 |
Manydimensional attribute spaces | 223 |
Relative probability functions | 228 |
Continuous probability sequences | 237 |
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 |
63 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 |
Attempts at a singlecase interpretation of probability | 366 |
The frequency interpretation of the probability of the single case | 372 |
The logical interpretation of probability | 378 |
The problem of a multivalued logic | 387 |
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 | 427 |
The probability of hypotheses 84 The various forms of induction in empirical science | 429 |
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 |
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Common terms and phrases
ability according antecedent probabilities applied assertion assume assumption attribute space average B₁ Bernoulli Bernoulli's theorem calculus of probability characterized combination concept of probability condition constructed convergence coördinated corresponding defined definition degree of probability derived determined disjunction dispersion domain of invariance elements enumeration equal equation example exclusive disjunctions existence expression finite follows formula frequency interpretation given holds horizontal sequences independent individual inductive inference instance interval inverse probability lattice limit mathematical mean value metalanguage method multiplication theorem mutual probabilities normal sequences notation observed obtain occur posit possible principle of indifference prob probability calculus probability function probability implication probability logic probability statements probability transfer problem propositional propositional function quences reference class regarded relation represents result rule of induction sections special theorem statistical superscript symbol tautologies term theorem of addition theorem of multiplication theory of probability tion truth tables truth values variable