## The Theory of Probability |

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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 COÖRDINATED 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 |

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488 | |

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### Common terms and phrases

ability according addition amounts applied assertion assume assumption attribute average axioms belongs calculus calculus of probability called combination concept concerning condition considerations considered constructed contains continuous convergence corresponding counted defined definition depends derived determined direction disjunction dispersion elements enumeration equal example existence expression fact finite follows formula frequency frequency interpretation function given holds horizontal illustrated implication includes independent individual induction inference infinite instance interpretation interval introduced justification kind knowledge known latter lattice laws leads limit logic mathematical means method normal sequences notation object observed obtain occur operations physical posit possible presented principle prob probability probability sequences problem proof propositional question reason reference regarded relation relative replaced represents requires respect restricted result rule satisfied selection sense sequence shows statement statistical symbol term theorem theory tion true truth variable write