Logic with a Probability Semantics

The present study is an extension of the topic introduced in Dr. Hailperin's Sentential Probability Logic, where the usual true-false semantics for logic is replaced with one based more on probability, and where values ranging from 0 to 1 are subject to probability axioms. Moreover, as the word "e;sentential"e; in the title of that work indicates, the language there under consideration was limited to sentences constructed from atomic (not inner logical components) sentences, by use of sentential connectives ("e;no,"e; "e;and,"e; "e;or,"e; etc.) but not including quantifiers ("e;for all,"e; "e;there is"e;). An initial introduction presents an overview of the book. In chapter one, Halperin presents a summary of results from his earlier book, some of which extends into this work. It also contains a novel treatment of the problem of combining evidence: how does one combine two items of interest for a conclusion-each of which separately impart a probability for the conclusion-so as to have a probability for the conclusion based on taking both of the two items of interest as evidence? Chapter two enlarges the Probability Logic from the first chapter in two respects: the language now includes quantifiers ("e;for all,"e; and "e;there is"e;) whose variables range over atomic sentences, not entities as with standard quantifier logic. (Hence its designation: ontological neutral logic.) A set of axioms for this logic is presented. A new sentential notion-the suppositional-in essence due to Thomas Bayes, is adjoined to this logic that later becomes the basis for creating a conditional probability logic.Chapter three opens with a set of four postulates for probability on ontologically neutral quantifier language. Many properties are derived and a fundamental theorem is proved, namely, for any probability model (assignment of probability values to all atomic sentences of the language) there will be a unique extension of the probability values to all closed sentences of the language.