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 "sentential" 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 ("no," "and," "or," etc.) but not including
quantifiers ("for all," "there is"). 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
("for all," and "there is") 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.
