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Logic with a Probability Semantics (Hardcover)
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Logic with a Probability Semantics (Hardcover)
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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.
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