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Studies in Logic


Logic and Conditional Probability

A Synthesis

Philip Calabrese

This monograph develops an algebra of Boolean fractions, (a|b) – ordered pairs of propositions or events – “a if b”, “event a given event b”. In nine chapters, the author shows that these conditional propositions (together with their associated instantiations or models):

Provide logical elements that better represent and more faithfully facilitate manipulation of certain and uncertain conditional information

Extend the Boole’s algebra of 2-valued statements to a 3-valued system that includes “inapplicable statements” - those whose condition may be false in some or all instances (examples, cases, models...)

Allow a definition of the probability of an arbitrary Boolean proposition

Non-trivially combine Boolean logic with standard conditional probability theory

Provide a complete and adequate development of the crucial 4th operation for Boolean logic, namely conditioning, including iterated conditioning

Provide an expanded theory of deduction defined in terms of the extended operations on the Boolean fractions

Admit a variety of deduction relations, and that the deductively closed sets generated by some initial set of conditionals can be calculated

Extend the ordinary function operations of sum, difference, product & quotient to real-valued functions with possibly different or overlapping domains of definition

Represent & simplify complex conditional statements in Bayesian expert systems used to calculate maximum information entropy solutions

Explicate the logic of quantum measurements by better expressing the changing conditions in quantum mechanics

25 October 2017


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