On representation and approximation of operations in Boolean algebras

Several universal approximation and universal representation results are known for non Boolean multi valued logics such as fuzzy logics In this paper we show that similar results can be proven for multi valued Boolean logics as well Introduction The study of Boolean algebras started when G Boole con sidered the simplest Boolean algebra the two valued algebra B f g ffalse trueg In this algebra the operation and negation a a have the direct logical meaning of or and and not It is known that in this Boolean algebra an arbitrary operation i e an arbitrary function B B B can be represented as a superposition of these three basic logical operations e g the implication a b can be represented as b a etc Logic is still one of the area of application of Boolean algebras but starting from the classical Kolmogorov s monograph Boolean algebras namely algebras of events became an important tool in another area foundations of probability In contrast to the Boolean algebra B in more complex Boolean algebras there are functions B B B which cannot be represented in terms of the three basic operations Until recently these functions were rarely used traditionally in probability applications only the standard operations and a were used or operations which can be explicitly described in terms of these three Some complex events can be easily described by using these three operations but some other complex events like a hypothetic conditional event a if b whose probability is equal to the conditional probability P ajb cannot be described via these three operations this result is due to Lewis for its detailed description see e g Chapter of Some researchers even thought that since we cannot get an expression for a conditional event by using the three basic operations of Boolean algebra we therefore cannot describe such events in Boolean algebra at all It was recently shown however see e g that if we use a new operation an operation which cannot be explicitly represented in terms of and a then it is possible to describe conditional events within the Boolean algebra formalism Namely to describe conditional events a if b we can consider instead of an individual event a potentially in nite sequence of similar events and interpret a if b as a is true in the rst moment of time in which b is true Comment Let us describe this representation in more formal terms for those readers who are familiar with the main notions of mathematical probability theory other readers can skip this comment In more formal terms instead of the original Boolean algebra B of all events measurable subsets of a algebra we consider the set IN of all in nite sequences of events i with a product measure this construction is standard in probability theory where it is used to formulate and prove limit theorems and the new Boolean algebra of all events on N Then the event a if b is interpreted as b a