A set F of Boolean functions is said to be functionally complete if every Boolean function is definable by combining functions in F. Post clarified when a set of Boolean functions is functionally complete (with respect to classical semantics). In this paper, by extending Post’s theorem, we clarify when a set of Boolean functions is functionally complete with respect to Kripke semantics.

The problem of enumerating the types of Boolean functions under the group of variable permutations and complementations was first stated by Jevons in the 1870s. but not solved in a satisfactory way until the work of Pólya in 1940. This paper explains the details of Pólya's solution, and also the history of the problem from the 1870s to the 1970s.

We describe sets of partial Boolean functions being closed under the operations of superposition. For any class A of total functions we define the set ????(A) consisting of all partial classes which contain precisely the functions of A as total functions. The cardinalities of such sets ????(A) can be finite or infinite. We state some general results on ????(A). In particular, we describe all 30 closed sets of partial Boolean functions which contain all monotone and zero-preserving total (...) class='Hi'>Boolean functions. (shrink)

Using Koszmider's strongly unbounded functions, we show the following consistency result: Suppose that κ, λ are infinite cardinals such that κ++ + ≤ λ, κ<κ = κ and 2κ = κ+, and η is an ordinal with κ+ ≤ η < κ++ and cf = κ+. Then, in some cardinal-preserving generic extension there is a superatomic Boolean algebra equation image such that equation image, equation image for every α < η and equation image. Especially, equation image and equation image (...) can be cardinal sequences of superatomic Boolean algebras. (shrink)