Sharpening Logical Independence

A set of elements of a Boolean algebra (BA) is called independent if it is not equal to any of its proper subsets (Tarski, 1956, p. 82). This basic notion of logical independence, typically used in the theory of deductive systems, is very weak. However, as Tarski remarks, there are two different directions by which this idea has been sharpened (ibid, p.36 n.). On one side a finite set of elements S belonging to a BA A is called “maximally independent” iff the principal filters generated by the elements of S do not have any element of A in common except 1. On the other side a finite set B of elements of a BA A is said to be “completely independent” iff B generates a free BA. The differences between the two notions and their philosophical relevance will be discussed. Complete independence can be further refined by the notion of (logical) separability. Given a Boolean algebra A, a set B of elements of A\{0, 1} is said to be separable in A iff there exist a family of subalgebras {Ai} (i ∈ I) of A such that A is the internal sum (or free product) of Ai and every Ai contains exactly one element of B. While separability is stronger than complete independence with respect to finite BAs, in infinite atomless BAs it comes down to complete independence (D. H. Fremlin, 2003 – unpublished results). A refinement of separability, stronger than complete independence in a general way, is here proposed.