A linear space is a system of points and lines such that any two distinct points determine a unique line; a Steiner k-system (for k 2) is a linear space such that each line has size exactly k. Clearly, as a two-sorted structure, no linear space can be strongly minimal. We formulate linear spaces in a (biinterpretable) vocabulary τ with a single ternary relation R. We prove that for every integer k there exist 2 ℵ 0 -many integer valued functions µ such that each µ determines a distinct strongly minimal Steiner k-system Gµ, whose algebraic closure geometry has all the properties of the ab initio Hrushovski construction. Thus each is a counterexample to the Zilber Trichotomy Conjecture.

A logical approach to Bell's Inequalities of quantum mechanics has been introduced by Abramsky and Hardy [2]. We point out that the logical Bell's Inequalities of [2] are provable in the probability logic of Fagin, Halpern and Megiddo [7]. Since it is now considered empirically established that quantum mechanics violates Bell's Inequalities, we introduce a modified probability logic, that we call quantum team logic, in which Bell's Inequalities are not provable, and prove a Completeness Theorem for this logic. For this end we generalise the team semantics of dependence logic [11] first to probabilistic team semantics, and then to what we call quantum team semantics.

We use sets of assignments, a.k.a. teams, and measures on them to define probabilities of first-order formulas in given data. We then axiomatise first-order properties of such probabilities and prove a completeness theorem for our axiomatisation. We use the Hardy-Weinberg Principle of biology and the Bell's Inequalities of quantum physics as examples.

We present a framework for studying the concept of independence in a general context covering database theory, algebra and model theory as special cases. We show that well-known axioms and rules of independence for making inferences concerning basic atomic independence statements are complete with respect to a variety of semantics. Our results show that the uses of independence concepts in as different areas as database theory, algebra, and model theory, can be completely characterized by the same axioms. We also consider concepts related to independence, such as dependence.