Uncertain Information

A formal theory of information should represent A consistent system of questions Aggregation or combination of information Focussing or Extraction of information Information algebra (Φ, D) cover these aspects: D a lattice of domains (questions) Φ a semigroup under combination ∀x ∈ D, φ ∈ Φ, φ ⇒x is the information in φ relating to x. Axiomatics Formal definitions and axioms: (Φ, D) is a two-sorted algebra with two binary operations: satisfying the following axioms: Φ a commutative semigroup under combination, D a lattice, In Φ, a partial order can be defined as follows: φ ≤ ψ, if φ ⊗ ψ = ψ. 1 This means that φ contains less information than ψ. 2 Alternatively: ψ implies φ. Important models of information algebra are induced by classical propositional and predicate logic, as well as by many nonclassical (but monotone) logics. A information system is a triple (L, C, S), where L a set of sentences C a consequence operator on L S a ∩-system of sublanguages of S The family of closed set C(X) forms an information algebra if, C satisfies some conditions. The information algebra induced by predicate logic is a reduct of cylindric algebras (proposed by Henkin, Tarski) and also of polyadic algebras (proposed by Halmos). It seems natural to model uncertain information in this framework by random variables taking values in an information algebra: 1 Main problem: What measurability requirements are to be imposed? 2 A possible solution is proposed here. If φ, ψ ∈ Φ are pieces of information asserted (proved, derived), then φ ⊗ ψ = φ ∨ ψ is also asserted, η ≤ φ implies that η is asserted. A subset of Φ satisfying these conditions is a ideal of Φ. There are two operations between ideals: