Model Theory of Measure Spaces and Probability Logic

Abstract

We study the model-theoretic aspects of a probability logic suited for talking about measure spaces. This nonclassical logic has a model theory rather different from that of classical predicate logic. In general, not every satisfiable set of sentences has a countable model, but we show that one can always build a model on the unit interval. Also, the probability logic under consideration is not compact. However, using ultraproducts we can prove a compactness theorem for a certain class of weak models.

Computational aspects of satisfiability in probability logic

“…This contrasts our result in § 7 that ε-satisfiability is 1 1 -hard for rational ε ∈ (0, 1), which completes our proof that ε-satisfiability is 1 1 -complete for such ε. Finally, in § 8 we use the results from section 6 to show that 0-logic is compact, contrasting an earlier result by Kuyper and Terwijn [8,Theorem 8.1] that ε-logic is not compact for rational ε ∈ (0, 1).…”

“…We recall some more definitions from [8]. D) is an ε-model (N , E) over the same language such that N is a submodel of M in the classical sense and such that E is a submeasure of D. We shall denote this by (N , E) ⊂ ε (M, D).…”

“…Theorem 2.13 ([8,Theorem 7.4]) Let L be a countable first-order language not containing equality or function symbols. Then, for all rationals 0 < ε 1 ≤ ε 0 ≤ 1, ε 0 -satisfiability many-one reduces to ε 1 -satisfiability.…”

“…Then, for all rationals 0 < ε 1 ≤ ε 0 ≤ 1, ε 0 -satisfiability many-one reduces to ε 1 -satisfiability. Remark 2.14 As discussed in [8], we can perform these reductions per quantifier: that is, we do not need to apply the reduction to the entire formula, but can apply it on a per-quantifier basis. For example, if we let f 0 be the reduction from 0-satisfiability to 1 2 -satisfiability and f 1 4 the reduction from 1-satisfiability to 1 2 -satisfiability, then for all formulas ϕ 0 , ϕ 1 2 , ϕ 1 4 the expression "there exists a probability model (M, D) such that (M, D) |= ε ϕ ε for all ε ∈ 0, 1 4 …”

“…[8,Theorem 5.9]. We can apply the construction above on [0, r ] and use AC to choose the values f i takes in the atoms; since there are only countably many atoms, this does not alter the measurability of the f i .…”