[Paper Review] A Logic for Arguing About Probabilities in Measure Teams

Tapani Hyttinen, Gianluca Paolini+1 more|arXiv (Cornell University)|Sep 6, 2015

Philosophy and History of Science5 references3 citations

TL;DR

This paper introduces a logic for reasoning about probabilities of first-order formulas within measure teams—structures combining teams of assignments with probability measures. It provides a complete axiomatization for first-order properties of these probabilities, illustrated through applications in genetics (Hardy-Weinberg principle) and quantum physics (Bell’s inequalities), proving completeness over infinite measure teams with respect to an ω-saturated model framework.

ABSTRACT

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.

Motivation & Objective

• To develop a formal logic for reasoning about probabilities of first-order formulas using teams of assignments equipped with probability measures.
• To axiomatize first-order properties of such probabilities and prove their completeness.
• To apply the framework to foundational principles in science, such as the Hardy-Weinberg equilibrium and Bell’s inequalities.
• To establish completeness over infinite measure teams, abstracting from finite empirical data.
• To unify frequency-based probability with team semantics and model-theoretic reasoning in a logical framework.

Proposed method

• Define measure teams as quadruples $(\Omega, \mathcal{F}, P, \tau)$, where $\tau$ maps a probability space to assignments in a structure $\mathcal{A}$, with measurability conditions on formula satisfaction.
• Use team semantics to interpret the probability of a first-order formula $\phi$ as the measure of assignments in $X$ satisfying $\phi$, i.e., $P(\{s \in \Omega \mid \mathcal{A} \models_s \phi\})$.
• Axiomatize properties of these probabilities using a sound and complete proof system, including axioms for complement, disjunction, and boundedness of probabilities.
• Prove completeness via a construction of a model $\mathcal{R}^X_Q$ from a countable set of formulas, using $\omega$-saturation to realize types over infinite sequences of assignments.
• Apply the framework to concrete scientific examples: the Hardy-Weinberg principle in genetics and Bell’s inequalities in quantum mechanics, showing they are derivable within the logic.
• Handle both finite and infinite measure teams, with completeness theorems established for infinite teams using $\omega$-saturated structures.

Experimental results

Research questions

• RQ1 Can a complete first-order logic be developed for reasoning about probabilities of formulas in teams with measures?
• RQ2 How can the frequency interpretation of probability be formally integrated into team semantics and first-order logic?
• RQ3 Are foundational scientific principles like the Hardy-Weinberg equilibrium and Bell’s inequalities derivable within this logical framework?
• RQ4 What model-theoretic conditions (e.g., $\omega$-saturation) are necessary and sufficient for completeness in this logic?
• RQ5 How does the logic handle the transition from finite data teams to idealized infinite measure teams?

Key findings

• The paper establishes a complete axiomatization for first-order properties of probabilities in measure teams, proving that syntactic derivability coincides with semantic entailment.
• The completeness theorem holds for infinite measure teams when the underlying structure is $\omega$-saturated, ensuring that all finitely satisfiable sets of probability constraints can be realized.
• The Hardy-Weinberg principle is formally captured and derivable in the logic, demonstrating its applicability to population genetics.
• Bell’s inequalities are shown to be expressible and provable within the framework, illustrating its relevance to quantum foundations.
• The logic extends propositional probability logic by allowing quantification and first-order structure, with a completeness result even when the set of formulas is finite and not necessarily positive bounded.
• A counterexample shows that positivity of the probability constraints is a necessary condition for completeness in the general case, highlighting the logical boundaries of the system.