In a 1978 paper published by David P. Ellerman and Gian-Carlo Rota, the duo discuss the relationship(s) exhibited by measure theory, probability theory, and logic paying special attention to how these concepts apply to describing the nature of quantification via an algebraic perspective.
I was hoping that someone who is familiar with the paper (or for someone whose research area would permit them an understanding) to elucidate its underlying concepts. In particular, in paragraph two on page 3 (of 21) in the attached pdf, the authors state
"It has long been known (e.g., Wright [12]) that there is an analogy between averaging operators such as the conditional expectation operators of probability theory and the algebraic (existential) quantifiers in Halmos" theory of polyadic algebras (Halmos [5]) or the cylindrifications used by Tarski and his co-workers in the theory of cylindric algebras (Henkin, Monk, and Tarski [7]). All these operators satisfy an averaging condition which has the general form: $A(f) \cdot A(g) = A \cdot (f \cdot A(g))$. We will provide some theoretical underpinning for this analogy by constructing the logical quantifiers using an abstract rendition of conditional expectation operators on a ring of simple random variables."
Can someone please elaborate upon how the logical quantifiers are related to conditional expectation operators as the term is used here. Additionally, can anyone point me towards some good resources for pursuing this line of inquiry further (lectures, videos, books, etc.)?
