Probabilities on First Order Models


It is known that set algebras corresponding to first order models (i.e., cylindric set algebras associated with first order interpretations) are \emph{not} $\sigma$-closed, but closed w.r.t. certain infima and suprema i.e., \[ łeft|\exists x \alpha\right|=\bigcup_{i\in\omega}łeft|\alpha(y_i)\right| \quad\text{and}\quad łeft|\forall x \alpha\right|=\bigcap_{i\in\omega}łeft|\alpha(y_i)\right| łeqno{(*)} \] for \emph{any} infinite subsequence $y_1,y_2,\ldots y_i,\ldots$ of the individuum variables in the language. We investigate probabilities defined on these set algebras and being continuous w.r.t. the suprema and infima in $(*)$. We can not use the usual technics, because these suprema and infima are not the usual unions and intersections of sets. These probabilities are interesting in computer science among others, because the probabilities of the quantifier-free formulas determine that of \emph{any} formula, and the probabilities of the former ones can be measured by statistical methods.