I've recently been looking at the literature on assigning probabilities to first-order formulae (not just propositional formulae). Following the treatment here, let $L$ be a first-order language, let $D$ be some domain, and let $L(D)$ be $L$ expanded by enough constants $a_1, a_2, \dots$ to name all elements of $D$. Gaifman showed in 1964 that if we are given a probability map from the quantifier-free sentences of $L(D)$ to $[0, 1]$, then there is a unique extension of that to a probability map from the sentences of $L(D)$ to $[0, 1]$ satisfying the following condition: $$ p(\exists x \phi(x)) = \mathrm{sup} \{p(\phi(a_{i_1}) \lor \cdots \lor \phi(a_{i_k})) \}$$ where the supremum is taken over all finite sets of constants.
My question is this: what is the motivation for imposing the Gaifman condition?
If I'm reading it right, then it identifies the probability of an existentially quantified claim with the maximal probability assigned to some disjunction of substitution-instances.
But don't we want to allow that the probability of an existentially quantified statement might be greater than the probability of any given finite disjunction (for, roughly, the same reasons that we allow that the probability of a disjunction might be higher than the probabilities of either of its disjuncts)?