Are there descriptive complexity representations of quantum complexity classes?
Gurevich in formulating the conjecture about a logic that could capture $\mathsf{P}$ requires the logic to be computable in two ways: (1) the set of sentences legally obtainable from the vocabulary $\sigma$ has to be computable, given $\sigma$; and (2) the satisfiability relation needs to be computable from $\sigma$, i.e., ordered pairs consisting of a finite structure $M$ and a sentence $\varphi$ such that all models isomorphic to $M$ satisfy $\varphi$.  Also, significantly for comparison with this randomized logic result, the vocabulary $\sigma$ has to be finite.  (A vocabulary is a set of constant symbols and relation symbols, for example, equals sign, less-than sign, $R_1,R_2,\ldots$)  This is a paraphrase of Definition 1.14 of this paper by Gurevich, which is reference [9] in the quote Kaveh gave.

The paper about BPP and randomized logic presents a significantly different framework.  It starts with a finite vocabulary $\sigma$, and then considers a probability space of all vocabularies that extend $\sigma$ with some disjoint vocabulary $\rho$.  So a formula is satisfiable in the new randomized logic if it is satisfiable in "enough" logics based on extensions of $\sigma$ by different $\rho$.  This is my butchering of Definition 1 in the Eickmeyer-Grohe paper linked to by Robin Kothari.  In particular, the vocabulary is not finite (well, each vocabulary is, but we have to consider infinitely many distinct vocabularies), the set of sentences of this logic is undecidable, and the notion of satisfiability is different from the one put forth by Gurevich.