Expanding my comment into an answer:
While there is a definite connection with probability in the propositional case per Berci's comment - treating each propositional atom as an independent coin-flip - this connection doesn't generalize well outside that context. For example, in first-order logic there is no obvious probability distribution on the "set" (or rather, any sufficiently large set) of structures. There is however an approach to "comparative truth" which applies to basically all logics and is fundamentally algebraic as opposed to probabilistic, namely the Lindenbaum algebra: the partial order given by equivalence classes of sentences modulo equivalence over $T$, with the ordering given by $T$-provability. The partial ordering corresponds exactly to "at least as true as (assuming $T$)."
Typically this partial order comes with additional structure due to the interaction between the syntax and the deduction relation. For example, in the case of classical propositional logic we get the structure of a Boolean algebra, in the case of intuitionistic propositional logic we get the structure of a Heyting algebra, classical first-order logic gives a cylindrical algebra, and so forth. Usually the phrase "Lindenbaum algebra" refers to the partial order equipped with such natural structure; arguably I should call the bare partial order the "Lindenbaum order."
It's worth noting however that there are serious algebraic issues here, not directly relevant to your question but foundational to the meaning of the Lindenbaum algebra.
The starting point is the observation that syntax may play poorly with the Lindenbaum algebra. As noted above, some syntactic structure translated to structure in the Lindenbaum algebra, but this is not true in general: for example, in reasonable propositional modal logics we can have $T\vdash \varphi\leftrightarrow\psi$ but $T\not\vdash (\Box \varphi)\leftrightarrow(\Box\psi)$. So while the set of sentences can be thought of as a free algebra in an appropriate sense, this doesn't interact well with deductions.
The interplay between the deduction relation and the syntactic algebra is quite nuanced; see Blok and Pigozzi's book or the first bit of Czelakowski's book for a good description of the situation. This all belongs to the general subject of algebraic logic.