Diana Manister wrote:

 > Since they were friends, I'm sure Eliot was acquainted with Russell's
 > work in logic, but whether Eliot had any interest or use for detailed
 > knowledge of quantification theory seems unlikely, except possibly
 > insofar as it refers to discourse which includes things that exist only
 > discursively, like mermaids.


In 1914 while Eliot was studying philosophy at Harvard he sat in on
Russell's course on logic and, with other students in the class, had
weekly teas with Russell (Mr. Apollinax).

As for mermaids: a snippet of "Bertrand Russell and T. S. Eliot: their 
dialogue" follows:
http://digitalcommons.mcmaster.ca/cgi/viewcontent.cgi?article=1574&context=russelljournal

    The reader of both men's works recognizes that Russell is claiming more
    than just a literary image: he is claiming what amounts to an influence
    towards a solipsistic theory of knowledge and a consequent world view.
    Russell's use of the words "imagination" and "hallucinations" recalls
    Eliot's discussion of unreal objects, and of universals and particulars,
    in Chapter 5 of Knowledge and Experience in the Philosophy of F. H.
    Bradley (the title under which his dissertation was finally published in
    1964; Russell is mentioned in this chapter). It is the question of
    unreal objects, a moot one in the epistemologist's theory of knowledge,
    that led Russell to formulate his theory of acquaintance-to which Eliot
    reacts so emphatically in his dissertation: "But I find the notion of
    acquaintance completely unsatisfactory ."3