  Cirquent calculus is a novel proof theory permitting component-sharing
between logical expressions. Using it, the predecessor article "Elementary-base
cirquent calculus I: Parallel and choice connectives" built the sound and
complete axiomatization CL16 of a propositional fragment of computability logic
(see http://www.csc.villanova.edu/~japaridz/CL/ ). The atoms of the language of
CL16 represent elementary, i.e., moveless, games, and the logical vocabulary
consists of negation, parallel connectives and choice connectives. The present
paper constructs the first-order version CL17 of CL16, also enjoying soundness
and completeness. The language of CL17 augments that of CL18 by including
choice quantifiers. Unlike classical predicate calculus, CL17 turns out to be
decidable.
