The paper begins with the presumption that the Kantian and positivist view
that logic is true independently of a subject matter is mistaken. In this (shortened) version of the
paper, Quine's criticisms of the view will be presupposed and only slightly enlarged upon. It is argued
apriori that full reflective understanding of logic and relatively elementary deductive reasoning is
rationally connected to commitment to mathematics and mathematical entities. The paper assumes that
mathematics is committed to mathematical entities; it does not argue this point. The argument centers
on connecting full reflective understanding of logic, and ultimately of relatively elementary deductive
reasoning itself, with mathematical commitments.
Of course, there are many conceptions of what counts as "logic". I do not assume a Fregean or a Russelian conception of logic. These conceptions already contain axioms that are committed to the existence of infinitely many mathematical objects. I rely on (and briefly try to motivate) any of a number of more modern conceptions whose axioms do not commit them to the existence of a specific number of objects (or, for free logics, to the existence of any objects at all). The main argument of the paper is also neutral as to whether second-order logic is counted "logic", or whether, on the contrary, "logic" is conceived as confined to something like first-order logic--as I believe that there are some non-Quinean reasons for doing.
It is emphasized that the objectively apriori connections between deductive reasoning and commitment to mathematics need not be accepted by or even comprehensible to a given deductive reasoner. The relevant connections emerged only slowly in the history of logic. But they can be recognized retrospectively as "implicit" in logic and in relatively elementary deductive reasoning. The paper tries to specify the relevant sense of "implicit".