Is logic part of the world's ultimate structure?

"Everyone faces the question of what is 'real' and what is the mere projection of our conceptual apparatus, of which issues are substantive and which are 'mere bookkeeping'. [...] These are questions of structure: how much structure is there in the world? Unless one is prepared to take the verificationist's easy way out, and say that 'theories are the same when empirically equivalent', one must face difficult questions about where to draw the line between objective structure and conceptual projection. The ontological realist draws the line in a certain place: part of the world's distinguished structure is its [logical] quantificational structure. Those who regard ontological realism as 'overly metaphysical' should remember that they too must draw a line.

And in fact, the ontological realist can give a pretty convincing argument for his choice of where to draw the line. Quine's (1948) criterion for ontological commitment is good as far as it goes: believe in those entities that your best theory says exists. But in trying to decide how much structure there is in the world, I can think of no better strategy than this extension of Quine's criterion: believe in as much structure as your best theory of the world posits. The structure posited by a theory corresponds to its primitive notions - its 'ideology' in Quine's (1951) terminology - which includes its logical notions as well as its predicates.

[...] [N]otice this: every serious theory of the world that anyone has ever considered, employs a [logical] quantificational apparatus, from physics to mathematics to the social sciences to folk theories. Quantificationalism is as indispensable as it gets. This is defeasible reason to think that we're onto something, that quantificational structure is part of the objective structure of the world, just as the success of spacetime physics gives us reason to believe in objective spacetime structure.

[...] If you remain unconvinced and skeptical of ontology, what are your options? First, you could reject the notion of objective structure altogether. I regard that as unthinkable. Second, you could reject the idea of structure as applied to logic. I regard that as unmotivated.

[...] There are [...] alternatives to ontological realism. [...] [I]f you [are] tempted by one of the alternatives, think about one final thing. Is your rejection of ontological realism based on the desire to make unanswerable questions go away, to avoid questions that resist direct empirical methods but are nevertheless not answerable by conceptual analysis? If so, none of [...] [the alternatives] will give you what you desire. None of them lets you bypass debate over the ultimate structure of the world. Far from it: each is simply an alternative proposal about what that structure is like. Given each proposal there remain substantive metaphysical questions, namely those that can be raised in what the proposal grants to be fundamental terms. Furthermore, the very assertion that the proposed variety of structure, as opposed to the quantificational structure [...], is part of reality's objective structure seems itself to be incapable of being established by either straightforward empirical means or conceptual analysis. In fact, even a 'negative' thesis such as quantifier variance itself is a claim about the extent of the world's structure, and as such is as epistemologically problematic as any thesis in first-order metaphysics. Quantifier variance is 'just more metaphysics'.

[...] The point of metaphysics is to discern the fundamental structure of the world. That requires choosing fundamental notions with which to describe the world. No one can avoid this choice. Other things being equal, it's good to choose a set of fundamental notions that make previously intractable questions evaporate. [...] But no other than a positivist can make all the hard questions evaporate. If nothing else, the choice of what notions are fundamental remains. There's no detour around the entirety of fundamental metaphysics."

Ted Sider, “Ontological Realism,” in Metametaphysics: New Essays on the Foundations of Ontology, eds. David J. Chalmers, David Manley and Ryan Wasserman (Oxford University Press, 2009): pp. 416-420.