Abstraction, truth, and free logic

Abstractionism is the view that Fregean abstraction principles underlie our knowledge of the existence of mathematical objects. It is often assumed that the abstractionist proof for the existence of such objects requires ‘negative free logic’ in which all atomic sentences with empty terms are false. I argue that while negative free logic is not indispensably needed for the proof of abstract existence, there is a motivation for it—along broadly Fregean lines. The standard motivation for negative semantics rests on the explanation of truth in terms of reference. This line of reasoning, however, is not available in a context in which the reference of abstract terms must be proved, and not presupposed. I reverse the direction of explanation, thereby offering a novel motivation, Truth Priority, for the use of negative semantics. Some of the implications of Truth Priority for the abstractionist conception of ontology and reference will also be explored.