 A particularly important and pressing case of the ontological implications of logic are logicist programs in the philosophy of mathematics, in particular Frege’s conception of logical objects and his philosophy of arithmetic. Frege and neo-Fregeans following him believe that arithmetic is logic (plus definitions) and that numbers are objects whose existence is implied by arithmetic. Thus in particular, logic implies the existence of certain objects, and numbers are among them. Frege’s position has been criticized as being untenable since logic has to be neutral about what there is. Thus mathematics, or even a part thereof, can’t be both logic and about objects. The inconsistency of Frege’s original formulation of his position sometimes has been taken to show this, but since consistent formulations of Frege’s philosophy of arithmetic have surfaced this last point is moot. Frege’s argument for numbers as objects and arithmetic as logic is probably the best known argument for logic implying the existence of entities. It has been very carefully investigated in recent years, but whether or not it succeeds is controversial. Followers of Frege defend it as the solution to major problems in the philosophy of mathematics; their critics find the argument flawed or even just a cheap trick that is obviously going nowhere. We will not discuss the details here, but a detailed presentation of the argument can be found in the entry on  Frege’s theorem and foundations of arithmetic  as well as (Rosen 1993), which gives a clear and readable presentation of the main argument of (Wright 1983), which in turn is partially responsible for a revival of Fregean ideas along these lines. Frege’s own version is in his classic (Frege 1884). A discussion of recent attempts to revive Frege can be found in (Hale and Wright 2001), (Boolos 1998) and (Fine 2002). A discussion of Frege’s and Kant’s conceptions of logic is in (MacFarlane 2002) which also contains many historical references.